
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00013543/00001
Material Information
 Title:
 Suboctave wavelet representations and applications for medical image processing
 Creator:
 Zong, Xuli, 1960
 Publication Date:
 1997
 Language:
 English
 Physical Description:
 xiv, 137 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Algorithms ( jstor )
Data smoothing ( jstor ) Dyadics ( jstor ) Image contrast ( jstor ) Image enhancement ( jstor ) Image processing ( jstor ) Low noise ( jstor ) Noise reduction ( jstor ) Signals ( jstor ) Wavelet analysis ( jstor ) Computer and Information Science and Engineering thesis, Ph.D ( lcsh ) Dissertations, Academic  Computer and Information Science and Engineering  UF ( lcsh ) Image processing  Digital techniques  Mathematical models ( lcsh ) Imaging systems in medicine  Mathematical models ( lcsh ) Wavelets (Mathematics) ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph.D.)University of Florida, 1997.
 Bibliography:
 Includes bibliographical references (leaves 131136).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Xuli Zong.
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 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 028641851 ( ALEPH )
38854628 ( OCLC )

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Full Text 
SUBOCTAVE WAVELET REPRESENTATIONS AND APPLICATIONS
FOR MEDICAL IMAGE PROCESSING
By
XULI ZONG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1997
Copyright 1997
by
Xuli Zong
ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr. Andrew Laine, for all his support and
thoughtful advice during my graduate study. I would also like to thank Drs. Ger
hard Ritter, Sartaj Sahni, Edward Geiser, and John Harris for serving on my thesis
committee. Their time and thoughtful suggestions are greatly appreciated.
I am very grateful to Prof. Arthur Hornsby of the Department of Soil and Water
Science, University of Florida, for providing my financial support from January 1991
through June 1994. I would also like to thank Prof. Edward Geiser and Prof. An
drew Laine for providing me a graduate research assistantship from September 1994
to December 1995 and from May 1996 to January 1997, respectively. I also want
to thank Dr. Dean Schoenfeld of the Robotics Lab in the Department of Nuclear
and Radiological Engineering, University of Florida, for providing my financial sup
port during the final period of my Ph.D. study. His effort for reviewing part of my
dissertation is very much appreciated.
Special thanks go to Dr. Anke MeyerBaese of the Department of Electrical and
Computer Engineering for her encouragement and constructive discussions with me.
I would like to thank members of the Image Processing Research Group for some
enjoyable moments, their help, and their friendship.
Finally, I would like to thank my parents and my relatives for their support and
constant encouragement, as well as my wife and my daughter for their understanding,
patience, and love which gave me the strength to fulfill my educational objectives.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
LIST OF TABLES .....
LIST OF FIGURES ....
ABSTRACT ........
CHAPTERS
1 INTRODUCTION .
1.1
1.2
1.3
1.4
M otivations .
Review of Related Methods .....................
Objectives of DeNoising and Enhancement .
Wavelet Based Approaches for DeNoising and Enhancement .
1.5 Organization of This Dissertation ...........
2 DENOISING AND ENHANCEMENT TECHNIQUES .
2.1 Introduction ......................
2.2 Noise Modeling ......................
2.2.1 Additive Noise Model .............
2.2.2 Approximate Speckle Noise Model ......
2.3 Uniform Wavelet Shrinkage Methods for DeNoising
2.3.1 Hard Thresholding ...............
2.3.2 Soft Thresholding ..............
2.4 Enhancement Techniques ..............
2.4.1 Enhancement by a Nonlinear Gain Function .
2.4.2 Enhancement by Generalized Adaptive Gain
3 DYADIC WAVELET REPRESENTATIONS .......
3.1 Discrete Dyadic Wavelet Transform ..........
3.1.1 OneDimensional Dyadic Wavelet Transform
3.1.2 TwoDimensional Dyadic Wavelet Transform
3.2 DWTBased DeNoising and Feature Enhancement .
viii
xii
xiii
. 11
. 14
. 14
. 16
. 19
. 20
. 20
. 21
. 22
. 24
. 26
. 27
. 29
. 35
3.2.1 Algorithm for Additive Noise Reduction and Enhancement
3.2.2 Algorithm for Speckle Reduction with Feature Enhancement
3.2.3 DWTBased DeNoising .
3.2.4 Regulating Threshold Selection through Scale Space .
3.2.5 DWTBased Enhancement with Noise Suppression .
3.3 Application Samples and Analysis .
3.3.1 Less Affection from PseudoGibbs Phenomena .
3.3.2 Additive Noise Reduction and Enhancement .
3.3.3 Speckle Reduction with Feature Enhancement .
3.4 Clinical Data Processing Study .
4 SUBOCTAVE WAVELET REPRESENTATIONS .. .. .
4.1 Introduction .
4.2 Continuous SubOctave Wavelet Transform .
4.2.1 OneDimensional SubOctave Wavelet Transform .
4.2.2 TwoDimensional SubOctave Wavelet, Transform .
4.3 Discrete SubOctave Wavelet Transform .
4.3.1 OneDimensional Discrete SubOctave Wavelet Transform
4.3.2 TwoDimensional Discrete SubOctave Wavelet Transform
4.4 SW TBased DeNoising .......................
4.5 SWTBased Enhancement with Noise Suppression .
4.6 Application Samples and Analysis .
5 PERFORMANCE MEASUREMENT AND COMPARATIVE STUDY
5.1
5.2
5.3
Performance Metric for Quantitative Measurements .
Quantitative Comparison of Signal/Image Restoration .
Quantitative Comparison of Image Enhancement .
6 OTHER APPLICATIONS OF WAVELET REPRESENTATIONS
6.1 Border Identification of Echocardiograms .
6.1.1 Overview of the Algorithm .
6.1.2 Multiscale Edge Detection .
6.1.3 Shape Modeling ...................
6.1.4 Boundary Contour Reconstruction .
6.1.5 Smoothing of a Closed Contour without Shrinkage
6.1.6 Sample Experimental Results .
6.2 Multiscale Segmentation of Masses .
6.2.1 Overview of the Metliod .
6.2.2 Feature Extraction ..................
6.2.3 Classification via a Radial Basis Neural Network .
6.2.4 Sample Experimental Results .
. .97
. 98
. .99
... 101
. .102
. .104
. .105
. 107
. 109
... 110
. 110
. 112
7 CONCLUSIONS .............................. 115
APPENDICES
A FIR FILTERS FOR COMPACT SUPPORT WAVELETS ....... 117
A.1 First Order Derivative Wavelets of Spline Smoothing Functions 117
A.2 Second Order Derivative Wavelets of Spline Smoothing Functions 122
B IMPLEMENTATION OF SUBOCTAVE WAVELET TRANSFORMS 125
B.1 OneDimensional SubOctave Wavelet Transform ......... .127
B.2 OneDimensional Inverse SubOctave Wavelet Transform ..... 128
B.3 TwoDimensional SubOctave Wavelet Transform ........ 129
B.4 TwoDimensional Inverse SubOctave Wavelet Transform ..... 129
REFERENCES .................... ............... 131
BIOGRAPHICAL SKETCH ........................... 137
LIST OF TABLES
3.1 Impulse responses of filters H(w), G(w), K(w), and L(w). 35
3.2 Quantitative measurements of manually defined borders. ...... 55
3.3 Quantitative measurements of interobserver mean border differences
in mm on original versus enhanced images, as shown in Figure 3.25. 58
4.1 Quantitative measurements of performance for denoising and feature
restoration. ......................... ....... 77
5.1 Quantitative measurements: Average Square Errors 119'912" for var
ious signal restoration methods. .... ... 89
5.2 Quantitative measurements: RISE for various denoising methods. .91
5.3 Comparison of contrast values obtained by traditional contrast stretch
ing (CST), unsharp masking (UNS), and multiscale nonlinear process
ing of suboctave wavelet transform (SWT) coefficients of a mammo
gram containing a mass lesion. ..... 93
5.4 Contrast improvement index (CII) for enhancement by traditional con
trast stretching (CST), unsharp masking (UNS), and multiscale non
linear processing of suboctave wavelet transform (SWT) coefficients
of a mammogram with a mass ...................... .94
5.5 Comparison of contrast values obtained by multiscale adaptive gain
processing of dyadic wavelet transform (DWT) and suboctave wavelet
transform (SWT) coefficients. Mammographic features: minute
microcalcification cluster (MMC), microcalcification cluster (MC),
spicular lesion (SL), circular (arterial) calcification (CC), and well
circumscribed mass (WCM) ................... .. .. 96
5.6 CII for enhancement by multiscale adaptive gain processing of dyadic
wavelet transform (DWT) and suboctave wavelet transform (SWT)
coefficients. Mammographic features: minute microcalcification clus
ter (MMC), microcalcification cluster (MC), spicular lesion (SL), cir
cular (arterial) calcification (CC), and wellcircumscribed mass (WCM). 96
LIST OF FIGURES
2.1 Thresholding methods: soft thresholding and hard thresholding. 21
2.2 A nonlinear gain function for feature enhancement with noise suppres
sion .... .... .. .. 22
2.3 A generalized adaptive gain function. ... 24
3.1 A 3level DWT decomposition and reconstruction of a 1D function. 29
3.2 A 3level DWT decomposition and reconstruction of a 2D function. 32
3.3 A 2D analysis filter bank. ........................ 33
3.4 A 2D synthesis filter bank. ................ ....... 34
3.5 Coefficient and energy distributions of signal "Blocks". 39
3.6 A sample scaling factor function. .... 41
3.7 PseudoGibbs phenomena. (a) Orthonormal wavelet transform of an
original signal and its noisy signal with added spike noise. (b) Pseudo
Gibbs phenomena after both hard thresholding and soft thresholding
denoising under an OWT..................... ... 42
3.8 Multiscale discrete wavelet transform of an original and noisy signals. 42
3.9 DWTbased reconstruction after (a) hard thresholding, (b) soft thresh
olding, and (c) soft thresholding with enhancement. ... 43
3.10 DeNoised and feature restored results of DWTbased algorithms; first
row: original signal, second row: noisy version, third row: denoised
only result, and fourth row: denoised and enhanced result signal. 46
3.11 DeNoising and enhancement. (a) Original signal. (b) Signal (a) with
added noise of 2.52dB. (c) Soft thresholding denoising only (11.07dB).
(d) DeNoising with enhancement (12.25dB). ... 47
viii
3.12 DeNoising and enhancement. (a) Original image. (b) Image (a) with
added noise of 2.5dB. (c) Soft thresholding denoising only (11.75dB).
(d) DeNoising with enhancement (14.87dB). .... 47
3.13 DeNoising and enhancement. (a) Original MRI image. (b) DeNoising
only. (c) DWTbased denoising with enhancement. ... 49
3.14 DeNoising and enhancement. (a) Original MRI image. (b) DeNoising
only. (c) DWTbased denoising with enhancement. ..... 49
3.15 An algorithm for speckle reduction and contrast enhancement. ... .49
3.16 Results of denoising and enhancement. (a) A noisy ED frame. (b)
Wavelet shrinkage denoising only method. (c) DWTbased denoising
and enhancement.............................. 50
3.17 Results of denoising and enhancement. (a) A noisy ES frame. (b)
Wavelet shrinkage denoising only method. (c) DWTbased denoising
and enhancement. ................... ......... 51
3.18 A generalized adaptive gain function for processing an echocardiogram
in Figure 3.17(a). .. .. .. .. .. .. .. ... ... .. 52
3.19 Results of various denoising methods. (a) Original image with speckle
noise. (b) Median filtering. (c) Extreme sharpening combined with
median filtering. (d) Homomorphic Wiener filtering. (e) Wavelet
shrinkage denoising only. (f) DWTbased denoising with enhance
m ent.. ... .. 53
3.20 Results of various denoising methods. (a) Original image with speckle
noise. (b) Median filtering. (c) Extreme sharpening combined with
median filtering. (d) Homomorphic Wiener filtering. (e) Wavelet
shrinkage denoising only method. (f) DWTbased denoising and en
hancem ent. .. 54
3.21 Area correlation between manually defined borders by two expert car
diologist observers. ........................... .. 56
3.22 Border difference variation on the original images. (a) Distribution of
Epi ED border differences. (b) Distribution of Epi ES border differ
ences. (c) Distribution of Endo ED border differences. (d) Distribu
tion of Endo ES border differences. The solid lines are the third order
polynomial fitting curves .......................... 57
3.23 Border difference variation on the enhanced images. (a) Distribution
of Epi ED border differences. (b) Distribution of Epi ES border differ
ences. (c) Distribution of Endo ED border differences. (d) Distribu
tion of Endo ES border differences. The solid lines are the third order
polynomial fitting curves .................... .. 59
3.24 Denoising and image enhancement: (a) An original ED frame; (b) An
original ES frame; (c) The enhanced ED frame; (c) The enhanced ES
fram e .. .. .. 60
3.25 Image and border display: (a) An original ED frame with manually
defined borders overlaid; (b) An original ES frame with manually
defined borders overlaid; (c) The enhanced ED frame with manually
defined borders overlaid; (c) The enhanced ES frame with manually
defined borders overlaid. Red and yellow borders represent the two
observers. ... 61
4.1 A 3level SWT decomposition and reconstruction diagram for a 1D
function .. 70
4.2 Divisions of the frequency bands under the SWT shown in Figure 4.1. 71
4.3 A 2level 4suboctave decomposition and reconstruction of a SWT. 72
4.4 Frequency plane tessellation and filter bank. (a) Division of the fre
quency plane for a 2level, 2suboctave analysis. (b) Its filter bank
along the horizontal direction ....................... 72
4.5 Smoothing, scaling, and wavelet functions for a SWT. These functions
are used for a 2suboctave analysis. .. 74
4.6 An example of level one analytic filters for 2 suboctave bands and
a lowpass band. The dashed curve is the frequency response of a
first order derivative approximation of a smoothing function and the
dashdot curve is the frequency response of a second order derivative
approximation. The solid curve is a scaling approximation at level one. 75
4.7 Denoised and restored features from the SWTbased algorithm. From
top to bottom: original signal; noisy signal; denoised signal; overlay
of original and denoised signal ... 78
4.8 Limitations of a DWT for characterizing bandlimited high frequency
features. .. .80
4.9 Denoised and enhanced results of a noisy "Doppler" signal under a
DWT (25.529dB) and a SWT (26.076dB). .... 81
4.10 Enhancement with noise suppression. (a) A low contrast image of RMI
model 156 phantom with simulated masses embedded. (b) Enhance
ment by traditional histogram equalization. (c) SWTbased enhance
ment with noise suppression. .. 82
4.11 Enhancement with noise suppression. (a) Area of a low contrast mam
mogram with a microcalcification cluster. (b) Best enhancement by
traditional unsharp masking. (c) SWTbased enhancement with noise
suppression. (d) SWTbased enhancement of a local region of interest
(ROI) with noise suppression. ..... 84
5.1 Enhancement results. (a) Area of a low contrast mammogram with
a mass. (b) Enhancement of (a) by traditional contrast stretching.
(c) Enhancement of (a) by traditional unsharp masking. (d) SWT
based enhancement of (a) with noise suppression. (e) The same area
of a low contrast mammogram contaminated with additive Gaussian
noise. (f) Enhancement of (e) by traditional contrast stretching. (g)
Enhancement of (e) by traditional unsharp masking. (h) SWTbased
enhancement with noise suppression. (i) Handsegmented mass and
ROI for quantitative measurements of performance. ... 92
5.2 Phantom enhancement results. (a) Phantom image. (b) Mammogram
M56 with blended phantom features. (c) Nonlinear enhancement un
der a DWT. (d) SWTbased enhancement with noise suppression. 95
6.1 The circular arc templates for matched filtering. ... 101
6.2 Dynamic shape modeling. ........................ 102
6.3 Connecting broken boundary segments. The first row shows four typ
ical cases showing the end points of two broken segments belong to a
large segment. The second row is the result after connecting the two
broken segments for each case. ..... 103
6.4 Attached point removal. The first row shows four typical cases with
attached points. The second row is the result after attached point
removal for each corresponding case. ... 104
6.5 Border identification of the LV from a shortaxis view. (a) An original
frame of the LV. (b) Edge maps detected using a DWT. (c) The center
point of the LV and extracted boundary segments. (d) Connected
boundary contours. (e) Contours in (d) overlaid with the original. (f)
Final estimated boundaries. ....................... 106
6.6 Local nonshrinking smoothness filtering of a closed contour. (a) The
smoothed contours. (b) Contours in (a) overlaid with the contours in
Figure 6.5(d) before smoothness filtering. ... 107
6.7 Border identification of an echocardiogram at ED. (a) An original
frame of the LV at ED. (b) The detected center point and endocardial
as well as epicardial boundaries overlaid with the original. 108
6.8 Border identification of a frame at ES from the same sequence of
echocardiograms as Figure 6.7. (a) An original frame of the LV at
ES. (b) The detected center point and boundaries overlaid with the
original. .......................... ....... 108
6.9 Network architecture, a threelayer resourceallocating neural network
of radial basis functions. ......................... 111
6.10 Test Images. First row: original ROI images; Second row: smoothed
and enhanced images; Third row: ideal segmentation results.
Columns: (ac) real mammograms, (d) a mathematical model. 113
6.11 Experimental results of image segmentation. Four test cases, one each
row, are shown. The first column (a) is an original image, column (b)
is smoothed and enhanced images, column (c) is the segmented result,
and column (d) is the result of a traditional statistical classifier. 114
A.1 (a) A cube spline function and its first and second order derivative
wavelets, and (b) the fourth order spline with its first and second
order derivatives. ............... .............. 124
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SUBOCTAVE WAVELET REPRESENTATIONS AND APPLICATIONS
FOR MEDICAL IMAGE PROCESSING
By
Xuli Zong
December 1997
Chairman: Dr. Andrew F. Laine
Major Department: Computer and Information Science and Engineering
This dissertation describes suboctave wavelet representations and presents appli
cations for medical image processing, including denoising and feature enhancement.
A suboctave wavelet representation is a generalization of a traditional octave dyadic
wavelet representation. In comparison to this transform, by finer divisions of each
octave into suboctave components, we demonstrate a superior ability to capture
transient activities in a signal or image. In addition, suboctave wavelet represen
tations allow us to characterize bandlimited features more efficiently. DeNoising
and enhancement are accomplished through techniques of minimizing noise energy
and nonlinear processing of suboctave coefficients to improve low contrast features.
We identify a class of suboctave wavelets that can be implemented through band
splitting techniques using FIR filters corresponding to a mother dyadic wavelet. The
methodology of suboctave based nonlinear processing with noise suplpression is ap
plied to enhance features significant to medical diagnosis of dense radiographs.
Xlli
In our preliminary studies we investigated several denoising and enhancement
algorithms. DeNoising under an orthonormal wavelet transform was shown to cause
artifacts, including pseudoGibbs phenomena. To avoid the problem, we adopt a
dyadic wavelet transform for denoising and enhancement. The advantage is that
less pseudoGibbs phenomena was shown in our experimental results. We devel
oped algorithms for reducing additive and multiplicative noise. The algorithm for
speckle reduction and contrast enhancement was applied to echocardiographic im
ages. Within a framework of multiscale wavelet analysis, we applied wavelet shrink
age techniques to eliminate noise while preserving the sharpness of salient features. In
addition, nonlinear processing of feature energy was carried out to improve contrast
within local structures and along object boundaries.
A study using a database of clinical echocardiographic images suggests that such
denoising and enhancement may improve the overall consistency and reliability of
myocardial borders as defined by expert observers. Comparative studies on quantita
tive measurements of experimental results between our algorithms and other methods
are presented. In addition, we applied wavelet representations under dyadic or sub
octave wavelet transforms to other medical image processing problems, such as border
identification and mass segmentation.
xiv
CHAPTER 1
INTRODUCTION
1.1 Motivations
Noise, artifacts, and low contrast can cause signal and image degradations during
data acquisition of many signals and images, especially in areas of medical image ap
plications. Different image modalities exhibit distinct types of degradation. Mammo
graphic images often exhibit low contrast while images formed with coherent energy,
such as ultrasound, suffer from speckle noise. Transmitted audio signals sometimes
have the problem of channeladded noise. These degradations not only lower audio
or visual quality, but also cause analysis and recognition algorithms to sometimes fail
to achieve their objectives.
Since poor image quality often makes feature extraction, analysis, recognition,
and quantitative measurements problematic and unreliable in various areas of sig
nal/image processing and computer vision, much research has been devoted to im
prove the quality of acquired signals and images degraded by these factors [30]. Data
restoration techniques are often targeted to reduce noise. Unlike noise, artifacts
sometimes comprise not only high frequency noise components, but also middleto
low frequency components which are very hard to differentiate from other typical
features in the spectrum of frequency contents. Simple denoising techniques will
have problems reducing artifacts, so applicationspecific methods often have to be
used to eliminate artifacts based on certain prior knowledge. Tle quality of low
contrast images can be improved through designated featiire (nhan'cement.
The promise of wavelet representations under dyadic or further suboctave wavelet
transforms and the need for improving degraded signals/images have motivated this
dissertation research. Reducing artifacts is not a major concern of this research. The
major task of this research is to develop reliable techniques for reducing noise and
enhancing salient features important to various application problems. The promising
denoising and feature enhancement techniques may improve the reliability and per
formance of signal/image processing and computer vision algorithms for highlevel
tasks, such as object detection or visual perception. In addition, we also show the
capability of wavelet representations for other medical imaging applications.
1.2 Review of Related Methods
Signal/image restoration and enhancement have been the focus of much research
in the areas of signal and image processing as well as computer vision. Several
denoising methods and image enhancement techniques have been developed and re
ported in the literature [30, 15, 26, 17, 62, 58, 68]. Most of these methods can be
roughly classified as spatial, statistical, and Fourierdomainbased. Many tradi
tional methods were reviewed by Jain [30]. These conventional techniques for image
restoration and enhancement have shown certain limitations of balancing the effect
of removing noise and enhancing features.
For denoising purposes, spatial and frequency domain smoothing methods often
not only reduce noise, but also smooth out high frequency components of wideband
features as a sideeffect. This is because the smoothing effect applies both noise and
high frequency components of features. Oversmoothing noise often causes certain
interested features being blurred. Recently multiscale and/or multiresolution wavelet
techniques for signal/image restoration have been reported, suggesting improved re
sults in signal/image quality [50, 46, 19, 9, 18, 59, 8, 65].
Mallat and Hwang [50] introduced a local maxima based method for removing
white noise. Their method analyzes the evolution of local maxima of the dyadic
wavelet transform cross scales and identifies the local maxima curves above a cer
tain measurement metric which more likely correspond to features than noise. The
denoised signal/image is then reconstructed based on the extracted local maxima
corresponding to significant feature singularity points. Lu et al. [46] further extended
the ideal of local maxima curves to the local maxima tree cross scales and employed
a different measurement metric to detect features from noise. Coifman and Majid
[9] developed a wavelet packetbased method for denoising signals. It is an iterative
method for extracting features based on the best wavelet packet basis, which removes
noise energy below a certain threshold.
Donoho and Johnstone [19, 20] presented thresholdingbased wavelet shrinkage
methods for noise reduction. These methods uniformly reduce noise coefficients below
a global threshold. Hard thresholding and soft thresholding have trade off between
preserving features and achieving smoothness. Soft thresholding denoising was fur
ther explained by Donoho [18] and proved that with high probability the denoised
signal is at least as smooth as the noisefree original in a wide variety of smoothness
measures and comes nearly as close (in the mean square sense) to the original as
any other estimated results. But the method still faces the problem of balancing the
removal of noise and signal details in order to get a better performance in terms of
visual quality and quantitative measurements. Thresholdingbased wavelet shrinkage
under an orthonormal wavelet transform has shown undesired sideeffects, including
pseudoGibbs phenomena [8]. In order to solve the problem, Coifman and Donoho
[8] developed a translationinvariant denoising method. Their method alleviates the
problem, but oscillations after denoising remain visible. Wei and Burrus [65] used
redundant wavelet representations to achieve translationinvariant effects for image
restoration when applied to various image coding schemes.
Most noise in reality is additive, but certain noise can not be characterized well
as additive noise, such as speckle noise. Speckle noise can be better approximated
as multiplicative noise [30]. Image formation under coherent waves results in a gran
ular pattern known as speckle. The granular pattern is correlated with the surface
roughness of an object being imaged. Goodman [25] presented an analysis of speckle
properties under coherent irradiance, such as laser and ultrasound. The primary
differences between laser and ultrasound speckle were pointed out by Abbott and
Thurstone [1] in terms of coherent interference and speckle production. For speckle
reduction, earlier techniques include temporal averaging [25, 1], median filtering,
and homomorphic Wiener filtering [30]. Homomorphic Wiener filtering is a method
which converts multiplicative noise into additive noise and applies Wiener lowpass
filtering to reduce noise. Similar to temporal averaging, one speckle reduction tech
nique [57] used frequency and/or angle diversity to generate multiple uncorrelated
syntheticaperture radar (SAR) images which were summed incoherently to reduce
speckle. Hokland and Taxt [28] reported a technique which decomposed a coherent
image into three components, one of which, called subresolvable quasiperiodic scat
ter, causes speckle noise. The component was eliminated by harmonic analysis and
processing.
In the last few years, several wavelet based techniques were developed for speckle
reduction. Moulin [54] introduced an algorithm based on the maximumlikelihood
principle and a wavelet regularization procedure on the logarithm of a radar image
to reduce speckle. Guo et al. [27] first reported a method based on wavelet shrinkage
for speckle reduction. In the method of Guo et al., wavelet shrinkage of a logarith
mically transformed image is applied for speckle reduction of SAR images. They
also proposed several approaches to combine data from polarization to achieve better
performance. We [71, 72] have developed a method for speckle reduction similar to
the one by Guo et al. [27]. The differences are that (a) noise is modeled as multi
plicative, taking a homomorphic approach to reduce the noise, (b) different wavelets
and multiscale overcomplete representations are employed in our approach, and (c)
an enhancement mechanism is incorporated into our denoising process. Thus, our
method can not only reduce speckle noise, but also enhance interesting features.
In the last two decades, many image enhancement methods have been published
in the literature. Several spatial and frequencybased techniques [11, 30, 24, 44, 58]
were developed for various image enhancement purposes. Contrast stretching, high
pass filtering, histogram modification methods are described in Jain [30]. Contrast.
stretching was an earlier technique for contrast enhancement [30]. This method has
limitations of selecting features based on local information for enhancement because
it is a global approach and the enhancement function is linear or piecewise linear.
Contrast stretching may also amplify noise when input data are corrupted by noise.
Some image enhancement schemes applied to medical image modalities have been
developed and studied in the literature [30, 58, 45, 39, 35]. Specifically, spatial and
frequencybased techniques [30, 24, 11, 44] have been developed for ultrasound image
enhancement. A statistical enhancement method, which used the local standard
deviation of a surrounding region centered around each pixel to replace its value to
enhance edges in ultrasound images, was reported by Geiser [23].
Conventional filteringbased techniques for denoising and image enhancement
have shown certain limited ability for removing noise without blurring features and
for enhancing contrast without amplifying noise because spatial and frequency rep
resentations can not separate features from noise well. In the last few years, many
techniques based on multiscale features, such as edges and object boundaries, have
achieved success for image enhancement in several application areas [32, 40, 41, 45].
Recently waveletbased nonlinear enhancement techniques have produced superior
results in medical image enhancement [39, 35, 73].
1.3 Objectives of DeNoising and Enhancement
To improve the quality of acquired signals and images degraded by noise and/or
low contrast, most traditional methods try either to reduce noise or to enhance fea
tures. At first glance, denoising and feature enhancement appear to be two conflict
ing objectives, especially to traditional methods for denoising or image enhancement.
However, they are simply two sides of the same coin. The purpose of denoising is to
eliminate noise, primarily in high frequency, while methods of feature enhancement
attempt to enhance specific signal details, including contrast enhancement. The dif
ference lies in the fact that features often occupy a wider frequency band than noise.
It is even more difficult to achieve both objectives when feature details are corrupted
by noise. Traditional spatial and filteringbased methods for denoising often reduce
noise at a price of blurring features while singlescale conventional methods for con
trast enhancement may amplify noise. The singlescale representation of a signal in
time (or pure frequency) is problematic when attempting to discriminate signal from
noise.
Because of the limited ability of traditional techniques for denoising or feature
enhancement, the two conflicting objectives can not be accomplished simultaneously
through earlier methods under spatial or Fourier representations with a single res
olution of frequency contents. When the two mechanisms, denoising and feature
enhancement, are combined under a framework of a new representation or platform
which helps to overcome the drawback of each mechanism when acting alone, we will
have a much better chance to fulfill the two objectives at the same time. Wavelet
transforms and wavelet theory can be one method for new representation and plat
form. This may be why wavelet representations have attracted more and more at
tention of researchers aiming at signal/image restoration and feature enhancement.
Recently wavelet based methods have shown promise in accomplishing the two ob
jectives at the same time because wavelet decomposition can fine tune frequency
resolution of signal details. We are able to treat distinct components of details at
finetocoarse scales differently to achieve desired effects of denoising and feature en
hancement. Algorithms have been developed under such a multiscale wavelet analysis
framework [71, 73].
1.4 Wavelet Based Approaches for DeNoising and Enhancement
Since Morlet and Grossmann [52] formulated the first wavelet decomposition,
wavelet theory [52, 12, 13, 14, 6, 47, 48, 49, 51, 64] has been developed and well
documented in the last 14 years. Some practical applications of the theory have been
developed, but more applications are still under the developing stage. There are many
choices of wavelets with different properties [12]. Denoising using some wavelets hav
ing oscillations may lead to certain unwanted and undesired sideeffects, for example
noiseinduced ripples and oscillations when reconstructed under incomplete coeffi
cients in the wavelet domain. This could be one of the major factors resulting in
artifacts, including the socalled pseudoGibbs phenomena in the neighborhood of
sharp variation points (singularities) after denoising (for details see Coifman and
Donoho [8]).
Orthonormal Wavelet Transform (OWT) and discrete Dyadic Wavelet Transform
(DWT) have been used in various application areas, such as data compression, edge
detection, texture analysis, noise reduction, and image enhancement. The compact
and local support of wavelets in spatial and frequency domains has been a valu
able property for characterizing features locally. This enable us to remove noise and
enhance features locally without affecting other features distant apart. In a prelim
inary implemented method, DWT has been adopted as our major analysis tool for
denoising and contrast enhancement [73]. The reasons are quite obvious. A DWT
with the first order derivative of a smoothing function as its basis wavelet can sep
arate noise energy from feature energy reasonably well in the wavelet domain. The
DWT also correlates prominent features in its multiscale representation, such as edges
and object boundaries. After experimental analysis and understanding of signal and
noise behaviors in scale space, we are able to find out which wavelet coefficients to
modify to enhance certain features of interest (FOI) based on simple thresholding
and nonlinear processing. The mother wavelet is a smoothing function and is anti
symmetric with few oscillations, which keeps us relatively free from the sideeffects
shown under OWT with a basis wavelet having slight oscillations itself. This effect
can be clearly seen from the denoised results under OWT and our denoised results
[73]. The filters used to perform the DWT have compact support of a few taps. The
DWT is a stable and overcomplete representation. DWT wavelet coefficients (WC)
have a more clear meaning that they are proportional to the signal magnitude or
image intensity changes (gradients) at certain scales. WCs reflect energy in a signal,
so we can rephrase that a DWT with the aforementioned wavelets is a process for
the diffusion of the energy of a signal and converting it into the energy of the signal
at different scales in its wavelet representation.
Even through the DWTbased algorithm [73] has produced better results than
denoising only methods for signal/image restoration, we have observed that a DWT
has limited ability to characterize bandlimited features, such as texture information,
speech or sound signals including ultrasound signals. To more reliably identify fea
tures through the timescale space, we formulate and implement a suboctave wavelet
transform, which is a generalization of the DWT. The suboctave wavelet transform
provides a means to visualize signal details in equaldivided suboctave frequency
bands and is shown to characterize signal details more effectively. The theoretical
development of a suboctave wavelet transform, FIR filter design, and efficient im
plementation are part of this thesis research. Further more, in this thesis, we are
developing a complete algorithm and quantitatively measure its performance which
will be compared to other techniques for denoising and/or feature enhancement.
In an approach developed during this research, we achieve denoising and fea
ture enhancement under a framework of multiscale suboctave wavelet analysis and
judicious nonlinear processing [42, 43]. We seek to eliminate noise while restoring
or enhancing salient features. Through multiscale representation by a discrete sub
octave wavelet transform (SWT) with first and second order derivative approxima
tions of a smoothing function as its basis wavelets, we can distinguish feature energy
from noise energy reasonably well. The objectives of denoising and feature enhance
ment are achieved by simultaneously lowering noise energy and raising feature energy
through designated nonlinear processing of wavelet coefficients in the transform do
main. Through parameterized processing, we are able to achieve a flexible control and
the potential to reduce speckle and restore (or even enhance) contrast along features,
such as object boundaries. As shown in our earlier work [73], this approach for speckle
reduction and contrast enhancement is less affected by pseudoGibbs phenomena [8].
1.5 Organization of This Dissertation
The rest of this dissertation is organized as follows. In Chapter 2, we review .solli
denoising and enhancement methods. We describe how to regulate thresholdblased
wavelet shrinkage through scale space and show how to design an enhancement func
tion with noise suppression. In Chapter 3, we present dyadic wavelet transform based
techniques for denoising and feature enhancement. Sample application results and
analysis are presented. In Chapter 4, we derive and formulate a suboctave wavelet
transform mathematically and show how it generalizes the dyadic wavelet transform.
The advantage of a suboctave representation over a dyadic wavelet representation
is presented. Sample application results are presented. In Chapter 5, we describe
how to quantitatively measure the performance of an algorithm for denoising and
enhancement. Some comparisons are made between the results of other published
methods, and our DWTbased as well as SWTbased methods. In Chapter 6, we
apply wavelet representations under dyadic and suboctave wavelet transforms to
other problems of medical image processing. Experimental results and analysis are
presented. This dissertation is concluded in Chapter 7. In Appendix A, we present
FIR filters used for dyadic and suboctave wavelet transforms. In Appendix B, we
introduce procedures for a fast implementation of a suboctave wavelet transform in
one and two dimensions.
CHAPTER 2
DENOISING AND ENHANCEMENT TECHNIQUES
In this chapter, we are going to overview the impact of noisy and low contrast sig
nals/images, and review some related methods for denoising and enhancement. We
also describe how to regulate the thresholdbased denoising techniques and design an
enhancement function with noise suppression. We then introduce the image restora
tion and enhancement techniques employed in our algorithms. The reason that we
put this chapter ahead of wavelet representations described in Chapters 3 and 4 is
that both our DWT and SWT based algorithms share image restoration and enhance
ment techniques introduced in this chapter. The advantage of this organization is
that we avoid describing the denoising and enhancement operators repeatedly when
presenting our DWT and SWT based algorithms for noise reduction and contrast
enhancement, so we can simply refer to the operators in this chapter.
2.1 Introduction
Signal and image degradations by noise and low contrast are frequent phenomena
of signal/image data acquisition, especially in medical imaging. Image degradations
have a significant impact on the performance of human experts and computerassisted
methods for medical diagnosis. For example, a human medical expert may fail to
capture some important information from a noisy and low contrast, image when per
forming medical diagnosis, especially when exhausted. A cardiologist may have to
perform border interpolation in order to identify myocardial borders when border
information is incomplete and corrupted by speckle noise and make decision based
11
on unreliable information. Noise and low contrast make it problematic for human
experts and computer algorithms to identify features of diagnostic importance in
medical imaging. In addition, noise and low contrast often make feature extraction,
analysis, and recognition algorithms unreliable, so improving the quality of acquired
medical images becomes necessary. Signal/image restoration and/or enhancement
are usually taken as the first step of a high level task of image processing and com
puter vision. For instance, in most image segmentation algorithms, image smoothing
is usually carried out as the first step (or preprocessing) for segmentation in order to
reduce noise interference on the performance of these algorithms.
Most traditional approaches for denoising have a singleminded objective which
is to reduce noise while minimizing the smoothing of features. Traditionally, noise
is frequently not a concern in feature enhancement algorithms. In the combined ap
proach developed during this thesis research, we will focus on two goals; (1) to remove
noise and (2) to enhance salient features to a desired degree. As part of this research,
we have implemented algorithms for removing additive and multiplicative noise re
spectively while enhancing prominent features at the same time. These algorithms
are primarily based on wavelet representations, wavelet shrinkage, and feature em
phasis. Wavelet shrinkage is a technique which uniformly reduces wavelet coefficients
through a certain operator, such as hard thresholding or soft thresholding. During
the process, small coefficients, mainly attributed to noise, are usually removed. For
feature enhancement, we revitalize low contrast FOI through feature emphasis (in
creasing the energy level for each of these features). When the noise level in a signal
or image is high, these algorithms are capable of not only removing noise, but also
restoring features to near their original quality and even enhancing certain FOI se
lectively. When the noise level is low, such as in a low contrast medical image, our
algorithms can enhance features with noise suppression to avoid amplifying noise.
The main ideas behind selected wavelet shrinkage and salient feature emphasis
encapsulate the two fundamental objectives of denoising and feature enhancement:
(a) Remove noise, but not features, and
(b) Enhance the features of interest, but not noise.
These are two conflicting objectives in the sense that both sharp features and noise lie
in high frequency of the spectrum. Noise is often smoothed out at the price of blurred
features left in a traditional denoising algorithm. On the other hand, enhancing cer
tain FOI corrupted by noise is more likely to amplify noise in an enhancementonly
technique without a noise suppression mechanism incorporated. This prevents tradi
tional algorithms from attempting to achieve both of the objectives simultaneously
because noise and features can not be distinguished well in spatial or Fourier repre
sentations. In Fourier domain, denoising is usually carried out through some type
of lowpass filtering in nature, including templatebased spatial averaging. On the
other hand, feature enhancement is accomplished under a certain type of highpass
filtering. They are in conflict with each other when performed on a single set of data
represented either in time or in frequency. From this analysis, it sounds like that some
type of bandpass filtering may be a choice. But in fact single bandpass filtering at
a frequency band has a very limited capability of removing noise and enhancing fea
tures. To achieve both the objectives, we need a suitable representation or platform
which can separate features from noise well and locally. Multiscale wavelet represen
tation developed by Mallat and Zhong [51] has shown promise in separating features
and noise through scale space. As outlined by Daubechies [13] as an example, we
formulated and implemented multiscale suboctave wavelet representation [42, 43], a
generalization of the dyadic wavelet representation, to further improve the capability
of characterizing features from noise. A dyadic wavelet. t ransform is Iriefly explained
in Chapter 3 while a suboctave wavelet transform is formulated and described in
Chapter 4.
2.2 Noise Modeling
Noise modeling is an important part of a noise reduction method and it affects
which kind of techniques should be used to reduce noise. Efficient noise models can
make denoising more effective. When a noise behavior is not fully understood or
still can not be completely explained, its accurate noise model is very difficult to
obtain. However, approximate noise models, such as speckle noise modeling, may be
used in such a case. Continuous noise modeling is of theoretical importance while
discrete noise models are more related to practical signal/image processing for noise
reduction. Through the sampling theory, a discrete noise model can be obtained
from sampling its corresponding continuous noise model with a sample rate (at least
Nyquist rate) to avoid aliasing effect.
2.2.1 Additive Noise Model
For some signal/image processing applications considered, such as simulated sig
nals and mammograms, noise is better approximated as an additive phenomenon. In
general, additive noise can be represented by the formula
f(x) = g(x) + %a(x), (2.1)
where g(x) is a desired unknown function. The function f(x) is a noisy observation
of g(x), 7a(x) is additive noise, and x is a vector of spatial locations or temporal
samples. By using vector notation, we unified the noise model for 1D, 2D, ..., ND
cases. For our signal/image processing, 1D and 2D noise models are what we are
interested in. Noise %a(x) is usually approximated as Gaussian white noise, so it
has zero mean (pI = 0) and a noise level a, the standard deviation of the Gaussian
function. For 1D signal processing, we discretize Equation (2.1) as
f(n) = g (n) + ra (n), (2.2)
where n Z, f(n) f(nT + s) (g(n) and ?7a(n) are similar), T is a sampling period,
and s is a sampling shift. For 2D image processing, Equation (2.1) is discretized as
f(m,n) = g(m, n) + ra (m,n), (2.3)
where (m,n) E Z2, f(m,n) = f(mT, + s, nTy + s,) (g(m,n) and rla(m,n) are
similar), T, and T, are sampling periods along horizontal and vertical directions, s,
and s, are sampling shifts.
For an additive noise model, there exist techniques based on mean squared error
or 11 norm optimization to remove noise. Such techniques include Donoho and John
stone's wavelet shrinkage techniques [19, 20, 18], Chen and Donoho's basis pursuit
denoising [5], Mallat and Hwang's localmaximabased method for removing white
noise [50], and wavelet packetbased denoising [9, 10].
By incorporating denoising and feature enhancement mechanisms within a frame
work of wavelet representations [73, 42], we seek to reduce noise and enhance contrast
without amplifying noise. We shall demonstrate that subtle features barely seen or
invisible in a mammogram, such as microcalcification clusters, spicular lesions, and
circular (arterial) calcifications, can be enhanced via wavelet representations and
judicious selection and modification of transform coefficients. Since our algorithm
treats noise and features independently, superior results were obtained for similar
data when compared to existing algorithms designed for denoising or enhancement
alone.
In our investigation, we studied hard thresholding and Donoho and Johnstone's
soft thresholding wavelet shrinkage [19, 18] for noise reduction. An advantage of soft
thresholding is that it can achieve smoothness while hard thresholding better pre
serves features. In our approach for accomplishing denoising and feature enhance
ment, we take advantage of both thresholding methods. Donoho and Johnstone's
soft thresholding method [19, 18] was developed on an orthonormal wavelet trans
form [12] primarily applied with Daubechies's Symmlet 8 basis wavelet. These results
showed some undesired sideeffects, from pseudoGibbs phenomena [8]. By using an
overcomplete wavelet representation and basis wavelets with fewer oscillations, a re
sult relatively free from such sideeffects after denoising was observed experimentally
on similar data sets. In our algorithm, we first adapt regularized soft thresholding
wavelet shrinkage to remove noise energy within the finer levels of scale (such as levels
1 and 2). We then apply to wavelet coefficients within the selected levels (such as
levels 3 and 4) of analysis a nonlinear gain with hard thresholding incorporated to
preserve features while removing small noise perturbations.
2.2.2 Approximate Speckle Noise Model
Coherent interfering cause speckle noise. An accurate and reliable model of the
noise is desirable for efficient speckle reduction. But it remains a difficult problem.
An approximate speckle noise model is formulated below. Here, our primary objective
is to accomplish speckle reduction for 2D digital echocardiograms, so we formulate
the noise model directly in two dimensions. The formulation of a onedimensional
noise model is similar.
Since speckle noise is not simply additive, Jain [30] presented a general model for
speckle noise as
f(x, y) = g(x, y) r7n (x, y) + ra (x, y), (2.4)
where g(x, y) is an unknown 2D function, such as a noisefree original image, to be
recovered, f(x, y) is a noisy observation of g(x, y), rq (x,y) and ra(x, y) are multi
plicative and additive noise respectively, x and y are the variables, such as spatial
locations, and (x, y) E R2. Since the effect of additive noise (such as sensor noise)
with level ad is considerably smaller than multiplicative noise (coherent interfering)
(,a(x,y)j2
proximated by
f(x, y) = g(x, y) 7(x, y). (2.5)
To separate the noise from the original image, we take a logarithmic transform on
both sides of Equation (2.5),
log(f(x, y)) = log(g(x, y)) + log(r, (x, y)). (2.6)
Equation (2.6) can also be rewritten as
f'(x, y) = g'(x, y) + l (x, y). (2.7)
Now we can approximate q (x, y) as additive white noise and may apply various
waveletbased approaches for additive noise reduction. With uniform sampling, we
obtain the discrete version of equation (2.7) as
f(m, n) = g'(m, n) + 7 (m, n),
(2.8)
where (m, n) E Z2, f'(m, n) = ft(mT + s,, nTy + Sy), T. and Ty are sampling peri
ods along horizontal and vertical directions, sx and s, are sampling shifts. Wavelet
representation and wavelet transforms will be presented in the next two chapters. To
describe how the denoising method works, here, we only need the fact that a wavelet
transform is a linear transformation, and we borrow its notation for a wavelet rep
resentation whose details are given in the following chapters. The symbol W is
represented as a wavelet transform, Wf as a wavelet coefficient at scale 23 (or level
j) and direction d (1 for horizontal and 2 for vertical), Sj is a scaling approximation
at a final level J. By the properties of a linear transform, we have
W[f (m, n)] = W[gt(m, n)] + W[ 7 (m, n)] (2.9)
after applying wavelet transform on the both sizes of Equation (2.8) where
W[f (m, n)] = {(Wf[f (m, n)])d=1,2, 1
W[g'(m, n)] = {( (g (m, n)])d=1,2,
W[ra (m,n)]= {( (7 (m,n)])d=1,2,1
Since noise lies in high frequency, it will reduce to near zero after a finite number
of repeated smoothings, so Sj[rlj7(m,n)] + 0. In fact, at most 5 level wavelet de
composition is needed to smooth out noise for most noise reduction applications we
conducted. This is why we only carry out speckle noise reduction through eliminat
ing noise energy d (W4 [ (rm, n)])2. For image restoration purposes, it is desirable
to recover W[gl(m, n)], the wavelet transform of a desired function g'(m, n), from
W[f'(m, n)] by reducing W[t7r(m, n)] in the wavelet domain. By taking the inverse
wavelet transform, we may be able to recover g((m, n) or a close approximation. For
noise reduction and feature enhancement, we want to increase further the sharpness
of features of interest, such as myocardial boundaries, through nonlinear stretching
for feature energy gain on signal details Wf[g'(m, n)].
Jain showed a similar homomorphic approach [30, pp. 313316] for speckle reduc
tion of images with undeformable objects through temporal averaging and homomor
phic Wiener filtering. The motion and deformable nature of human hearts through
time prevents us from getting the same status of the left ventricle for multiple frames.
Because we treat noise and feature components differently, we are able to produce
a result that is superior to denoising only algorithms. We show that our algorithm
is capable of not only reducing noise, but also enhancing features of diagnostic im
portance, such as myocardial boundaries in 2D echocardiograms obtained from the
parasternal shortaxis view.
2.3 Uniform Wavelet Shrinkage Methods for DeNoising
Thresholdbased denoising is a simple and efficient technique for noise reduction
when being applied within a framework of wavelet representations which can separate
features from noise. Hard thresholding has long been used as a useful tool, includ
ing denoising. Soft thresholding wavelet shrinkage for denoising was developed by
Donoho and Johnstone [19]. Hard thresholding and soft thresholding have trade off
between preserving features and achieving smoothness. When features in the wavelet
domain can be clearly distinguished from noise, applying hard thresholding wavelet
shrinkage can achieve a better result than soft thresholding. When it is not the case
and smoothness is more desirable, soft thresholding should be the choice.
2.3.1 Hard Thresholding
A hard thresholding operation can be expressed as
(x) = TH(v(x),t) = v(x)(Iv(x)I > t)+,
(2.10)
where t is a threshold value, x E D where D is the domain of v(x), and u(x) is the
result of hard thresholding and has the same sign as v(x) if nonzero. The meaning
of (Iv(x) > t)+ is defined by the expression
(Iv(x)I>t)+ =
0
if Iv(x)>I t,
otherwise.
2.3.2 Soft Thresholdiny
Soft thresholding [19, 18] is a nonlinear operator and can be described by
u(x) = Ts(v(x), t) = sign(v(x)) (Iv(x) t)+,
(2.11)
where threshold parameter t is proportional to the noise level and x E D, the domain
of v(x), and u(x) is the result of soft thresholding and has the same sign as v(x) if
nonzero. The expression (Iv(x) t)+ is defined as
(Iv(x) Iv(x)l t if v(x)j >t,
0 otherwise.
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1
0.8 0.6 0.4 0.2
0 0.2 0.4
v(x) T
0.6 0.8 1
Figure 2.1. Thresholding methods: soft thresholding and hard thresholding.
The function sign(v) is defined as
1
sign(v) = 1
0
if v > 0,
if v < 0,
otherwise.
Figure 2.1 shows a soft thresholding operation compared with hard thresholding.
2.4 Enhancement Techniques
In this section, we describe how to design an enhancement function with noise
suppression incorporated. Several choices of enhancement functions are presented.
Analysis and discussion of the reasons for our design philosophy are also included.
Soft Thresholding vs Hard Thresholding
7
*  Soft Thresholding
 ofHard Thresholding
.
7

I 
,
,
   =
7 '
Nonlinear Enhancement Function: T1=0.1, T2=0.2, T3=0.85, alpha = 0.4
, .
IZ 0 ...... ......... ... .....
I I
0.2
0.4 / : I I
II I
/ / I I
0.6
0.8
1 i I
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
v T1 T2 T3
Figure 2.2. A nonlinear gain function for feature enhancement with noise suppression.
2.4.1 Enhancement by a Nonlinear Gain Function
In the design of an enhancement function, we try to accomplish the two tasks of an
effective enhancement; (a) enhance features selectively and efficiently and (b) avoid
amplifying noise. An enhancement operator with noise suppression is desirable and
can be a choice for achieving the two aforementioned tasks. Since we can not fulfill
the tasks satisfactorily in the original or Fourier representation of a signal or image,
this leads us to look at other representations through some kinds of transformation.
Through our study and experiments, we observed that dyadic wavelet representations
have shown a great promise for separating features from noise. Therefore, we can
apply the kind of enhancement functions, which will be introduced momentarily,
to enhance FOI. We thereafter generalize dyadic wavelet representations to produce
suboctave wavelet representations for characterizing bandlimited features frequently
seen in medical images more efficiently.
A parameterized nonlinear gain function, which is targeted to accomplish the two
tasks of an effective enhancement, can be formulated as
0 if Ivi < T1,
ENL(v) = ssign(v) (T2 + T ((Ivl T2)/T) ) if T2 < Ivi < T3, (2.12)
s v otherwise,
where v E [1, 1], 0 < a < 1, T = (T3 T2), s is a positive scaling factor which
is used to adjust the overall energy of a processed image. Parameters T1, T2, and
T3 are selected values. For each input value v less than T1, the small coefficient is
more likely resulted from noise where v is a normalized coefficient. For input value
v greater than T3, the contrast of the corresponding feature of v is already relatively
high. No special treatment for the coefficient is needed, so we only do linear scaling
which is needed to keep the enhancement function from becoming decreasing, which
may cause artifacts. The normalized coefficients within the range between T2 and
T3 are what we would like to enhance because their contrast is relatively low and
our features of interest have the corresponding coefficients in this range. Thus, we
nonlinearly stretch their energy gain to revitalize these features. The range between
T1 and T2 is considered as a risk area. Both noise and features may have components
in this range, so we try to avoid amplifying noise by simply linear scaling and treated
this range similar to the area of values greater than T3. Figure 2.2 shows a sample
enhancement function. This enhancement operator is less flexible than the operator
in Section 2.4.2, but it is computationally more efficient. The operator can serve as
a choice if speed is a concern.
Generalized Adaptive Gain: C=10, B=0.35, T1=0.1, T2=0.2, T3=0.9
0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8
v T1 T2 T3
Figure 2.3. A generalized adaptive gain function.
2.4.2 Enhancement by Generalized Adaptive Gain
In the last few years, several waveletbased enhancement techniques have been de
veloped [46, 45, 37, 39]. Adaptive gain through nonlinear processing [37, 34] has been
used to enhance features in digital mammograms. The adaptive gain through non
linear processing is further generalized [42] to incorporate hard thresholding in order
to avoid amplifying noise and to remove small noise perturbations. The generalized
adaptive gain (GAG) nonlinear operator is defined as
0 if lvi < T1,
EGAG(V) q [T2 + a (sigm(c(u b)) sigm(c(u + b)))] if T2 < lv < T3, (2.13)
s v otherwise,
where v E [1, 1], a = (T3 T2) a, q = s sign(v), u = (vI T2)/(T3 T2), b E (0, 1),
0 < T1 < T2 < T3 < 1, c is a gain factor, s is a scaling factor which is used to adjust
the overall energy of a processed image. Parameter a can be computed by
1
a= (2.14)
a sigm(c(1 b)) sigm(c(1 + b))' (2
1
sigm(v) = + (2.15)
1 + e
Parameters T1, T2, and Ta are selected values. When T1 = T2 = 0 and Ta = 1,
the expression is equivalent to the adaptive gain nonlinear function used previously
[37, 34]. The interval [T2, T3] serves as a sliding window for feature selectivity. The
slide can be adjusted to emphasize coefficients within a specific range of variation.
To increase the overall energy of a processed image, we assign s a value greater than
one (s > 1). Similarly we may reduce the energy of a processed image by letting
s < 1. When s = 1, the scaling factor s does not contribute to the overall energy
change, and makes the overall operator equivalent to the operator reported by Laine
and Zong [42]. Thus, by selecting gain values and local windows of energy, we may
achieve a more desirable enhancement. Figure 2.3 presents an adaptive gain function
with feature selectivity.
CHAPTER 3
DYADIC WAVELET REPRESENTATIONS
In this chapter, we describe multiscale wavelet transforms at dyadic scales adopted
in our preliminary algorithms for denoising and enhancement as well as edge detec
tion. Waveletbased denoising and enhancement are presented in this chapter while
edge detection through wavelet maximum representation will be described in Chap
ter 6. Since we use a dyadic wavelet transform primarily for discrete signal and
digital image processing, the transformation is presented in the discrete domain from
an implementation point of view. For our purposes, we are only interested in the
overcomplete (redundant) representation under a finitelevel discrete dyadic wavelet
transform. For more theoretical work and continuous dyadic wavelet transforms, Mal
lat and Zhong have presented indepth details [51, 69]. DWTbased algorithms for
signal/image processing are developed on dyadic wavelet representations described
in this chapter, image restoration and enhancement operators presented in the last
chapter.
3.1 Discrete Dyadic Wavelet Transform
The discrete dyadic wavelet transform developed by Mallat and Zhong [51] has
been previously applied to areas, including edge detection, texture analysis, noise
reduction, and image enhancement. Multiscale representation under a dyadic wavelet
transform provides a useful framework for characterizing features in terms of sharp
variation points. For denoising and enhancement purposes, compactly supported
wavelets can be utilized to eliminate noise and sharpen contrast within structures
26
and along object boundaries without affecting distant features [39, 35, 73] because
wavelet transforms localize feature representations.
3.1.1 OneDimensional Dyadic Wavelet Transform
First, we describe the discrete dyadic wavelet transform in one dimension and
then extend it to two dimensions. For discrete signal and digital image processing,
only a finitelevel discrete dyadic wavelet transform is usually needed for practical
applications. A Jlevel discrete dyadic wavelet transform of a 1D discrete function
f(n) E 12(Z) can be represented as
W[f (n)] = {(VW[f(n)])i
where W [f(n)] is a wavelet coefficient at scale 2j (or level j), location n E Z, Sj[f(n)]
is a coarse scale approximation at the final level J and position n. Wavelet coefficient
Wj[f(n)] and scaling approximation Sj[f(n)] at level j can be defined as
+00
Wj[f(n)] = f 2* J(n) f(n')2 (n n'), (3.2)
7'=o
+00
Sj[f(n)] = f (n) E= f(n')p2j(n n'), (3.3)
n'=oo
respectively, where 2(n) = n) () and 2() = j () are analysis wavelets
and scaling functions dilated at scale 23. The inverse discrete dyadic wavelet trans
form can be represented as W1 (W1[W[f(n)]]). For a perfect decomposition and
reconstruction, we have
J
f(n) = WY[W[f(n)]] = Sj[f (n)] 2 (n) + W )[f ] (n),] 72(n), (3.4)
j=1
where (p2 (n) = p2J (n). In order to get a perfect reconstruction of a 1D discrete
function, analysis and synthesis wavelets and the scaling function should satisfy
J
I(w)12 = (2jw)'(2jw) + k1(2Jw)12
j=1
+oo
= (2 w)7(2 w). (3.5)
j=1
The finitelevel dyadic wavelet decomposition in (3.1) forms a complete representa
tion for a Jlevel dyadic wavelet transform. For a particular class of dyadic wavelets,
such as the first order derivatives of spline smoothing functions, the finitelevel di
rect and inverse discrete dyadic wavelet transform of a 1D discrete function can be
implemented in terms of three filters, H, G, and K. The three filters should satisfy
the following condition
IH(w) 2 + G(w)K(w) = 1, (3.6)
where H(w) is a low pass filter, G(w) and K(w) are high pass filters.
The dyadic wavelet decomposition in Equation (3.1) can be formulated in terms
of the following recursive relations between two consecutive levels j and j + 1 in the
Fourier domain as
Wj+l[f(w)] = G(2jw)Sj[f(w)], (3.7)
i+i[f(w))] = H(2jw)S [f(w)], (3.8)
where j > 0, and So[f(w)] := f(w). The reconstruction So[f(w)] from a dyadic
wavelet decomposition can be represented recursively as (j changes from J 1 to 0)
Sj[f(w)] = Wj+l[f(w)]K(23w) + Sj+4[f(w)]H(2jw),
(3.9)
Figure 3.1. A 3level DWT decomposition and reconstruction of a 1D function.
where H is the complex conjugate of H. The DWT decomposition and reconstruction
based on the above recursive relations are shown as a block diagram in Figure 3.1.
The process of wavelet decomposition is referred to as wavelet analysis while the
wavelet reconstruction process is sometimes called wavelet synthesis.
3.1.2 TwoDimensional Dyadic Wavelet Transform
In the rest of this section, we shall present discrete dyadic wavelet analysis and
synthesis of a 2D discrete function (image). The decomposition will produce both
hightomiddle frequency signal details (wavelet coefficients) and a low frequency
scaling approximation (scaling coefficients) of an image at some final level of analysis.
Similarly a finitelevel discrete dyadic wavelet transform is desirable for our digital
image processing. A Jlevel discrete dyadic wavelet transform of a 2D discrete
function f(na, ny) E 12(Z2) can be represented as
W[f(n,,n,)] = {(Wf [f (n, ny)])d=1,2,1
F ] w II:
(3.10)
where W [f(nz, ny)] is a wavelet coefficient at scale 23 (or level j), position (n.x, n),
and spatial orientation d (1 for horizontal and 2 for vertical), S [f(nx, ny)] is a coarse
scale approximation at the final level J and position (nx, ny). Wavelet coefficient
Wf[f(n,, ny)] and scaling approximation Sj[f(nx, ny)] at level j can be defined as
+00 +00
Wf[f (nx, ny)]= f *d (n,ny)= n f(m', n')V ( m', n n), (3.11)
m'=oo n'=00
+00 +00
Sj[f(nz, n,)] = f (pj (n, ny)= E f(m', n')c (m m', n n'),(3.12)
m'=o n'=oo
respectively, where cd4(nI,n,) = pd(nj, ) and 2(n, ny) = ) are
analysis wavelets and scaling functions dilated at scale 2, and d = 1, 2 represents
horizontal or vertical spatial orientation. In our approach for denoising and enhance
ment, we are interested in some basis wavelets which are the first order derivatives
of continuous smoothing functions V(x, y); thus, l'(nx, ny) and 2(nx, ny) can be
formulated as
(nx, ny) (x,y) 2(nny) = 9(xy) (3.13)
y=ny y=ny
Convolution with dilated 1 (nx, ny) and 2 (nx, ny) produces sharp variations along
horizontal and vertical directions for salient features. In wavelet frame represen
tations, we can employ a different synthesis basis wavelet yd(nx, ny) for the recon
struction of the original 2D discrete function. The inverse discrete dyadic wavelet
transform can be represented as W' (W1[W[f(nx, ny)]]). For a perfect decompo
sition and reconstruction,
f(n, ny) = W' [W[f(nx, n)]]
J 2
= Sj[f(n, ny)] *( 2(nx, ny) + Wf [f (nz, ny)] 72 (nx, ny), (3.14)
j=l d=l
where (~J (n, ny) = P2(nt, ny). In order to get a perfect reconstruction of a 2D
discrete function, analysis and synthesis wavelets, the scaling function should satisfy
J 2
I (wx, W )2 = E d(2Jwx, 2jY)d(2Jw, 2jw) + 1I(2jwu, 2JWy) 2. (3.15)
j=l d=l
The finitelevel dyadic wavelet decomposition in (3.10) generates a complete repre
sentation for a Jlevel dyadic wavelet transform. For a particular class of 2D dyadic
wavelets, such as the first order directional derivatives of spline smoothing functions,
Mallat and Zhong [51] showed that the finitelevel direct and inverse dyadic wavelet
transform of a 2D discrete function can be implemented in terms of four 1D filters,
H, G, K, and L. The four filters should satisfy the following perfect decomposition
and reconstruction conditions
H(w)12 + G(w)K(w) 1, (3.16)
1 + IH(w)2 2
L(w) = (3.17)
2
Mallat and Zhong [51] also showed how to design 1D finite impulse response (FIR)
filters, H, G, K, and L, for a 2D wavelet transform.
Similar to the 1D case, a 2D dyadic wavelet decomposition in Equation (3.10) can
be formulated in terms of the following recursive relations between two consecutive
levels j and j + 1 in the Fourier domain as
WT+/f[f (uy, w)] = G(2jwu) S[f (w, Wy)], (3.18)
W+l [f (w ay)] = G(2Wy)S [f (wx, Wy)], (3.19)
S+l[f(wx, y)] = H(2jw2)H(2j, )Sj[f(w,w y)],
(3.20)
1 I
Wif) :
Figure 3.2. A 3level DWT decomposition and reconstruction of a 2D function.
where j > 0, and o[f(wx, wy)] := f(w,, wy), the Fourier transform of f(nx, ny). The
reconstruction S0[f(wx, wy)] from a dyadic wavelet decomposition can be represented
recursively as
S [f(wX, wy)] = Wi+1l[f(wX, wy)]K(2jwx)L(2jwy) + W^? [f(w,, Wy)]L(2jwx)K(2jwy)
+Sj+l[f(wx, wy)]HT(2jx)H(2 j y), (3.21)
where H is again the complex conjugate of H. A 2D DWT decomposition and
reconstruction based on the above recursive relations are shown as a functional block
diagram in Figure 3.2 for J = 3. For a pair of 2D analysis and synthesis filter banks
shown in Figures 3.3 and 3.4, reconstructed f*(nx, ny) is equal to f(nx, ny) when no
processing is performed on V[f(n,, ny)]. The 2D analysis and synthesis filter banks
in Figures 3.3 and 3.4 are constructed using FIR filters shown in Table 3.1 where, for
instance, H(w) = En h(n)ein.
Profiles
j=1
j=2
j2
j=3
dccap (j=3)
0 4
 2 0 3
' It
05 i
04
02I
3 2 A 1 2 3
03
02
2 I
0 i
05
3 2 0 I 2 3
09
2
OI
07
04
01
a
Figure 3.3. A 2D analysis filter bank.
d=l
d=2
........
Profiles d=1 d=2
j=0 ~ 
o4
0 o
o01
0=2
05
o4
02
01
071
decap (j=3) 1 , a 1 2
Figure 3.4. A 2D synthesis filter bank.
Table 3.1. Impulse responses of filters H(w), G(w), K(w), and L(w).
3.2 DWTBased DeNoising and Feature Enhancement
In this section, we first introduce algorithms for noise reduction and feature
restoration or enhancement based on an additive noise model presented in Section
2.2.1 and for speckle reduction with feature enhancement based on an approximate
speckle noise model formulated in Section 2.2.2. We then describe the methods and
present formulation for denoising and enhancement based on the operators intro
duced in Sections 2.3 and 2.4.
3.2.1 Algorithm for Additive Noise Reduction and Enhancement
Because denoising and enhancement techniques are incorporated into a frame
work of wavelet representations under dyadic wavelet transforms, our algorithm for
noise reduction and contrast enhancement consists of four major steps. In these
steps, parameterized denoising and enhancement operators are utilized. The sample
experimental results are shown for these operations. The parameters can be fine
tuned to achieve two distinct purposes. One is for denoising with feature restoration
while the other is for image enhancement with noise suppression. They are related
in certain sense, such as removing noise and improving the quality of features. These
FIR filters for m = 4 and c = 2
n h (n) g(n) k(n) l(n)
4 0.001953125
3 0.00390625 0.015625
2 0.0625 0.03515625 0.0546875
1 0.25 1.0 0.14453125 0.109375
0 0.375 1.0 0.36328125 0.63671875
1 0.25 0.36328125 0.109375
2 0.0625 0.14453125 0.0546875
3 0.03515625 0.015625
4 0.00390625 0.001953125
methods are designed to remove noise with feature restoration or enhancement in an
additive noise model. The four steps of a DWTbased denoising and enhancement
method are listed as follows:
1. Carry out a DWT to obtain a complete representation of noisy data in the
wavelet domain.
2. Shrink transform coefficients within the finer scales to partially remove noise.
3. Emphasize features through a nonlinear pointwise operator to increase energy
among features within a specific range of variation.
4. Perform an inverse DWT and reconstruct the signal/image.
Unlike Donoho and Johnstone's methods [19] for denoising, an advantage of this
method is that it also applies feature enhancement to further improve the performance
of signal/image restoration. This algorithm has an ability to suppress noise (without
amplifying noise) when applied for contrast enhancement compared to enhancement
only methods.
3.2.2 Algorithm for Speckle Reduction with Feature Enhancement
Speckle noise was modeled as approximate multiplicative noise in Section 2.2.2.
Similar to the method in Jain [30], we apply a homomorphic approach to reducing
speckle noise. The algorithm consists of six major steps. The six steps of a DWT
based denoising and enhancement method for the speckle noise model are listed as
follows:
1. Perform a logarithmic transform to convert an image containing multiplicative
noise into an image with additive noise.
2. Carry out a DWT and obtain a complete representation of the logtransformed
image in the transform domain.
3. Shrink coefficients within the finer scales to partially remove noise energy.
4. Emphasize features through nonlinear pointwise processing to increase the
energy among features within a specific range of variation.
5. Perform an inverse DWT and reconstruct the denoised and enhanced image so
that it approximates its noisefree original in log scale with features enhanced.
6. Finally, perform an exponential transform on the reconstructed image to con
vert it from log scale to linear scale. The resulting image is now denoised and
enhanced.
This method takes a similar homomorphic transform to convert multiplicative noise
into additive noise. Unlike Jain's method [30], we incorporate a feature enhancement
mechanism into the noise reduction procedure to sharpen blurred features (feature
restoration or enhancement) after denoising.
Wavelet representations under discrete dyadic wavelet transforms were described
in Section 3.1 for both one and two dimensions. These denoising and contrast
enhancement schemes are based on wavelet shrinkage and feature emphasis on top
of the wavelet representations. Wavelet shrinkage is a technique to uniformly reduce
wavelet coefficients in order to remove noise coefficients for the purpose of denoising.
Feature emphasis, on the other hand, is trying to increase the magnitudes of feature's
coefficients to gain energy for low contrast features. Below, we describe how to
perform DWTbased denoising and enhancement.
3.2.3 DWTBased DeNoising
Since dyadic wavelet transforms with first order derivatives of smoothing functions
as basis wavelets can efficiently identify features with sharp variation, we are able
to achieve the objective of noise reduction through either hard thresholding or soft
thresholding. Hard thresholding preserves features better while soft thresholding can
achieve the effect of smoothness.
Here we describe thresholdbased denoising in two dimensions. One dimensional
case is similar. To achieve the purpose of denoising through hard thresholding, we
can modify DWT coefficients for noise reduction by
W fd,* [f(n, ny)] = MfTH (W[f (nx, ny)]/Mf, (3.22)
Mf = max(Wj [f (n, ny)] ), (3.23)
where d = 1,2, j = 1,...,k, k < J, and tf is, in general, a threshold related to noise
level and scale. Parameter tJ can be directionally related if we have orientation prefer
ence. TH is the hard thresholding operator presented in Section 2.3.1. The threshold
tj should be selected to possibly remove most noise coefficients while preserving fea
ture coefficients. Selection of thresholds in [71, 73] was trialanderror based. The
selection can be guided by examining the histogram and energy distribution of coef
ficients. Wavelet transforms generate a small number of large coefficients carrying a
significant amount of energy, especially from fine to coarse scales, for sharp variation
points while producing a large number of small coefficients mostly corresponding to
noise. Thresholds decrease from fine to coarse scales because noise energy is smoothed
out through repeated smoothings (scaling) by low pass filtering. This point is made
clear, as shown in Figure 3.5 for Donoho and Johnstone's synthetic signal "Blocks".
The guideline is to select decreasing thresholds which can remove a great number of
small coefficients carrying most noise energy and keep a limited number of large coef
ficients for feature energy. Thresholds through fine to coarse levels may be regulated
by a decreasing function which will be discussed momentarily.
The Histogram of Wavelet Coefficients
An Aifl.
500
g400
S300
S200
100
Wavelet Coefficient Magnitude
2 4
Wavelet Coefficient Magnitude
Selected Threshold at Level 2
04
0 2 4 6
Wavelet Coefficient Magnitude
1
600
400
I :o
0 2 4
Wavelet Coefficient Magnitude
The Energy of Wavelet Coefficients
Selected Threshold at Level 1
60
i40
20
0
0 2 4 6
Wavelet Coefficient Magnitude
i nn.
80
60
S40
20
0 2 4 6
Wavelet Coefficient Magnitude
Selected Threshold at Level 2
0 o
!0
0 2 4 6
Wavelet Coefficient Magnitude
0 2 4 6
Wavelet Coefficient Magnitude
Figure 3.5. Coefficient and energy distributions of signal "Blocks".
300
o200
100
600
4400
200
Selected Threshold at Level 4
Selected Threshold at Level 3
LO~
When noise level is high, hard thresholding denoising may not be able to achieve
overall smoothness. If this is the case, we can carry out soft thresholding based de
noising. For reducing noise and achieving smoothness effect, we can modify DWT
coefficients by
wd*[ f(nx, n,)] = M Ts(W [f (n, ny)]/M), t) (3.24)
Mf = max(IWd[f (n:, ny)] ), (3.25)
Snx ny
where d = 1, 2, j = 1, ..., k, k < J, and tj is a threshold usually related to noise level
and scale. Ts is the soft thresholding operator presented in Section 2.3.2. Thresholds
can be selected similarly based on the above discussion. Wavelet coefficients are
normalized to the range between 1 to 1 before thresholding operations.
3.2.4 Regulating Threshold Selection through Scale Space
Donoho and Johnstone's method of soft thresholding uses a single global threshold
[19, 20] under orthonormal wavelet transforms. Since noise coefficients under a DWT
have average decay through finetocoarse scales, we can regulate both soft and hard
thresholding by applying coefficient dependent thresholds at different scales. The
regulated threshold tf can be computed through a linearly decreasing function
S Tmaz a(j 1)) d if Tm (j 1) > Tin,3.26)
Tmin a otherwise,
where Jd is the standard deviation, a is a decreasing factor between two consecutive
levels, Tmax is a maximum factor related to af while Tmin is a minimum factor,
1 < j < J, and d E {1, 2}. When the noise level in original corrupted data is
unknown, some methods use the standard deviation to approximate the noise level, so
Tmax
Tmin 
1 J Level
Figure 3.6. A sample scaling factor function.
the thresholds are related to ao. Figure 3.6 shows a sample scaling factor function for
the computation of regulated thresholds. Our denoising algorithms are implemented
in a way that a, Tmax, and Tmin can be interactively tuned to achieve distinct effect
of noise reduction.
3.2.5 DWTBased Enhancement with Noise Suppression
Through either a nonlinear gain function or generalized adaptive gain nonlin
ear operator, we can achieve the effect of contrast enhancement for certain FOI by
processing DWT coefficients as
W'* [f (n, ny)] = M Eop(W[f (nx, ny)]/M), (3.27)
M = max(IW d[f(nx,ny)]l), (3.28)
t n .n .
where position (nx, ny) E D, the domain of f(n,, ny), d = 1, 2, j E {k,..., J}, and
1 < k < J. The enhancement operator Eop can be ENL or EG .; presented in
Section 2.4. Since these operators are defined on input values between 1 to 1, we
normalize wavelet coefficients before applying the operators.
0 02 0.4 0.6 0.8 1
(c). WCs of Orignal Box
8
7
6
5
4
0 0.2 0.4 0.6 0.8 1
(b). Noised Box
0 0.2 04 0.6 0.8 1
(d). WCs of Noised Box
8
7
6
4
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 I
(c). HardThresholded WCs of Noised Box
8
7 I
6
5
4
0 0.2 0.4 0,6 0.8 I
(b) Reconstoicied after Soflibresholdoig
(b). Reconstructed after SoffThresholding
30
20
10
0 0.2 04 0.6 08
(d). SoftThresholded WCs of Noised Box
8
7
6
5
4
0 0.2 0.4 0.6 0.8 1
Figure 3.7. PseudoGibbs phenomena. (a) Orthonormal wavelet transform of an orig
inal signal and its noisy signal with added spike noise. (b) PseudoGibbs phenomena
after both hard thresholding and soft thresholding denoising under an OWT.
20 20
10 10
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(a) Original Signal
(b) Noisy Signal
(c) Original DWT Coefficients (d) Noisy DWT Coefficients
Figure 3.8. Multiscale discrete wavelet transform of an original and noisy signals.
10.
0lo ,
S 01 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10[ ''A'
10
0 0.1 0.2 0.3 0.4 0.5 0S6 0.7 0.8 0.9 1
20
10
0 0.2 0 0.4 0.5 0.6 0.7 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
0 0.1 0.2 0 3 0.4 0.5 0.6 0 7 0.8 0.9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0o.5 0.6 0.7 0.8 0.9 1
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 1
Figure 3.9. DWTbased reconstruction after (a) hard thresholding, (b) soft thresh
olding, and (c) soft thresholding with enhancement.
The Enhanced Signal and the Processed DWT Coefficients
20
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 018 0.9 1
5 I 1
5[ I I i
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5s
0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 0.9 1
5
J "~
0 0.1 02 03 04 0.5 0.6 0.7 0.8 0.9 1
20
The Enhanced Signal and the Processed DWT Coefficients
20
10
0 A
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5
0 0.1 0.2 0 3 0.4 0.5 0 6 0.7 0.8 0.9 1
50 01 02 03 04 05 06 07 08 0.9 1
0 0.1 0.2 03 0.4 0.5 04 07 0.8 0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The Enhanced Signal and the Processed DWT Coefficints
20
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5
0 01 02 03 04 05 06 0,7 08 09
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I
0 0.1 02 03 04 05 06 07 08 09
0 0
5 [ I I v
0 O.A 0.2 0.3 0.4 0.5 0,6 0.7 0.8 0.9 1

3.3 Application Samples and Analysis
In this section, we present the experimental results of our algorithms on some
sample signals and images. First, we show that these algorithms are less affected
from pseudoGibbs phenomena through a simple and illustrative experiment. Our
results are compared to Donoho and Johnstone's results to show this effect. We
then present denoising and enhancement results for additive noise and speckle noise
models.
3.3.1 Less Affection from PseudoGibbs Phenomena
Coifman and Donoho [8] showed that both hard and soft thresholding denoising
under an orthonormal wavelet transform produced undesired sideeffects, including
pseudoGibbs phenomena. To solve the problem, they [8] presented translation
invariant denoising methods to overcome the artifacts partially caused by the lack
of shiftinvariance of an OWT. Their methods alleviated the problem by making it
less obvious, but oscillations after denoising remained visible. Several experimental
results showed that our algorithms were less affected from pseudoGibbs phenomena
[73]. We have used a simple and intuitive synthetic signal to demonstrate this effect of
our algorithms when compared to Donoho et al.'s methods. Our experimental results
on Donoho and Johnstone's four synthetic signals also demonstrate this point.
Figures 3.7, 3.8, and 3.9 are used to show that our methods are relatively free
from pseudoGibbs phenomena. We generated a synthetic signal to illustrate what
may cause the sideeffect and how our methods can basically avoid it. Figures 3.7(a)
shows an original signal, its noisy version with added spike noise, and the orthonor
mal wavelet coefficients of the original and noisy signals. Figure 3.7(b) shows the
effect of pseudoGibbs phenomena under Donoho et al.'s hard thresholding and soft
thresholding methods through WaveLab (a software package from Donoho's research
group). Notice that a feature of sharp variation produces not only large coefficients
but also small coefficients under an OWT. The small coefficients are removed under
both hard and soft thresholding methods. A typical orthonormal wavelet usually has
at least certain oscillations [12, 64] in order to satisfy the (admissibility) condition of
an orthonormal wavelet
f +00 1 )(W) 12
/o (I dw < oo,
where #(w) = f+ /i(x)eixwdx. In the spatial domain, the corresponding contin
uous wavelet function O(x) has sufficient decay and satisfy
f +00
J (x)dx = 0.
OO
Figures 3.9(a) and 3.9(b) present our denoising results under regulated hard
thresholding and soft thresholding. Both methods remove the noise without causing
pseudoGibbs phenomena, but soft thresholding also smoothes features (step edges)
a little bit. The features are basically restored through our enhancement mechanism
in figure 3.9(c). This experiment is used to illustrate the fact that the results of our
algorithm are less affected from the sideeffect (pseudoGibbs phenomena) compared
to the results from Donoho et al.'s methods.
3.3.2 Additive Noise Reduction and Enhancement
Based on an additive noise model, here, we present application results of de
noising and enhancement for synthetic signals and medical images. The first part is
targeted for signal/image restoration while the second part is for enhancement with
noise suppression.
20 8 1 1
*:^U3Wnj
1400 600 80 1000 1200 0i0'  1800 200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
2 0 0 1 1 1
10 1 1 1 1
10t  i  i   i i i  i i
0 200 400 600 800 1000 1200 1400 1600 1800 2000
20 0 0 2 0
10 
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
30
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
4 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0  I I j
0 200 400 600 800 1000 1200 1400 1600 1800 2000
40 o I I I I I I 0
(b) "Bumps"
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
40 I
0 0 0 A 0 0 6
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
(c) "HeaviSine" (d) "Doppler"
Figure 3.10. DeNoised and feature restored results of DWTbased algorithms; first
row: original signal, second row: noisy version, third row: denoised only result, and
fourth row: denoised and enhanced result signal.
DeNoising with Feature Restoration
In this part of experiments, our denoising and enhancement techniques are pri
marily used for signal/image restoration. To achieve this objective, we want to reduce
noise and restore salient features. Since noise reduction usually causes features to be
blurred, our enhancement methods are deployed to sharpen the blurred features.
Figures 3.10 displays denoised with feature restored results of our DWTbased
algorithms. First rows of figure 3.10(a)(d) are original signals, second rows are noisy
signals, third rows are denoised only results, and the last rows are denoised with
Figure 3.11. DeNoising and enhancement. (a) Original signal. (b) Signal (a) with
added noise of 2.52dB. (c) Soft thresholding denoising only (11.07dB). (d) DeNoising
with enhancement (12.25dB).
(b) (c)
Figure 3.12. DeNoising and enhancement. (a) Original image. (b) Image (a) with
added noise of 2.5dB. (c) Soft thresholding denoising only (11.75dB). (d) DeNoising
with enhancement (14.87dB).
2
0
0 50 100 150 200 250
0 50 100 150 200 250
0 50 100 150 200 250
0 50 100 150 200 250
feature restored result signals. Our results are pretty close to Coifman and Donoho's
new and best results produced by the full cyclespinning translationinvariant de
noising algorithm which is computationally more complex. For their "Bumps" and
"Doppler" test signals, our results are better than their best results. When processing
a noised signal or image with lowcontrast, this algorithm can be used to boost up
the contrast through adding external energy to its signal energy by specifying a larger
gain factor.
Figures 3.11 and 3.12 show our denoising and enhancement results on Mallat
and Hwang's test signal and image. The signal and image are corrupted by noise
of higher levels, but we get better results (higher recovery SNR) than Mallat and
Hwang's results on the same signal and image with less noise.
Enhancement with Noise Suppression
In this part of experiments, we try to achieve contrast enhancement without
amplifying noise. Figures 3.13 and 3.14 show the denoising and enhancement results
on two MRI head images with unknown noise level. The experimental results of de
noised and enhanced images from our algorithm are visibly and quantitatively better
than the results from the thresholdingbased methods alone for denoising, especially
for high level noise.
3.3.3 Speckle Reduction with Feature Enhancement
Speckle reduction and contrast enhancement can be accomplished in the trans
form domain by judicious multiscale nonlinear processing of wavelet coefficients
(Wd[f(nx, ny)])d=1,2, 1
approximate speckle noise model in Section 2.2.2, we can usually separate the noise
component from a desired function. Wavelet transforms help to further distinguish
signal from noise in the spatialscale space.
(b) (c)
Figure 3.13. DeNoising and enhancement. (a) Original MRI image.
only. (c) DWTbased denoising with enhancement.
(b) DeNoising
(b) (c)
Figure 3.14. DeNoising and enhancement. (a) Original MRI image. (b) DeNoising
only. (c) DWTbased denoising with enhancement.
Figure 3.15. An algorithm for speckle reduction and contrast enhancement.
(a) (b) (c)
Figure 3.16. Results of denoising and enhancement. (a) A noisy ED frame. (b)
Wavelet shrinkage denoising only method. (c) DWTbased denoising and enhance
ment.
Our multiscale homomorphic algorithm, as shown in Figure 3.15, for speckle re
duction and feature enhancement was tested on echocardiograms of varying quality.
These image sequences were acquired from the parasternal shortaxis view. Figures
3.16 and 3.17 show the results of denoising with or without feature enhancement
on end diastolic (ED) and end systolic (ES) frames. The speckled original frames
are shown first. Results from wavelet shrinkage denoising only and denoising with
enhancement are shown in the Figures 3.16(b) and 3.16(c) respectively. Figure 3.18
shows a nonlinear operator for enhancing the image in Figure 3.17(a). This operator
looks much different from Figure 2.3 because of the log transform effect. Experimen
tal results are also compared with other speckle reduction techniques, such as median
filtering, extreme sharpening combined with median filtering [11, 44], homomorphic
Wiener filtering, and a wavelet shrinkage denoising [19, 18]. Figures 3.19 and 3.20
show sample results of the above mentioned methods on two typical frames from two
different echocardiographic sequences with the left ventricle as the region of interest.
Figure 3.19(a) is sample noisy image. The result of median filtering with a 5x5
mask is shown in Figure 3.19(b). Figure 3.19(c) displays sample result of extreme
sharpening combined with median filtering. The result from homomorphic Wiener
(a) (b) (c)
Figure 3.17. Results of denoising and enhancement. (a) A noisy ES frame. (b)
Wavelet shrinkage denoising only method. (c) DWTbased denoising and enhance
ment.
filtering is shown in Figure 3.19(d). The last two images in Figures 3.19(e) and
3.19(f), display the results from wavelet shrinkage denoising only and our denoising
and enhancement algorithms. The algorithm produces smoothness inside a uniform
region and improves contrast along structure and object boundaries in addition to
speckle reduction. The denoised and enhanced results of noisy echocardiographic
images from this algorithm appear to outperform the results from soft thresholding
denoising alone. Our current algorithm is implemented such that most parameters
are set dynamically for adaptive denoising and feature enhancement.
3.4 Clinical Data Processing Study
A study of clinical image processing was conducted to investigate the effect of
denoising on the consistency and reliability to manually defined borders of the left
ventricle in 2D shortaxis echocardiographic images [70]. Experimental results in
dicate the algorithm is promising. Myocardial borders manually defined by expert
observers exhibit less variation after denoising. It seems that in echocardiograms,
where no real borders are clearly visible and incomplete, expert borders usually in
dicate a close range where real borders may occur. When two expert borders agree
Generalized Adaptive Gain
I I II
0.8 ii
0.6 
0.4 
0.2 
0.6 ii
I I I
II I
1 0.8 0.6 0.4 0.2 O 0.2 0.4 0.6 0.8
T2 T3
Figure 3.18. A generalized adaptive gain function for processing an echocardiogram
in Figure 3.17(a).
with each other, the range of real borders is more likely limited around the two expert
borders. The study of clinical image processing shows that denoising and feature
enhancement help to improve the consistency and reliability of manually defined
borders by expert observers.
The set of test images in our study of clinical image processing was selected
from an echocardiographic database exhibiting diverse image quality. Sixty systolic
sequences of 2D shortaxis echocardiographic images were selected. Half of the test
images were rated as good quality while the rest were considered as poor quality.
For more details about how these echocardiographic sequences were acquired, we
refer the reader to Wilson and Geiser [66]. Statistical results have shown that there
is some improvement in consistency and reliability for manually defined borders by
expert observers examining denoised images compared to their original noisy images.
Quantitative measurements were calculated in terms of the mean of absolute border
differences (MDistDiff) in distance (mm) and the mean of border area differences
(d)
Figure 3.19. Results of various denoising methods. (a) Original image with speckle
noise. (b) Median filtering. (c) Extreme sharpening combined with median filtering.
(d) Homomorphic Wiener filtering. (e) Wavelet shrinkage denoising only. (f) DWT
based denoising with enhancement.
(a) (b)
(d) (e)
Figure 3.20. Results of various denoising methods. (a) Original image with speckle
noise. (b) Median filtering. (c) Extreme sharpening combined with median filtering.
(d) Homomorphic Wiener filtering. (e) Wavelet shrinkage denoising only method.
(f) DWTbased denoising and enhancement.
Table 3.2. Quantitative measurements of manually defined borders.
All Test Images Good Images Poor Images
Ori vs Enh Ori vs Enh Ori vs Enh
MDistDiff Endo (in mm) 2.1040 1.8168 1.5972 1.5322 2.6118 2.1014
Epi (in mm) 1.7846 1.6743 1.3979 1.5886 2.1713 1.7601
MAreaDiff Endo (in cm2) 2.3731 1.8893 1.6597 1.4543 3.0865 2.2058
Epi (in cm2) 2.5676 2.0799 1.5823 1.9540 3.5528 2.3243
(MAreaDiff) in cm2. The border difference was measured by its close approximation
in 64 radial directional difference from an estimated center [66] of the left ventricle.
Manually defined borders by experts looking at poor images were improved more
than those of good images after denoising. The statistical results of quantitative
measurements of two sets of expert manually defined borders are shown in Table 3.2.
The statistical computation results listed under the column "Ori" are the quantitative
measurements between two sets of expert borders on the original speckled images
while the results under the column "Enh" are based on the denoised and enhanced
images. It is worth mentioning that a single set of denoising and enhancement
parameters were used to process all the test echocardiographic images used in this
study. We suggest that a single value set of parameters should be enough for de
noising and enhancing a class of images with a similar noise pattern and selected
features.
Figure 3.21 shows the correlation between the areas delineated by the two expert
observers. The four diagrams in Figure 3.21(a) present the correlation of ED Epi
(epicardial) border areas, ES Epi border areas, ED Endo (endocardial) border areas,
and ES Endo border areas on the original noisy images. The four diagrams in Fig
ure 3.21(b) show similar results for the denoised images with features restored or
enhanced. The solid lines in the figure are the linear regression lines, while the dash
and dotted lines are ideal regression lines. From the diagrams, it is clear that the
56
ED Epi Area Origial Image ES Epi Ar Oiginal Image ED Epi Arem DeNoised Image ES Epi Area DNoised Image
S0 80 
60
b ++ 60
So o
Observer 1 Observer I Observer I Observer I
ED Endo Area Original Image ES Endo Area Original Image ED Endo Ara DcNoised Image ES Endo Area DeNoised Image
40 30 40 30 ,
30 30 20
S20 15 20 15
0 + +0 310
10 10 +
5 5
0 00 01
o2   , I Q2 '.   o  
0 10 20 30 40 0 10 20 30 0 10 20 30 40 0 10 20 30
Observer I Observer I Observer I Observer
(a) (b)
Figure 3.21. Area correlation between manually defined borders by two expert car
diologist observers.
points which represent the two expert border areas on the same denoised image are,
in general, more toward the ideal regression line. Additional improvement can be seen
on the Endo area correlation for the denoised images. For most echocardiograms in
the study, there is usually less Endo border information than Epi border information.
Noisy border information (low signaltonoise ratio (SNR)) affects border interpola
tion by human observers for the manually defined borders. After denoising, Endo
border information in terms of SNR is improved, so the expert border areas tend to
agree with each other, especially ES Endo areas. The statistical computation results
shown in Table 3.2, show evidence for this analysis.
Figure 3.22 shows the distributions of mean border differences on the original
images; (a) the distribution of Epi ED border differences, (b) the distribution of Epi
ES border differences, (c) the distribution of Endo ED border differences, and (d) the
distribution of Endo ES border differences. Figures 3.23(a)(d) show the distributions
of mean border differences on the enhanced images, similar to Figures 3.22(a)(d).
The solid lines in Figures 3.22 and 3.23 are the third order polynomial fitting curves in
a leastsquares sense. With the same scale for both Figures 3.22 and 3.23, Figure 3.23
Epi ED Border Difference Distribution (Graph Date: 24Jul97)
S10 20 30 40
Index of Original Image Sequences
50 60
Endo ED Border Difference Distribution (Graph Date: 24Jul97)
+
6
Dalhdot Line: Mean
Dashed Line: Mean / 2SD
5 Solid Line Regression Curve
S + +
4 + + + + +
3 4 + + +.+
2.7 .. .. .^^ ^ 
++ +
+ ++ + + + +4+
++ + ++ +
I + +
0
Border Difference Mean 217
1 Boler Difference SD 18
ii
S10 20 30 40
Index of Original Image Sequences
50 60
5 Dashdot Line Mean
Dashed Line Mean 4/ 2SD +
Solid Line Regression Crve +
4 
+ +
+ + +
3 + +
+ +
2 .+ +. . 4   
+ + ^ + + +
2 + .+..+ + +A
+ + + +
0
Border Differnce Mean 1 92
Border Difference SD 102
2tIi
0 10 20 30 40
Index of Original Image Sequences
Figure 3.22.
Border difference variation on the original images. (a) Distribution
of Epi ED border differences. (b) Distribution of Epi ES border differences. (c)
Distribution of Endo ED border differences. (d) Distribution of Endo ES border
differences. The solid lines are the third order polynomial fitting curves.
Dashdo. iUe Mean
S Dashed Line Mean 2SD
Solid Line Regression Crve
03
3             .
+ +
0 +2 4 +
2 + + +
++
+ + + + +
044
BoRd Difference Mean = 1.65
Border Difference SD = 0.74
1
0 10 20 30 40 50 60
Index of Original Image Sequences
(b)
Endo ES Border Difference Distrbution (Graph Date: 24Jul97)
6
Dashdn Line Mean
5 Dashed ne Mean l 2SD +
Solid Line: Regression Curve
E 4
3:   :.4
I ..+ ... ++ + + .
S+ + +
S+ +
4 .+4 + + +
0
+, + 4+
+ + +
+4+ 4 + +
Border Difference Mean = 204
1 Border Differenc SD I 13
50 60
Epi ES Border Difference Distribution (Graph Date: 24Jul97)
Table 3.3. Quantitative measurements of interobserver mean border differences in
mm on original versus enhanced images, as shown in Figure 3.25.
EpiED EpiES EndoED EndoES
Ori 4.7 4.0 6.3 4.6
Enh 1.4 2.3 1.2 1.3
shows that border distance differences for enhanced images have smaller means and
standard deviations than the corresponding differences for the original noisy images
as shown in Figure 3.22.
Figure 3.24 shows an example of denoising and image enhancement. Figure 3.25
shows the same images as Figures 3.24 with two expert manuallydefined borders
overlaid. Significant overall improvement on the agreement of two expert borders is
visible from the overlaid borders of the enhanced images compared to the original
images, as shown visually in Figure 3.25 and quantitatively in Table 3.3. The Endo
borders have more improvement than the Epi borders based on quantitative mea
surements and visual appearance. Statistical analysis shows improvement in terms of
the mean of absolute border differences and mean border area differences of denoised
images compared to their original images. From the overall statistical analysis, the
greatest impact is on the expert borders drawn on images with poor image quality,
such as the images in Figure 3.24.
Epi ED Border Difference Distribution (Graph Date: 29Jul97)
) 10 20 30 40
Index of Enhanced Image Sequences
0 10 20 30 40
Index of Enhanced Image Sequences
50 60
50 60
10 20 30 40
Index of Enhanced Image Sequences
0 10 20 30 40
Index of Enhanced Image Sequences
Figure 3.23. Border difference variation on the enhanced images. (a) Distribution
of Epi ED border differences. (b) Distribution of Epi ES border differences. (c)
Distribution of Endo ED border differences. (d) Distribution of Endo ES border
differences. The solid lines are the third order polynomial fitting curves.
Dashdat Line Mean
Duhed Line: Mean ./ 2SD +
+ Solid Lie: Regresion Curve
S ++ +
+ +
+ + + + +
...+ +_ + + + +
+ +
+ 4 4 + + 4+4 +
 
+ +
Border Difference Mean = 183
Border Difference SD = 088
4 Dashdo Line; Mean
Dalid Line: Mean / 2SD
Solid Line Regresion Curve
+ ++
+ ++ + 4+
+ + + +
+
+ ++ + + + ++ + +++++,
0 Border Difference Mea 1.52
Border Difference SD 0.53
1
2
Endo ES Border Difference Distribution (Graph Date: 29Jul97)
Endo ED Border Difference Distribution (Graph Date: 29Jul97)
6
Dashdot Line: Mean
Dulhed Line: Man +/ 2SD
5 Solid Line: Regresson Clv +
    
34 +
+ + ++ + + ++
++ +
+++
+ + + ++ +
+ 4 ++
Border Difference Mean 207
Border Difference SD = 1.09
50 6C
50 60
Dashdot Line Mean
Dashed Line: Mean r 2SD
Solid Line: Regrssion Curve
  
+
+ ++ + + +4 + +
+ + +
+++ + 
Border Difference Man 1.57
Boder Difference SD 0; 69
Epi ES Border Difference Distribution (Graph Date: 29Jul97)
(a) Original ED (b) Original ES
(c) Enhanced ED (d) Enhanced ES
Figure 3.24. Denoising and image enhancement: (a) An original ED frame; (b) An
original ES frame; (c) The enhanced ED frame; (c) The enhanced ES frame.
(a) Original ED (b) Original ES
(c) Enhanced ED (d) Enhanced ES
Figure 3.25. Image and border display: (a) An original ED frame with manually
defined borders overlaid; (b) An original ES frame with manuallydefined borders
overlaid; (c) The enhanced ED frame with manuallydefined borders overlaid; (c)
The enhanced ES frame with manuallydefined borders overlaid. Red and yellow
borders represent the two observers.
CHAPTER 4
SUBOCTAVE WAVELET REPRESENTATIONS
In this chapter, we introduce suboctave wavelet transforms. A suboctave wavelet
transform is a generalization of a traditional dyadic wavelet transform and we use
an example to show the advantage of suboctave representations over dyadic wavelet
representations for characterizing bandlimited features frequently seen in medical
imaging. We formulate both continuous and discrete suboctave representations in
one and two dimensions.
4.1 Introduction
Our DWT based algorithms for denoising and enhancement have achieved im
proved performance compared to other published methods. Through our analysis
and experiments, we observed that a DWT has a limited ability to characterize fea
tures, such as texture, and subtle features of importance in mammographic images.
The traditional DWT is an octavebased transform where scales increase as powers
of two [51]. However, the best representation of a signal's details may exist be
tween two consecutive levels of scale within a DWT [36]. To more reliably isolate
noise and identify features through scale space, we designed a multiscale suboctave
wavelet transform (SWT), which generalizes the DWT. A suboctave wavelet trans
form provides a means to represent. details within suboctave frequency bands of
equallyspaced divisions within each octave frequency band. The theoretical devel
opment of a suboctave wavelet transform and its efficient implementation was briefly
described by Laine and Zong [42], and later explained and extended [43].
62
The initial motivation for us to explore suboctave decomposition was that we had
observed the limitation of dyadic wavelet transforms for characterizing bandlimited
features and sought better frequency resolution for detecting such subtle features. A
dyadic wavelet transform is an octavebased transformation, where scales increase as
powers of two. Daubechies [13] introduced the generalization and extension of wavelet
decomposition and reconstruction under the context of orthonormal wavelet trans
forms by subband splitting and presented early examples. An extension and gener
alization of dyadic wavelet transforms is multiscale suboctave wavelet decomposition
and reconstruction. Both Daubechies's methods and our techniques for suboctave
wavelet transforms have achieved similar (better) frequency localization. However,
we are primarily interested in overcomplete (redundant) wavelet representations, a
generalization of dyadic wavelet representations. Most orthonormal wavelet bases
have the effect of decorrelating features while dyadic wavelet bases correlate salient
features through scales, which is what we are most interested in for enhancement
purposes. In the rest of the chapter, we present the mathematical formulation of
suboctave wavelet bases in both space and frequency domains. The decomposition
and reconstruction procedure is carried out in terms of filter bank theory and band
splitting techniques.
4.2 Continuous SubOctave Wavelet Transform
Through a wavelet transform, a function (input signal) can be represented by
its projection onto a family of wavelet bases for decomposition and possibly perfect
reconstruction. If the family of wavelets bases {pn(x)} is complete and orthonormal,
a wavelet transform with critical sampling is usually referred to as an orthonormal
wavelet transform [13]. A Haar wavelet is a simple example of an orthonormal wavelet.
However, the Haar wavelet is a discontinuous function and is not localized in the
frequency domain. The analysis filters {H, G} for computing an orthonormal wavelet
transform must satisfy the following design constraints
IH(w)12 + IG(w) 2 = 1,
H(w) G(w) = H(w) G(w) = 0. (4.1)
If the family of bases {'n((x)} is complete and linearly independent, but not
orthonormal, the wavelet transform is called a biorthogonal wavelet transform.
Biorthogonal wavelets have dual basis functions. More generally, if the family of
wavelets { n(x)} is not linearly independent (redundant) and overcomplete, they
may form a wavelet frame representation [64].
For a dyadic wavelet transform, the orthonormal constraint is relaxed, so we can
have distinct decomposition and reconstruction wavelets as long as the corresponding
lowpass filter H(w) and highpass filters G(w), K(w) satisfy [51]
IH(w)12 + G(w)K(w) = 1. (4.2)
The above discussion is for one dimension functions. It can easily extend to two
dimensions through a few well known methods [13, 51]. In this section, we focus on
continuous suboctave wavelet transforms. We first discuss onedimensional multi
scale suboctave wavelet transforms and corresponding suboctave wavelet represen
tations. We then introduce twodimensional suboctave wavelet transforms (SWT)
and 2D suboctave wavelet representations.
4.2.1 OneDimensional SubOctave Wavelet Transform
If we further divide an octave frequency band into M equallyspaced suboctave
bands (here, M is limited to a power of two), then M wavelet bases can be used to
capture the detail information of a signal in each suboctave frequency band. The
M wavelet functions are represented as )m(x) E L2(R), where m = {1, 2,.., M},
L2(R) denotes the space of measurable, squareintegrable 1D real functions. An M
suboctave wavelet transform of a 1D function f(x) L2(R) at scale 2i (level j)
and location x, and for an mth suboctave frequency band is defined as
W f(x) = f ')b(x) = f(t)Vt (x t)dt, (4.3)
where M (x) = 1 'm( ) is the dilation of the mth wavelet basis {m(x), at scale 2j,
m = {1, 2, ..., M}, and j E Z. In the frequency domain, we can write
WV f(w) = f (w) m(2'w), (4.4)
by taking the Fourier transform of Equation (4.3). A scaling approximation of a 1D
function f(x) is defined as
Sf (z) = f pj (x). (4.5)
To provide a more flexible choice for the M basis wavelets, we impose that the wavelet
functions satisfy a wavelet frame condition (similar to Mallat and Zhong [51])
+oo M
A < E E Ihm(2Jw) 2 < B,
j=oo m=l
where A and B are positive constants and w E R. In the spatial domain, we have
+oo M
AIIf(x)112 < IIWf(x)12 BIIf(xl2.
j=oo m=1
The function f(x) can be recovered from its suboctave wavelet transform by the
formula
+00 M +00 M
f(x)= E E Wj, f 'y7()= E E f* 1 *7 y(x ), (4.6)
j=oo m=l j=oo m=l
where y7"(x) is the mth synthesis wavelet for the inverse suboctave wavelet trans
form. The set of frequency response of {I/J(x) I m = 1,2, ..., M} together at any scale
23 are required to capture the details within an octave frequency band. Finally, for
perfect reconstruction of f(x), analysis wavelets OM(x) and synthesis wavelets y2({x)
should satisfy
+00 M
E E (2jw) Im (2) = 1. (4.7)
j=oo m=1
Equation (4.7) can be obtained by taking the Fourier transform on both sides of
Equation (4.6). To ensure exact reconstruction, the frequency axis is covered by
both analysis and synthesis wavelets. Thus, the wavelets Tm(x) can be any functions
whose Fourier transforms satisfy Equation (4.7). There are certainly many choices
for analysis and synthesis wavelets that satisfy Equation (4.7). For denoising and
feature enhancement purposes, we are interested in the class of wavelets which are
an approximation to the first or second order derivatives of a smoothing function,
such as spline functions of any order. A 1D suboctave wavelet transform can be
easily extended to 2D by computing suboctave wavelet coefficients along horizontal
and vertical directions [51], as explained next in the following section. Extensions to
higher dimensions are straight forward and analogous.
4.2.2 TwoDimensional SubOctave Wavelet Transform
Daubechies described two ways to extend 1D orthonormal wavelet transforms
to two dimensions [13]. Here we adopt the way of dyadic wavelet extension to two
dimensions introduced by Mallat and Zhong [51] by computing suboctave wavelet
coefficients along horizontal and vertical directions. An M suboctave wavelet trans
form of a 2D function f(x, y) E L2(R2) at scale 2j (level j) and location (x, y), for
the mth suboctave frequency band is defined as
Wdmf(x, y) = f m(, y),
(4.8)
where mdm Y(x, y) = f1d'm(,, ), d = {12} (for horizontal and vertical directions),
m = {1, 2,..., M}, and j E Z. L2(R2) denotes the space of measurable, square
integrable 2D functions. In the Fourier domain, Equation (4.8) simply becomes
WT'I ,mf(Wx, w) = f/(Wz, W) ?dm(2JWx, 2JY).
(4.9)
The function f(x, y) can be recovered from its 2D suboctave wavelet transform by
the formula
+oo 2 M
f(x,y) = E W~m f*m(Xy)
j=oo d=l m=l
+00 2 M
= E EE *'im m m", ).
j=oo d=1 m=1
For perfect reconstruction, M2 (x, y) and 72j(x, y) must satisfy
+00 2 M
E E E dm(2jw,, 2jyw) d'm(2jw, 2jwy) = 1.
j=oo d=1 m=1
(4.10)
(4.11)
This exact reconstruction condition is obtained by taking the Fourier transform of
Equation (4.10).
4.3 Discrete SubOctave Wavelet Transform
Continuous wavelet transforms are useful to demonstrate the properties of wavelet
decomposition and reconstruction and are helpful for theoretical approval of the per
fect reconstruction while discrete wavelet transforms are practical important for dis
crete signal and digital image processing. The transform parameters in a suboctave
wavelet transform are continuous variables. This results in a highly redundant rep
resentation. It is possible to discretize these parameters and still achieve perfect
reconstruction [64]. For digital image processing, the suboctave wavelet transform
of a discrete function can be carried out through uniform sampling of a continu
ous suboctave wavelet transform. Below, we describe the discrete formulation of a
suboctave wavelet transform.
4.3.1 OneDimensional Discrete SubOctave Wavelet Transform
In the discrete domain, scales are also discrete and limited by the finest scale of
one unit. A suboctave wavelet transform can similarly be decomposed into dyadic
scales and we can get a perfect reconstruction through its corresponding inverse
suboctave wavelet transform. In general, a function can be decomposed into fine
tocoarse dyadicc) scales by its convolution with dilated wavelets { 2j (x)}jEZ. This
can be done through repeated smoothings (low pass filtering) and detail finding (high
pass filtering). In the discrete domain, because of the limitation of the finest scale,
scales have to be greater than or equal to 1, so we let
+oo M
I(w) 2 = E E m(2w) m(2iw). (4.12)
j=1 m=l1
Thus, for a Jlevel discrete suboctave wavelet transform, we can write
J M
i(w)12 = E E em(24w) im(2ji) + I(2Jw)12. (4.13)
j=l m=1
The notation {Wjl f f(n), Sjf(n) Ij = 1, 2,..., J, and m = 1,2,..., M} is defined as
the wavelet representation of a discrete function f(n) under a Jlevel discrete M
suboctave wavelet transform. Wjf (n) and Sjf(n) are uniform samplings of their
continuous counterparts respectively.
If we let J = 1, then Equation (4.13) becomes
M
Ig(w)12 = (2w) m(2w) + (2w)12. (4.14)
m=r
Let us assume that for each basis wavelet pair Qm(w) and (m(w), there exists a pair of
corresponding functions Gm(w) and Km(w) which need to be determined and (with
certain temporal shift t) satisfy
Om(2w) = (w) Gm(w) e w,
im(2w) = #(w) Km(w) et.
And, for scaling function ((w), there exists a function H(w) which satisfies
b(2w) = <'(w) H(w) eit.
Replacing hm(2w), fm(2w), and ((2w) in Equation (4.14), we obtain a fundamental
relation for suboctave wavelet transforms (SWT); that is,
M
IH(w) 2 + Gm(w) Km(w) = 1. (4.15)
m=l
Figure 4.1. A 3level SWT decomposition and reconstruction diagram for a 1D
Figure 4.1. A 3level SWT decomposition and reconstruction diagram for a 1D
function.
The discrete suboctave wavelet transform of a function f(n) C 12(Z) can be
implemented by using the following recursive relations between two consecutive levels
j and j +1
W4J+lf(w) = Sjf(w) Gm(2'w), (4.16)
Si+lf(w) = if (w) H(2jw), (4.17)
where j > 0, 1 < m < M, and Sof(w) = f(w). And, the reconstruction Sof(W)
from a suboctave wavelet decomposition can be implemented through the recursive
relation
M
Sjf(w) = Sj+lf(w) H(23w) + W Tlf (w) K m(2iw), (4.18)
m=l
low frequency high frequency
SM 3 W2 W1
3 2 1 1 1 1
S M ww
3 3 2 1W2 2 2 2
W3 W3W3W3
Figure 4.2. Divisions of the frequency bands under the SWT shown in Figure 4.1.
where H is the complex conjugate of H. A threelevel M suboctave wavelet decom
position and reconstruction process based on the above recursive relations is shown in
Figure 4.1. The corresponding divisions of frequency bands are depicted graphically
in Figure 4.2. In general, for an M suboctave analysis and synthesis, we require
M pairs of corresponding basis wavelets. A SWT is a multiwavelet transform with
a single scaling function [60, 61]. When M is a power of 2, we can carry out de
composition and reconstruction using a set of FIR filters corresponding to a single
pair of basis wavelets. Figure 4.3 presents a filter bank for carrying out a 2level
4 suboctave decomposition and reconstruction using FIR filters corresponding to a
single pair or two pairs of basis wavelets. This describes a more general way to do the
suboctave decomposition where finetocoarse octave decomposition and suboctave
band splitting can be carried out through two sets of different FIR filters. It reduces
to the case [42] when H = Hs and G = G, where H, and G, are used for suboctave
band splitting.
4.3.2 TwoDimensional Discrete SubOctave Wavelet Transform
For the decomposition of a 2D discrete function, we let the frequency response
of a scaling function be defined in the formula
+oo 2 M
I '(WX, y)2 = d,(2j y) dm(2j, 2jWy) (4.19)
j=l d=l m=l
Figure 4.3. A 2level 4sub tae decomposition and reconstruction of a SWT.
Figure 4.3. A 2level 4suboctave decomposition an
The Fiwcr Bank for a 2Levd. 2Sub~Oav WaVC1l Tra'sfrm
0W4
H....)x 1 0
0.2
 3 2 Frequecy
(a) (b)
Figure 4.4. Frequency plane tessellation and filter bank. (a) Division of the frequency
plane for a 2level, 2suboctae analysis. (b) Its filter bank along the horizontal
direction.
For a Jlevel 2D discrete suboctave wavelet transform, we can formulate
J 2 M
(W, y) 2 E d, m(2JW, 2Jy) +d'm(2jW, 2jW) + 1(2jw, 2wy) 2
j=1 d=l m=l
(4.20)
If the scaling approximation of a function f(x, y) at scale 2i is represented by
Sjf(x, Y) = f 2 (x, y), (4.21)
then { Wdfmf(nz, ny), Sjf(n,, ny) Id = 1,2, j = 1,.., J, and m = 1,.., M } is called
the wavelet representation of a discrete function f (n, ny) for a 2D Jlevel discrete
M suboctave analysis. In general, cp(x, y) is a 2D scaling function and Cd'(x, y)
and 7d'm(x, y) are the mth directional analysis and synthesis wavelets. There are
many choices of these functions that satisfy Equation (4.20). Similar to the way 2D
wavelets are constructed using 1D wavelets [51], we use tensor products to construct
2D suboctave wavelets using 1D suboctave wavelets. Thus we can write
,l'm(2jx, 2jwy) = ?m(2jwx)0(2JlWy), (4.22)
2,m(2jws, 2jwy) = b(2Jlwx) m(2wy), (4.23)
o(23wx, 23wy) = b(2Jwx)2)(2Jwy). (4.24)
Through this construction, we implemented a 2D suboctave wavelet transform using
1D convolution with FIR filters of the 1D suboctave wavelet transform previously
described. Figure 4.4(a) shows the division of the frequency plane under a 2level
SWT where M = 2. Figure 4.4(b) shows the corresponding filter bank along the
horizontal direction where the curve shown in red corresponds to the analytic fil
ter 'i(2w, 2wy) (WI'1) projected along the wx axis, the black curve for W'2, the
Spline Smoothing Function First Order Derivative Approximation
1 0 1 1 0 1
Second Order Derivative Approximation Cubic Spline Scaling Function
2
1 1.2
0    0.8
1 0.6
0.4
2
0.2
3 0
1 0 1 1 0 1
Figure 4.5. Smoothing, scaling, and wavelet functions for a SWT. These functions
are used for a 2suboctave analysis.
magenta curve for W'1 the blue curve for W21'2, and the green one for S2. A 2D
suboctave wavelet transform can be implemented by 1D convolution with FIR filters
along horizontal and vertical directions. The details along the diagonal directions are
embedded in the details along horizontal and vertical directions. In Figure 4.5, a
fourthorder spline smoothing function, its first and second derivative approximation
as suboctave wavelets, and the corresponding scaling function are shown.
A dyadic wavelet transform can be a special case of a suboctave wavelet trans
form with M = 1. As an example, a discrete 2suboctave wavelet transform is shown
to divide the details of an octave band into details of 2 suboctave bands. As shown
in Figure 4.6, one suboctave can be used for detecting local maxima while the other
suboctave band identifies zerocrossings. The dashed curve corresponds to the fre
quency response of a first order derivative approximation of a smoothing function
and the dashdot curve shows the frequency response of a second order derivative
1 2 3 4 5 6
Figure 4.6. An example of level one analytic filters for 2 suboctave bands and a
lowpass band. The dashed curve is the frequency response of a first order deriva
tive approximation of a smoothing function and the dashdot curve is the frequency
response of a second order derivative approximation. The solid curve is a scaling
approximation at level one.
approximation. The solid curve is a scaling approximation for analysis at level one.
Thus, these analysis filters take advantage of both local maxima and zerocrossing
representations to characterize local features emergent within each scale.
4.4 SWTBased DeNoising
Our experiments showed that a SWT with first and second order derivative ap
proximation of a smoothing function as its basis wavelets, separated coefficients best
characterized by feature energy from coefficients characterized by noise energy.
Suboctave wavelet coefficients can be modified by hard thresholding for noise
reduction by
Wfr*f(x) = Mdm TH (W mf(x)/M. tmm) (4.25)
Mjm = max(IWfmf(x)) (4.26)
where d = {1, 2} (omitted for 1D signals), j = {1,..., k}, and k < J, m = {1,..., M},
and td4m is (in general) a threshold related to noise and scale. The processed result
wdm,'*f(x) is a modified coefficient. Position x denotes n for 1D signal processing
and (n,, ny) for 2D image noise reduction.
SWT coefficients can be processed through soft thresholding wavelet shrinkage
for noise reduction by
Wf,*f(x) = M s(Wm w f(x)/M d tm), (4.27)
M2zdm = max(lW,'mf(x)), (4.28)
where d = {1, 2} (omitted for 1D signals), j = {1, ..., k}, k < J, m = {1,..., M}, and
tdim is a noise and scalerelated threshold. Again, the result Wf,'m*f(x) is a processed
coefficient. Similarly, position x denotes n for 1D signal processing and (n,, ny) for
2D image noise reduction. Recall that Donoho and Johnstone's soft thresholding
method used a single global threshold [19]. However, since noise coefficients under
a SWT have average decay through finetocoarse scales, we regulate soft threshold
ing wavelet shrinkage by applying coefficientdependent thresholds decreasing across
scales [72]. When features and noise can be clearly separated at the finer levels of
scale, applying hard thresholding instead of soft thresholding may further improve
performance. However, hard thresholding may fail to smooth noise if the noise is
strong locally.
Table 4.1. Quantitative measurements of performance for denoising and feature
restoration.
Method or Measurement Blocks Bumps HeaviSine Doppler
Noisy Signal: ao,/a, 6.856 6.735 6.895 7.017
Noisy Signal: 10 oglo(au/a,) (dB) 16.726 16.567 16.771 16.923
Restored Result: as/a, 27.258 20.988 35.453 20.129
Restored Result: 10 logio(ar2/a) (dB) 28.779 26.439 30.993 26.076
4.5 SWTBased Enhancement with Noise Suppression
Through a nonlinear enhancement function or a generalized adaptive gain opera
tor, SWT coefficients were modified for contrast enhancement by
wd,m,* f (n, nZ) d m EOP (4.29)
Mm = max(Wdm f(n, ny) ), (4.30)
2rx iny
where d = {1,2}, 1 < m < M, and 1 < j < J. The pointwise operator's output,
Wjm,* f(n,, ny) is simply a processed coefficient. The enhancement operator Eop
can be ENL or EGAG presented in Section 2.4. Since these operators are defined
on input values between 1 to 1, we normalize suboctave wavelet coefficients before
applying the operators.
4.6 Application Samples and Analysis
In this section, we present several experimental results of SWT based denoising
and enhancement for both 1D signal and 2D images. First we present experimen
tal results based on a 1D suboctave wavelet transform. We show the denoising
capability of our method with feature restoration compared to existing denoising
methods.
20I
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
1 0
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10 I,, ,
10 i
0 200 400 600 800 1000 1200 1400 1600 1800 2000
20
0 200 400 600 800 1000 1200 1400 1600 1800 2000
(a) "Blocks"
1
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10 1 1 1 1
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
(c) "HeaviSine"
20
0 200 400 600 800 1000 1200 1400 1600 1800 2000
40 2 4 I I 0 0 A
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
to0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
(d) "Doppler"
Figure 4.7. Denoised and restored features from the SWTbased algorithm. From
top to bottom: original signal; noisy signal; denoised signal; overlay of original and
denoised signal.
Experimental results of SWTbased denoising are shown in Figure 4.7. In the
fourth plot shown in each Figure 4.7(a)(d), the denoised results are overlaid with
their corresponding original signals. Table 4.1 shows quantitative measurements of
each result shown in Figures 4.7. In comparison to previously published methods
processing the same signals [8], results of the SWTbased method are noticeably im
proved and basically free from artifacts, pseudoGibbs phenomena. In the next exper
iment, we show the limitations of a traditional DWT for characterizing bandlimited
high frequency features. Figures 4.8(a)(b) show the original and noisy "Doppler"
signals with their corresponding 5level DWT and a coarse scale approximation. The
finest scale bandlimited features (see the second plot in Figure 4.8(a)) are weak and
obscured when noise is present (see the second plot in Figure 4.8(b)). These high fre
quency bandlimited features are lost in a soft thresholding denoising method, shown
in Figure 4.9(b). Figures 4.8(c)(f) show 2suboctave wavelet transforms of the orig
inal and noisy "Doppler" signals. Figures 4.8(c) and 4.8(e) show first suboctave
coefficients while Figures 4.8(d) and 4.8(f) display second suboctave coefficients.
Figure 4.9 shows denoised and enhanced results of noisy "Doppler" under a DWT
and a SWT. The loss of high frequency bandlimited features, made the result from
the DWTbased method less attractive than the SWTbased method, for processing
medical images, such as microcalcification in mammograms.
Next, experimental results of enhancement with noise suppression of mammo
graphic images via a 2D suboctave wavelet transform are presented. We demon
strate the advantage of enhancement without amplifying noise, including background
noise [37]. In the first experiment for image enhancement, we attempt to enhance
the visibility of a radiographic image of a RNII (Radiation Measurements Inc., Mid
dleton, WI) model 156 phantom with simulated masses embedded. Figure 4.10(a)
presents a low contrast image of the RMI model 156 phantom with simulated masses
I 200 400 600 800 1000 1200 1400 1600 1800 2000
(a)
The Original Signal and Its First SubOctave Coefficients ofa SWT
30
5
5n nA I n'? np (a n tig n.6 or 0,2 np I
A A Al Al Ad 06 0A7 0 2 A A
5
5 n 3 a, n I. n o> atn n ,o
A Al Al AY A A's A 0,6 aI AS 0a
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(c)
The Noisy Signal and Its First SubOctave Coefficients of a SWT
', , ,
s ." .. .
,10 o o,. o,: o o. o,
A 01 02 0A 0 AAS A A 0A7 A 09
0
5
0
j:^y\T .^..^ .
5
t i nj i7 nI nd nai nA n7 nV no
A Al a A n A A Aft nl A o
1  9 T A n ( m A nt i t il Or 0. n O
0 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 1
The Original Signal and Its DWT Coefficients
i= i Al ,l Sy i lfyn 11(p i~ fn i tft n 5 (5 (
0 P> n 5 (5 an iAyn i1(5 11n ijpn1 lA 1if
I.
4o
Figure 4.8. Limitations of a DWT for characterizing bandlimited high frequency
features.
A119 APO 60 i90a in iDPO 12p lnW14 1111 i2C5
20
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
(b)
The Original Signal and Its Second SubOctave Coefficients ofa SWT
S0 n 2 0al an d A 5 n Aft 7 naA A. no
n n ni ,7 n  ns n,. n7 n, no
j'^, i yA ,
5
5 fi nf l, nA nQ nA n n no
0
s
0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I
0
(d)
The Noisy Signal and IIs Second SubOctave Coedfcients ofa SWT
l fi .k n HA f, H 7 R "a 
5
I nit 0 nA nA nA nAA 17 ; A n
5
A l ni 03 nA n AS 7 n nA o A
5
0
A A A, A A\ n A Ao 7 A AS a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
30
10
5
A At1 0.2 0,3 0.4 A.5 Alk 0l AS AS9
The Noisy Signal and Its DWT Coefficients
, 290 490 60 2i90 i CP i4PO i*0
4 100 1ai NOi Wi 1poi i12 14f i )ii) 14M I p
4 1 12 AA 61490 1 590lop1 1 i*0 itA t /po I"p
4 5129 4151 5111 5155 11 '1115 1(11 151n 15 1 i l I
40~*v '? *" "Uaip painiln*
(a)
The Enhanced Signal and the Processed First SubOctave Coefficients of a SWT
5 ,
01 U Ad nAi f n nA n n no
0 0.1 0,2 0,3 0.4 0.5 06 07 08 0.9 1
 I A 11.1 n a n q miA ni 0. no
5
a 0i 0n 2 ni an 5 a i ni no
I 0.1 0,2 0.3 0.4 0.5 0,6 03 08 0.9 1
Figure 4.9. Denoised and enhanced results of a noisy "Doppler" signal under a DWT
(25.529dB) and a SWT (26.076dB).
The Original Signal and Its DWT Coefficients
h unn m\ 0 a i8 in 41 n lan4i WO 14
Q I I I 
_J (BlHd\^ 
i I I i Ql Am  p 1In
0 200 400 600 800 1000 1200 600 8 20
0 200 400 600 800 1000 1200 1400 1600 1800 2000
The Enhanced Signal and the Processed DWT Coefficients
t h awm an a0 i n9n 9i9 1in iain 1i4 ?tyl i
I I I I I I ; W I I I I1
a tim m an anm iian ai un9i in 1 1iii n lP 1iP n
10 2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
The Enhanced Signal and the Processed Second SubOctave Coefficients of a SWT
10
0 1 n 2 (11 a, ii an nA n1 7 ni0 no
s
5
5
a nm n, ni nA n ni, m A A a no
1 5 (
5
1 i 02 ii n( 1 i in n "a
0 01 0.2 0.3 04 0.5 0.6 0.7 08 09
(a) (b) (c)
Figure 4.10. Enhancement with noise suppression. (a) A low contrast image of RMI
model 156 phantom with simulated masses embedded. (b) Enhancement by tradi
tional histogram equalization. (c) SWTbased enhancement with noise suppression.
of distinct sizes. Figure 4.10(b) shows enhancement by traditional histogram equal
ization. SWTbased enhancement with noise suppression is shown in Figure 4.10(c).
Unsharp masking, a popular technique for enhancing radiographs, failed to enhance
barely seen masses (low frequency features) in Figure 4.10(a). In comparison to tra
ditional histogram equalization, the SWTbased method enhanced the visibility of
masses without amplifying noise.
In the next experiment, we enhance low contrast mammographic images contain
ing a microcalcification cluster. Figure 4.11(a) shows a region from a low contrast
mammogram containing a distinct microcalcification cluster. Next, enhancement by
traditional unsharp masking is presented in Figure 4.11(b). Figure 4.11(c) shows the
result of SWTbased enhancement with noise suppression. In practice, a radiologist
may want to view certain suspicious areas of low contrast within a large digital mam
mogram for potential breast lesions with a magnifier to improve visibility of an area.
In Figure 4.11(d), we try to provide a similar function by improving the visibility of a
local region of interest (ROI) through SWTbased enhancement with noise suppres
sion. The region within the black square (120x120) is enhanced. Note that the area
does not have to be a square, but a rectangle. Traditional unsharp masking shown
83
in Figure 4.11(b) enhances the area of the microcalcification cluster slightly but also
amplifies noise. As shown in Figures 4.11(c) and 4.11(d), enhancement under a SWT
makes barely seen microcalcification clusters more visible without amplifying noise.
84
(c) (d)
Figure 4.11. Enhancement with noise suppression. (a) Area of a low contrast mam
mogram with a microcalcification cluster (b) Best enhancement by traditional un
sharp masking (c) SWTbased enhancement with noise suppression (d) SWTbased
enhancement of a local region of interest (ROI) with noise suppression.
,3 .P,...
sharp m i. () w.(d)
enhancemen.t"of a loca region of interest (1101) "wthno p .
": }i "" .. ' "' ,, '
.. .. ( .. .,.. .., : . .. ... ; _.,.,
Figue 411.Enhacemnt ithnois supresio. (a Ara o a uw cntrst ani
,,,ra r ,,it ,: i r c l if c t o _: : .: .. ,. ,.. ..._:!_._,:.:' .:..i... : ";_ : ,# iI ...... ..
sharp ~ ~ ~ ~ ~ mak"g (c ";k. se ,nemn ','h nos sup.sso ,'t< .W
en a.mnto , %,, reo ,f _ner s R I .it ,os ,: .. _.... ....
CHAPTER 5
PERFORMANCE MEASUREMENT AND COMPARATIVE STUDY
In this chapter, we present several measures of relevant quantitative metric for
evaluating an algorithm's performance and show a few comparative studies of quan
titative measurements between our algorithms and other published methods. In
Section 5.1, a few quantitative measurements used in this dissertation are described
and formulated mathematically. Section 5.2 shows the quantitative results of our
algorithms and other relevant methods for signal/image restoration. In Section 5.3,
we present quantitative measurements of image enhancement among our developed
algorithms and a few other related methods. Earlier chapters demonstrated visual
quality of denoised as well as enhanced results (visual performance) while this chap
ter focuses on quantitative performance.
5.1 Performance Metric for Quantitative Measurements
The quality of a noisy signal/image is often measured by the ratio of signal vari
ance to noise variance (or signal energy level of variation to noise energy level) using
a log scale. The quantity is called signaltonoise ratio. A signal/image is more likely
severely degraded when noise level is high (low signaltonoise ratio). We have used
the quantitative term signaltonoise ratio when displaying our earlier experimental
results. A formal definition of signaltonoise ratio is given below.
Signaltonoise ratio (SNR) is defined as
2
SNR = 10loglo ( q),
an
where a2 and a are signal variance and the average noise energy (average least
squares error between a signal and its noisefree original version), respectively [30].
Here, we denote an ideal (original) signal as g and the restored version from its noise
signal as g*. For 1D signals, a2 is defined as
2 1
ao = NE(g() g)2
n=1
1N
N = N g (n),
n=1
and 2 is defined by
= 1 g* (n) g(n))2.
n=1
For 2D images, a 2 is defined as
1 M N
a2 = MN E(g(m, n) 9)2,
1 M N
= MN~E Eg(m,n)
m=l n=1
while a,2 is defined by
1 M N
a =MN Z: (g*(m,n)g(m,n))2
S Nm=1 n=1
The term SNR was frequently used in many noise reduction algorithms for quan
titative measurements of performance and has also been applied to measure our
denoising methods. The quality improvement of a signal/image can be measured by
an improved (higher) SNR compared to the SNR of its noisy version.
The quantitative measurements of average square errors were used to measure the
performance of noise reduction algorithms [19, 20, 8]. For our comparative studies, we

Full Text 
PAGE 1
SUBOCTAVE WAVELET REPRESE TATIO S AND APPLICATIO S FOR MEDICAL IMAGE PROCESSING By XULI ZO G A DISSERTATION PRESENTED TO THE GR D TE SCHO L OF THE U IVERSITY OF FLORIDA I PARTI L FULFILL i vIE T OF THE REQUIREMENTS FOR THE DEGREE F DOCTOR OF PHILOS PH"\ U IVER ITY OF FL RID 1997
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Copyright 1997 by Xu li Zong
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ACK OWLEDGEMENTS I would lik e to thank m y a dvisor Dr. Andrew Lain e, for a ll hi s s upport and t hou g h t ful advice durin g m y gra du ate st ud y. I would a l so lik e to thank D rs Ger hard Ritt e r Sartaj Sahni Edward Geiser, and John Harris for se rvin g on my thesi co mmitt ee Th e ir time and thoughtful s ug gest ion s a r e g r eat l y appreciated. I am very grateful to Prof. Arthur Hornsby of the D e partm e nt of Soil and Water Science niv ers it y of Florida for providing m y finan c i a l s upport from Janu ary 1991 through Jun e 1994. I would also lik e to thank Prof. Edward Geiser and Prof. dr ew Lain e for providing m e a graduate r esea r c h ass i sta ntship from September 199 4 to D ece mb e r 199 5 and from May 1996 to Janu a r y 1997 r es p ect iv e l y. I a l o want to thank Dr. D ea n Schoenfeld of the Roboti cs Lab in the D e partm nt o f N ucl ea r a nd R a diolo g i ca l Engineering University of Florid a, for providin g m y fin a n c i a l up port during the final p e riod of m y Ph.D. study. His e ffort for r ev i ew ing p a rt of m y di sse r tat ion i v r y mu c h appreciated. Sp c i a l thank go to Dr. Anke MeyerBae e of the D e p a rtm e nt of Electri a l and Computer Engine rin g for h e r e n co urag e m e nt a nd co n st ru ct iv di c u i o n with m I would lik e to thank m e mb r of the Im age Pro ces in g R s arc h Group f r m njo ya bl e mom nt th ir h e lp a nd their fri nd s hip. Finally I would lik to thank m y p ar nt an d my rel t i v f r t h ir upp r and co n sta nt e n co ur ag ment as w 11 a my wif nd m y daught r for t h ir und r andin patienc a nd lov whi h gav m th tr n gt h t fulfill m du a ional bje tiY lll
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ACK OWLEDGEME TS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTERS 1 I TRODUCTION TABLE OF CONTENTS lll Vlll Xll Xlll 1 1.1 Motivations . 1 1. 2 Review of Related Methods 2 1.3 Objectives of DeNoising and Enhancement 6 1.4 Wavelet Based Approaches for DeNoising and Enhancement 7 1.5 Organization of This Dissertation . . . . . 9 2 DEOISI GA DE HANCEMENT TECHNIQUES 2.1 Introduction . 2.2 oise Modeling 2.2.1 Additive oise Model 2.2.2 Approximate Speckle Noise Model 2.3 Uniform Wavelet Shrinkage Methods for DeNoising 2.3.1 Hard Thresholding 2.3.2 Soft Thresholding .............. 2.4 Enhancement Techniques . . . . . . .. 2.4.1 Enhancement by a Nonlinear Gain Function 2.4 .2 Enhancement by Generalized Adaptive Gain 3 DYADIC WAVELET REPRESE TATIONS 3.1 Discrete Dyadic Wavelet Transform . 3.1.1 OneDimensional Dyadic Wavelet Transform 3.1.2 TwoDimensional Dyadic Wavelet Transform 3.2 DWTBased DeNoising and Feature Enhancement lV 11 11 14 14 16 19 20 20 21 22 24 26 26 27 29 35
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3.2.1 A l gor i thm for Additiv oise Reduction and Enhancement 35 3.2.2 A l gor ithm for Spe kl Reduction with F ature Enhancement 36 3.2.3 DWTBased DeNoising . . . . . . . . 37 3.2.4 Regulating Threshold Se l ction through Scale Space 40 3.2.5 DWTBased En h anc ment with oi e Suppres ion 41 3.3 App li cat ion Samples and Ana l ys i s . . . . . 44 3.3 .1 Less Affection from PseudoGibb Phenomena. 44 3.3 2 Addit i ve No i se R duction and Enhancem nt .. 3.3 3 Speckl R duction w i th Featur Enhanc ment 3.4 C lini ca l Data Processing Study . . . . 4 SUBOCTAVE WAVELET REPRESE TATIONS 45 48 51 62 4.1 In troduct i on . . . . . . . . . 62 4.2 Continuous SubOctave Wavelet Transform 63 4.2.1 OneDimensional S ubO ctave Wave l t Tran form 64 4.2.2 TwoDimensional S ubO ctave Wavelet Tran form 66 4.3 Di scret SubOctave Wavelet Transform . . . . 68 4.3.1 OneDimen i ona l Discrete Sub O ctave Wave l et Transform 68 4.3.2 TwoDimensional Discrete Sub Octave Wave l et Transform 71 4 4 SWTBased Deo i sing . . . . . . . 75 4.5 SWTBased Enhancement with Noi e Suppression 77 4.6 App li cation Samples and Ana l ys i s . . . . 77 5 PERFORMA CE MEASUREME TA D COMPARAT I VE ST D Y 85 5.1 Performance Metr i c for Quantitative Measurement .. 5.2 Quant i tat i ve Comparison of Signal/Image Restoration 5.3 Quantitative Compar i son o f Im age Enhancem nt 85 89 91 6 OTHER APPL I CAT IO NS OF WAVELET REPRESE T TIO S 97 6. 1 Border Id ntification of Ec h ocardiogram 6 .1.1 Ov rview of t h e A l gor i thm 6 .1. 2 Mu l t i ca l e Edge D t ct i on .... 6.1.3 Shape Mode lin g . . . . 6.1.4 Boundary Contour R on truction 6.1.5 Smoot hin g of a C l ed C ntour w i thout hrink g 6.1.6 Samp l e Exp r im ntal R ult 6.2 Mu l t i a l S gm ntation f Ma 6.2.1 Ov rview f th M thod ... 6.2.2 6.2.3 6.2.4 F at ur Extr t i on . . . C l a ifi at i n v i a a Radi 1 B Sampl Exp rim ntal R ult ural t, rk 97 9 99 101 102 10 10 107 109 110 11 11
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7 CO CLUSIO S . . . . . . . . . . . . . . . 115 APPENDICES A FIR FILTERS FOR COMPACT SUPPORT WAVELETS 117 A.1 First Order Derivative Wavelets of Spline Smoothing Functions 117 A.2 Second Order Derivative Wavelets of Spline Smoothing Functions 122 B IMPLEME TATIO OF SUBOCTAVE WAVELET TRA SFORMS 125 B. l OneDimensional SubOctave Wavelet Transform . 127 B.2 OneDimensional Inv erse SubOctave Wavelet Transform 128 B.3 TwoDimensional SubOctave Wavelet Transform . . 129 B.4 TwoDimensional Inverse SubOctave Wavelet Transform 129 REFERE CES ....... BIOGRAPHICAL SKETCH Vl 131 137
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3 .1 3.2 3.3 4 .1 5. 1 5 .2 5 .3 5.4 5 5 LIST OF T A BLE S Impul se r es p o n ses o f fil ters H (w) G(w) K (w) a nd L (w). Qu a n t i ta ti ve m eas ur e m e n ts o f m a nu a ll y d e fin e d b o rd e r s. Q u a n t i ta ti ve m eas ur e m nt s o f in te ro b se r ve r m ea n b o rd e r diff e r e n ces in mm o n o ri g in a l vers u s e nh a n ce d im age as s h ow n in Fi g ur e 3.25 . Q u a n t i tat i ve m eas ur e m e n ts o f p e rform a n ce for d e n o i s in g a nd feat ur e r es t o r at i o n . . . . . . . . . . . . . . . . ll g *g l/ 2 Qu a n t it a ti v m eas ur e m e n ts : Average Squ a r e E r ro r s N 2 N for va ri o u s s i g n a l r e torat i o n m et h o d s . . . . . . . . .... Qu a n t i tat i ve m eas ur e m e nt s : R M S E fo r va ri o u s d e n o i s in g m et h o d s. Co mp a ri so n o f co n trast va lu es o b ta in e d b y tra di t i o n a l co n trast t r tc h in g (C ST ), un s h a rp m as kin g (U S ), a nd mul t i sca l e n o nlin ea r p roce s in g o f s uboc t a v e wa v e l e t tra n s form ( SWT ) coe ffi c i e n ts o f a m a mm o gra m co n ta inin g a m ass l es i o n. . . . . . . . . . . .. Co n trast imp rove m e n t ind ex (C II ) fo r e nh a n ce m e n t by t r a di t i o n a l co tra s t st r etc hin g (C ST ), un s h a rp m as kin g (U S), a nd multi sca l e n o n lin ea r p rocess in g o f s uboctave wav l et tra n sfo rm (SW T ) co ffi i n t o f a m a mm og r a m wi t h a m a s . . . . . . . . . . . Co mp a ri so n o f co n trast va lu s obta in e d b y m ul t i s a l e ada p t i v ga in pro s in g o f d ya di c wav l et tra n sfo rm ( D W T ) a nd u bo tav wav 1 tra n form ( SW T) co ffi c i n ts. Ma mm ogra phi c f at ur : m inu t mi c ro a lcifi ca ti o n clu st r (MMC) mi ro a l ifi ati n lu t r ( 1 ) pi c ul a r 1 i o n (S L ), c ir ul a r ( a r t ri a l ) a l ifi at i on ( ) a n d w 11 ir um c rib d m a (WCM). . . . . . . .... 5 6 C II for nh a n c m nt b y mul t i a l a d apt i ve g i n pr f d ad i wav 1 t t r n sfo rm ( D vV T ) a nd u bv av 1 t tran f rm ( \\ T) o ffi i n t fa mm ogra phi f at ur : m i n u t mi r a l i fi a i n lu t r (MMC) mi r alc i fi at i n cl u t r ( I ) p i ul ar 1 i n ( L) ul a r ( a rt ri a l ) a l ifi at i n ( C) and w 11 i r u m r i b d n n ( \ I). 11 35 55 58 77 89 9 1 93 94 6
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LIST OF FIGURES 2.1 Thresholding methods: soft thresholding and hard thresholding. . 21 2.2 A nonlinear gain function for feature enhancement with noise suppression. . . . . . . . . . 22 2.3 A generalized adaptive gain function. 24 3.1 A 3level DWT decomposition and reconstruction of a 1D function. 29 3.2 A 3level DWT decomposition and reconstruction of a 2D function. 32 3.3 A 2D analysis filter bank. 3.4 A 2D synthesis filter bank. 3.5 Coefficient and energy distributions of signal Blocks 3.6 A sample scaling factor function. 33 34 39 41 3.7 PseudoGibbs phenomena. (a) Orthonormal wavelet transform of an original signal and its noisy signal with added spike noise. (b) Pseudo Gibbs phenomena after both hard thresholding and soft thresholding denoising under an OWT. . . . . . . . . . . . . 42 3.8 Multiscale discrete wavelet transform of an original and noisy signals. 42 3.9 DWTbased reconstruction after (a) hard thresholding (b) soft thresh olding and ( c) soft thresholding with enhancement. . . . . . 43 3.10 Deoised and feature restored results of DWTbased algorithms; first row: original signal second row: noisy version third row: denoised only result and fourth row: denoised and enhanced result signal. . 46 3.11 DeNoising and enhancement. ( a) Original signal. (b) Signal ( a) with added noise of 2.52dB. (c) Soft thresholding denoising only (11.07dB). (d) DeNoising with enhancement (12.25dB). . . . . . . . 47 Vlll
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3.12 D oi in g and en h ancement. (a) Or i gina l image. (b) Im age (a) with add d noise of 2.5dB. ( c) Soft t hr esho ldin g denoising on l y (11. 75dB). ( d) Deoising with enhancement (14.87dB). . . . . . . . 47 3.13 Deoi ing and en h ancem nt. ( a) Original MRI imag (b) De01smg on l y. (c) DWTbased denoising with enhancement .... .. 3 .1 4 DeNoising and enhancement. (a) Original MR I im age. (b) De0 1 smg on l y. (c) DWTbased denoising with en h ancement ...... 3.15 An a l gorithm for spec kl e reduction and contrast enhancement. 3.16 R su l t of denoising and en h ancement. (a) A noi sy ED frame. (b) Wavelet shr ink age denoising on l y method. ( c) DWTbased denoising 49 49 49 and enhancement. . . . . . . . . . . . . . 50 3.17 Results of denoising a nd e nh anceme nt (a) A noisy ES frame. (b) Wavelet s hrink age denoi ing on l y method. (c) DWTbased denoising and en h ancement. . . . . . . . . . . . . . . 5 1 3.18 A genera li zed adapt iv e gain function for processing a n ec h ocardiogram in Figure 3.17( a). . . . . . . . . . . . . . . 52 3.19 Result of var i ous denoising methods. (a) Original image with sp ck l e noise. (b) Median filt e rin g. ( c) Extreme s h arpen in g comb in ed with median filtering. ( d) Homomorphic Wiener filtering. ( ) Wavelet s hrink age denoising on l y. (f) DWTb a ed denoising with enhanc ment .................. 3.20 Results of var i ous d noi in g m t h ods. (a) Original imag with p ckl noi se. (b) Median filtering. ( c) Extreme sharpen in g com bin d with m dian filtering. ( d) Homomorphic Wi n r filtering. ( e) Wavel t shr ink ag denoising on l y method. (f) DWTba ed denoi ing and n53 h ancement. . . . . . . . . . . . . . . . . 54 3.21 Ar a corre l at i o n b tw n manually d fin d borders by two xp rt ardiologist ob rvers. . . . . . . . . . . . . . 56 3.22 Bord r diff r nee var i ation on the or i g in a l imag (a) Di tribution f Ep i ED bord r diff r n (b) Di tribution of Epi ES bord r diff ences. ( ) Di tr ibu tion f Endo ED border diff r n e ( d) Di tri tion of Endo ES b rd r diff r n Th li d lin ar th third rd r polynomial fitting ur v s. . . . . . . . . . . . . 7 lX
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3.23 Border difference variation on the enhanced images. ( a) Distribution of Epi ED border differences. (b) Distribution of Epi ES border differ ences. (c) Distribution of Endo ED border differences. (d) Distribu tion of Endo ES border differences. The solid lines are the third order polynomial fitting curves. . . . 3.24 Denoising and image enhancement: (a) An original ED frame ; (b) An original ES frame; ( c) The enhanced ED frame; ( c) The enhanced ES 59 frame. . . . . . . . . . . . . . . . . . 60 3.25 Image and border display: (a) An original ED frame with manually defined borders overlaid; (b) An original ES frame with manually defined borders overlaid; ( c) The enhanced ED frame with manually defined borders overlaid; ( c) The enhanced ES frame with manually defined borders overlaid. Red and yellow borders represent the two observers. . . . . . . . . . . . . . . . . 61 4.1 A 3level SWT decomposition and reconstruction diagram for a 1D function. . . . . . . . . . . . . . . . . . 70 4.2 Divisions of the frequency bands under the SWT shown in Figure 4.1. 71 4.3 A 2level 4suboctave decomposition and reconstruction of a SWT. 72 4.4 Frequency plane tessellation and filter bank. (a) Division of the frequency plane for a 2level, 2suboctave analysis. (b) Its filter bank along the horizontal direction. . . . . . . . . . . . 72 4.5 Smoothing, scaling, and wavelet functions for a SWT. These functions are used for a 2suboctave analysis. . . . . . . . . . 74 4.6 An example of level one analytic filters for 2 suboctave bands and a lowpass band. The dashed curve is the frequency response of a first order derivative approximation of a smoothing function and the dashdot curve is the frequency response of a second order derivative approximation. The solid curve is a scaling approximation at level one. 75 4.7 Denoised and restored features from the SWTbased algorithm. From top to bottom: original signal; noisy signal; denoised signal overlay of original and denoised signal. . . . . . . . . . . 78 4.8 Limitations of a DWT for characterizing bandlimited high frequency features. . . . . . . . . . . . . . . . . . 80 4.9 Denoised and enhanced results of a noisy Doppler signal under a DWT (25.529dB) and a SWT (26.076dB). . . . . . . . 81 X
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4. 10 E nh a n ce m n t w i t h n o i se s uppr ess i o n. (a) A l ow co n t r ast im age o f R n m o d 1 1 5 6 ph a ntom with s imul ate d m asse e m be dd d. ( b ) E nh a n ce m e n t b y tra di t i o n a l hi stogra m e qu a li zat i o n (c) SW Tb ase d e nh a n ce m n t wi t h n o i se s uppr ess ion . . . . . . . . . . . . 82 4. 11 E nh a n ce m e n t wi t h n o i se s uppr ess i o n. (a) A r ea o f a l ow co ntr a t m a m og ram with a mi c ro ca lcifi ca ti o n clu ste r. ( b ) B est e nh a n ce m e n t by tra di t i o n a l un s h a rp m as kin g. ( c) SW Tbase d e nh a n ce m e n t w i t h n o i se s uppr ess i o n. ( d) SWTb ase d e nh a n ce m e n t o f a l oca l r eg i o n o f in terest ( RO I ) wi t h noi se s uppr ess i o n . . . . . . . . . . . 84 5 .1 Enh a n ce m e nt r es ul ts ( a) A r ea o f a l ow co n trast m a mm ogra m w i t h a m ass. ( b ) Enh a n ce m e n t o f ( a) b y tra di t i o n a l co n trast st r etc hin g (c) Enh a n ce m e n t o f (a ) b y t ra diti o n a l un s h a rp m as kin g. ( d ) SW b ase d e nh a n ce m e nt o f (a) w i t h n o i se s uppr ess i o n (e) Th e sa m e a r ea o f a l ow co n t r ast m a mm ogra m co n ta min ate d w i t h a ddi t i ve Ga u ss i a n n o i se. ( f ) E nh a n ce m e nt o f ( e) b y tra di t i o n a l co n trast st r etc hin g. (g) E nh a n ce m e n t o f (e) b y tra di t i o n a l un s h a rp m as kin g. ( h ) S W Tb ased e nh a n ce m e n t w i t h noi se s uppr ess i o n. ( i ) H a ndseg m e nt e d m ass a nd ROI for qu a n t i tat iv e m eas ur e m e n ts o f p er form a n ce . . . . . 92 5.2 Ph a n to m e nh a n ce m e nt r es ul t (a) Ph a n to m im age. ( b ) Ma mm ogra m M5 6 w i t h bl e nd e d ph a n to m feat ur es. ( c) o nlin ea r e nh a n ce m e n t und er a DWT. ( d ) SWTb ase d e nh a n ce m e n t w i t h n o i se s uppr ess i o n. 95 6. 1 T h e c ir c ul a r a r c te mpl ates for m atc h e d fil te rin g 101 6.2 D y n a mi c s h a p e m o d e lin g. 6.3 Co nn ect in g b ro k e n b o und a r y seg m e n t Th e fir t row s h ow fou r t p i ca l cases s h ow in g th e e nd p o int s o f t wo b ro k n seg m e n t b l o n g to a l a r ge seg m e nt. Th e seco nd row i s th e r es ul t a f te r co nn ect in g t h e tw 10 2 b ro k e n seg m e n ts for eac h ca . . . . . . . . . . 1 03 6.4 Attac h d p o in t r e m ova l. Th fi rst row s h ows fo u r typ i a l a w i t h attac h d p o in t Th e s co ncl row i t h e r ul t aft r atta h d po in t r m ova l fo r eac h co rr s p o ndin g ca . . . . . . . 10.J 6 .5 B o rd e r id e n t ifi at i o n o f t h e L V fr o m a h ortax i (a) n ri g in a l fr a m o f t h e L V. ( b ) E d g m a p s d t t d u in g a D\ T. ( ) T h nt r p o in t f t h L V a nd ext r a t d b o un clar gm n t ( cl) n n t cl b o un d ry o n to ur (e) Co n t u r in (cl) v rl a i cl w i t h th r i i n 1. (f) F in a l e t i mat d b o un dar i . . . . . . . . . . . . l X l
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6.6 Local nonshrinking smoothness filtering of a closed contour. (a) The smoothed contours. (b) Contours in ( a) overlaid with the contours in Figure 6.5(d) before smoothness filtering. . . . . . . . . 107 6.7 Border identification of an echocardiogram at ED. (a) An original frame of the LV at ED. (b) The detected center point and endocardial as well as epicardial boundaries overlaid with the original. . . 108 6.8 Border identification of a frame at ES from the same sequence of echocardiograms as Figure 6.7. (a) An original frame of the LV at ES. (b) The detected center point and boundaries overlaid with the original. . . . . . . . . . . . . . . . . . 108 6.9 etwork architecture, a threelayer resourceallocating neural network of radial basis functions. . . . . . . . . . . . . 111 6.10 Test Images. First row: original ROI images; Second row: smoothed and enhanced images ; Third row: ideal segmentation results. Columns: (ac) real mammograms (d) a mathematical model. . . 113 6.11 Experimental results of image segmentation. Four test cases, one each row are shown. The first column (a) is an original image column (b) is smoothed and enhanced images, column (c) is the segmented result and column ( d) is the result of a traditional statistical classifier. . 114 A. l ( a) A cube spline function and its first and second order derivative wavelets, and (b) the fourth order spline with its first and second order derivatives. . . . . . . . . . . . . . . 124 Xll
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Abstract of Dissertation Presented to the Graduate School of the niversity of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SUBOCTAVE WAVELET REPRESE TATIO SA D APPLICATIO S FOR MEDICAL IMAGE PROCESSING Chairman: Dr. Andrew F. Laine By Xuli Zong December 1997 Major Department: Computer and Information Science and Engineering This dissertation describes suboctave wavelet representations and presents appli cations for medical image processing including denoising and f ature nhanc m nt. A suboctave wavelet representation is a generalization of a traditional octave d adic wavelet repre entation. In comparison to this transform by finer divi ion of ach octave into suboctave components we demonstrate a superior abilit to apture transient activities in a signal or image. In addition subo tave wavel t repr tations allow us to characterize bandlimit d f atur mor ffi i ntl D 01 mg and nhan m nt ar accompli h d through t hniqu of minimizing noi n rg and nonlin ar pro W id ntify a las ing of uboctav o ffi i nt to impr v low n ra t f atur that an b impl m nt d thr u h b n plitting t hniqu u ing FIR filt r rr p n ing t a moth r d h m thodology of u buppr ~ppli d to nhan f atur ignifi nt t m di al dia n i of d n ra Ii raph Xlll
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In our preliminary studies we investigated everal denoi ing and enhancement algorithms. Deoising under an orthonormal wavelet transform was shown to cause artifacts including pseudoGibbs phenomena. To avoid the problem we adopt a dyadic wavelet transform for denoising and enhancement. The advantage is that le s pseudoGibbs phenomena was shown in our experimental results. We devel oped algorithms for reducing additive and multiplicative noise. The algorithm for speckle reduction and contrast enhancement was applied to echocardiographic im ages. Within a framework of multiscale wavelet analysis we applied wavelet shrink age techniques to eliminate noise while preserving the sharpness of salient features. In addition nonlinear processing of feature energy was carried out to improve contrast within local structures and along object boundaries. A study using a database of clinical echocardiographic images suggests that such denoising and enhancement may improve the overall consistency and reliability of myocardial borders as defined by expert observers. Comparative studies on quantita tive measurements of experimental results between our algorithms and other methods are presented. In addition we applied wavelet representations under dyadic or sub octave wavelet transforms to other medical image processing problems such as border identification and mass segmentation. XlV
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C H A PT E R 1 I N TR O D UC TI ON 1.1 Mot i vat i o n o i e a r t i facts a nd l ow co n trast ca n ca u se s i g n a l a nd im age d egra d atio n s durin g data ac qui s i t i o n o f m a n y s i g n a l a nd im ages es p ec i a ll y in a r eas o f m e di ca l im ag a pli cat i o n s Diff e r e nt im age m o d a liti es ex hibi t di st in ct ty p es o f d eg r a d at i o n. Ma mm o gra phi c im ages o f te n ex hibi t l ow co n t r ast w hil e im ages fo rm e d w i t h co h ere n t e n ergy s u c h as ul traso und s uff e r fr o m s p ec kl n o i se. Tr a n s mi tte d a udi o s i g n a l s so m et im es h ave t h p ro bl e m o f c h a nn e la dd e d n o i se Th ese d eg r adat i o n s n ot o nl y l owe r a u d i o o r v i s u a l qu a li ty, bu t a l so ca u se a n a l ys i s a nd r ecog ni t i o n a l go ri t hm s to so m t im s fa il to ac hi e v e t h e ir o bj ct iv es. S in ce p oo r im age qu a li ty o f te n m a k es feat ur ex t rac ti o n a n a l y i s r cog niti o n a nd qu a n t i ta ti ve m eas ur e m e n ts probl m at i c a nd unr li ab l in va ri o u s a r as o f i g n a l / im age p rocess in g a n d co m p u te r v i s i o n mu c h r esea r c h h as b e n d evoted to i p rove t h e qu a li ty o f ac qui r d s i g n a l s a nd im ages d egra d e d b y t h ese factor [3 0 ). D ata restorat i o n tec hniqu s are o f t n ta r g t e d to r e du ce n o i se U nlik e n o i art i fact o m et im es co mpri n ot o nl y hi g h fr q u n y n o i e co m po n n t b u t a l m i dd l to l ow fr qu n cy co m po n n t w hi h ar ve r y h a r d to d i ff r n t i at fr m t h r p i a l f at ur in t h s p e t rum o f fr qu n y o n te n t i mp l h n i qu i ll h ave p ro bl e m r e du in g a r t i fa t a ppli t i np i fi m t h d u d to limin at a r t i fa t b a d n rta i n pr i r k n w l l Th q u :t li t) f 1 o n t r a t im ag a n b i mprov d t h r u g h d i n at d f at ur nh a n c m n 1
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2 T h e pro mi se o f wave l et represe n tat i o n s und e r d ya di c o r fur t h e r s uboctave wave l et tra n sforms a n d t h e n eed for i mprov in g d egra d e d s i g n a l s/ im ages h ave mot i vate d t hi s d i ssertat i o n r esea r c h. R ed u c in g art i facts i s n ot a m a j o r co n ce rn o f t hi s r esea r c h. Th e ma j or tas k o f t hi s resea r c h i s to deve l o p r e li a bl e tec hniqu es fo r r e du c in g n o i se a n d e nh a n c in g sa li e n t feat ur es i m p o r ta n t to va ri o u s a ppli cat i o n pro b l e m s. Th e pro mi s in g de n o i s in g a n d feat ur e e nh a n ce m e n t tec hniqu es m ay imp rove t h e re li a bili ty a nd p e fo r ma n ce o f s i g n a l / im age p rocess in g a nd co mpu te r v i s i o n a l go ri t hm s for hi g hl eve l tas k s s u c h as o bj ect d e t ect i o n o r v i s u a l p e r ce pti o n In add i t i o n we a l so s h ow t h e capab ili ty o f wave l et r e pr ese n tat i o n s for ot h e r m e di ca l im ag in g a ppli cat i o n s 1 2 R ev i ew o f R e l ate d Met h o d s S i g n a l / im age restorat i o n a nd e nh a n ce m e n t h ave bee n t h e foc u s o f mu c h r esea r c h m t h e areas o f s i g n a l a nd im age p rocess in g as w e ll as co m p u ter vi s i o n. S evera l d e n o i s in g m et h o d s a nd im age e nh a n ce m e n t tec hniqu es h ave bee n deve l ope d a nd re p o r te d in t h e li tera tur e [3 0 1 5, 2 6 17 62 58 68 ] Most o f t h ese m et h o d s ca n be ro u g hl y class ifi e d as s p at i a l, stat i st i ca l, a nd F o u r i e rdo m a inbased. Ma n y trad ti o n a l met h ods we r e r ev i ewed b y J a in [3 0 ] Th ese co nv e n t i o n a l tec hniqu es fo r im age r estorat i o n a nd e nh a n ceme n t h ave s h ow n ce r ta in limi tat i o n s o f ba l a n c in g t h e e ff ect o f r emov in g n o i se a nd e nh a n c in g feat ur es Fo r de n o i s in g pu r p oses s p at i a l a nd fr e qu e n cy d o m a in smoot hin g m et h o d s o f te n n ot on l y re du ce n o i se, b u t a l so s m oot h o u t hi g h fr eq u e n cy co mp o n e n ts o f w id e ba nd f eat ur es as a s id e e ff ect. Thi s i s b eca u se t h e s m oot hin g e ff ect a ppli es bot h n o i se a nd hi g h fr eq u e n cy co mp o n e n ts o f feat ur es. Ove rs m oot hin g n o i se o f te n ca u ses ce r ta in in tereste d f eat ur es be in g blurr e d. R ece n t l y mul t i sca l e a nd/ or mul t ir eso lu t i o n wave l et tec hni q u es fo r s i g n a l / im age r estorat i o n h ave b ee n r e p o r ted s u ggest in g imp rove d re s ul ts in s i g n a l/im age q u a li ty [ 5 0 46 1 9 9, 1 8, 59, 8, 65 ]
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3 Mallat and Hwang (50] introduced a local maxima based method for removing white noise. Their method ana l yz s the evo lu tion of l oca l maxima of the dyadic wav l et transform cros sca l es and identifies the l oca l maxima curves above a cer tain m asurement metric which more likely correspond to features than noise. The denoised signa l/im age is then reconstructed based on the extracted local maxima corresponding to s i gnificant feature sing ul ar it y points. Lu et al. (46] further extended the ideal of l oca l maxima curves to the lo ca l maxima tree cross sca l es and emp l oyed a different measurement metric to detect features from noise. Coifman a nd Majid [9] developed a wavelet packetba ed method for denoising signa l s. It is an iterative method for extracting feat ur es based on the best wavelet packet basis, which removes noise energy below a certa in thres h o ld. Donoho and Johnstone [19 20] presented thresho ldin gbased wavelet shrinkage methods for noise reduction. These methods uniformly reduce noise coeffic i ents b l ow a g l oba l t hr esho ld. Hard thresholding and soft thresho ldin g have trade off between preserving features and achiev in g smoothness. Soft thresho ldin g denoising was fur ther exp l a in ed by Donoho [18] and proved that with high probability the d n ois d s i gna l i s at l east as smooth as the noisefree origina l in a wid variety of smoothnes measures and comes nearly as close (in the mean square s nse) to the orig in a l a any other estimated results. But th method sti ll face th problem of ba l ancing th removal of noise and ignal d tai l s in order to g t a b tter p rforman in t rm of visual quality and quantitativ measur m nt Thre holdingba d w v 1 t hrinkag under an orthonormal wav 1 t tran f rm ha h wn und ir d in luding ps udoGibb ph n m na [8]. In rd r to l v th pr b l m ifm n and n h [8] d ve l op d a tran l at i oninv riant d noi ing m th d. Th ir m th l 11 problem but o illati ns aft r d n i ing r m in i ibl v i and urru [ ] u
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4 redundant wavelet representations to achieve translationinvariant effects for image restoration when applied to various image coding schemes. Most noise in reality is additive but certain noise can not be characterized well as additive noise, such as speckle noise. Speckle noise can be better approximated as multiplicative noise [30]. Image formation under coherent waves results in a gran ular pattern known as speckle. The granular pattern is correlated with the surface roughness of an object being imaged. Goodman [25] presented an analysis of speckle properties under coherent irradiance, such as laser and ultrasound. The primary differences between laser and ultrasound speckle were pointed out by Abbott and Thurstone [1] in terms of coherent interference and speckle production. For speckle reduction earlier techniques include temporal averaging [25 1], median filtering and homomorphic Wiener filtering [30]. Homomorphic Wiener filtering is a method which converts multiplicative noise into additive noise and applies Wiener lowpass filtering to reduce noise. Similar to temporal averaging, one speckle reduction tech nique [57] used frequency and/ or angle diversity to generate multiple uncorrelated syntheticaperture radar (SAR) images which were summed incoherently to reduce speckle. Hokland and Taxt [28] reported a technique which decomposed a coherent image into three components, one of which, called subresolvable quasiperiodic scat ter causes speckle noise. The component was eliminated by harmonic analysis and processing. In the last few years several wavelet based techniques were developed for speckle reduction. Moulin [54] introduced an algorithm based on the maximumlikelihood principle and a wavelet regularization procedure on the logarithm of a radar image to reduce speckle. Guo e t al. [27] first reported a method based on wavelet shrinkage for speckle reduction. In the method of Guo et al., wavelet shrinkage of a logarith mically transformed image is applied for speckle reduction of SAR images. They
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5 also propo d s v ral approach e s to c ombine dat a from polarization to achieve b ette r performance. We [71 72] hav developed a method for spec kl e reduction simi l ar to the one by Guo e t al. [27]. The diff ere nc es are that ( a) noi se i s modeled as multi plicative, taking a homomorphi c approach to reduce the noise (b) different wavel e ts and multiscale overcomplete r eprese ntation s are emp lo yed in our approach and ( c) an en han ceme nt mechanism i s in corpo rat ed into our denoising process. Thu our method ca n not only r e du ce spec kl e nois e, but also enhance int e r est ing features. In the la st two de ca d es, many imag e e nhan ce m e nt m et hod s hav e been publish e d in the lit e ratur e. Several spatial and fr eq u e n cy ba se d techniques [11 30 24, 44 58] were d eve lop e d for various imag e e nhan ceme nt purposes. Contrast stretc hin g high pas s filt e ring hi stogram modifi cat ion m et hod s are described in J a in [ 30]. Contrast stretching was an ea rli e r technique for contrast e nhan ce m e nt [30]. This m et hod ha s limitation s of selecting features ba se d on lo ca l information for e nhan ce m e nt because it i s a global approach and the e nhan ce m e nt fun ct ion i s lin ea r or piecewise lin ea r. Contrast stretc hin g ma y also amp lif y nois e when input d ata are cor rupt e d by noise. Some ima ge e nhan ce m e nt sc h emes applied to m e di ca l im age modaliti es h ave b ee n d eve lop ed and studied in the lit e ratur e [30 58, 45 39 35]. Specifically s p at i a l a nd fr e qu e ncyba se d techniques [30 24, 11 44 ] hav e b ee n developed for ultra so und im age e nhan ce m e nt A stat i st i ca l e nhan ceme nt method which u sed th lo ca l tandard d ev iation of a s urrounding r eg ion ce nt ere d around eac h pixel to replace i t va lu to e nhan ce e dg es in ultrasound imag es, was r ported b y Geiser [23]. Conventional filt e ringbas e d techniques for denoising and im ag nhan m nt hav e s hown ce rt a in limit e d ability for removing noi se without blurring atur a nd for e nhan c ing co ntrast without amp lif y in g noi se b cau patial and fr qu n r r ese ntation s ca n not s parat featur from noi w 11. In th 1 t f w ar many techniqu s based on multiscal featur s, s u h a dg and bj t b undni h a v
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6 ac hi eve d s u ccess fo r im age e nh a n ce m e nt in severa l a ppli cat i o n a r eas [32, 4 0 4 1 45) R e ce n t l y wave l et b ase d n o nlin ea r e nh a n ce m e n t tec hniqu es h ave produ ce d s up er i o r r es ul ts in m e di ca l im age e nh a n ce m e n t [ 3 9 3 5 7 3). 1. 3 Obj ect i ves o f D e o i s in g a nd Enh a n ce m e n t To imp rove t h e q u a lit y o f ac quir e d s i g n a l s a nd im ages d egra d e d b y n o i se a n d / o r l ow co n trast m os t traditi o n a l m e th o d s t r y e ith e r t o r e du ce n o i se o r t o e nh a n ce fea t u res At fir s t g l a n ce d enoi s in g a nd fe at ur e e nh a n ce m e n t a pp ea r to b e two co nfli ct in g o bj ect i ves es p ec i a ll y t o t raditi o n a l m e th o d s for d e n o i s in g o r im age e nh a n ce m e n t. H oweve r t h ey a r e s impl y tw o s id es o f t h e sa m e co in. Th e purp ose o f d e n o i s in g i s to e limin ate n o i se, prim a ril y in hi g h fr e qu e n cy w hil e m et h o d s o f feat ur e e nh a n ce m e n t atte mp t to e nh a n ce s p ec ifi c s i g n a l d eta il s, includin g co nt rast e nh a n ce m e nt. Th e dif fe r e n ce li es in t h e fa ct th a t fe at ur es o f te n occ up y a w id e r fr e qu e n cy b a nd t h a n n o i se. It i s eve n m o r e diffi c ul t to ac hi eve b o th o bj ect i ves w h e n feat ur e d eta il s a r e co rrup te d b y n o i se. T ra di t i o n a l s p at i a l a nd fil te rin g b ase d m et h o d s fo r d e n o i s in g o f te n r e du ce n o i se at a pr i ce o f blurrin g feat ur es w hil e s in g l e sca l e co n ve n t i o n a l m et h o d s fo r co trast e nh a n ce m e n t m ay a mplif y n o i se. Th e s in g l e sca l e r e pr ese n tat i o n o f a s i g n a l in t im e ( or pu re fr e qu e n cy) i s p ro bl e m at i c w h e n atte m pt in g to di sc rimin ate s i g n a l fr om n 01se. B eca u se o f t h e limi te d a bilit y o f tra di t i o n a l tec hniqu es for d e n o i s in g or feat ur e e nh a n ceme n t, t h e two co nfli ct in g o bj ect i ves ca n n ot b e acco mpli s h e d s imul ta n eo u s l y t h ro u g h ea rli er m e th o d s und e r s p at i a l o r F o uri e r r e pr ese n tat i o n s w i t h a s in g l e r es o lu t i o n o f fr eq u e n cy co n te n ts W h e n t h e two m ec h a ni s m s, d e n o i s in g a nd feat ur e e nh a n ce m e n t a r e co mbin e d und e r a fr a m e w o rk o f a n ew r e pr ese nt a ti o n o r pl atfo r m w hi c h h e lp s to ove r co m e t h e d rawbac k o f eac h m ec h a ni s m w h e n ac tin g a l o n e we w ill h ave a m u c h bette r c h a n ce to fulfill t h e t w o o bj ect i ves at t h e sa m e t im e. W ave l et
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7 tra n fo rm a nd wav l e t th e or y ca n b e o n m e th o d fo r n ew r e pr ese n tat i o n a nd p l at for m Thi s may be w h y wave l et r e pr s n tat i o n s h ave at t racte d m o r e a nd more at t n t i o n o f resea r c h e r s a imin g at s i g n a l/im age r estorat i o n a nd feat ur e nh a n c m e n t R ece n t l y w a v e l et b ase d m e th o d s h ave s hown promi se in acc ompli s hin g t h e t w o o j ect i v s at t h e a m e tim e b eca u se w ave l et d eco mp os i t i o n ca n fin e t un e fr eq u e n cy r eso lu t i o n of s i g n a l d e t a il s. W e a r e a bl e t o t r eat di st in c t co mp o n e nt s o f d eta il s at fin e to c o a r se sca l es diff e r e ntl y t o ac hi e v e d es ir e d e ff ects o f d e n o i s in g a nd f eat ur e e h a n ce m e nt. A l go ri t hm s h a v e b ee n d e v e l o p e d und e r s u c h a multi sca l e wave l et a n a l ys i s fr a m e w o rk [71 73] 1. 4 W a v l e t B ase d A ppro ac h es for D e o i s in g a nd Enh a n ce m e n t Sin ce Mo rl et a nd Gro ss m a nn [ 5 2] formul a t e d t h e first wave l et d eco mp os i t i o n w a v e l e t th e or y [5 2 12 13 1 4, 6 4 7 4 8 4 9 5 1 6 4 ] h as b ee n d eve l o p e d a nd we ll d oc um e nt e d in t h e l as t 1 4 yea r s Som e p ract i ca l a ppli ca ti o n s o f t h e t h eo r y h ave b ee n d eve l o p e d bu t m o r e a ppli ca ti o n s a r e st ill und e r th e d eve l o pin g stage Th e r e a r m a n y c h o i ces o f w ave l ets w ith diff e r e n t prop e r t i es [12]. D e n o i s in g u s in g so m e wave l ts h av in g osc ill at i o n s m ay l ea d to c r ta in unw a n te d a nd und es ir e d s id ff ect fo r xamp l n o i se indu ce d rippl s a nd osc ill a ti o n s w h e n r eco n s tru cte d und r in com pl o ffii n ts in th e w ave l t d o m a in Thi co uld b e o n o f t h e m a j o r fa tor r ul t i ng i n a r t i facts, in ludin g t h oca ll e d p se ud o ibb ph n o m e n a in h n i g h b r h d f h a rp va ri at i o n p in t ( in g ul a ri t i ) a f t r d n i in g (fo r d ta il i fma n an d D n h o [ 8]). Or t h o n o rm a l W av l t Tr a n form ( WT) a nd d i av l t Tran f rm ( D WT) h ave b n u d in var i o u a ppli d d t ti o n t ext ur n a l y i n o i r du t i n a n d i m nh a n m n t. h mp t
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8 and local support of wavelets in spatial and frequency domains has been a valu able property for characterizing features locally. This enable us to remove noise and enhance features locally without affecting other features distant apart. In a prelim inary implemented method DWT has been adopted as our major analysis tool for denoising and contrast enhancement [73]. The reasons are quite obvious. A DWT with the first order derivative of a smoothing function as its basis wavelet can sep arate noise energy from feature energy reasonably well in the wavelet domain. The DWT also correlates prominent features in its multiscale representation, such as edges and object boundaries. After experimental analysis and understanding of signal and noise behaviors in scale space we are able to find out which wavelet coefficients to modify to enhance certain features of interest (FOI) based on simple thresholding and nonlinear processing. The mother wavelet is a smoothing function and is anti symmetric with few oscillations which keeps us relatively free from the sideeffects shown under OWT with a basis wavelet having slight oscillations itself. This effect can be clearly seen from the denoised results under OWT and our denoised results [73]. The filters used to perform the DWT have compact support of a few taps. The DWT is a stable and overcomplete representation. DWT wavelet coefficients (WC) have a more clear meaning that they are proportional to the signal magnitude or image intensity changes (gradients) at certain scales. WCs reflect energy in a signal so we can rephrase that a DWT with the aforementioned wavelets is a process for the diffusion of the energy of a signal and converting it into the energy of the signal at different scales in its wavelet representation. Even through the DWTbased algorithm [73] has produced better results than denoising only methods for signal/image restoration we have observed that a DWT has limited ability to characterize bandlimited features such as texture information
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9 s p eec h o r so und s i g n a l includin g ultra so und s i g n a l s. T o m o r e r e li a bl y id e n t if y fea t ur e t h ro u g h t h e t im e sca l e s p ace, we formul a t e a nd impl e m e nt a s uboctave wave l et tra n s form whi c h i s a ge n e rali z ation o f th e DWT Th e s uboctave wave l et t ra n sfor m provid es a m ea n s to v i s u a li ze s i g n a l d eta il s in e qu a ldi v id e d s uboctave fr e qu e n cy b a nd s a nd i s s h o wn t o c h a r ac t e ri ze s i g n a l d e t a il s m o r e e ff ect i ve l y Th e t h eo r et i ca l d eve lopm e nt of a s ubo c t a v e w a v e l e t tra n s form FIR fil te r d es i g n a nd e ffi c i e n t im pl e m e ntation a r e p a r t o f thi s th es i s r esea r c h Fur t h e r mor e, in t hi s t h es i s, we a r e d e v e lopin g a c ompl ete a l g orithm a nd qu a ntit a tiv e l y m e a s ur e it s p e rform a n ce w hi c h will b e co mp a r e d t o o th e r t ec hniqu es for d e noi s in g a nd/ o r fe at ur e e nh a n ce m e n t In a n a ppro ac h d e v e lop e d durin g thi s r esea r c h w e ac hi e v e d e n o i s in g a nd fea tu re e nh a n ce m e nt und e r a fr a m e work o f multi sca l e s ubo c t a v e w a v e l et a n a l ys i s a nd judi c iou s n o nlin ea r pro cess in g [4 2, 4 3]. W e see k to e limin ate n o i se w hil e r esto rin g o r e nh a n c in g sa li e nt fe a tur e s Throu g h multi sca l e r e pr ese nt a tion b y a di sc r e t e s ub oc t ave w ave l et tran s form (SWT ) w ith firs t a nd seco nd o rd e r d e riv at i ve a pp rox im a tion s o f a s m oot hin g fun c tion as it s b as i s w a v e l e t s, we ca n di st in g ui s h feat ur e e n ergy from n o i se e n e r gy r easo n a bl y w e ll. Th e o bj ect i ves o f d e n o i s in g a nd f eat ur e e nh a n ce m e n t a r e ac hi eve d b y imult a n e ou s l y l owe rin g n o i se e n ergy a nd ra i s in g f eat ur e e n ergy throu g h d es i g n a t e d n o nlin ea r pro cess in g o f wav l e t coe ffi c i e nt in th e tra n sfor m d m a in Throu g h p ara m ete ri ze d pro cess in g, w e a r e a bl e to ac hi v a fl ex i b l co n tro l a n d th p o t e nti a l t o r e du c s p ec kl a nd r es t o r e ( o r eve n nh a n ce) o n tra t a l o n g f at ur s u c h as obj c t bound a ri es A hown in our ea rli r w o rk [73], t hi a ppr r e du ti o n a nd co ntr a t e nh a n e m e nt i 1 s a ffi t d b y p ud i bb ph n m n a [ ]. 1. 5 Or ga ni zat i o n o f T hi Di Th r e t of thi di rt at i o n i o r ga ni z d a f 11 w In h ap r 2 m d n o i in g a nd nh a n m n t m t h d W d r i b h w t r ul a h r h 1 1 ba I
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10 wave l e t shrinkage through scale space and show how to design an enhancement func tion with noise suppression In Chapter 3 we present dyadic wavelet transform based techniques for denoising and feature enhanceme nt Sample app li cat i on r esu l ts and ana l ysis are presented. In Chapter 4 we derive and formulate a s ubo ctave wavelet transform mathematically and s how how it ge n era liz es the dyadic wavelet transform. The advantage of a suboctave representation over a dyadic wavelet representation is presented. Sample app li cat i on results are presented In C h apter 5, we describe how to quantitatively measure the performance of an a l gor i thm for denoising and e nh ancement. Some compar i sons are made between t h e results of other published methods and our DWTbased as we ll as SWTbased methods. In Chapter 6 we app l y wavelet representations under dyadic and s uboctave wavelet transforms to other problems of medical im age processing. Experimental results and ana l ysis are presented. This dissertation i s concluded in Chapter 7. In Append ix A, we present FIR filters used for dyadic and sub o ctave wavelet transforms. In Appendix B we in troduce procedures for a fast impl ementat ion o f a sub octave wave l et transform in one and two dimensions.
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CHAPTER 2 DENOISI G A D ENHANCEME T TECH IQUES In this c hapt e r we are going to overview the impa ct of noi sy and low co ntra st s i g nals/im ages and r ev iew so m e related m et hod s for denoising a nd en h anceme nt We also d escr ib e how to regulate the thresholdbased d e noi s in g techniques and design a n e nhan ce m e nt fun ct ion with nois e suppression. W e then introdu ce the ima ge restora tion a nd e nh a n ce ment techniques e mplo ye d in our a lgorithm s. Th e r easo n that we put this c hapt e r a h ea d of wavelet r ep r ese ntation s d esc rib e d in Chapters 3 a nd 4 is that both our DWT and SWT b ase d algorithms s har e imag e r estorat ion a nd e nh a n ce m e nt techniques introdu ce d in this c hapt e r. Th e advantage of this organization i s that we avo id d esc ribing the d e noi s in g a nd e nhan ce m e nt operators repeatedly when pr ese nting our DWT and SWT based a l go rithm s for nois e reduction a nd co nt ra t e nhan ce m e nt so we ca n s impl y r efe r to the operators in this c h apter. 2.1 Introdu ct ion Signal a nd im age d egra dation s by noi se a nd low co ntra st a r e fr qu nt ph n om na of signal/image data acquisition es p ec i a ll y in m di a l im ag in g. Im g d gradati n hav e a s ignifi ant impa ct on th p rformanc of hum a n m et hod for m di ca l dia g no s i s. For xamp l a hum n m di a l xp rt may fa il t ca ptur e om import a nt inform at ion from a n i y and l ow ntra t imag wh n p rforming m di a l di ag no s i p i a ll y wh n x h a u t d. ard i l ogi t ma) ha p e rform b rd r int e rpol at ion in ord r to id ntif y my ard i a l b rd r wh n b rd r information i in ompl t a nd orrupt d b p kl noi and m k d i i on ba d 11
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12 on unreliable information. oise and low contrast make it problematic for human experts and computer algorithms to identify features of diagnostic importance in medical imaging. In addition noise and low contrast often make feature extraction analysis and recognition algorithms unreliable so improving the quality of acquired medical images becomes necessary. Signal/ image restoration and/ or enhancement are usually taken as the first step of a high level task of image processing and com puter vision. For instance in most image segmentation algorithms image smoothing is usually carried out as the first step ( or preprocessing) for segmentation in order to reduce noise interference on the performance of these algorithms. Most traditional approaches for denoising have a singleminded objective which is to reduce noise while minimizing the smoothing of features. Traditionally noise is frequently not a concern in feature enhancement algorithms. In the combined ap proach developed during this thesis research, we will focus on two goals; (1) to remove noise and (2) to enhance salient features to a desired degree. As part of this research we have implemented algorithms for removing additive and multiplicative noise re spectively while enhancing prominent features at the same time. These algorithms are primarily based on wavelet representations wavelet shrinkage and feature em phasis. Wavelet shrinkage is a technique which uniformly reduces wavelet coefficients through a certain operator such as hard thresholding or soft thresholding. During the process small coefficients mainly attributed to noise are usually removed. For feature enhancement, we revitalize low contrast FOI through feature emphasis (in creasing the energy level for each of these features). When the noise level in a signal or image is high these algorithms are capable of not only removing noise, but also restoring features to near their original quality and even enhancing certain FOI se lectively. When the noise level is low such as in a low contrast medical image our algorithms can enhance features with noise suppression to avoid amplifying noise.
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1 3 Th e m a in id as b e hind se l ec t d w a v e l et s hrink age a nd sa li n t f eat ur e e mph as i s e n ca p s ul ate th e tw o fund a m e nt a l obj ect i ves o f d e n o i s in g a nd feat ur e e nh a n ce m e n t: (a) R e m ove n o i se bu t no t feat ur es a nd ( b ) Enh a n ce th e fe a tur es of in te r est but n o t noi se Th ese a r e t w o co nfli c tin g obj ec ti ves in th e se n se th a t b ot h s h a rp fe a tur es a nd n o i se li e in hi g h fr e qu e n cy of th e s p ect rum. N oi se i s o ft e n s mo ot h e d o ut at th e pri ce o f blurr e d fe a tur es l e f t in a tradition a l d e noi s in g a l go ri t hm. On th e oth e r h a nd e nh a n c in g ce t a in FOI c orrupt e d b y noi se i s mor e lik e l y to a mplif y noi se in a n e nh a n ce m e ntonl y tec hniqu e without a noi se s uppr ess ion m ec h a ni s m in co rporat e d Thi s pr eve nt s tradi tion a l a l go rithm s from at t e mptin g t o ac hi eve b o th of t h e o bj ect i ves s imul ta n eo u s l y b eca u se n o i se a nd fe a tur e s ca n not b e di s tin g ui s h e d w e ll in s p at i a l o r F o uri e r r e pr e se nt a tion s In Fouri e r dom a in d e noi s in g i s u s u a ll y ca rri e d out throu g h s om e t y p e o f lowp ass fil te rin g in n a tur e including t e mpl a t e b ase d s p a ti a l averag in g On t h e oth e r h a nd fea tur e e nh a n ce m e n t i s acc ompli s h e d und e r a ce r ta in ty p e o f hi g hp ass filt e rin g Th ey a r e in c onfli c t with eac h oth e r wh e n p e rform e d o n a s in g l e set o f d ata r e pr ese nt e d e ith e r in tim e o r in fr e qu e n cy. From thi s a n a l ys i s it s ound lik e t h at so m t y p e o f b a ndp ass filt e rin g m ay b e a c hoi ce Bu t in fact s in g l e b a ndp as fil t rin g at a fr e qu e n cy b a nd h as a ve r y limit e d ca p a bili ty of r e m ov in g n o i s a nd nh a n c in g f tu res. T o ac hi eve both th e o bj ct iv es, w e n ee d a s ui ta bl r pr n tat i o n o r pl atfo r m whi c h ca n p ara t e fe a tur es fr o m noi se we ll a nd l o a ll y M ul t i a l wav 1 t r p r n ta tion d v l o p d b y M a ll a t a nd Zh o n g [5 1 ] h a h ow n pro mi m p a r a i n f at u r a nd n o i t hrou g h sca l e s p a A o u t lin d b y D a u b hi [ 1 3] n mp l w formul a t d a nd impl m nt d multi a l of h a r a t ri z in g f t ur fr m n i d tav wav 1 t r p r fo rt h r im pr n [42 4 l a t h i pa i l i t t ra n f r m i bri fl y X J l in l
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14 in Chapter 3 while a suboctave wavelet transform is formulated and described in Chapter 4. 2.2 Noise Modeling oise modeling is an important part of a noise reduction method and it affects which kind of techniques should be used to reduce noise. Efficient noise models can make denoising more effective. When a noise behavior is not fully understood or still can not be completely explained its accurate noise model is very difficult to obtain. However approximate noise models such as speckle noise modeling may be used in such a case. Continuous noise modeling is of theoretical importance while discrete noise models are more related to practical signal/image processing for noise reduction. Through the sampling theory, a discrete noise model can be obtained from sampling its corresponding continuous noise model with a sample rate (at least Nyquist rate) to avoid aliasing effect. 2.2.1 Additive Noise Model For some signal/image processing applications considered such as simulated sig nals and mammograms noise is better approximated as an additive phenomenon. In general additive noise can be represented by the formula f(x) g(x) + TJa (x) (2.1) where g(x) is a desired unknown function. The function J(x) is a noisy observation of g(x) T/a (x) is additive noise and x is a vector of spatial locations or temporal samples. By using vector notation, we unified the noise model for 1D 2D ... ND cases. For our signal/image processing, 1D and 2D noise models are what we are interested in. oise Tia (x) is usually approximated as Gaussian white noise so it
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15 has z ro mean = 0) and a noise l evel a the standard deviation of the Gaussian fun tion. For 1D signa l proc ssing, we discretize Equation (2.1) as J(n) g(n) + TJa(n), (2.2) where n E Z f (n) f (nT + s) (g(n) and TJa(n) are simi l ar), Tis a samp lin g period and s is a samp lin g shift For 2D image processing Equation (2.1) i s discretized as J(m n) = g(m n) + TJa(m n), (2.3) where (m n) E z 2, J(m n) J(mT x + sx, nTy + sy) (g(m, n) and TJa(m n) are sim il ar), T x and Ty are samp lin g periods along horizontal and vertical directions sx and Sy are samp lin g shifts. For an add itiv e noise model there exist techniques based on mean squared error or l 1 norm optim i zation to remove noise. Such techniques include Donoho and John stone's wavelet shr ink age techniques [19, 20 18], Chen and Donoho 's basis pursuit denoising [5], Mallat and Hwang 's lo ca lm aximabased m et hod for removing whit e nois e [50], and wavelet packetbased denoising [9, 10]. By in co rporating denoi ing and feature e nhan ce ment mechanisms within a fram work of wavelet r pr ese ntations [73 42], w seek to r duce noi and e nhan ontra t without amp lif ying noise. W sha ll d monstrat e that ubtl f atur bar l y n r invi sibl in a mammogram uch as mi ro a l ifi ation c ir c ular (art e rial) alcification ca n b nhan d via wav 1 t judi iou s l ection nd modifi ation of tran form ffi i nt treat noi e and f atur s ind p ndentl y, up ri r r pi ular 1 10n and nt ti n a nd 111 ur a l rithm bt a in d f r imil ar
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16 data when compared to existing algorithms designed for denoising or enhancement alone. In our investigation we studied hard thresholding and Donoho and Johnstone s soft thresholding wavelet shrinkage [19 18] for noise reduction. An advantage of soft thresholding is that it can achieve smoothness while hard thresholding better pre serves features In our approach for accomplishing denoising and feature enhance ment we take advantage of both thresholding methods. Donoho and Johnstone s soft thr e sholding method [19 18] was developed on an orthonormal wavelet trans form [12] primarily applied with Daubechies s Symmlet 8 basis wavelet. These results showed some undesired sideeffects from pseudoGibbs phenomena [8]. By using an overcomplete wavelet representation and basis wavelets with fewer oscillations a re sult relatively free from such sideeffects after denoising was observed experimentally on similar data sets. In our algorithm we first adapt regularized soft thresholding wavelet shrinkage to remove noise energy within the finer levels of scale (such as levels 1 and 2). We then apply to wavelet coefficients within the selected levels (such as levels 3 and 4) of analysis a nonlinear gain with hard thresholding incorporated to preserve features while removing small noise perturbations. 2.2.2 Approximate Speckle oise Model Coherent interfering cause speckle noise. An accurate and reliable model of the noise is desirable for efficient speckle reduction. But it remains a difficult problem. An approximate speckle noise model is formulated below. Here, our primary objective is to accomplish speckle reduction for 2D digital echocardiograms so we formulate the noise model directly in two dimensions. The formulation of a onedimensional noise model is similar
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17 Sin p kl noise is not s impl y additiv Jain [30] presented a genera l model for speck l e noi a J(x y) = g( x, y) rJm( x, y) + rJa( x, y) (2.4) w h ere g( x, y) is an unkn ow n 2D function s u ch as a noisefree or i ginal image to b recov red J( x y) i s a noisy obs rvation of g( x, y) rJm(x y) and rJa(x y) are multi plicative and add i t i ve noise respectively x and y are t h e variables such as spatia l l ocat i ons and ( x y) E R 2 Since the effect of add i tive noise (suc h as sensor noise) with l eve l Cia is cons id erab l y sma ll er than multiplicative noise ( co h erent in terfering) (IITJ 0 ( x, y)ll 2 << llg( x, y)rJm(x y)ll2) in echocard i ograms Equation (2.4) can be ap proximated by f ( X, Y) = g ( X, Y) T/m ( X Y) (2.5) To separate t h e noise from the or i g in a l im age we take a lo gar ithmi c transform on both s id es of Equation (2.5) log(J(x y)) = log(g( x, y)) + log(rJm(x y)). (2.6) Equatio n (2.6) can a l so be rewritten as /( x y) = i( x, y) + rJi( x, y). (2.7) Now w can approx im ate rJi( x, y) as additiv whit noi e and ma app l van u wav l etbased approac h for addit i v noi r du tion. With uniform mpling obtain t h e di rete v r i n f quation (2. 7) a f1(m n) = i(m n) + rJi(m n) ( ~ )
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18 where (m n) E Z 2 f1(m n) J1(mT x + s x, nTy + sy) T x and Ty are sampling peri ods along horizontal and vertical directions s x and Sy are sampling shifts. Wavelet representation and wavelet transforms will be presented in the next two chapters. To describe how the denoising method works here we only need the fact that a wavelet transform is a linear transformation, and we borrow its notation for a wavelet rep resentation whose details are given in the following chapters. The symbol W is represented as a wavelet transform, Wf as a wavelet coefficient at scale 2J ( or level j) and direction d (1 for horizontal and 2 for vertical) S 1 is a scaling approximation at a final level J. By the properties of a linear transform we have W[/(m n)] = W[g1(m n)] + W[77i(m n)] after applying wavelet transform on the both sizes of Equation (2.8) where W[/(m n)] = {(W/[/(m, n)])d=l 2 l S j S J S1[f1(m n)]}, W[g1(m n)] = {(W/[g1(m n)])d=l 2 l S j S J S1[i(m n)]} W[77i (m n)] = { (W/[TJi(m n)])d=l 2 l S j S J S1[TJi (m n)]}. (2 9) Since noise lies in high frequency it will reduce to near zero after a finite number of repeated smoothings so S 1 [77i(m n)] + 0. In fact at most 5 level wavelet de composition is needed to smooth out noise for most noise reduction applications we conducted. This is why we only carry out speckle noise reduction through eliminat ing noise energy E (Wf [TJi ( m n) ])2. For image restoration purposes it is desirable to recover W[g1 ( m n)] the wavelet transform of a desired function g1 ( m n) from W[j1 ( m n)] by reducing W[77i ( m n)] in the wavelet domain. By taking the inverse wavelet transform we may be able to recover gl(m, n) or a close approximation. For
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19 noi e r d u ct i o n a nd feat ur e e nh a n ce m ent, we want to in c r ease furt h er t h e s h arpness of feat ur es o f in terest, s u c h as m yoca rdi a l bo und a ri es, t h ro u g h n o nlin ea r st r etc hi ng for f at u re e n ergy ga in o n s i g n a l d eta il s Wf [g 1 ( m, n)] Ja in s h owe d a s imil a r h o m o m o rphi c a pp roac h [ 3 0 pp 3 1 3 3 1 6] for spec kl e re du c t i o n o f i mages w i t h und efo rm a bl e o bj ects t h ro u g h te mp ora l averag in g a n d h omomor phi c W i e n e r fil te rin g. Th e m ot i o n a nd d e form a bl e n at ur e o f hum a n h earts t h ro u g h t im e pr eve n ts u s fr o m gett in g t h e sa m e s t a tu s o f t h e l e ft ve n t ri cle fo r mul t ipl e fr a m es. B eca u se we t r eat n o i se a nd feat ur e com p o n e n ts diff e r e n t l y, we are ab l e to prod u ce a r es ul t t h at i s s u pe ri o r to d e n o i s in g o nl y a l go rithm s. We s h ow t h at o ur a l go ri t hm i s capab l e o f n ot o nl y r e du c in g n o i se, bu t a l so e nh a n c in g feat u res o f di ag n os ti c im p o r ta n ce, s u c h as m yoca rdi a l b o und a ri es in 2 D ec h oca rdi ogra m s o b ta in e d fr o m t h e p araste rn a l s h o r t ax i s v i ew 2 3 U nifo r m W ave l et S hrink age Met h o d s for D e 01s m g Th res h o ldb ase d d e n o i s in g i s a s impl e a nd e ffi c i e nt t ec hniqu e fo r n o i se r e du ct i on w h e n b e in g a ppli e d w i t hin a fr a m ewo rk o f wave l et r e pr ese n tat i o n s w hi c h ca n se p arate feat ur es fr o m n o i se. H a r d t hr es h o ldin g h as l o n g bee n u se d as a u se ful too l in cl u d in g d e n o i s in g. So f t t hr es h o ldin g wave l et s hrink age fo r d e n o i s i ng was deve l oped b D o n o h o a nd J o hn sto n e [ 1 9 ]. H a rd t hr es h o ldin g a nd so f t t hr es h o l d in g h ave trade off b etwee n pr ese r v in g feat ur es a n d ac hi ev in g s m oot hn ess W h e n feat ur s in t h wav 1 t d o m a in ca n b e cl ea rl y di st in g ui s h e d fr o m n o i se app l y in g h ard t hr h o l ding wa s h r ink age ca n ac hi eve a better res ul t t h a n so f t t hr s h o l d in g. W h n i t i not th a a nd m oot hn ess i m o r e d e i rab l e, so f t t hr h o l d i ng h o ul d b t h h i
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20 2.3.1 Hard Thresholding A hard thresholding operation can be expressed as u ( x) = TH ( v ( x) t) = v ( x) (Iv ( x) I 2 t) +, (2.10) where t is a threshold value, x E D where D is the domain of v( x) and u(x) is the result of hard thresholding and has the same sign as v( x) if nonzero. The meaning of (lv(x) I 2 t)+ is defined by the expression { 1 if lv(x)I 2 t, (lv(x) I 2 t)+ = 0 otherwise. 2.3.2 Soft Thresholding Soft thresholding [19 18] is a nonlinear operator and can be described by u(x) = Ts( v( x) t) = sign( v( x)) (lv(x) I t)+ (2.11) where threshold parameter tis proportional to the noise level and x E D the domain of v( x) and u(x) is the result of soft thresholding and has the same sign as v( x) if nonzero. The expression (lv(x) I t)+ is defined as { lv(x)I t (lv(x) I t)+ = 0 if lv(x) I > t, otherwise.
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0 8 0 6 0.4 0.2 Soft Thresholding vs Hard Thresholding I I I I I Soft Thresho l ding Hard Thresholding / / / / / / 0 . . . . . . .. .... ,, . . . . . . . . . ... ... J 0 .2,0.4 ../ / 0 6,..: o 8 ..1 / / / / / / / / / / / / / / / / / 0 8 0 6 0.4 0.2 0 0.2 v( x) 0.4 T 0 6 / / / / / / / / / / 0 8 Figur e 2 .1. Thr es holding m et hods: soft thresho ldin g and hard thresho l ding. Th e fun ct i o n s ign( v) i s d e fin e d as s i gn(v) = 1 if V > 0 1 if V < 0 0 otherwise. Figure 2.1 s how s a so ft thresho ldin g operation co mp a r e d with h ar d thre h o l ding. 2.4 Enhancement Techniqu s 2 1 In this s ct ion we d escr ib e how to design a n e nh a n ce m nt fun ct i n w i t h n i s uppr e s ion in co rpor ate d. Sev ra l c h o i c o f nhanc ment fun tion ar pr nt d Ana l ys i s a nd di sc u s ion of the r aso n for o ur de i gn philo phy ar a l in lu d d.
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0.8 0 6 0.4 0 2 ....J z 0 I t.Ll 0 2 0.4 0 6 0 8 1 1 Nonlinear Enhancement Function: Tl=O l T2=0 2 T3=0.85, alpha= 0.4 / / 0 8 0 6 0.4 0 2 0 0.2 v Tl T2 / 0.4 0 6 0 8 T3 22 Figure 2.2 A nonlinear gain function for feature enhancement with noise suppression. 2.4.1 Enhancement by a Nonlinear Gain Function In the design of an enhancement function we try to accomplish the two tasks of an effective enhancement; ( a) enhance features selectively and efficiently and (b) avoid amplifying noise. An enhancement operator with noise suppression is desirable and can be a choice for achieving the two aforementioned tasks. Sin c e we can not fulfill the tasks satisfactorily in the original or Fourier representation of a signal or image this l e ads us to look at other representations through some kinds of transformation. Through our study and experiments we observed that dyadic wavelet representations have shown a great promise for separating features from noise. Therefore we can apply the kind of enhancement functions which will be introduced momentarily to e nhance FOI. We thereafter generalize dyadic wavelet representations to produce subo c tave wavelet representations for characterizing bandlimited features frequently seen in medical images more efficiently.
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23 A p ara m e t e ri ze d n o nlin ea r ga in fun ct i o n w hi c h i s ta r gete d to accom pli s h t h e two tas k s o f a n e ff ect i ve e nh a n ce m e n t, ca n be formul a t e d as 0 if l v l < Ti ENL(v) = s s i g n( v) ( T2 + T ((lv l T2)/T) 0 ) if T 2 lv l T 3, (2. 1 2) sv o th e rwi se, w h e r e v E [1 1 ], 0 < a < 1 T = ( T3 T 2), s i s a p os i t i ve sca lin g facto r w hi c h i s u se d t o a dju s t th e o v era ll e n e r gy o f a pro cesse d im age P ara m e t e r s T i, T 2 a nd T 3 a r e se l ecte d v a lu es F o r eac h input v a lu e v l ess t h a n T i, th e s m a ll coe ffi c i e n t i s m o r e lik e l y r es ult e d from n o i se wh e r e v i s a n o rm a li ze d coe ffi c i e n t F o r inpu t v a lu e v g r eate r t h a n T 3 t h e co n tras t o f th e co rr es p o ndin g fea tur e o f v i s a lr ea d y r e l a ti ve l y hi g h No s p ec i a l tr eat m e nt for t h e c o e ffi c i e nt i s n ee d e d so w e o nl y d o lin ea r sca lin g whi c h i s n ee d e d t o k ee p th e e nh a n ce m e n t fun c tion from b eco min g d ec r eas in g, w hi c h m ay ca u se a rtifa c t s Th e norm a li ze d coe ffi c i e nt s within th e ran ge b e tw ee n T 2 a nd T 3 a r e wh a t w e w o uld lik e to e nh a n ce b eca u se th e ir co nt ras t i s r e l at i ve l y l ow a nd o ur fe a tur es o f int e r es t h a v e th e co rr es p o ndin g coe ffi c i e n ts in thi s ran ge Thu s, we n o nlin ea rl y s tr e t c h th e ir e n e r gy ga in t o r ev it a li ze t h ese feat ur es. Th e ra n ge b etwee n T i a nd T 2 i s co n s id e r e d as a ri s k a r ea. B ot h n o i se a nd fea tur es m ay h ave co mp o n n ts in t hi s ran ge, so w e tr y t o a v o id a mplif y in g n o i se b y s impl y lin ea r sca lin g a nd tr ated thi s r a n ge s imil a r to th e a r ea o f va lu es g r eate r t h a n T 3 Fi g ur e 2.2 h ow a amp l e nh a n ce m e nt fun ct i o n. Thi s e nh a n ce m e n t o p erato r i s l ess fl x ibl e t h a n t h in Sect i o n 2.4 2, but i t i s co mpu tat i o n a ll y m o r e ffi c i e n t Th op r ator an rv a a c hoi ce if s p ee d i s a co n ce rn
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Generalized Adaptive Gain : C=10 8=0 35 T1 =0 1 T2=0.2 T3=0 9 ('.)
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t h e overa ll e n ergy o f a pro c ss d im age. P a r a m ete r a ca n b e co mpu te d by 1 a= s i g m (c( l b) ) s i g m (c( l + b )) 1 s i g m (v ) = 1 + eV 25 (2 .1 4) (2. 1 5) P ara m ete r s T 1 T 2 a nd T 3 a r e se l ecte d va lu es. W h e n T 1 = T 2 = 0 and T 3 = l t h e ex pr ess i o n i s e qui va l e n t to t h e a d a p t i ve ga in n o nlin ea r fun c ti o n u se d p rev i o u s l y [3 7 34). Th e in te r va l [T 2 T 3 ] se r ves as a s li d in g w ind ow fo r feat ur e se l ect i v i ty. T h e s lid e ca n b e a dju ste d to e mph as i ze coe ffi c i e n ts w i t hin a s p ec ifi c ra n ge o f va ri at i o n T o in crease t h e overa ll e n e r gy o f a p rocesse d im age we ass i g n s a va lu e greater t h an o n e ( s > l ) S imil a rl y we m ay r e du ce t h e e n ergy o f a p rocesse d im age by l ett in g s < l. W h e n s = l t h e sca lin g fa cto r s d oes n ot co n t ribu te to t h e overa ll e n r gy c h a n ge a nd m a k es t h e overa ll o p e r ato r e qui va l e n t to t h e o p erato r re p o r te d b y L a in e a nd Z o n g [42). T hu s by se l ect in g ga in va lu es a nd l oca l w in dows o f e n e r gy we m ay ac hi ev a m ore d es i ra bl e e nh a n ce m e n t Fi g ur e 2 3 pr ese n ts a n a d apt i ve ga in fun ct i on w i t h feat ur e e l ect i v i ty.
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CHAPTER 3 DYADIC WAVELET REPRESE TATIO S In this chapter, we describe multiscale wavelet transforms at dyadic scales adopted in our preliminary algorithms for denoising and enhancement as well as edge detec tion. Waveletbased de noising and enhancement are presented in this chapter while edge detection through wavelet maximum representation will be described in Chap ter 6. Since we use a dyadic wavelet transform primarily for discrete signal and digital image processing the transformation is presented in the discrete domain from an implementation point of view. For our purposes we are only interested in the overcomplete (redundant) representation under a finitelevel discrete dyadic wavelet transform. For more theoretical work and continuous dyadic wavelet transforms, Mal lat and Zhong have presented indepth details [51, 69]. DWTbased algorithms for signal/image processing are developed on dyadic wavelet representations described in this chapter, image restoration and enhancement operators presented in the last chapter. 3.1 Discrete Dyadic Wavelet Transform The discrete dyadic wavelet transform developed by Mallat and Zhong [51] has been previously applied to areas, including edge detection texture analysis, noise reduction and image enhancement. Multiscale representation under a dyadic wavelet transform provides a useful framework for characterizing features in terms of sharp variation points. For denoising and enhancement purposes compactly supported wavelets can be utilized to eliminate noise and sharpen contrast within structures 26
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27 and along obj ct boundarie without affi ting distant features [39 35 73] becaus wav let transform l oca li z feature r pr ntations. 3.1.1 OneDimensional Dyadic Wavelet Tran form Fir t we describe the discrete dyadic wave l et transform in one dimension and then extend it to two dimensions. For discrete signa l and digital image processing only a finitelevel discrete dyadic wavelet transform is usually needed for practical app li cations A Jlevel discrete dyadic wavelet transform of a 1D discrete function f (n) E l 2 ( Z ) can be represented as (3.1) where vVj[f (n)] is a wavelet coeffic i ent at ca l e 2j (or l eve l j) lo cation n E Z SJ[J(n)] is a coarse sca l e approx im ation at the final 1 vel J and position n. Wav l et co fficient W j [J ( n)] and sca lin g approx im at ion Sj [J ( n)] at l eve l j can be defined as +oo Wj[f (n)] = f 'l/J 2 i (n) = L J(n') 'l/J 2 i (n n ) (3.2) n' =+ Sj[J(n)] = f <{) 2 i (n) = L J(n 1 )
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28 where
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f Wt [f] G(w) 1~1 K (w) Hew) ' W)fJ j G(2w) 1~ : 1 K (2w) W3[f] j .., G (4w) 1,ci K (4w) Hc2w) SifJ j H(2w) ~, Sjf] H (4w) t,i H (4w) H(w) Figure 3.1. A 3level DWT d eco mpo s ition and r eco n st ru ct ion of a 1D fun ct i on. 29 where fl i s the complex co njugat e of H Th e DWT d eco mposition a nd r eco n st ru ct i on b ase d on the above r ec ur s iv e r e l a tion s are s hown as a block di agra m in Figur e 3. 1. Th e pro cess of wavelet d eco mposition is r efe rr e d to as wavelet a n a l ys i s while th wavelet r eco n st ru ct ion pro cess i s so m et im es ca ll e d wavelet sy nth s i 3.1.2 TwoDimensional D ya di c Wavelet Transform In the rest of this sect ion we s h a ll pr ese nt di c r ete d yad i c wav e l e t a n a l y i and sy nth es i of a 2D di sc r ete fun ct ion (image). Th e d eco mpo s ition will produc bo h hi g htomiddl e fr eq u e n cy ign a l d eta il s ( wavelet c oefficients) and a l ow fr qu n sca lin g approx imation (scaling coe ffi c i e nt s) of a n im age at some final 1 1 of a nah i Similarly a finit e l e v 1 discr t dyadic wavel e t tran form i s d e ir a bl f r ur di i ta l im ag proce s in g. A J1 v e l di r t d yad i wav 1 t tran form of 2D di r fun tion f (n x, ny) E l 2 (Z 2 ) a n b r pr nt d C .10)
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3 0 where Wf [J(n x, ny)] i s a wave l et coe ffi c i e n t at sca l e 2i (or l eve l j), pos i t i o n (n x, ny) and spat i a l orie n tat i o n d ( 1 for h or i zo n ta l a nd 2 fo r ve r t i ca l ) S 1 [f (nx, ny)] i s a coar e sca l e approx i mat i o n at t h e fi na l l eve l J a nd p os i t i o n ( n x ny). Wave l et coe ffi c i e n t Wf [J(n x ny)] a nd sca lin g approx i mat i o n Sj[J(nx ny)] at l eve l j ca n be d e fin ed as + oo +oo Wf [J(n x, ny)] = J '1/J g i (nx, ny) = L L J (m n ) 'l/J g i (m m n n ) (3 .11 ) m'=oo n'=oo + oo +oo Sj[J(n x, ny)] = f Cf)2 i (nx ny) = L L J (m n )cp 2 i (m m n n ) (3. 1 2) m'=oo n'= oo a n a l ys i s wave l ets a nd sca lin g fun ct i o n s dil ated at sca l e 2i a nd d = 1 2 r eprese n ts h or i zo n ta l o r ve r t i ca l s p at i a l o ri e n tat i o n. In o ur a ppro ac h for d e n o i s in g a nd e nh a n ce me n t we are in te r este d in so m e bas i s wave l ets w hi c h a r e t h e fir st o rd e r d e ri vat i ves o f co n t inu o u s s m oot hin g fun ct i ons 'l9(x, y); t hu s 'l/; 1 (nx ny) a n d 'l/; 2 (n x, ny) can be fo r mul ated as x=nx y=n y n / ,2( ) = 8iJ(x y) 'f' nx, ny 8y x = nx y = n y (3 .1 3) Co n vo lu t i o n w i t h dil ate d 'l/; 1 (n x ny) a nd 'l/; 2 (n x, ny) pro du ces s h arp va ri at i o n s a l o n g h or i zo n ta l a nd ve r t i ca l di rect i o n s for sa li e n t feat ur es In wave l et fr a m e r e p rese tat i o n s we ca n e m pl oy a diff e r e n t sy n t h es i s b as i s wave l et "/ ( n x, ny) fo r t h e reco st ru ct i o n o f t h e o ri g in a l 2 D di sc r ete fun ct i o n Th e in verse di screte dya di c wave l et tra n sfor m ca n b e r e pr ese n te d as w 1 ( w 1 [ W [J(nx, ny)]]). F o r a p e r fect decompo s i t i o n a n d reco n st ru ct i o n J 2 S1[J(n x ny)] ip2J (n x, ny) + LL Wf [ J(nx, ny)] "f g i (n x, ny) (3. 1 4) j=ld=l
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3 1 w h ere cj; 2 J (n x ny) = cp 2 J (n x, ny)In o rd e r to get a p e r fect r eco n st ru ct i o n o f a 2 D d i sc r ete fun ct i o n a n a l ys i s a nd sy nth es i s wave l ets t h e sca lin g fun ct i o n s h o ul d satis f y J 2 l cp (w x, wy) l 2 = LL ,,Ji (2jw x, 2jwyf l (2jw x, 2jwy) + l cp (2 1 w x, 2 1 wy) l 2 (3. 1 5) j = l d = l Th e fini te l eve l d ya di c w ave l e t d eco mp os i t i o n in (3. 10 ) ge n erates a co m p l et e r e pr e se n tat i o n for a Jl eve l d ya di c w ave l e t t ra n s form F o r a p a r t i c ul a r class o f 2 D d yad i c wave l e t s s u c h as t h e fir st o rd e r dir ect i o n a l d e ri vat i ves o f s plin e s m oot hin g fun c tio n s Ma ll at a nd Zh o n g [ 5 1] s h o w e d t h at t h e fini te l eve l dir ect a nd in verse d ya di c wave l et tra n s form o f a 2 D di sc r ete fun ct i o n ca n b e impl e m e n te d in te rm s o f four 1D fil te r s H G K a nd L Th e four filt e r s s h o uld sat i s f y t h e foll ow in g p e r fect d eco mp os i t i o n a nd r eco n st ru ct i o n co nditi o n s IH( w)l 2 + G(w) K (w) = 1 L( w) = 1 + IH (w)l 2 2 (3. 1 6) (3. 1 7) Ma ll a t a nd Zh o n g [ 5 1] a l so s h owe d h ow to d es i g n 1D fini te impul se r espo n s e (F I R) fil te r s H G K a nd L fo r a 2 D w ave l et tra n sfo rm S imil a r to t h e 1D case a 2 D d ya di c wave l et deco mp os i t i o n in Eq u atio n (3 .1 0) a n b e formul ate d in te rm s o f t h e foll o win g rec ur s i ve r l at i o n s betw n two on u t i I ve l s j a nd j + 1 in t h e F o uri e r d o m a in as ( 3 1 ) ( 1 ) C ~ )
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32 I G( wx) w, [f] K (wx) L (wy J 2 w, [f] L (wx) Gr wyJ Kr wy J f I f W 2 [f] K( 2wx) G( 2wx) L (2wyJ 2 W 2 [f] L (2wx) Gr 2wyJ K(2 wy) H( wxJ I /j__ (wx) win: K(4 wx) Hrw yJ G(4 wx) I L (4wyJ H (wyJ I 2 I wp1; H( 2wx) G(4w y J L(4 wx) H( 2wx) s, [f] I K(4 wyJ H( 2wyJ I Hr 2wyJ H (4wx) s 3 [f1 : H (4wx) s 2 [fl H(4 wyJ H r4wyJ Figure 3.2. A 3l eve l DWT decomposition and reconstruction of a 2 D function. where j 0 and S 0 [f (wx, wy)] := ](wx, wy), the Fourier transform of J(n x, ny)The reconstruction S 0 [f (wx, wy)] from a dyadic wave l et decomposition ca n be represented r ec ursivel y as (3.21) where fI is again the comp l ex conjugate of H. A 2 D DWT decomposition and reconstruction based on the above recursive relations are shown as a functional b l ock diagram in F i gure 3.2 for J = 3 For a pair of 2 D ana l ysis and synthesis filter banks shown in Figures 3.3 and 3.4, reconstructed J*(nx, ny) is equa l to J(nx, ny) when no pro cess ing is performed on W[f (nx, ny)]. The 2 D ana l ysis and synthesis filter banks in Figures 3.3 and 3.4 are co nstru cte d using FIR filters shown in Tab l e 3.1 where, for instanc e, H(w) = I:n h(n) eiwn
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33 Profiles d=l d=2 I 02 j=l 0 __, 0 I ' : 1 (', I\ I I I I I I I I I \ I I \ 02 I j=2 0 __, 2 0 I 2 ' 1 09 \ \ j=3 0 J \ __, _ 0 d e ca p (j=3) ,__, ~2 _,_ Figur e 3.3. A 2D ana l y is filt r b a nk.
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j=l j=2 j=3 decap (j=3) Profiles :: ~\' \ I I o ~ \ I I \ / o, l \ I o, t \ : 0 f \ / o '''J J 2 1 0 1 2 3 o, f f 03 1 0 2 ..I ./ o l 3 09 [ 08 0 7 [ 06 t 04 o, l 02 O J 3 o, f o, f 0 7 1 o, [ o, f o, f 2 _, r, ,\ I \ I \ I \ I \ I \ I I I I \ I \ I I I II 1 0 I\ I\ I I\ : \ I \ I \ I I I I / I 0 I I I I I I I I I I \ I i :, ~ ] 2 I j 2 o, ~~ ~ ..J 2 1 0 d=l Figure 3.4. A 2D synthesis filter bank. 34 d=2
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35 Table 3.1. Impulse responses of filters H(w), G(w) K(w) and L(w). FIR filters for m = 4 and c = 2 n h(n) g(n) k(n) l ( n) 4 0.001953125 3 0.00390625 0.015625 2 0.0625 0.03515625 0.0546875 1 0.25 1.0 0.14453125 0.109375 0 0.375 1.0 0.36328125 0.63671875 1 0.25 0.36328125 0.109375 2 0.0625 0.14453125 0.0546875 3 0.03515625 0.015625 4 0.00390625 0.001953125 3.2 DWTBased DeNoising and Feature Enhancement In this section, we first introduce algorithms for noise reduction and feature restoration or enhancement based on an additive noise model presented in Section 2.2.1 and for speckle reduction with feature enhancement based on an approximate speckle noise model formulated in Section 2.2.2. We then describe the methods and present formulation for denoising and enhancement based on the operators intro duced in Sections 2.3 and 2.4. 3.2.1 Algorithm for Additive Noise Reduction and Enhancement Because denoising and enhancement techniques are incorporated into a fram work of wavelet representations under dyadic wavelet transforms, our algorithm for noise reduction and contrast enhancement consists of four major t ps. In th steps, parameterized denoising and enhancement operators are utiliz d. Th ampl exp rimental re ults are shown for these operations. Th param t r an b fin tuned to achieve two di tinct purpo s. On i ford noi ing with f atur r torati n whil th other i for imag nhanc m nt with noi uppr 1011. Th r la d in certain sens su h as r moving noi e and impr ving th qualit f tur h
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36 methods are designed to remove noise with feature restoration or enhancement in an additive noise model. The four steps of a DWTbased denoising and enhancement method are listed as follows: 1. Carry out a DWT to obtain a complete representation of noisy data in the wavelet domain. 2. Shrink transform coefficients within the finer scales to partially remove noise. 3. Emphasize features through a nonlinear pointwise operator to increase energy among features within a specific range of variation. 4. Perform an inverse DWT and reconstruct the signal/image. Unlike Donoho and Johnstone s methods [19] for denoising an advantage of this method is that it also applies feature enhancement to further improve the performance of signal/image restoration. This algorithm has an ability to suppress noise ( without amplifying noise) when applied for contrast enhancement compared to enhancement only methods. 3.2.2 Algorithm for Speckle Reduction with Feature Enhancement Speckle noise was modeled as approximate multiplicative noise in Section 2.2.2. Similar to the method in Jain [30], we apply a homomorphic approach to reducing speckle noise. The algorithm consists of six major steps. The six steps of a DWT based denoising and enhancement method for the speckle noise model are listed as follows: 1. Perform a logarithmic transform to convert an image containing multiplicative noise into an image with additive noise. 2. Carry out a DWT and obtain a complete representation of the logtransformed image in the transform domain.
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37 3. Shrink coeffic i e nt s within the fin er sca l es to partially remove noise energ 4. Em ph as i ze f eat ur es through nonlin ear pointwise processmg to mcrease the e n ergy a mong features within a spec ifi c range of variation. 5. Perform a n inv erse DWT a nd r eco nstru ct t h e denoised and en h anced image so that it approximates it s nois e fr ee original in l og sca l e with features enhanced 6. Finally, p e rform an ex pon e ntial transform on the r eco nstru cted im age to con vert it from lo g sca l e to lin ea r sca l e The r es ultin g im age i s now denoised and e nh a n ce d This m et hod takes a s imilar homomorphic transform to co nv ert multipli cat i ve noise into a dditiv e noi se. Unlike Jain 's m et hod [30], we in co rporat e a feature e nh a n ce m e nt m ec h a nism into the nois e r e du ct ion pro ced ur e to s h ar p e n blurred features (feature r estorat ion or e nhan ce m e nt) after d e noi s ing Wavelet r e pr ese nt a tion s und e r dis c r ete d ya di c wavelet transforms were described m Section 3.1 for both one and two dim e n s ions. Th ese d e noi s ing a nd co ntra st e nh a n ce m e nt sc h e m es a r e ba se d on wavelet s hrink age a nd feature e mph as i s o n top o f the wavelet representations. Wavelet shrinkage i s a technique to uniforml y r ed u ce wavelet coe ffi c i e nt s in order to r e mov e noi se coeffic i e nt s for the purpose of d e n o i s in g. Feature e mph as i s, on the other h a nd is trying to in crease the m ag nitud es of feature s coe ffi c i e nt s to ga in e n e r gy for low co ntra st f eat ur es. B e low we describe how to perform DWTba sed denoising a nd e nh a n ceme nt. 3.2.3 DWTB ased D eNo i s ing Since dyadic wavelet transforms with first order derivativ s of moothing fun tion as basis wavelets ca n e ffi c i e ntl y id e ntif y features with harp var i ation w ar ab l to ac hi eve the objective of noi e reduction through i th r h ard thr h ldi ng r ft
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38 thresholding. Hard thresholding preserves features better while soft thresholding can achieve the effect of smoothness. Here we describe thresholdbased denoising in two dimensions. One dimensional case i similar. To achieve the purpose of denoising through hard thresholding we can modify DWT coefficients for noise reduction by (3.22) Mf = max(IW/[J(n x, ny)]I), n x, ny (3.23) where d = 1 2 j = 1, ... k, k J and tj is, in general a threshold related to noise level and scale. Parameter tj can be directionally related if we have orientation prefer ence. TH is the hard thresholding operator presented in Section 2.3.1. The threshold tj should be selected to possibly remove most noise coefficients while preserving fea ture coefficients. Selection of thresholds in [71, 73] was trialanderror based. The selection can be guided by examining the histogram and energy distribution of coef ficients. Wavelet transforms generate a small number of large coefficients carrying a significant amount of energy, especially from fine to coarse scales for sharp variation points while producing a large number of small coefficients mostly corresponding to noise. Thresholds decrease from fine to coarse scales because noise energy is smoothed out through repeated smoothings (scaling) by low pass filtering This point is made clear as shown in Figure 3.5 for Donoho and Johnstone s synthetic signal "Blocks The guideline is to select decreasing thresholds which can remove a great number of small coefficients carrying most noise energy and keep a limited number of large coef ficients for feature energy. Thresholds through fine to coarse levels may be regulated by a decreasing function which will be discussed momentarily.
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The H istogram of W ave l et Coefficients 400,~ 300 Selec 1 ed Thre sho ld a l Level I 100 o..__ _,_,_~~~' 0 2 4 6 W ave l et Coefficient Magnitud e 800,~ ~ 600 Selec1ed Thre s h o l d al Level 3 0 2 4 W ave l e t Coeffic i e nt M ag nitud e 6 (a) 600,~., 500 E 400 "' Seleeled Threshold al Level 2 t100 :i: 200 100 0 0 2 4 6 Wa ve l e t Coefficie nt Ma g nitud e 1000 800 E 600 "' Selecled Thre sho ld al Level 4 co ~ (~ :c 400 200 0 \ 0 2 4 6 W ave l et Coefficie nt M ag nitud e The Energy of Wav e let Coefficients 80~~~~~ >. Oil 60 i3 40 C UJ 20 2 4 6 Wa vele t Coeffic i e nt M agni tud e 100,~~~ 8 0 >60 e_JJ 0) C UJ 40 20 Selected Thre s h o ld al Level 3 2 4 6 W ave l e t Coefficie nt M ag nitud e (b) 80 ~~~ >. Oil 60 i3 40 C UJ 20 Selected Thresh o ld at Level 2 0 ..___.___.__.__'''''''''''''.L..WU....U >. e_JJ 0) C UJ 0 2 4 6 W ave l e t Coefficie nt M ag nitud e 1 50 ,~~ Sele c ted Thresh o ld al Level 4 100 6 Wav e l e t Coefficient Magnitud e Figur 3.5. Co ffici e nt and e n r gy di tribution of i g n a l Bl 39
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40 When noise level is high hard thresholding de noi ing may not be able to achieve ov e rall smoothness. If this is the case we can carry out soft thresholding ba ed de noising. For reducing noi e and achieving smoothness effect we can modify DWT coefficients by (3.24) Mf = max(IW/[f (nx ny)]I), n x, n y (3.25) where d = l 2 j = l ... k k ::; J and tff is a threshold usually related to noise level and scale. Ts is the soft thresholding operator presented in Section 2.3.2. Threshold can be selected similarly based on the above discussion. Wavelet coefficients are normalized to the range between 1 to 1 before thresholding operations. 3.2.4 Regulating Threshold Selection through Scale Space Donoho and Johnstone s method of soft thresholding uses a single global threshold [19 20] under orthonormal wavelet transforms. Since noise coefficients under a DWT have average decay through finetocoarse scales, we can regulate both soft and hard thresholding by applying coefficient dependent thresholds at different scales. The regulated threshold tff can be computed through a linearly decreasing function tf = { (Tma x a(j 1)) oJ if Tma x a(j 1) > Tm i n T m i n oJ otherwise (3.26) where CYJ is the standard deviation, a is a decreasing factor between two consecutive levels T ma x is a maximum factor related to CYJ while T m i n is a minimum factor 1 ::; j ::; J and d E {1 2}. When the noise level in original corrupted data i unknown some methods use the standard deviation to approximate the noise level so
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4 1 Tm ax Tmin 1 J Level Fi g ur e 3 6. A sa mpl e sca lin g facto r fun ct i o n t h e t hr es h o l ds a r e r e l ate d to a f. Fi g ur e 3 6 s h ows a sa mpl e sca lin g facto r fun ct i o n fo r t h e co mpu tat i o n o f r eg ul ate d t hr es h o ld s Our d e n o i s in g a l gor i t hm s a r e impl e m e n te d in a way t h at a Tmax a nd T m i n ca n b e in teract i ve l y t un e d to ac hi eve di st in ct e ff ect o f n o i se r e du c ti o n 3 2 5 DWTB ase d Enh a n ce m e n t wi t h No i se S upp ress i o n Th ro u g h e i t h e r a n o nlin ea r ga in fun ct i o n o r ge n era li zed a d a p t i v e ga in n o nlin ea r o p erato r we ca n ac hi eve t h e e ff ect o f co n trast e nh a n ce m e n t fo r ce r ta in FO I b process in g DWT coe ffi c i e n ts as (3.27 ) Nlf = m ax(IW/ [f (n x, ny)]I) n x, n y (3 2 ) w h re p os i t i o n (n x, ny) E D t h e do m a in o f f(n x ny) d = l 2 j E {k J} and 1 :S k '.S J. T h e e nh a n ce m n t o p rato r Eop an b E L r E cri pr e n d in S ct i o n 2.4. S in e t h ese o p rator ar d fin d on in p u t a l u b t n 1 t 1 w n o rm a li z wav 1 t o ffi i n t b for app l y in g t h p r a t r
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42 (a) O n g m a l B ox ( b ) No o se d B ox (a) R cco n s lnl c lcd after H a rdThr cs h o l din g ( b ) R cco nsLru c t cd a ft er S o ftThr cs h o l d in g JO, JO ,, J O,~ 20 20 20 20 1 0 1 0 1 0 1 0 0 1 0~~ 10~~ 0 0 2 0 .4 0 6 0 8 I O 0 2 0 4 0 6 0 8 I (c) WC s o f Ori g n al B ox ( d ) WCs o f N o i se d B ox ( c ) H ar dThr cs h o ld c d W Cs o f No i se d B ox ( d ) S o ftThr es h o ld ed W Cs o f Noosed B ox 8, s ~ 4 0 0 2 0 4 0 6 0 8 I 0 2 0 4 0 6 0 8 I 0 2 0 .4 0 6 0 8 I 4~~ 0 0 2 0 4 0 6 0 8 I (a) (b) Fi g ur e 3.7. P se ud o Gibb s ph e nom e na. ( a ) Orth o norm a l w ave l et tran s form of a n o ri g in a l s i g n a l a nd it s n o is y s i g n a l with a dd e d s pik e noi se (b) P se ud o Gibb s ph e nom e n a a ft e r b ot h h a rd thr es holding a nd s oft thr es holdin g d e noi s ing und e r a n OWT. :: t : : :i: : 'I: : : l :t : : :i: '. : I : : : l 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 (a) Or i g in a l S i g n a l (b) No i sy Signa l J : : : 1 : : : i: : l J :': : 1 : ; : I: < l 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 J : : : A : : : v: : : l J : :v: < l 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 J : : : A : : : v : : : l J : : : A : : : v : : : l 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 09 1 : t : : Z \ : : l :t : : 7 : \ : : l 0 0 1 0.2 0 3 0 4 0 5 0 6 0 7 0 8 09 1 0 0 1 0 2 0 3 04 0 5 0 6 0 .7 08 0 9 1 (c) O ri gina l DWT Coeffic i ents ( d ) No i sy D W T Co e fficients Fi g ur e 3 .8. M ul t i sca l e di sc r ete wa v e l e t tran s form of a n origin a l a nd n o i sy s i g n a l s
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43 The E nh a n ced Sig nal a nd th e Pr ocessed DWT Cocffic 1 cnlS : : : I : : : I : : : 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 I ~t A j 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 Jt A V j 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 ~t .A V j 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 I : : / : : : \ : : : 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 (a) The En h a n ced S i g n a l and th e Proc ess ed DWT Coefficie n ts The Enhanced Signa l and t.he Proc essed D\Vf Coe ffic1cnlS : : : J : : : 1 : : : 1 0 0 1 0 2 0 3 0.4 0 5 0 6 0. 7 0 8 0 9 I jt V j :! ":' "' "' A "' ":' V j 0 0.1 0 2 0.3 0.4 0 5 0 6 0 7 0 8 0 9 I ~t /\ V j ~1 ", "' > O' ";' < "' j 0 0 1 0 2 0.3 04 0.5 06 0 7 0 8 0 9 1 0 0 1 0 2 0 J O 4 0 5 0 6 0 7 0 8 0 9 I ( b) (c) Figure 3.9. DWTbas ed r co n st ru ct ion aft r (a) h ar d thr h o ldin g (b) oft thr olding and ( c) so ft thresholding with e nhan m nt.
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44 3.3 Application Samples and Analysis In this section we present the experimental results of our algorithms on some sample signals and images. First we show that these algorithms are less affected from pseudoGibbs phenomena through a simple and illustrative experiment. Our results are compared to Donoho and Johnstone s results to show this effect We then present denoising and enhancement results for additive noise and speckle noise models. 3.3.1 Less Affection from PseudoGibbs Phenomena Coifman and Donoho [8] showed that both hard and soft thresholding denoising under an orthonormal wavelet transform produced undesired sideeffects, including pseudoGibbs phenomena. To solve the problem they [8] presented translation invariant denoising methods to overcome the artifacts partially caused by the lack of shiftinvariance of an OWT. Their methods alleviated the problem by making it less obvious but oscillations after denoising remained visible. Several experimental results showed that our algorithms were less affected from pseudoGibbs phenomena [73]. We have used a simple and intuitive synthetic signal to demonstrate this effect of our algorithms when compared to Donoho et al. s methods. Our experimental results on Donoho and Johnstone s four synthetic signals also demonstrate this point Figures 3. 7 3.8 and 3.9 are used to show that our methods are relatively free from pseudoGibbs phenomena. We generated a synthetic signal to illustrate what may cause the sideeffect and how our methods can basically avoid it. Figures 3.7(a) shows an original signal its noisy version with added spike noise and the orthonor mal wavelet coefficients of the original and noisy signals. Figure 3. 7(b) shows the effect of pseudoGibbs phenomena under Donoho e t al. s hard thresholding and soft thresholding methods through WaveLab (a software package from Donoho s research
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45 gro up ). ot i ce t h at a feature of s h arp var i atio n produces not o nl y l arge coeffic i ents but a l so sma ll coe ffi c i e nt s und e r a n OWT. The sma ll coeffic i ents are removed under both h a rd a nd so ft thresholding m et h ods. A typ i ca l ort h ono rm a l wavelet u sua ll y has at l east certa in oscillations [12, 64] in order to satisfy t h e (admissibility) cond i tion of a n orthonormal wavelet j +oo l~(w)l 2 I I dw < oo, 00 W where '!f (w) = J i;: 'l/J (x)ei xw dx. In the spat i a l domain the co rr espo ndin g contin uou s wave l et fun ct i o n 'ljJ ( x) h as s uffi c i e nt d ecay a nd sat i s f y r +oo J 00 'l/J (x )dx = 0 Figures 3.9( a) a nd 3.9(b) pr ese nt o ur d e noi s in g r es ult s und er regulated h ard t hr es h o ldin g a nd so ft thresholding. Both m et hod s r e mov e t h e n o i se without ca u s in g p se ud o Gibb s ph e n ome n a, but so ft thresholding a l so s m oot h es features (step edges) a littl e bit. The features a r e basically restored through o ur e nh a n ce m e nt m ec h an i sm in fi g ur e 3.9(c). This expe rim e n t i s u sed to illu strate the fact t h at t h e results of our a l gor ithm a r e l ess a ff ected from t h e s id ee ff ect (pse ud o Gibb s phenomena) compar d to the results fr o m Don o ho et al. 's m et h ods. 3.3.2 Additive No i se Reduction a nd Enhancement B ased o n an add itiv e n o i se model h ere, we present app li cat i on re ul t of de noi s in g a nd e nh a n cement for sy n t h et i c s i g n a l s and medical im ag Th fir t part i targeted for s i gna l/im age restoration while th se ond part i for enhan m nt wi h n o i se s uppr es i o n
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46 ldlri : u : : c:75 :l t ili I~: : OJ : :l 0 200 400 600 800 IOOO 1200 1 400 1 600 1 800 2000 0 200 IOO 600 800 1000 1200 14 00 1600 1 800 2000 '.l Pir:1J'~ ... 1r, .. ~~.1=5.. : .. ;.J t]J:L: : .: ] 0 200 400 600 800 1 000 1200 1 400 1 600 1 800 2000 0 200 400 600 800 1 000 1 200 14 00 1 600 1 800 2000 ~Rtij : : : :i t ]j[1: : u : :l 0 200 400 600 800 1 000 1 200 1 400 1 600 1 800 2000 0 200 400 600 800 1 000 1 200 14 00 1 600 1 800 2000 ldt i. : u : c:75 : :l : LilJc:1 : : : :l 0 200 400 600 800 IOOO 1200 1400 1600 1800 2000 0 200 400 600 800 1 000 1 200 1 400 1600 1 800 2000 ( a) Blo c ks ( b ) Bumps _::~ 0 200 400 600 800 I 000 1 200 1 400 1 600 I 800 2000 0 200 4 00 600 800 I 000 1 200 1400 I 600 I 800 2000 t::s::z:s;.a _:: ~ 0 200 400 600 800 1 000 1 200 1 400 1600 1 800 2000 0 200 4 00 600 800 1000 1 200 14 00 1 600 1 800 2000 _:: ~ 0 200 400 600 800 1 000 1 200 1 400 1 600 1 800 2000 200 400 600 800 1 000 1 200 1 IOO 1 600 1 800 2000 : :~ 0 200 IOO 600 800 IOOO 1 200 1 IOO 1600 1800 2000 200 IOO 600 800 1 000 1200 1 IOO 1 600 1 800 2000 (c) Hea v i S in e" ( d ) Doppl e r Figure 3.10. DeNoised and feature restored results of DWTbased algorithms ; first row: original signal second row: noisy version third row: denoised only result and fourth row: denoised and enhanced result signal. Deoising with Feature Restoration In this part of experiments our denoising and enhancement techniques are pri marily used for signal/image restoration To achieve this objective we want to reduce noise and restore salient features. Since noise reduction usually causes features to be blurred our enhancement methods are deployed to sharpen the blurred features. Figures 3.10 displays denoised with feature restored results of our DWTbased algorithms. First rows of figure 3.10( a)( d) are original ignals second rows are noisy ignals third rows are denoised only results and the last rows are de noised with
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47 ;~ 0 50 100 150 2 00 250 :~;rs;A/?5iiJ 0 5 0 100 150 2 00 250 ;~ 0 50 100 1 50 200 250 ;~ 0 50 100 1 50 2 00 250 Figur e 3.11. D e Noising and e nhan ce m e nt. ( a) Original signal. (b) Signal ( a) with added nois e of 2.52dB. (c) Soft thresholding d e noising only (11.07dB). (d) D e 01smg with e nhan ce m e nt (12.25dB). (a) ( b ) (c) (d) Figur 3.12. D eNo i s in g an d e nh a n ceme nt. (a) Original im ag (b) Im ag (a) \\i h added noi s of 2.5dB. (c) Soft thresholding d e noi in g on l (11.7 5 dB) (d) D with e nhan cement (14.87dB).
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48 feature restored result signals. Our results are pretty close to Coifman and Donoho s new and best results produced by the full cyclespinning translationinvariant de noising algorithm which is computationally more complex. For their Bumps and Doppler test signals our results are better than their best results. When processing a noised signal or image with lowcontrast this algorithm can be used to boost up the contrast through adding external energy to its signal energy by specifying a larger gain factor. Figures 3.11 and 3.12 show our denoising and enhancement results on Mallat and Hwang s test signal and image. The signal and image are corrupted by noise of higher levels but we get better results (higher recovery SNR) than Mallat and Hwang s results on the same signal and image with less noise. Enhancement with Noise Suppression In this part of experiments, we try to achieve contrast enhancement without amplifying noise Figures 3 13 and 3.14 show the denoising and enhancement results on two MRI head images with unknown noise level. The experimental results of de noised and enhanced images from our algorithm are visibly and quantitatively better than the results from the thresholdingbased methods alone for denoising especially for high level noise. 3.3.3 Speckle Reduction with Feature Enhancement Speckle reduction and contrast enhancement can be accomplished in the trans form domain by judicious multiscale nonlinear processing of wavelet coefficients (Wf[f (n x, ny)])d=i 2 1 5: j 5: J obtained under dyadic wavelet transforms. Through the approximate speckle noise model in Section 2.2.2 we can usually separate the noise component from a desired function. Wavelet transforms help to further distinguish signal from noise in the spatialscale space.
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49 (a) ( b ) (c) Fi g ur e 3.13 D e N oi s in g a nd e nh a n ce m e n t ( a) Ori g in a l M RI im age. ( b ) D e 01 s m g o nl y (c) DWTb ase d d e n o i s in g w i t h e nh a n ce m e n t. (a) ( b ) ( c) Fi g ur e 3 .1 4 D e o i s in g a nd e nh a n ce m e n t. ( a ) Ori g in a l M RI im age. ( b ) D eNo i in g o nl y. ( c) DWTb ase d d e n o i s in g wi t h e nh a n ce m e nt. Log DWT DeN Enh IDWT Exp F i g ur 3 .1 5 A n a l gor i t hm fo r p c kl r d u tion and ntra t n h an m nt.
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50 (a) (b) (c) Figure 3 16. Results of denoising and enhancement. (a) A noisy ED frame. (b) Wavelet shrinkage denoising only method. (c) DWTbased denoising and enhance ment. Our multiscale homomorphic algorithm as shown in Figure 3.15, for speckle re duction and feature enhancement was tested on echocardiograms of varying quality. These image sequences were acquired from the parasternal shortaxis view. Figures 3.16 and 3.17 show the results of denoising with or without feature enhancement on end diastolic (ED) and end systolic (ES) frames. The speckled original frames are shown first. Results from wavelet shrinkage denoising only and denoising with enhancement are shown in the Figures 3.16(b) and 3 16(c) respectively. Figure 3.18 shows a nonlinear operator for enhancing the image in Figure 3.l 7(a). This operator looks much different from Figure 2.3 because of the log transform effect. Experimen tal results are also compared with other speckle reduction techniques such as median filtering extreme sharpening combined with median filtering [11, 44], homomorphic Wiener filtering and a wavelet shrinkage denoising [19 18]. Figures 3.19 and 3.20 show sample results of the above mentioned methods on two typical frames from two different echocardiographic sequences with the left ventricle as the region of interest. Figure 3.19( a) is sample noisy image. The result of median filtering with a 5 x 5 mask is shown in Figure 3.19(b). Figure 3.19(c) displays sample result of extreme sharpening combined with median filtering. The result from homomorphic Wiener
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51 (a) (b) (c) Figure 3.17. Results of denoising and enhancement. (a) A noisy ES frame. (b) Wavelet shrinkage denoising only method. (c) DWTbased denoising and enhance ment filtering is shown m Figure 3.19(d). The last two images in Figures 3.19(e) and 3.19(f), display the results from wavelet shrinkage denoising only and our denoising and enhancement algorithms. The algorithm produces smoothness inside a uniform region and improves contrast along structure and object boundaries in addition to speckle reduction. The denoised and enhanced results of noisy echocardiographic images from this algorithm appear to outperform the results from soft thresholding denoising alone. Our current algorithm is impl e mented such that most param eters are set dynamically for adaptive denoising and feature enhancement. 3.4 Clinical Data Processing Study A study of clinical image processing was conducted to inv est igat e the ff ect of denoising on the consistency and r e liability to manually d e fin d bord r s of th 1 ft ventricle in 2D shortaxis echocardiographic imag es [70]. Experimental r ult in dicate the algorithm is promising. Myocardial bord rs m a nuall y d fin d b xp rt observers exhibit l ess variation after d enoi s ing It m that in h o ardiogram wh e re no real bord rs ar cl arly visibl and in ompl t xp rt bord r u u a ll in dicate a close range wher real bord r may o ur. Wh n two P rt b rd r agr
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Generalized Adaptive Gain 0 8 0.6 0.4 0 .2 0 0.2 0.4 0 .6 0 8 I I 0 8 0 6 0.4 0 2 II II II II II II II II 111 '11 I ii I I I 1 I! 111 111 I ii 1 11 I II /0.4 0 6 0 8 T3 52 Figure 3.18. A generalized adaptive gain function for processing an echocardiogram in Figure 3.17( a). with each other, the range of real borders is more likely limited around the two expert borders. The study of clinical image processing shows that denoising and feature enhancement help to improve the consistency and reliability of manually defined borders by expert observers. The set of test images in our study of clinical image processmg was selected from an ec hocardiographic database exhibiting diverse image quality. Sixty systolic sequences of 2D shortaxis echocardiographic images were selected. Half of the test images were rated as good quality while the rest were considered as poor quality For more details about how these echocardiographic sequences were acquired, we refer the reader to Wilson and Geiser [66]. Statistical results have shown that there is some improvement in consistency and reliability for manually defined borders by expert observers examining denoised images compared to their original noisy images. Quantitative measurements were calculated in terms of the mean of absolute border differences (MDistDiff) in distance (mm) and the mean of border area differences
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53 (a) (b) (c) (d) (e) ( f) Figure 3.19. R es ults of various denoising methods. (a) Original im age with p kl nois e. (b) Median filt er ing. ( c) Extreme s h arpen in g comb in ed with median filt ring. ( d) Homomorphi c Wiener filtering. ( e) Wavelet s hrink age denoising on l (f) D based denoising with e nh a n cement.
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54 (a) (b) ( c) ( d ) (e) ( f ) Fi g ur e 3 .2 0 R es ult s o f v a riou s d e noi s ing m e thod s ( a ) Origin a l im a g e with s p ec kl e n o i se ( b ) Me di a n fil te rin g ( c) E xt r e m e s h a rp e nin g c ombin e d with m e di a n filt e rin g. ( d ) H o m o m o rphi c Wi e n e r filt e rin g. ( e ) W a v e l e t s hrink age d e n o i s in g o nl y m et h o d. ( f ) D W Tb ase d d e n o i s in g a nd e nh a n ce m e n t.
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55 Table 3.2. Quantitative m eas ur eme nt s of manually defined borders. All Test Im ages Good Im ages Poor Im ages Ori vs Enh Ori vs Enh Ori vs Enh M Di st Diff Endo (in mm) 2.1040 1.8168 1. 5972 1. 5322 2.6118 2. 101 4 Epi (in mm) 1.78 46 1.67 43 1.3979 1. 5886 2.1713 1. 7601 MArea Diff Endo (in cm 2 ) 2.3731 1.8893 1.6 597 1 .4543 3.0865 2 2058 Epi (in cm 2 ) 2.5676 2.0799 1. 5823 1.9 54 0 3.5528 2 3243 (MAreaDiff) in cm 2 Th e border diff ere n ce was m eas ur e d b y it s close approximation in 64 radial directional difference fr om an est imat ed ce nt e r [66) of t h e l eft ventr icl e. Manually d e fin ed borders by ex p e rts looking at poor imag es were improv ed mor e than those of good im ages after d e noi s ing Th e stat i st i ca l r es ult s of quantitative m easure m e nt s of two sets of ex p e rt manuall y d e fin ed borders a r e s hown in Table 3.2. The stat i st i ca l comp ut at ion results li ste d und e r the co lumn "Or i a r e t h e quantitative measurements between two sets of expe rt bord ers on the original spec kl ed im ages while the r es ult s und e r the co lumn Enh are ba se d on the d e noi se d a nd e nh a n ce d im ages. It i s worth mentioning that a s ingl e set of d e noi s in g a nd e nh ancement parameters were u sed to pro cess all the test ec ho ca rdio grap hi c im ages u sed in this st ud y. We s ug gest that a s in g l e value set of p aramete r s s hould be e n o u g h for de noi s in g a nd e nh a n c in g a class of im ages with a s imil ar noi e pattern and e l cted features. Figure 3.21 s h ows the corre l at ion b tween th areas delineated by th two xpert obse rv ers. The four di ag r ams in Figur 3.21(a) pr s nt the orr l at i o n of ED Epi ( ep i card i a l) bord r a r ea ES Epi bord r ar as ED Endo ( ndo ard i a l ) b rd r ar a a nd ES Endo bord r ar as on t h e original noi y im ag Th ur diagram in Fig ur 3.2l(b) s how imilar r ult s for t h d noi d im ag w i th f tur e nh an ed. Th lid lin in t h figur ar th lin ar r gr i on lin hil h d h a nd dotted lin ar id a l r gr i on lin Fr m th d i agram it i I ar h at th
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80 60 40 20 ED Ep 1 Area Ongmal Image O"' 0 20 40 60 80 40 "'JO ~20 1 0 0 Observer I ED End o Arca Original Image 0 10 20 30 40 Observer I (a) 60 40 20 0 30 25 ES Ep1 Arca Onginal Lrn agc 20 40 60 Observer I ES Endo Arca Original Ima ge + 10 20 30 Observer I 80 60 40 20 40 "'30 20 ED Epi Area OcNotscd Lmage 20 40 60 80 Observer I ED Endo Arca DcN o iscd Im age 1 0 20 30 40 Observe r I 56 ES Ep Arca DcN01sed Im age 60 40 20 0 0 20 40 60 25 ~20 1 5 i3 10 ( b) Observer I ES Endo Arca DcNoiscd Ima ge 10 20 30 Observer I Figure 3.21. Area correlation between manually defined borders by two expert car diologist observers. points which represent the two expert border areas on the same denoised image are, in general, more toward the ideal regression line. Additional improvement ca n be seen on the Endo area correlation for the denoised images. For most echocardiograms in the study, there is usually less Endo border information than Epi border information. oisy border information (low signaltonoise ratio (SNR)) affects border interpola tion by human observers for the manuall y defined borders After denoising Endo bord e r information in terms of S R is improved so the expert bord e r areas tend to agree with each other, especially ES Endo areas. The statistical computation results shown in Table 3.2 show evidence for this analysis. Figur e 3.22 shows the distributions of mean border differences on the original imag es; ( a) the distribution of Epi ED bord e r differences (b) the distribution of Epi ES bord e r differences (c) the distribution of Endo ED border differences and (d) the distribution of Endo ES border differences Figur es 3.23( a)( d) show the distributions of m ea n border differences on the enhanced images similar to Figures 3.22(a)(d). Th e solid lines in Figur es 3.22 and 3.23 are the third order polynomial fitting c urves in a leastsquares sense. With the same scale for both Figures 3.22 and 3.23, Figure 3.23
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"' Cl w Q. w Ep 1 ED B orde r D 1ff erc n cc Dislribu11 o n (Gra ph Dale : 24Jul 97) D ashdot Lrnc M c:an Dashed Lrnc : M c.an 1/ 2SO S o hd Lmc Rcgrcss o n Curve __________________________ + ________ + __ ,__ 2 +_ + I 0 I 2 0 I + ++ + + ++ + 1 0 20 30 + + B orde r Diff crc n cC' M eli n I 92 B o ni e r D1fferc11ce S D I 02 40 50 Index ofOng m a l Ima ge Sequenc es (a) Endo ED B o rde r D 1ff erc n cc Di s tribut i o n ( Grap h Date : 2 4 Ju l 97) Oashdot Lmc M ean D:ul1 cJ Lmc M ean 1 / 2S 0 Soli d L111c Rc~r css on Curve + + ++ 60 ~ + + + + ++ + + ++ + + +. +_ .... + + + + +++ B order O1ff c r cn c M c1111 2 17 R ordc r 01frcrcn c S D I 18 2~~~ ___ _.,_ ____ .._ ___ ''' 0 1 0 20 30 40 50 60 In dex o r OnginaJ Im age S e qu e nces (c) 57 E p1 ES B o rder D 1 ffercncc Oi s tribuu o n (Graph Dale 24Ju l 97) e 4 .. D ashdot Lmc M ea n D ashed L mc Me an +/ !SD S o hd Linc R egress i on Curve cS ] "' "' w e.. w ] "' 2 + + I + +.:i" ++ + + + + + + _ _ __ B onlcr Diff erence M ean = I 65 B ordt'r D1ff cren c S O = 0 74 ++ + + 2~ ~~~~~~ 0 1 0 20 30 40 50 Ind ex o f Ongm a l lm agc Sequences ( b) E nd o ES B orde r Difference D1 s tnbu11 o n ( Graph Date : 24Jul97 ) Dashdot Lrnc Me an O ashc
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58 Table 3.3. Quantitative measurements of interobserver mean border differences in mm on original versus enhanced images as shown in Figure 3.25. EpiED EpiES EndoED EndoES Ori 4.7 4 .0 6.3 4.6 Enh 1.4 2.3 1.2 1.3 shows that border distance differences for enhanced images have sma ll er means and standard deviations than t h e corresponding differences for the original noisy images as shown in Figure 3.22. Figure 3.24 shows an examp l e of denoising and image enhancement. Figure 3.25 shows the same images as Figures 3.24 with two expert manuallydefined borders overlaid. Significant overall improvement on the agreement of two expert borders is visible from the overlaid borders of the enhanced im ages compared to the origina l images, as shown visually in Figure 3.25 and quantitatively in Table 3.3. The Endo borders have more improvement than the Epi borders based on quantitative mea surements and visua l appearance. Statistical ana l ysis shows improvement in terms of the mean of abso lu te border differences and mean border area differences of denoised im ages compared to their original images. From the overa ll stat i stica l analysis the greatest impact is on the expert borders drawn on images with poor image quality such as the images in Figure 3.24.
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Ep 1 ED B o rder Diff e ren ce D1 s uibuu o n ( Graph Date : 29J u l 97) Dashdot Li nc M ean Dashed Lmc M ean t/ 25 0 + E 4 + So hd Lmc : R cgrc:ss 1 on Curve .. + f" 8 ,5 t3 2 ++ + ] + ++.+ a, Cl UJ c. UJ "' Cl I ++++ B o rder D iffc r c:ncc Mean ::. I 83 B onier Diff erence S D = 0 88 + 2~~~'~~~~ 0 10 20 30 40 50 60 Ind ex o f En h a n ced Im age Seq u e n ces (a) E nd o ED Bord e r O 1 ffcr c n ce Di s tnbuu o n (Grap h Da t e : 29Jul97) D as hdot Linc '. M ea n D ashed Lrn c ; M ea n t / 2S D Soli d Lin c ; R cg 1 css10n Curve + + ~~ + + + 2 + + + + + + + + + UJ I ++ + + + + + + ++ 0 I B o nier D,ff crcncc: M ean 2 07 B o nier Diff erence S D I 09 2~'''''LI 0 1 0 20 30 40 50 60 Inde x o f E nhan ced Im age Sequ e nc es ( c) Ep1 ES B or der D1fferen c c Di s tribution ( Graph Dat e : 29Jul 97) Da.shdolL1nc M ean Dashed Lull: : M ean ti 2S D Solid Line : RcgrctS10n Curv i: 59 ~ ~+ + + ++ ++ .g 2 I + +_ J_ + + ~~+++~+~~+~++~~+ +~+~ +~ ~+ ~+~+~ ~+~+ ++ ~ +==+ I + ++ + + + + + ++ +++ + + + Q. + UJ I B o rder D1ffcrcn~ M ean = 1 52 B order Di ffe r ence S O ::. 0 H 2''''L....__ _._ ___ ___.__ ___ ____.., e 4 e i5 0 1 0 20 30 40 50 60 Index o f Enhan ce d Im age Sequences (b) E nd o ES Border DiITcrcncc Oislribu1i o n ( Graph Date : 29Jul 97) Oashdot L11lC M ean Dashed Lin c M c:in t / 25 D Solid Linc Rcrrcss1 0 11 Curve + .+_ _t.,. t.+_ +] 2 + + ++........ a, T ."+...~~ I +++ ++++++ ++++++ i +~ + +
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60 (a) Original ED (b) Original ES ( c) Enhanced ED ( d) Enhanced ES Figure 3.24. Denoising and image enhancement: (a) An original ED frame ; (b) An original ES frame ; (c) The enhanced ED frame; (c) The enhanced ES frame.
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61 (a) Original ED (b) Original ES ( c) Enhanced ED (d) Enhan d ES Figur 3.25. Im age and bord r di play: (a) An original ED fram wi h m a nuall defined bord rs ov rlaid; (b) An origina l ES fram with manuall d fin d b rd r overlaid; ( ) Th nhan d ED fram with manuall d fin d b rd r rl a id ( ) The enhanc d ES fram with manuall d fin d b rd r ov rlaid R d and 11 v bord rs r pr ent th tw
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CHAPTER 4 SUBOCTAVE WAVELET REPRESE TATIO S In this chapter, we introduce suboctave wavelet transforms A suboctave wavelet transform is a generalization of a traditional dyadic wavelet transform and we use an example to show the advantage of suboctave representations over dyadic wavelet representations for characterizing bandlimited features frequently seen in medical imaging. We formulate both continuous and discrete sub octave representations in one and two dimensions. 4.1 Introduction Our DWT based algorithms for denoising and enhancement have achieved im proved performance compared to other published methods. Through our analysis and experiments we observed that a DWT has a limited ability to characterize fea tures such as texture and subtle features of importance in mammographic images. The traditional DWT is an octave b ased transform where scales increase as powers of two [51]. However, the best representation of a signal s details may exist be tween two consecutive levels of scale within a DWT [36]. To more reliably isolate noise and identify features through scale space we designed a multiscale sub octave wavelet transform (SWT), which generalizes the DWT. A suboctave wavelet trans form provides a means to represent details within suboctave frequency bands of equallyspaced divisions within each octave frequency band. The theoretical devel opment of a suboctave wavelet transform and its efficient implementation was briefly described by Laine and Zong [42], and later explained and extended [43]. 62
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63 Th e initi a l motiv a ti o n for u s to ex pl ore s u b octave d eco mp os i t i o n was t h at we h ad o b serve d t h e limi tat i o n o f d ya di c wave l et tra n s form s fo r c h aracter i z in g ba nd limi te d feat ur es a nd so u g ht b e tt e r fr e qu e n cy r eso luti o n fo r d etect in g s u c h s ub t l e f eat ur es. A d ya di c wave l e t tra n s form i s a n octa v e b ase d t r a n sfor m at i o n w h ere sca l es in c r ease as p owe r s o f two. D a ub ec hi es [13] in t rodu ce d th e ge n era li za ti o n a nd exte n s i o n o f wave l et d eco mp os i t i o n a nd r eco n st ru c ti o n und e r t h e co n text o f orth o n o rm a l wave l et tra n s fo rm s b y s ubb a nd s plittin g a nd pr ese n te d ea rl y exa mpl es. A n exte n s i o n a nd ge n er a li zat i o n o f d ya di c w a v e l e t t ra n s form s i s multi sca l e s uboc t a v e w ave l et d eco mp os i t i o n a nd r eco n s tru ct i o n. Bo t h D a ub ec hi es's m et hod s a nd o ur t ec hniqu es for s u b octave wa v e l et t ra n s form s h ave ac hi eve d s imil a r (b e tt e r) fr e qu e n cy l oca li za ti o n H oweve r we a r e prim a ril y int e r este d in ove r co mpl e t e (r e dund a nt) wa v e l et r e pr ese n tat i o n s, a ge n e rali za ti o n o f d ya di c wave l et r e pr ese nt a ti o n s Most o r t h o n o rm a l wave l et b ases h ave th e e ff ec t o f d eco rr e l at in g fea tur es w hil e d ya di c wave l et b ases co rr e l ate sa li e n t fea tur es t hrou g h sca l es, whi c h i s wh a t w e a r e m ost in te r este d in for e nh a n ce m e n t purp oses In th e r es t o f th e c h a p te r we pr ese n t t h e m at h e m at i ca l fo rmul at i o n o f s uboctave w a v e l e t b ases in b ot h s p ace a nd fr e qu e n cy d o m a in s. Th e d eco m pos i t i o n a nd r eco n s tru ct i o n pro ce dur e i s ca rri e d o u t in te rm s o f fil te r b a nk t h eo r y a nd ba n d s pli tt in g t ec hniqu es. 4.2 Co n t inu o u s S ubO ctave W ave l et T ra n sfo rm Th ro u g h a w ave l e t t r a n sfo rm a fun ct i o n ( inpu t s i g n a l ) a n b r pr n t d b i ts p ro j ec ti o n o n to a famil y o f wave l et b as s fo r deco mp os i t i o n a n d po i b l p rt t r eco n s tru ct i o n If th e fa mil y o f wave l ets b ase { ?/J n( x )} i o m p l t and o r t h n orma l a w a v e l e t t ra n s form wi t h c ri t i ca l sa mplin g i u u a ll r rr d to a an orthonormal wave l e t t rans f orm [ 1 3 ] A H aar wave l t i a im p l xamp l of an rt h norma l v av 1 t H o w v r t h H aar wav I t i s a di sco n t inu o u fu n t i n nd i n t 1 a li z d i n h
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64 frequency domain. The analysis filters { H G} for computing an orthonormal wavelet transform must satisfy the following design constraints IH(w)l 2 + IG(w)l 2 = 1, H(w) G(w) = fl(w) G(w) = 0. (4.1) If the family of bases { 'I/J n ( x )} is complete and linearly independent but not orthonormal the wavelet transform is called a biorthogonal wav e l e t transform. Biorthogonal wavelets have dual basis functions More generally if the family of wavelets { 'I/J n(x)} is not linearly independent (redundant) and overcomplete they may form a wav e l e t frame representation [64]. For a dyadic wavelet transform the orthonormal constraint is relaxed so we can have distinct decomposition and reconstruction wavelets as long as the corresponding lowpass filter H(w) and highpass filters G(w) K(w) satisfy [51] IH(w)l 2 + G(w)K(w) = 1. (4.2) The above discussion is for one dimension functions. It can easily extend to two dimensions through a few well known methods [13 51]. In this section we focus on continuous suboctave wavelet transforms. We first discuss onedimensional multi scale suboctave wavelet transforms and corresponding suboctave wavelet represen tations. We then introduce twodimensional suboctave wavelet transforms (SWT) and 2D suboctave wavelet representations. 4.2.1 OneDimensional SubOctave Wavelet Transform If we further divide an octave frequency band into M equallyspaced suboctave bands (here M is limited to a power of two) then M wavelet bases can be used to
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65 capture the detail information of a signa l in each suboctave frequency band. The NI wave l et functions are represented as 'l/J m( x ) E L 2 (R) where m = {1 2 . M} L 2 ( R ) denotes the space of measurable squareintegrab l e 1D real functions. An M suboctave wave l et transform of a 1D function f (x) E L 2 (R) at sca l e 2J (level j) and l ocat i on x, and for an mth suboctave frequency band is defined as r +oo Wt f ( x ) = f 'l/J ;J( x ) = } 00 f (t) 'lj; ;J( x t)dt (4.3) where 'l/J ;J ( x ) = 2 ~ 'lj; m ( {J i s the dilation of the mth wavelet basis 'lj; m ( x ) at sca l e 2J m = {1 2 ... M } and j E Z. In the frequency domain, we can write wr f (w) (4.4) by taking the Fourier transform of Equation ( 4.3). A sca lin g approx im ation of a 1D function f ( x ) is defined as (4.5) To provide a more fl ex ibl e c h oice for the M basis wave l ets, we impose that th wave l et functions sat i sfy a wavelet frame cond i t i on (s imil ar to Ma ll at and Zhong [51]) + oo M A < L L l ~ m(2jw)l2 < B j =oo m = l where A and B are positive constants and w E R In th spatia l domain w hav + oo M Allf(x)ii2 < L L llvVFf( x )i!2 < Bllf( x )i!2. j = oo m = l
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66 The function f ( x) can be recovered from its suboctave wavelet transform by the formula +oo M +oo M J( x) = I: I: wr 1 ,;J (x) = I: I: J '1/J;J ,;J ( x), (4.6) j=oo m=l j=oo m=l where ,m(x ) is the mth synthesis wavelet for the inverse suboctave wavelet trans form. The set of frequency response of { '1/J;J (x) Im= 1 2 ... M} together at any scale 2j are required to capture the details within an octave frequency band. Finally for perfect reconstruction of f ( x ) analysis wavelets '1/;:{J ( x) and synthesis wavelets ,;J ( x) should satisfy +oo M L L ,JJm (2jw) 'Ym (2jw) 1. (4.7) j=oo m=l Equation ( 4. 7) can be obtained by taking the Fourier transform on both sides of Equation ( 4.6). To ensure exact reconstruction, the frequency axis is covered by both analysis and synthesis wavelets. Thus the wavelets ,m ( x ) can be any functions whose Fourier transforms satisfy Equation ( 4. 7). There are certainly many choices for analysis and synthesis wavelets that satisfy Equation ( 4. 7). For denoising and feature enhancement purposes we are interested in the class of wavelets which are an approximation to the first or second order derivatives of a smoothing function such as spline functions of any order. A 1D suboctave wavelet transform can be eas ily extended to 2D by computing suboctave wavelet coefficients along horizontal and vertical directions [51], as explained next in the following section. Extensions to higher dimensions are straight forward and analogous. 4.2.2 TwoDimensional SubOctave Wavelet Transform Daubechies described two ways to extend 1D orthonormal wavelet transforms to two dimensions [13]. Here we adopt the way of dyadic wavelet extension to two dimensions introduced by Mallat and Zhong [51] by computing suboctave wavelet
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67 co fficients a l ong h or i zo n ta l a nd ve r tica l directions. An M suboctave wave l e t trans form of a 2D function f ( x y) E L 2 ( R 2 ) at sca l e 2j (level j) and l ocat i on ( x, y) for the mth suboctave frequency band is defined as (4 8) where 'I/J 1 ; m( x, y) = 2 t 'I/J d m(;, ffe) d = {1 2} (for horizontal and ve r tica l directions) m = {1 2 ... M } and j E Z. L 2 (R 2 ) denotes t h e space o f measurable square in tegrab l e 2D fun ctio n s. In the Fourier domain Equation ( 4.8) s impl y becomes (4.9) The fun ct i on f (x, y) ca n be recovered from it s 2D s uboctave wavelet transform by the formula +oo 2 M f(x y) Wd mj d m( ) D D D j '"'f 2 j x Y j =oo d = l m = l + oo 2 M f ~ 1 ,d m d m( ) D D D 'f/ 2 i '"Y 2 j x, Y ( 4.10) j =oo d = l m = l For perfect reconstruction 'I/J ;J(x, y) and ry ;J( x, y) must satisfy + oo 2 M L L L ,(/; d m(2jw x, 2jwy) 1 d m(2jw x, 2jwy) 1. (4.11) j =oo d = l m = l This exact reconstruction cond i t i on is obtained by taking th Fouri r tran form of Equation (4.10).
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68 4.3 Discrete SubOctave Wavelet Transform Continuou wavelet transforms are useful to demonstrate the properties of wavelet decomposition and reconstruction and are helpful for theoretical approval of the per fect reconstruction while discrete wavelet transforms are practical important for dis crete signal and digital image processing. The transform parameters in a suboctave wavelet transform are continuous variables. This results in a highly redundant rep resentation. It is possible to discretize these parameters and still achieve perfect reconstruction [64]. For digital image processing the suboctave wavelet transform of a discrete function can be carried out through uniform sampling of a continu ous suboctave wavelet transform. Below we describe the discrete formulation of a suboctave wavelet transform. 4.3.1 OneDimensional Discrete Sub Octave Wavelet Transform In the discrete domain, scales are also discrete and limited by the finest scale of one unit. A suboctave wavelet transform can similarly be decomposed into dyadic scales and we can get a perfect reconstruction through its corresponding inverse suboctave wavelet transform. In general a function can be decomposed into fine tocoarse (dyadic) scales by its convolution with dilated wavelets { 1/J 2 i ( x )}j E Z This can be done through repeated smoothings (low pass filtering) and detail finding (high pass filtering). In the discrete domain because of the limitation of the finest scale scales have to be greater than or equal to 1 so we let + oo M / rp (w)/2 = L L ~ m(2jw) :y m(2jw). ( 4.12) j = l m=l
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69 Thu for a ]level discrete suboctave wavelet transform we can write J M l cp (w)l 2 = L L ,J; m(2jwf y m(2jw) + l cp (2 1 w)l 2 ( 4 13) j = l m = l The notation { wr J(n) S 1 J(n) lj = 1 2 ... J and m = 1 2, ... M} is defined as the wavelet representation of a discrete function J ( n) under a ]level discrete M SU boctave wavelet transform. wr J ( n) and s J J ( n) are uniform samplings of their continuous counterparts respectively. If we let J = 1 then Equation (4.13) becomes M [
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f w' I w~ w~ 70 f w~ w~ Figure 4.1. A 3leve l SWT decomposition and r econstruction diagram for a 1D function. The discrete suboctave wave l et transform of a function f ( n) E l 2 ( Z ) can be implemented by using the following recursive relations between two consecutive l evels j and j + 1 (4.16) ( 4.17) where j 2:: 0, 1 ::; m ::; M, and S 0 J(w) = ](w). And, the reconstruction S 0 J(w) from a sub octave wavelet decomposition can be implemented through the recursive relation M Sjf(w) = Sj+1f(w) H(2jw) + L W]~ 1 f(w) Km(2jw), ( 4.18) m=l
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low frequency . . E',> ... HHIH . . 3 2 I WWW 2 2 2 M WI 71 high frequency . . . . 3 2 I WI WI WI Figure 4.2. Divi s ion s of the fr eq u e n cy bands und er t h e SWT s hown in Figure 4 .1. where fl i s the com pl ex co nju gate of H A threelevel M s uboctave wavelet decom position a nd r eco n st ru ct ion process based on the above recursive r e l at i o n s i s s h ow n in Fi g ur e 4 .1 The co rr es p o ndin g divi s ion s of frequency bands a r e d ep i cted gra phi ca ll y in Fi g ur e 4.2. In ge n era l for a n M s ubo ctave a n a l ys i s and sy nth es i s, we require M p a ir s of co rr espo ndin g basis wavelets. A SWT i s a multiw ave l et transform with a s in g l e sca ling fun ct ion [60, 61]. When M i s a pow er of 2 we ca n ca rr y o ut de co mpo s ition a nd reconstruction u s ing a set of FIR fil ters co rr espo ndin g to a s in g l e pair o f basis wavelets. Fi g ur e 4.3 presents a filter bank for ca rr y ing o ut a 2 l eve l 4 s ubo ctave decomposition a nd reconstruction u s in g FIR filt ers cor r espo ndin g to a s in g l e pair or two pairs o f basis wavelets. This describes a more genera l way to do the s uboctave decomposition where fin e t o coarse octave decomposition and s uboctave band sp littin g ca n be carr i ed o ut through two sets o f different FIR filters. It reduces to the case [42] when H = H s a nd G = G s where H s and Gs are u sed for s ub octave band plitting. 4.3.2 TwoDimensional Discr te SubOctave Wav 1 For t h e d compo i t i on o f a 2 D discret fun ct i o n w l e t the fr qu n r p n of a sca lin g fun t i on be defined in th formula +oo 2 M l cp (w x,wy )l 2 = LL L ,,j;d m(2jwx, 2jwy) 1 d m(2iw 2jwy)( .1 ) j = l d = l m = l
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Figure 4.3. A 2level 4suboctave decomposition and reconstruction of a SWT. : J,2 : w, : 2 2 :W, : W2,I I rr : (a) (J).X 7t 3 Th e Filler Bank fo r a 2u: ve l 2S ubOclave W ave l e t Tran sfo rm 2 1 0 F re qu ency (b) 72 Figur e 4. 4 Fr e quency plane tessellation and filter bank. ( a) Division of the frequency plane for a 2lev e l 2s ubo c tave analysis (b) Its filter bank along the horizontal dir ec tion.
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73 For a ]level 2D discrete s ubo ctave wave l et transform we can formulate J 2 M J cp (w x ,wy)J 2 = LL L ,,j; d m(2jw x 2jwy) t m(2jw x, 2jwy) + J cp (2 1 w x, 2 1 wy)J 2 j = l d = l m = l (4.20) If t h e sca lin g approx im at ion of a fun ct i on J ( x, y) at sca l e 2i i s represented by Sjf (x y) = J (f) 2 j ( x y) ( 4.2 1 ) then { wf'm J (n x, ny) S1f (n x ny) Jd = 1 2 j = l . J and m = 1 .. M} i s ca ll ed the wavelet representation of a di sc r ete fun ct i on f ( n x, ny) for a 2D Ilevel discrete M s uboctave a n a l ys i s. In general, cp ( x, y) i s a 2D sca lin g fun ct i o n a nd 'lj; d m( x, y) a nd ry d m ( x y) a r e t h e mt h dir ect ion a l a n a l ys i s a nd sy n t h es i s wave l ets. There are many c h o i ces o f t h ese fun ct ion s that sat i s f y Equation (4.2 0 ) Similar to the way 2 D wave l ets are constructed u s in g 1D wave l ets [51], we u se te n sor products to construct 2D s uboctave wavelets u s in g 1D s uboctave wavelets. Thus we can wr i te ,,J; 1 m(2j W x, 2j wy) = ,,J; m (2j W x ) cp (2j lwy) ,,J; 2 m(2iw x, 2jwy) = cp (2j lw x ) ,,J; m(2jwy) ( 4.22) ( 4 23) ( 4 .24) Through t hi s construct i on we impl emented a 2D s uboctave wav l et tran form u in g 1D convo lu t i on w i t h FIR filters of t h e 1D u boctave wave l et tran form pr i u 1 describ d. Figure 4.4( a) s h ows t h e division of t h fr qu n y p l an und e r a 21 SWT wh re M = 2. Figur 4.4(b) h ow th orr ponding filt r b nk a l on h h or i zonta l direction wh r t h e c ur v h own in r d orr p nd t th a nal y ti filA 1 1 ter 'I/J 1 1 (2w x, 2wy) (W 1 ) pro j t d a l ong th Wx ax1 th b l a k ur f rll' 1 2 th l
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0 8 0 6 0.4 0 2 Spline Smoothing Function 1 0 Second Order D erivative Approximation 2~~~~ 0 1 2 3 1 0 First Order Derivative Approximation 1 5~~~~ 0 5 0 0.5 1 1 5~~~~~ 1 0 Cubic Spline Scaling Function 1.2 0 8 0 6 0.4 0 2 0 1 0 74 Figure 4.5. Smoothing, scaling and wavelet functions for a SWT. These functions are used for a 2suboctave analysis. magenta curve for Wi 1 the blue curve for Wi' 2 and the green one for S 2 A 2D suboctave wavelet transform can be implemented by 1D convolution with FIR filters along horizontal and vertical directions. The details along the diagonal directions are embedded in the details along horizontal and vertical directions. In Figure 4.5 a fourthorder spline smoothing function its first and second derivative approximation as suboctave wavelets, and the corresponding scaling function are shown. A dyadic wavelet transform can be a special case of a suboctave wavelet trans form with M = 1. As an example, a discrete 2suboctave wavelet transform is shown to divide the details of an octave band into details of 2 suboctave bands. As shown in Figure 4.6, one sub octave can be used for detecting local maxima while the other suboctave band identifies zerocrossings. The dashed curve corresponds to the fre quency response of a first order derivative approximation of a smoothing function and the dashdot curve shows the frequency response of a second order derivative
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75 0 8 ,. / / \ 0 6 / \ 0.4 ,. .L. \ ,.,,.. 0 2 ,. / \ / / \ / / / / / \ I \ ,. 0 I / \ I \ / 0 2 \ / \ I 0.4 I I \ I 0 6 \ < / I / I 0 8 I \ I / 1 _,.,. 0 2 3 4 5 6 Figure 4.6. An example of level one analytic filters for 2 suboctave bands and a lowpass band. The dashed curve is the fr e quency r es ponse of a first order deriva tive approximation of a smoothing function and the dashdot curve is the frequency response of a second order derivative approximation. The solid curve is a scaling approximation at level one. approximation. The solid curve is a scaling approximation for analysis at lev 1 one. Thus these analysis filters take advantage of both local maxima and zerocrossing representations to characterize local features emergent within eac h cale. 4.4 SWTBased Deoising Our ex p er iments showed that a SWT with first and second ord r d rivativ ap proximation of a smoothing function as its basis wavelets separat d co fficient b t characteriz d by featur nergy from coefficients characterized by noi n rgy. Suboctav wavelet coefficients an b modifi d by hard thr h ldin g f r noi r du tion by ( .2 ) ( 4.2 )
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76 where d = {1 2} (omitted for 1D signals) j = {1, ... k} and k:::;; J, m = {1 ... M} and t~ ; m is (in general) a threshold related to noise and scale. The processed result wf'm ,* f (x) is a modified coefficient. Position x denotes n for 1D signal processing and (n x, ny) for 2D image noise reduction. SWT coefficients can be processed through soft thresholding wavelet shrinkage for noise reduction by ( 4.27) (4.28) where d = {1 2} (omitted for 1D signals) j = {1 ... k} k:::;; J, m = {1 ... M} and t~ ; m is a noise and scalerelated threshold. Again the result wf'm ,* f (x) is a processed coefficient. Similarly position x denotes n for 1D signal processing and (n x, ny) for 2D image noise reduction Recall that Donoho and Johnstone s soft thresholding method used a single global threshold [19]. However, since noise coefficients under a SWT have average decay through finetocoarse scales we regulate soft threshold ing wavelet shrinkage by applying coefficientdependent thresholds decreasing across scales [72]. When features and noise can be clearly separated at the finer levels of scale applying hard thresholding instead of soft thresholding may further improve performance. However hard thresholding may fail to smooth noise if the noise is strong locally.
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77 Table 4.1. Quantitative measurements of performance for denoising and feature restoration. Method or Measurement Blocks Bumps HeaviSine Doppler Noisy Signal: a 5 /ar, 6.856 6.735 6.895 7.017 Noisy Signal: 10log 10 (a ;/ a;) (dB) 16.726 16 .567 16.771 16.923 Restored Result: a 8 /ar, 27.258 20.988 35.453 20.129 Restored Result: 10log 10 (a;/a;) (dB) 28.779 26.439 30.993 26.076 4.5 SWTBased Enhancement with Noise Suppression Through a nonlinear enhancement function or a generalized adaptive gain opera tor, SWT coefficients were modified for contrast e nhancement by ( wd mf(n n )) wd m ,* J( ) = Md m E j x, y J nx, ny 21 OP Md,m 21 ( 4.29) ( 4.30) where d = {l 2}, 1 m M, and 1 j J. The pointwise operator s output wf'm,* J(n x, ny) is simply a processed coe fficient. The e nhancem e nt operator Eop can be ENL or EcAc presented in Section 2.4. Since these operators are defined on input values between 1 to 1 we normalize suboctave wavelet coefficient before applying the operators. 4.6 Application Samples and Analysis In this section, we present several ex p rimental r s ult s of SWT ba d d n i ing and e nhan c m nt for both 1D ignal and 2D imag es. Fir t w pr nt xp nm tal r ults based on a 1D suboctave wav 1 t tran form. W how th d n 1 mg capability of our m thod with f atur r storation ompar d to xi ting d n 1 m g m thod
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78 ldb : :u : : C75 : :l :LiliJCL : : : :l 0 200 IOO 600 800 1 000 1200 1400 1 600 1 800 2000 0 200 400 600 800 1 000 1 200 1400 1600 1800 2000 lJ:v ~ ... 1r. ... ~.e5 ... : ... ;.J : LiliJCL : : Ou .: :l 0 200 IOO 600 800 1 000 1200 1400 1600 1800 2000 0 200 400 600 800 1 000 1200 1 4 00 1600 1800 2000 ldti :u : : C75 : :l :LiliJCL : : : :l 0 200 400 600 800 1 000 1200 1400 1 600 1 800 2000 0 200 400 600 800 1 000 1200 1400 1 600 1 800 2000 1 ;ncu : :u : : c75 : :i :LiliJCL : : : :i 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1 200 14 00 1 600 1800 2000 (a) Blo c ks (b) Bumps _:r:s::;=s:2 0 200 400 600 800 1000 1200 1 400 1600 1800 2000 0 200 400 600 800 I 000 1200 1400 1 600 1800 2000 _:: ~ _::~ 0 200 1()() 600 800 I 000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1 IOO 1600 1800 2000 :~:~ 10'''''''''''10'''''''''''" 0 200 400 600 800 1 000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1 400 1600 1800 2000 :r :s~ _:: ~ 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 4 00 600 800 1 000 1 200 1 400 1 600 1800 2000 ( c) H eav iSin e" ( d) Doppl er Figure 4.7. Denois ed and r estored features from the SWTbased algorithm. From top to bottom: original signal; noisy signal; denoised signal overlay of original and denoised s ignal.
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79 Experimental results of SWTbased denoising are shown in Figur e 4.7. In the fourth plot shown in each Figure 4. 7( a)( d) the denoised results are overlaid with their corresponding original signals. Table 4.1 shows quantitative measurem e nts of each result shown in Figures 4.7. In comparison to previously published methods processing the same signals [8], results of the SWTbased method are noticeably im proved and basically free from artifacts, pseudoGibbs phenomena. In the next exper iment we show the limitations of a traditional DWT for characterizing bandlimited high frequency features. Figures 4.8( a)(b) show the original and noisy Doppler signals with their corresponding 5level DWT and a coarse scale approximation. The finest scale bandlimited features (see the second plot in Figure 4.8(a)) are weak and obscured when noise is present (see the second plot in Figure 4.8(b)). These high fre quency bandlimited features are lost in a soft thresholding denoising method shown in Figure 4.9(b). Figures 4.8( c )(f) show 2suboctave wavelet transforms of the orig inal and noisy Doppler signals. Figures 4.8(c) and 4.8(e) show first suboctave coefficients while Figures 4.8(d) and 4.8(f) display second suboctave coefficients. Figure 4.9 shows denoised and enhanced results of noisy Doppler under a DWT and a SWT. The loss of high frequency bandlimited features made the result from the DWTbased method less attractive than the SWTbased method for pro ce sing medical images such as microcalcification in mammograms. Next, experimental results of enhancement with noi e suppr s10n of mammographic images via a 2D suboctave wavelet transform ar pr s nt d. d montrate the advantage of enhancement without amplifying noi in luding ba kground noise [37]. In the first exp riment for imag nhan m nt w att mpt t nh an the visibility of a radiographic imag e of a RMI (Radiation f a ur m n In Iiddleton WI) model 156 phantom with imulat ~ d ma mb d l d. Figur .lO(a) pr sents a low contra t imag of the RMI mod 1 1 6 phant m wi h imul a d ma
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The Ongma l S,gnaJ and I ts DWT Coeff 1 c1cnts 10~~;:;;;::;~;:;;:;;::;::r..:::::::=:::::::~r:i 0 IO'''..__''=..__''..____.__, (a) The Original Signal and lls Firsl SubOctave Coefficie n ts or a SWT (c) The Noisy S1gnaJ and Its F u 'Sl SubOctave Coefficie n ts of a SWT J ~''''. j ~11, ::: .. ,;, .:~. 11 o:a I ::5.i I:; 11 ,::: 1,1::, I ; j ~ 0:? 0~ : o:a 0:5 o; 0:2 o:~ j~ 0:3 0:4 0 : 5 0:6 0:2 0:8 0:9 j~ o:a 0:5 0:6 0: 2 0 :8 0:9 JHl~":4 0:5 0:6 O :J 0:8 0:9 J 0 0. 1 0 2 0 3 0 4 05 0 6 0 7 0 8 0 9 I (e) 80 The No i sy Signa l and Its DWT Coefficients (b) The Orig inal Signa l and I ts Sec a nd SubOctave Coefficie n ts o r a SWT (d) The No i sy Signal and Its Second SubOctave Coeffic i ents of a SWT J 0 0 1 0 2 0 3 0 4 0.5 0 6 0 7 0 8 0 9 I (f) Figure 4.8. Limit at ions of a DWT for c hara cte rizing bandlimit e d high fr eq u e n cy features.
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The Original Signal a nd I ts DWT Coefficients 1or.r:;:;::;::::~rr:::::==::::::~ 0 IO''''=''=.,_____,____..___''' ( a ) The Enh anced Signal and the P rocessed First SubOctave Coefficients o f a SWT ( c ) The Enhanced S1gnaJ and the Pr ocessed OWT Coefficients (b) 10 0 IO"'''"~''~'_.___'" (d) 8 1 Fi g ur e 4. 9. D e noi se d a nd e nh a n ce d r es ul ts o f a n o i sy D o ppl r s i g n a l un d r a D WT (2 5.5 29dB) a nd a SWT ( 2 6 076dB)
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82 (a) (b) (c) Figure 4.10. Enhancement with noise suppression. (a) A low contrast image of RMI model 156 phantom with simulated masses embedded. (b) Enhancement by tradi tional histogram equalization. (c) SWTbased enhancement with noise suppression. of distinct sizes. Figure 4 l0(b) shows enhancement by traditional histogram equal ization. SWTbased enhancement with noise suppression is shown in Figure 4.l0(c). Unsharp masking a popular technique for enhancing radiographs, failed to enhance barely seen masses (low frequency features) in Figure 4.l0(a). In comparison to tra ditional histogram equalization the SWTbased method enhanced the visibility of masses without amplifying noise. In the next experiment we enhance low contrast mammographic images contain ing a microcalcification cluster. Figure 4.ll(a) shows a region from a low contrast mammogram containing a distinct microcalcification cluster. Next, enhancement by traditional unsharp masking is presented in Figure 4 ll(b). Figure 4.ll(c) shows the result of SWTbased enhancement with noise suppression. In practice a radiologist may want to view certain suspicious areas of low contrast within a large digital mam mogram for potential breast lesions with a magnifier to improve visibility of an area. In Figure 4 .11 ( d) we try to provide a similar function by improving the visibility of a local region of interest (ROI) through SWTbased enhancement with noise suppres sion. The region within the black square (120xl20) is enhanced. Note that the area does not have to be a square but a rectangle. Traditional unsharp masking shown
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83 in Figure 4.ll(b) enhances the area of the microcalcification cluster slightly but also amplifies noise. As shown in Figures 4.ll(c) and 4.ll(d), enhancement under a SWT makes barely seen microcalcification clusters more visible without amplifying noise.
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84 (a) (b) (c) (d) Figure 4.11. Enhancement with noise suppression. (a) Area of a low contrast mam mogram with a microcalcification cluster. (b) Best enhancement by traditional un sharp masking. (c) SWTbased enhancement with noise suppression. (d) SWTbased enhancement of a local region of interest (ROI) with noise suppression.
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CHAPTER 5 PERFORMANCE MEASUREMENT AND COMPARATIVE STUDY In this chapter, we present several measures of relevant quantitative metric for evaluating an algorithm's performance and show a few comparative studies of quan titative measurements between our algorithms and other published methods In Section 5.1, a few quantitative measurements used in this dissertation are described and formulated mathematically. Section 5.2 shows the quantitative results of our algorithms and other relevant methods for signal/image restoration. In Section 5.3 we present quantitative measurements of image enhancement among our developed algorithms and a few other related methods. Earlier chapters demonstrated visual quality of denoised as well as enhanced results (visual performance) while this chap ter focuses on quantitative performance. 5.1 Performance Metric for Quantitative Measurements The quality of a noisy signal/image is often measured by the ratio of ignal vari ance to noise variance ( or ignal energy level of variation to noi e energy l ev 1) u ing a log cale. The quantity is called signaltonoi ratio. A signal/imag i mor lik 1 sever ly degraded when noi e level is high (low ignaltonoi rati ) W h av u d the quantitativ term ignaltonoise ratio wh n di playing ur arli r xp rim ntal r ults. A formal definition of signaltonoi ratio i giv n b 1 w. Signaltonoi eratio (SNR) i d find a
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86 where a; and a~ are signal variance and the average noise energy (average l east squares error between a signa l and i ts noisefree orig in a l version) respectively [30]. Here we denote an ideal ( origina l ) signa l as g and the restored version from its noise signal as g For 1D signals, a; is defined as 2 1 2 as = N 6 (g ( n) g) n=l 1 N g = N L g(n), n=l and a~ is defined by 1 N a~ N L (g*(n) g(n))2. n=l For 2 D images, a; is defined as 1 M N a; MN L L (g(m,n) g)2, m=l n=l 1 M N g= MN L L g(m n) m=l n=l while a~ is defined by 1 M N a~ MN L L (g*(m, n) g(m, n))2. m=l n=l The term SN R was frequently used in many noise reduction a l gorithms for quan titative measurements of performance and has also been app li ed to measure our de n oising methods. The quality improvement of a signa l/im age can be measured by an improved (higher) SN R compared to the SN R of i ts noisy version. The quantitative measurements of average sq u are errors were used to measure the performance of noise reduct i on algorithms [19, 20, 8] For our comparative studies, we
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87 use the same measurements to compare the denoising performance between our algo rithms and other published methods such as Donoho and Johnstone 's thresholding based wavelet shrinkage, Coifman and Majid's wavelet packetbased denoising [9], and Chen and Donoho's basis pursuit denoising [5]. Average square error (ASE) for 1D signal recovery is defined as ASE= Ilg glltN N where llf 11~ N = ";/ 1 (J(n)) 2 Average square error (ASE) for 2D image restoration is defined by ASE= Ilg* gllt{M,N} MN Another often used metric for measuring an algorithm's performance is the root mean squared error [8] We also use the quantity for measuring our algorithms performance compared to Coifman and Donoho's methods. The root mean squared error (RM SE) for 1D signal restoration can be defined as RMSE= ";/ =l (g*(n) g(n))2 N For 2D image proc ess ing the root mean squared e rror (RMS E) for imag r toration is defin e d by RMSE= ":;[ = 1 ";/ = 1 (g*(m, n) g(m n)) 2 MN For imag e nhanc m e nt co ntra t ha b n u d a maJ r m a ur m nt f r an algorithm' p rforman e In ord r to p rform quantitativ mp ri n haY appli d th am quantitativ m asur in ludin g a v r i n f th opti a l d finiti n
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88 of contrast introduced by Morrow e t al [53] and a contrast improvement index [39]. The contrast C of an object of interest is defined by C= where ""i, 1 is the mean graylevel (intensity) value of a particular object in a region of interest called the foreground and ""i,b is the mean intensity value of a surrounding region called the background. The definition of C above takes the absolute value of the definition by Laine et al. [39]. The reason is that sometimes an object of interest may have lower average intensity than its surrounding area and the techniques introduced here can be applied to other image processing applications. We avoid having a negative value for the contrast of an object by using this expression. A contrast improvement index (CII) can be used as a quantitative measure of contrast improvement and is defined by CII = CProcessed Coriginal l where CProcessed and Coriginal are the contrasts for a region of interest in the enhanced and original images respectively. We use contrast and CII for comparing the perfor mance of a few image enhancement methods in Section 5.3. The performance of related algorithms, such as signal/image restoration and im age enhancement can be relatively compared using similar quantitative measure ments. The quantities allow comparison of the relative performance of the algorithms. In the next two sections we will evaluate the performance of several related methods for noise reduction and image enhancement.
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89 Table 5.1. Quantitative measurements: Average Square Errors llg*!lltN for various signal restoration methods. I Method or Measurement I Blo c ks I Bumps I HeaviSin e I Doppler I llfll tN/N 81.211 57.665 52.893 50.348 with noise 1.047 0.937 1.008 0.999 ideal wavelets 0 097 0.111 0.028 0.042 threshold ,\ tv 0.395 0.496 0.059 0.152 threshold (2 log N) 1 / 2 0.874 1.058 0.076 0.324 WP coherent denoising 0.433 0.532 0.133 0.257 xwpl and WB 0.081 0.182 0.056 0.126 xwpl and BOB 0.081 0.239 0.084 0.104 BPDNJumpWave 0.394 0 .76 1 0.072 0.282 BPDNHS 0.060 failed 0.100 0.319 DWT soft thresholding 0.149 0 .274 0.049 0.209 DWT denoising and enhancement 0.039 0.138 0.042 0 137 SWT denoising and enhancement 0.065 0.111 0.038 0 120 5.2 Quantitative Comparison of Signal/Image Restoration This section presents quantitative measurements for comparing the p e rforman ce of several algorithms aimed at data restoration. In Chapter 3, we have us ed SN R to show an algorithm's quantitative performance in a few experiments while showing its visual performance such as in Figures 3.11 and 3.12. Tabl e 4.1 in Chapter 4 has also shown SN R for noisy signals and restored r es ults after noise reduction and feature enhancement. Table 5.1 shows the quantitative m easur m nt s of our algorithm performance com pared to Donoho and Johnston e' thr holdba d d noising methods Coifman and Majid's wavelet packet bas d method a w 11 a Chen and Donoho s basis pursuit based methods. The quantity llflltN/N in Tabl 5. 1 i s th av rage signal n rg p r pix 1 wh r llf lltN = ~ := 1 J(n) 2 Th second row of th tabl how av rag rr r f th noi y signal to it nois fr original. Th r ult s in th third r ,;i. r pr du d b r id al wav 1 t coefficient e lection using th po iti n inf rm ti n f n fr a 1 t
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90 coefficients (in Donoho and Johnstone's term, with the help of an oracle which was called ideal wavelets"). The method is basically using wavelet coefficients of the noisefree signal to select and scale the corresponding noisy wavelet coefficients based on the noise level in order to remove noise components. The next two rows are the results of Donoho s wavelet coefficient thresholding methods under two different thresholds [19]. The row under WP coherent denoising" shows the results of Coif man's adapted waveform analysis and denoising [9]. The results of "xwpl and WB and "xwpl and BOB were produced by xwpl with "Wavelet Basis and Best Of Bases" algorithms respectively. The results of BPDNJumpWave and BPD HS are from Chen and Donoho s paper [5] (Figures 5.4 and 5.5) with the dictionaries of Jump+ Wave and HeaviSide ", respectively. The results of DWT soft thresh olding" were obtained through multiscale DWT and soft thresholding methods. The results of DWT denoising and enhancement" are our earlier results via wavelet shrinkage and feature emphasis of DWT coefficients. The last row shows the results from our suboctave wavelet based algorithm for denoising and enhancement. In the table, we include the results from other published methods only for comparison purposes. Table 5.2 shows the quantitative measurements of performance among our devel oped algorithms Coifman and Donoho s "TranslationInvariant Deoising methods [8], and Coifman s wavelet packet based methods [9]. The first seven rows in Table 5.2 are the results of Coifman and Donoho s thresholding and different translation invariant denoising methods. The next three rows are from Coifman's wavelet packet based methods similar to Table 5.1 but in a different measure. The next two rows are our earlier DWTbased results which are the same as Table 5.1 in a different quantitative metric The last row shows the results from our suboctave wavelet based algorithm for denoising and enhancement. These results are measured based
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91 Table 5.2. Quantitative measurements: RMSE for various denoising methods. I fethod I Blocks I Bumps I H eaviS in e I Doppler I Traditional, S8 0.9433 1.0040 0.2898 0.5352 Cycle Spine, S8 0.8901 0.8909 0.2830 0.4745 Fully TI, S8 0 .8458 0.8733 0.2855 0.4554 Soft, Haar 0.6574 1.2291 0.6086 1.0881 Soft, Haar TI 0.4817 0.8703 0.2247 0.7339 Hard Haar 0.2835 0.7038 0.4312 0.7383 Hard Haar TI 0.1708 0.3966 0.1818 0.3893 WP coherent denoising 0.6584 0.7299 0.3660 0.5070 xwpl and WB 0.2847 0.4273 0.2378 0.3563 xwpl and BOB 0.2847 0.4895 0.2903 0.3237 DWT soft thresholding 0 .33 19 0.5236 0.2215 0.4521 DWT denoising and e nhancement 0.1819 0 .3728 0.2050 0.3696 SWT denoising and enhancement 0.2567 0 .3334 0.1974 0.3477 on Root Mean Squared Errors" (RMSE). Coifman and Donoho [8] claimed they were using RMSE, but their results were computed using the root of overall square errors inst ead (probably caused by a programming error). Thus, the difference be tween the first seven rows in Table 5.2 and results in Coifman and Donoho [8] i s that each quantitative measure here is equal to the corresponding result in Coifman and Donoho [8] divided by v'2Q48. 5.3 Quantitative Comparison of Image Enhancement In this section, we compare the quantitative performance of imag e en h anc ment of a few r e l ated methods. Figures 4.10 and 4.11 have s hown th visual p rformanc of imag e e nh ancement with noise suppression und er suboctave wav l et tran form In th n ext f w comparat iv e studies, imag e en han c m nt i arr i d out on l ow on trast mammographic imag es conta ining a ma s, or imulat d mamm graphi f tur s. Contra t en han c m nt with noi uppr sion for mammogr phi 1mag "ith or without add itiv e noi is s hown in Figur 5.1. Figur .1 ( a) h ,~ an ar a fa 1 w contrast mammogram with a lobular ma B au th ntra t i p
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92 ( a) (b) (c) ( d) (e) (f) (g) (h) (i) Figure 5.1. Enhancement results. (a) Area of a low contrast mammogram with a mass. (b) Enhancement of (a) by traditional contrast stretching. (c) Enhancement of (a) by traditional unsharp masking. (d) SWTbased enhancement of (a) with noise suppression. (e) The same area of a low contrast mammogram contaminated with additive Gaussian noise. (f) Enhancement of ( e) by traditional contrast stretching. (g) Enhancement of ( e) by traditional unsharp masking. (h) SWTbased enhance ment with noise suppression. (i) Handsegmented mass and ROI for quantitative measurements of performance.
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93 T a bl e 5 3. C omp a ri so n of c ontra s t v a lu es o bt a in e d b y traditi o n a l co n trast st r etc hin g (C ST ), un s h a rp m as kin g ( UN S ), a nd multi sca l e n o nlin ea r p rocess in g o f s uboctave wa v e l e t tra n s form (SWT) c o e ffi c i e nt s o f a m a mmo gra m co nt a inin g a m ass l es i o n Contrast values for CST, UNS and SWT enhancement techniques M e thod ORI CST UN S SWT Fir s t row 0 01920 0 09736 0.08 3 90 0.119 54 S ec ond row 0 01918 0 0929 5 0.07 8 09 0 12 8 0 2 i s h a rdl y vi s ibl e Fi g ur e 5. 1 (b) s h o w s im a g e e nh a n ce m e nt b y t ra diti o n a l co n tras t s tr e t c hin g ( sca lin g th e l o w co ntra s t im age lin ea rl y t o th e full di s pl ay ra n ge o f g r ey l e v e ls). Fi g ur e 5 .l( c ) s h o w s e nh a n ce m e nt of Fi g ur e 5 .l( a ) throu g h un s h a rp m as in g Fi g ur e 5 .l(d) s how s t h e e nh a n ce d r es ult o f Fi g ur e 5. l( a ) throu g h SWTb ase d e nh a n ce m e nt with noi se s uppr ess i o n To d e mon s trat e th e c ap a bilit y o f t h e m et h o d for c ontra s t e nh a n ce m e nt with n o i se s uppr ess i o n w e a dd e d G a u ss i a n n o i se ( 1 4 dB ) t o Fi g ur e 5. l( a ) a nd ge n e rat e d a n o i sy im age s hown in Fi g ur e 5. l (e) Th e a ddi t i ve n o i se b eco m es mor e vi s ibl e in it s lin ea rl y e nh a n ce d im age, t hrou g h t ra diti o n a l co t ras t s tr e t c hin g, s h o wn in Fi g ur e 5 .l(f) Fi g ur e 5 .l( g ) s h o w s th e e nh a n ce d im age throu g h un s h a rp m as kin g U n s h a rp m as kin g e nh a n ce d th e m ass a nd a l so a mplifi e d th e noi se Fi g ur e 5 .l(h) s how s th e e nhan ce d im age t hrou g h SWTb ase d e nh a n ce m e n t with noi se s uppr e s s i o n a nd th e r es ult i s pr e tt y cl ose t o t h e e nh a n ce d im age in Fi g ur e 5. l(d ) T o es tim a t e t h e p e rform a n ce o f v a riou s e nh a n ce m e n t tec hni q u e q u a n t i ta ti ve l y, w e m a nu a ll y seg m e n te d th e m ass a nd ROI ( t h e a r ea w i t h t h e bl ac k q u ar ) s h o wn in Fi g ur e 5 .1 ( i ) Th e qu a n t i tat i ve m eas u re m e n ts o f p e rform a n c fo r nh a n m e nt t ec hniqu es pr ese nt e d in Fi g ur e 5. 1 a r e s h ow n in T a bl es 5.3 a n d 5.4 ba d o n co ntr as t v a lu es a nd co nt ra t imp rove m e n t indi s, w hi h w r d fin d and xp l a i n d in S ect i o n 5 1.
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94 Table 5.4. Contrast impro vement index (CII) for e nhan cement by traditional contrast stretch in g (CST) unsharp masking (UNS) a nd multiscale nonlin ea r processing of suboctave wave l et transform (SWT) coeffic i e nts of a mammogram with a m ass. Contrast Improvement Index for CST, UNS, and SWT enhancement techniques Method ORI CST UNS SWT First row 1.00000 5. 06997 4.36908 6.22501 Second row 0.99863 4.84 037 4. 06680 6.66665 Mathemat i ca l models of phantoms were previously co n st ru cted in Lain e et al. [39]. Here we adopt the same models, s hown in Figure 5.2(a), to validate these e h ancement tec hniqu es quantitatively a nd to compare the suboctave based m et hod to previous dyadic wavelet processing m et hod s for improving co ntr ast [39]. Th e mod e l s includ ed mammographic features s u c h as microcalcifications cy lindri ca l a nd sp i c u l ar findin gs, and lobul ar masses. To directly measure the difference of p erfor man ce between our enhancement techniques and t h e methods r epo rt ed in Lain e e t al. [39], we u sed t h e same blended imag e s h ow n in Figure 5. 2 (b). For mor e details about how t hi s im age was co n structed, we r efe r the reader to L a in e et al. [39]. Enhancement results for a blended im age with s imul ated m ammogra phi c features are s hown in Figure 5.2 Figure 5 .2 (a) s how s a phantom im age with the mammo graph i c f eatures. These feat ur es are blended with a mammogram M56, as s hown in Figure 5 .2 (b) For co mp ar ison purposes Figure 5.2(c) shows the r es ult a ft er process in g the blended m ammogram in Figure 5.2(b) under a DWTb ased method reported in Laine e t a l. [39]. Figure 5 2(d) i s the e nh a n ced result produced via the developed SWTbased enhancement tech niqu es with noi se s uppr ess ion T ab l es 5 5 a nd 5.6 s h ow t h e contrast va lu es a nd C II of the mammographic features displayed in Figure 5 .2 (a). From both v i s u a l appeara n ce a nd quantitative measurements we observe that the result (see Figure 5.2( d)) obtained v i a the SWTbased method i s overa ll super ior
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95 (a) (b) (c) (d) Figure 5.2. Phantom nhancement results. (a) Phantom imag (b) ammo g r a m M56 with bl nded phantom f atures. (c) onlin ar nhan m nt und r a D T (d) SWTbased enhancement with noise uppres ion.
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96 Table 5.5. Comparison of contrast values obtained by multiscale adaptive gain pro cessing of dyadic wavelet transform (DWT) and suboctave wavelet transform (SWT) coefficients. Mammographic features: minute microcalcification cluster (MMC) mi crocalcification cluster (MC), spicular lesion (SL), circular (arterial) calcification (CC) and wellcircumscribed mass (WCM). Contrast values for DWT and SWT enhancement techniques Feature MMC MC SL cc WCM Original 0 050 0 033 0.024 0.037 0.012 DWT 0.198 0.204 0.165 0.214 0.157 SWT 0 342 0 253 0.240 0.292 0.112 Table 5.6. CII for enhancement by multiscale adaptive gain processing of dyadic wavelet transform (DWT) and suboctave wavelet transform (SWT) coefficients. Mammographic features: minute microcalcification cluster (MMC), microcalcifica tion cluster (MC) spicular lesion (SL), circular (arterial) calcification (CC), and wellcircumscribed mass (WCM). Contrast Improvement Index ( CII) for DWT and SWT enhancement techniques Feature MMC MC SL cc WCM DWT 3.929 6.234 6.861 5.730 13.576 SWT 6.786 7.746 9.983 7.818 9.676 to that of traditional DWTbased methods of analysis. Experimental results have shown that the multiscale suboctave wavelet processing algorithm can make more obvious unseen or barely seen features of a mammogram without requiring additional radiation.
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CHAPTER 6 OTHER APPLICATIONS OF WAVELET REPRESENTATIO NS Thi s c hapt e r d esc rib es two other a ppli cat ion s of wavelet representations und er dyadic or s ubo ctave wavelet transforms to medical im age processing problems s u c h as border id e ntifi cat i o n a nd ma ss segme nt at ion. The co nt e nt s cove r ed h e r e are pri m a ril y based on part o f two published papers [38, 74]. At the sa m e time we h ave a dd ed a lit t l e mor e details to the two publi s h e d methods. Section 6.1 presents a method for m yoca rdi a l boundar y d etect i o n of ec ho ca rdiographi c im ages while Sec tion 6.2 describes a technique of im age seg m e nt at i o n for m a mmo grap hi c im ages 6.1 Bord e r Id e ntifi cat ion of Echocardiograms In this sect ion we present an a l go rithm for e ndo ca rdi a l a nd e pi ca rdi a l border id e ntifi cat ion of the l e ft ventricle in 2D s hortax i s ec ho car dio grap hi c im ages. For clarity in this di sse rt a tion we are going to ca ll the a l gor ithm HBD ( H eart Boundary D etect ion). The approac h r e li es on s h ape modeling o f e ndo ca rdi a l a nd ep i cardia l boundaries a nd promin e nt border informati o n extract ion from ec h ocard i ograp hi im age seq u e n ces. Th e HBD a lgorithm for m yocard i a l border id ent ifi at i on on i t of four major steps; waveletbased edge detection border segm nt extracti n bord r r eco n truction, a nd boundary smoot hin g Wavel t maximum repre ntation of dg dynamic h ap modeling a nd match d filt e rin g t hniqu e are utiliz d to d rmm the ce nt e r point of th 1 ft ve ntri 1 and arry o u t f atur xtra ti n f b rd r eg m c nt s to b tt r approx im at e nd o ard i a l and p 1 ard i a l b und ar i HBD rn st im at the c nt r point o f th I ft v n tr i l and al d t rmin b th n l ar lial 97
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98 and epicardial boundaries [38]. Myocardial boundary identification is autonomous requiring no human input for initial estimation of boundary locations. Sample ex perimental results are shown for endocardial and epicardial border identification in 2D shortaxis echocardiograms in Section 6.1.6. 6 .1.1 Overview of the Algorithm Automatic identification of myocardial boundaries of the left ventricle (LV) has been a focus of attention for many researchers employing computational methods to assist cardiologists in screening clinical heart diseases and to provide quantitative information for medical diagnosis. Many methods have been developed. Several approaches for boundary detection of the LV have shown partial success in either identifying the endocardial boundary [7 16, 22 63], the center point of the LV and partial boundary points [66 67], or epicardial boundary detection [21]. In these ( and most other) approaches human knowledge of the approximate shape of endocardial and epicardial boundaries provided by an expert and/or a computer generated center point of the LV have been used successfully for boundary detection. Thus, modeling and analysis of the LV shape [4] have played important roles in previous boundary detection methods. The methods behind our HBD algorithm are straightforward. Current technology provides ultrasonic images embedded with a considerable amount of speckle noise and signal dropout. The consequence is that it is difficult to identify closed contours of endocardial and epicardial boundaries of the LV which are well delineated within an image sequence. A priori knowledge about the general shape of the LV has been successfully used in some boundary detection algorithms. Our implementation combines highlevel shape descriptors and primitive boundary features to reconstruct
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99 clo ed boundary estimates. Four major steps are needed to carry out detection of endocardial and epicardial boundaries of the LV; (1) Multiscale wavelet analysis for edge detection (2) Matched filtering for feature extraction (3) Boundary contour reconstruction, and ( 4) Reconstructed boundary contour smoothing. In our HBD algorithm we first use a DWT (Dyadic Wavelet Transform) to find edge information for each ultrasound image through local maxima determination [51]. The shape of the LV is then defined by a dynamically adjustable model [66, 67]. Using shape modeling, we then carry out shape matched filtering [66] to determine the center point of the LV and extract important boundary segments. The extracted boundary segments are then connected and smoothed to obtain an approximation of endocardial and epicardial boundaries of the LV. We describe each step of the HBD algorithm below. 6.1.2 Multiscale Edge D tection Local maxima representations of a DWT [51, 40, 71] are applied for multis ale dge detection. M ul tiscale edges at 1 vel j can be computed by first computing th modulu Mj and angl Aj of dyadic wavel t d ompo ition Wf and W } (horizontal and v rtical omponents of wavelet co ffi i nt ) a follow
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100 where A 1 (n x, ny) is approximated to the closest orientation defined over an eight pixel neighborhood ( eight orientations). At level j we define the local dyadic wavelet maxima (edge points) 1 (n x, ny) as ! M 1 (n x, ny) if M 1 (n x ny) is maximum along direction A(n x ny), 1 (n x, ny) 0 otherwise. The local maxima of dyadic wavelet coefficients are located along gradient direction A(n x, ny) at each point (n x, ny)Multiscale edges at level j are then obtained by finding 1 greater than T 1 a small threshold; ! 1 (n x, ny) if 1 (n x, ny) > T 1 E 1 (n x, ny) 0 otherwise. We use the small threshold to get rid of some isolated points of noise perturbation. An advantage provided by this waveletbased edge detection method for echocar diograms is that the edge information at some middle level of the spatialfrequency space provides good estimated locations of heart boundaries with less noise for more reliable feature extraction. This is because myocardial boundaries usually show up as features of finetocoarse scales Since we are interested in large structures such as endocardial and epicardial boundaries we discard the fine detail edges and focus only on edge maps within the middle levels of scale. The level of scale identified depends on selected basis wavelets applied for a DWT and the structure size relative to the support of each wavelet. For a basis wavelet edge maps at level 3 provided more coherent structural information on our test echocardiographic images. However for another basis edges at level 4 demonstrated a better compromise. A small threshold 4.0 was applied to discard any edge points induced by speckle or background noise
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101 Low Circular Arc T e mplat e Upper Cir c ular Ar c T e mpl a t e Fi g ur e 6 1. Th e c ir c ul a r a r c te mpl a t es for m a t c h e d filt e rin g. 6 1.3 Sh a p e Mo d e ling As p a rt o f th e o p e rati o n s within thi s s t e p a p a ir o f c ir c ul a r a r c te mpl ates we r e ge n e rat e d a nd s h a p e m a t c h e d filt e rin g wa s p e rform e d t o es tim a t e th e ce nt e r p o in t o f th e L V u s in g th e e d ge m a p ob ta in e d from S te p 1. Thi s was acco mpli s h e d by fi rst u s in g a n a v e ra ge s i ze d c ir c ul a r ar c t e mpl a t e to find t h e po s t e ri o r e pi ca rdi a l b o und a r y seg m e nt s a nd d e t e rmin e th e p o t e nti a l ce nt e r p o int o f th e l owe r h a lf o f t h e L V. Fi g ur e 6 1 s h o w s c ir c ul a r a r c t e mpl ates for m a t c h e d filt e rin g. Ne x t, a n upp e r c ir c ul a r arc t e mpl a t e with th e sa m e s iz e as th e low e r c ir c ul a r a r c t e mpl ate w as ge n erate d a n d u se d t o find th e a nt e ri o r e pi ca rdi a l bound a r y seg m e n ts a nd to d ete rmin e t h e p ote n t i a l ce n te r p o in t o f th e upp e r h a lf o f th e L V. Th e seco nd m atc h e d fil te rin g was ca r r i e d o ut n ea r t h e l o w e r p o t e nti a l ce nt e r p o in t s in ce th e p oste ri o r e pi ca rdi a l b o r d r i s o n o f th e m os t r e li a bl e fea tur es found o n 2D s h o rtax i s ec h oca rdi og r a m s [ 66 67] T h es tim ate d ce nt e r p o int of th e L V w as id e n t ifi e d as t h e ce n te r p o in t o f t h e lin linkin g th e t w o p ote nti a l ce n te r p o in ts gme n t A ft e r est im at in g t h e ce n te r p o in t a nd ro u g h l oca ti o n o f ant ri o r a n d po t ri r e pi ca rdi a l b o rd r s, w e t h e n est im ate d wa ll t hi kn s a n d o n t ru t d w n n tr i e llipti cs h a p e d ba nd m as k s fo r extract in g b ot h ndo ardia l and p i ard i a l b un dar s g m e n ts. Thi s w as accom pli h e d v i a a im p l ma k i ng p r at i n T h b:t n d w i dt h o f ac h m as k s p c ifi dh ow m u h d format i n f m ard iu m va ll wa :i. 11 l d uri n
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102 Ideal Epi Border EndoMask EpiMask Figure 6.2. Dynamic shape modeling. feature extraction. Postprocessing after masking allowed us to remove extracted noise edge points which may include weak boundary segments. Thus, length thresholding was applied on all extracted segments. The removal of weak boundary segments gen erally did not affect boundary reconstruction as long as enough boundary segments remained (at least a boundary segment in each quadrant). Figure 6.2 shows the dynamically adjustable shape model. For the test images used in our experiments the default settings for the radius of the circular arc template Epi band width wall thickness and Endo band width were 75, 10, 25 and 8 respectively for a typical ED frame. These parameters need to be adjusted for a frame at ES a different ED frame or a frame between ES and ED for a better approximation of endocardial and epicardial boundaries. 6.1.4 Boundary Contour Reconstruction N e xt boundary segments obtained from the previous steps were connected to re construct clos e d boundary contours. First a preprocessing procedure was performed to c onn e ct brok e n boundary segm e nts belonging to large boundary segments via an
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103 1 1 1 V / V / I"/ 1 1 / I"/ / 1 1 1 1 1 1 / / V / 1 / 1 I"1 / 1 I"1 / 1 / 1 1 1 Figure 6 3 Connecting broken boundary segments. The first row shows four typical cases showing the end points of two broken segments belong to a large segme nt. The second row is the result after connecting the two broken segments for each case. 8neighbor connection and excessive points attached to a boundary segment were removed. Figure 6.3 shows the templates used to perform the operation of connect ing broken boundary segments. In the figure, a number 1" represents a point on a boundary segment while a blank means that the point is not on any boundary segments. The highlighted pixel is l ocated at the point of an operation. An attached point was an edge point whose removal would not break the boundary segm nt to which it was attached Figure 6.4 shows how the attached point removal op ration was carried out. During boundary contour reconstruction we avoided mislinking two n ighbor boundary segments by examining their re l ative locations orientation and di tan To avoid linkin g an endocardia l boundary segment to an pi ardial boundar ment ndocardial border segments w re labeled distinctly from pi a rdial bord r segm nts during feature extraction in the pr vious tep e u d a ir ular r r la tiv to the cent r point of the LV in t ad fa traight lin to int rpolat b tw n th corr sponding nd point of ach pair f tw n ighb r b undar gm nt If h link
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104 1 1 1 1 1 / '\ V '\ V V '\ 1 1 "\ / 1 / 1 1 / ~ 1 / 1 1 1 1 1 1 1 1 1 1 / '\ V '\ V V '\ 1 1 '\ / "\ / "\ / '\ / 1 1 1 1 1 Figure 6.4. Attached point removal. The first row shows four typical cases with attached points. The second row is the result after attached point removal for each corresponding case. procedure successfully connected all boundary segments we obtained closed endo cardial and epicardial boundary contours. If it failed to connect boundary segments because of their positions relative to each other or insufficient boundary segments, then only partial boundary points were realized. If a wrong connection ( along the expected boundary contour) occurred error between the detected boundary and the expected boundary was large and we observed that the detected contour was dis torted slightly relative to the predicted boundary contour. 6 1.5 Smoothing of a Closed Contour without Shrinkage It was necessary to smooth the reconstructed boundary contours for a better approximation of the endocardial and epicardial boundaries of the LV. Two local smoothing filters were investigated. One was a window spatialaveraging smoothness filter [30] and the other was a nonshrinking local reproducible smoothness filter [55]. The first carried out traditional spatial averaging with distinct window sizes and was satisfactory for subtle smoothing. However when more smoothing was required it shrunk the contour slightly. The second method applied was local reproducible
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105 moothing without shrinkage [55]. Both smoothing algorithms were implemented for comparative purposes. The nonshrinking local smoothness filter was selected for smoothing the endo cardial and epicardial boundaries of the LV. As pointed out by Oliensis [55], this local smoothness filter has several advantages over other methods such as Gaussian filtering: (1) It is local, (2) It introduces no shrinkage of curves, (3) The filter can be easily implemented and smoothness of contours can be controlled by a single param eter. This local smoothness filter is similar to a nonlinear lowpass filter. For more details about how the filter was designed and how to adjust parameters to achieve a distinct smoothness effect, we refer the reader to Oliensis [55]. 6.1.6 Sample Experimental Results The test images used in this study included several sequences of 2D shortaxis echocardiographic images of the LV. Each image was originally in polar format with a matrix size of 128x512 and 256 gray levels. These images were first converted to Cartesian coordinates and regions containing the entire LV boundary area were cropped to a size of 256x256. Figure 6.5. shows sample results from the four major steps of our HBD algorithm for endocardial and epicardial boundary detection of the LV in a 2D shortaxis echocardiogram. The original frame of the LV is given in Figure 6.5(a) which i near end diastole. Edge maps detected using a DWT at patialfr qu n y lev 1 3 are hown in Figure 6.5(b). The result off atur xtraction for d t rmining th nt r point of the LV as well as xtracting boundary gm nts with noi dg p int removed i pr sented in Figur 6.5(c). Figur 6.5(d) h w tw lo d ntour aft r boundary contour r on truction. Th d nt ur in Figur 6 5(d) ar with the original imag and pr nt d in Figur 6. ( ) t h w that th r n ru d
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106 (a) (b) (c) ( d) (e) (f) Figure 6.5. Border identification of the LV from a shortaxis view. (a) An original frame of the LV. (b) Edge maps detected using a DWT. (c) The center point of the LV and extracted boundary segments. ( d) Connected boundary contours. ( e) Contours in ( d) overlaid with the original. (f) Final estimated boundaries.
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107 (a) (b) Figure 6.6. Lo ca l nons hrinking smoot hn ess fil te ring of a closed co ntour. ( a) The s mooth ed co ntour s (b) Contours in (a) overlaid with t h e co ntour s in Figure 6.5(d) b efo r e s moothn ess filt e ring. e ndo ca rdi a l a nd ep i car di a l boundaries o f t h e LV fit the im age well a lth o u g h the s h ape of the boundary co ntours ap p ea r l ess natural compa r ed to real h ea r t boundaries The final s mooth e d boundary co ntour s overlaid with the original im age are s h ow n in Figure 6.5(f). Th e advantage of n o ns hrinkin g through lo ca l smoot hin g of a closed co ntour i s s hown in Figure 6.6. Di st in ct smoot hin g e ff ects ca n be ac hi eved by simp l y ad ju st ing t h e smoot hn ess factor. A noth er samp l e r es ul t of boundary detection on a fram e from a different seq u e n ce of ec h ocardiog r a m s at e nd diastole (E D ) is hown in Figure 6 .7 Figure 6.8 provides th result im age of the boundary det ct i on for a fr ame at e nd systo l e (ES) from the same s qu nee as Figur 6. 7. Th two boundar detection results are ove rl a id w i t h th ir corresponding or i gina l im age to d e mon trat typi a l a l gor ithm p rforman e. 6.2 In t hi s t i on, w pre nt an approa h for imag egm ntati n u mg ubta wav 1 t r pr ntati n and a dynami r our allo ating n ur a l n et rk [74]. F r a
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108 (a) (b) Figure 6.7. Border identification of an echocardiogram at ED. (a) An original frame of the LV at ED. (b) The detected center point and endocardial as well as epicardial boundaries overlaid with the original. (a) (b) Figure 6.8. Border identification of a frame at ES from the same sequence of echocar diograms as Figure 6.7. (a) An original frame of the LV at ES. (b) The detected center point and boundaries overlaid with the original.
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109 imilar r ason we call this algorithm MSM (Multiscale Segmentation of Ma es) The fSM algorithm is applied to identify regions of masses in mammographic images of varied degrees of perceptual difficulty. Each mammographic image is first decomposed into overcomplete wavelet representations of suboctave frequency bands. A feature vector for each pixel through the scale space is constructed from fine to coarse scales. The feature vectors are used to drive a neural network classifier of dynamic resource allocation for segmentation of masses. 6.2.1 Overview of the Method Texture information has been previously applied for image segmentation [29, 31 3, 2, 33]. Such methods perform best when texture information is a discriminating factor among different regions of interest in an image. In comparison to edgebased approaches for object detection [38], this method is a regionbased image segmen tation algorithm relying upon multiscale representations. Segmentation consists of four major steps. The first step is to preprocess an image for moothing noi e and enhancing low spatialfrequency features for segmentation [42]. The second step is feature extraction which is done through transforming an image (ROI) into sub octave spatialfrequency bands. ext feature vectors are constructed and us d to drive a neural network classifier for image segm ntation. Finally we postpro es the segmented regions obtained to remove isolated noi e segment Suboctav wavelet representations hav an improved capability of chara t nzmg subtle (bandlimited) features frequ ntly een in mammographic imag [42 43]. radial ba i network of dynamic re our allo ation wa hown t hav r adaptation and g n ralization in a f atur spa [56]. Exp rim ntal r ult al ng ith tatistical analysis ar partially om par d t a tradi ti nal la ifi r. F at ur xtra tion and la ification ar di cu d b 1 w.
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110 6.2 2 Feature Extraction In the approach for mass segmentation, feature extraction is accomplished by processing and combining information from each suboctave frequency band under a suboctave wavelet transform introduced in Chapter 4. A feature vector (pattern) is denoted as where J and M are selected values for a Jlevel Msuboctave wavelet transform L is the length of a feature vector, and x is a pixel spatial location. Element Pk is defined as Pk= wk g(WF J(x)) where 1 k < L k = (j l)M + m, and wk is a parameter (weight factor) and g() may be a function such as thresholding. Finally, element PL= WL g(S 1 f(x)) where S 1 f( x ) is the coarse scale approximation of J( x ). 6.2.3 Classification via a Radial Basis eural Network The neural network used for image segmentation of masses is a resourceallocating neural network having a threelayer architecture as described by Platt [56]. It al locates a new computational unit whenever an unusual pattern is presented to the network shown in Figure 6.9. This network forms compact representations yet learns easily and rapidly. The network can be used at any time in the learning process and the learning patterns do not have to be repeated. The units in this network respond to only a local region of the space of its input values. The network learns by allocating new units and adjusting the parameters of the existing units. If the network performs poorly on a presented pattern then a new
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F Input layer Radial basis functions Output layer 111 Figur 6.9. N twork archit tur a thr el ayer resourceallocating neural n twork of radial basi fun tion
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112 unit i allocated in an attempt to correct the response to the presented pattern. If the network performs well on a presented pattern then the network parameters are updated using the standard method of LMS (Least Mean Square) gradient descent. 6.2.4 Sample Experimental Results In this study we first test the network classifier for its adaptation and generaliza tion in the feature space with simple experiments. Several cropped images (regions of interest ROI) were used to train and/or test the neural network for segmentation performance. These experiments produced promising preliminary results. Sample training and test images are shown in Figure 6.10, where the first row shows original ROI images, the second row shows smoothed and enhanced versions, the third row presents "ideal segmentation results for each image. Several test images have been used to test the neural network trained through the three patterns. Figure 6.11 shows experimental results of mass segmentation within cropped regions. There are four displays for each test image. The first column is an original the second is a smoothed and enhanced version, the third shows the segmented result The fourth column is the result of a statistical classification based on local mean and standard deviation shown here for comparison. An improvement is quite visible for the segmented results produced from our MSM algorithm.
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* i~ ._, (a) 113 (b) ( ) (d) Figure 6.10. Test Imag s. First row: original ROI image ; S ond row: mooth d and enhanced images ; Third row: ideal eg mentation re ult Column : (a) r a l mammograms ( d) a mathematical mod 1.
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114 > a (a) (b) (c) (d) Figure 6.11. Experimental results of image segmentation. Four test cases one each row are shown. The first column (a) is an original image column (b) is smoothed and enhanced images column (c) is the segmented result and column (d) is the result of a traditional statistical classifier.
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CHAPTER 7 CO CLUSIO S In this dissertation, we presented a new overcomplete wavelet representation un der a suboctave wavelet transform and demonstrated the technology through ap plications primarily for medical image processing. A multiscale suboctave wavelet transform was designed and formulated as a generalization of a dyadic wavelet trans form. We studied orthonormal wavelet transforms and dyadic wavelet transforms for denoising and feature enhancement. Our experiments show d that dyadic wavelet transforms were more suitable for accomplishing the objectives of noise reduction and contrast enhancement while being less affected from pseudoGibbs phenomena. A suboctave wavelet transform further improved the ability for characterizing band limited features for both denoising and feature enhancement. Additive and speckle noise models were presented for noise reduction problems encountered in ultra ound imaging applications. We developed regulated thresholding and features 1 t d hancement techniques, as well as algorithms for sp ckle reduction of echo ardiogram and for image enhancement with nois suppression. W also presented exp rimental r sults and conducted p rf rman mparati studi to show re ults of our dev lop d algorithm in t rm f i ual qualit nd quantitativ m asurem nts. Exp rim nts show d that our alg rithms w r apabl of not only redu ing noi but al nhan ing f t ur f diagno ti imp rtan u has myocardial boundari in 2D h ardiogr m quir d fr m th vi w. A study using a
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116 of manually defined borders by expert observers. In addition wavelet representa tions under dyadic or suboctave wavelet transforms have been used in other medical image processing problems such as border identification and mass segmentation in mammograms. Possible future directions of this research include applications of suboctave wavelet representations to other signal/image analysis such as moving object de tection (automatic target detection). An optimized denoising and enhancement of the developed methods will require further study.
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APPE DIX A FIR FILTERS FOR COMPACT SUPPORT WAVELETS The design of FIR filters for a discrete wavelet transform is presented in this ap pendix. For our purposes, we are interested in 1D mother wavelets equal to first or second order derivatives of a spline smoothing function. Wavelet frame represen tations under a dyadic wavelet transform have more relaxed constraints. Under the constraint in Equation ( 4.2), we have more flexibility to choose scaling functions and mother wavelets with symmetric/antisymmetric or other desired features. These FIR filters can be used to construct the corresponding scaling function, analysis and syn thesis wavelets. The decomposition and reconstruction of a discrete function through dilation and translation of a mother wavelet form a wavelet frame representation. The design and construction of FIR filters, scaling and wavelet functions in this ap pendix concentrate on the onedimensional case. Twodimensional extensions can be accomplished through tensor product, as outlined by Mallat and Zhong [51]. A.1 First Order Derivative Wavelets of Spline Smoothing Functions FIR filters for constructing a 1D wavelet ( the first order derivative of a splin moothing function of any order) ar pr s nt d in thi s ction. Th FIR filt r implement a more general cla s of compact support wav let which in lud allat and Zhong dyadi wavelets [51]. When two con traints in Equation (4.1) are r la d to one con traint in Equation ( 4.2), th dilation and tran lation f u h ba i w velet form wav 1 t fram H(w) i a lowpa filt r f r on tru in a plin caling fun tion. G(w) and I<(w) ar highpa filt r f r th d mp iti n nd 117
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118 reconstruction of a dyadic wavelet transform. In addition to FIR filters for a DWT smoothing functions 0(w) scaling functions rp (w) and wavelet functions ,,f; (w) and i (w) are also presented below. Filters presented in this appendix provide more choices for dyadic wavelets of varied degree of smoothness. For the class of FIR filters introduced in this dissertation Equation (90) in Mallat and Zhong [51) becomes +oo W rp (w) = ei(m % 2)~ IT H(2n) n=l (A.1) where m 1 is the order of a spline scaling function cp ( x ) and m 2: 1. The symbol % is a modulus operator; where m % 2 is equal to O if m is even and 1 if m is odd. For a spline scaling function of order m 1 the corresponding H ( w) can be represented as (A.2) where H(w) is the transfer function of a FIR lowpass filter and m 2: 1. We can easily verify that The first order derivative wavelet of a spline smoothing function G(w) can be for mulated as G(w) (A.4)
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119 where G(w) is a highpass filter and c is a constant with one of the following valu : 1 if g(x) is equ iv a l ent to an operator for computing moving difference c = 2 if g ( x ) is for gradient operator 4 if it i s for Mallat's c hoi ce when m is odd. To satisfy the constra int s described in Mallat and Zhong [51], K(w) and L(w) can be derived g ivin g ( ) 1 IH(w)l 2 1 cos 2 m(~) _ iei~ w ml w Kw G(w) w ( ) in() L cos 21 (), and c i eiz s in c 2 j=O 2 (A.5) 1 + cos 2 m C=') L(w) = 2 2 (A .6 ) Filter L(w) is used to perform a 2D dyadic wavelet transform. The orresponding smooth in g function, analysis wavelet sca ling function, and synthesis wave l t ar represented by [ (wT + l 0(w) sm 4 4 (~) ~ (w) ( ) sm 4 [ (w) rl 4 'l w (~) (p (w) [ s in( ~) r (~) ,y (w) (iw) [ 4c in~ ~) rl "f:1 L) j = O o 2j (w ). 4 For edg d tection diff rent ampling hift (fi l t r pha ) n d a unt for id ntifying th a urate lo ati n of dg at diff r nt ( .7) ( ) ( .9) ( .10) b tak n int abl .1 ( a l ulat d u ing m = 4 and = 2) h w a t f FIR filt r f r a ampl wa, mput d ba d on th f rmulati n in thi pp n lix.
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120 Now we wa nt to verify a few things about the designed scaling, analysis and synthesis wavelet fun ct ions. Th ese fun ct ions were constructed using FIR filters which satisfy the co nstraint in Equation (3.6). Furthermore scaling, analysis and synthesis wavelet fun ct ion s satisfy cp(w) ei(m%2) ';f H(~) cp(~), (A.11) ;/; (w) w W W ei4 G(2) cp(2), (A.12) i (w) w W W (A.13) ei 4 K ( ) cp ( ) 2 2 In the n ext two propositions we prov e that scaling analysis and synthesis wavelet fun ct ions satisfy the dyadic wavelet decomposition and re co nstruction constraint a nd the designed function ;/; ( w) is ind ee d a d ya di c wavelet. In the proof we us e these fun ct ions alone without the construction constraints. Th e r e for e, it proves that the co nstru cted fun ct ions are co rr ect and are inde e d what we want for a d ya dic wavelet transform. Propo sition A .1 1 Th e constructe d sca l ing) analysis and synthesis wavelet funct ions through FIR filters designed in th is appendix satis fy the dyad ic wavelet d ecomposition and reconstruction constraint J +oo l cp (w) 1 2 = L ;/; (2jw ) i (2jw) + l cp (2 1 w) 1 2 = L ;/; (2jw ) i (2jw). (A.14) j=l j=l Proof: In the proof we d er ive it from the right hand side of Equation (A.14) to the l e ft s id e. First we simplify the multiplication term using (A.8) and (A.10) ;/;(w ) i (w)
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[ s in( ~) l 2m [ s in Cf) l 2m (~) (~) The right h a nd side of ( A .14) is + oo L ,,f; (2jw ) ,y (2jw) j = l Th e r e for e, we get the proof. Propo sition A.1.2 Th e constructed fun ction ,,f; (w) wave l e t 121 [ (w) ] m + l ( i w) st ~~J is indeed a Proof: We ca n ignor e the co n sta nt p art (by l ett ing c = 4) without lo in g ge n era li ty. [ (w) ] m+l The fun ct ion s1 ~~J i s eq uiv a l e nt to a sy mm et ri c spline fun ct ion (0( x )) of ord r [ ( w ) ] m+l min the s p at ial domain. Fun ct ion 'l/J (w) = (iw) s1 ~ ~ J is the Fouri r tran form of the fir st order derivative of the fun ct ion 0( x). The fir st ord r derivati of a plin sy mm etr i c fun ct ion i s a nti sy mm et ri c, so we h ave 'l/J ( x) = 'l/J ( x ). B au of th ca n ce ll at ion of 'ljJ ( x) on the right s id e of the x ax i with 'ljJ ( x ) on th 1 ft id r eac h po s itiv e x, we a n get [ : 'l/J ( x )d x = 0. Furth rmor the fun ti n 'l/J (x) ha s a omp t upp rt with uffi i nt d a (fa r de ay with a l arg r m). Th r f r th d i n d ( ) i a v a 1
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122 A.2 Second Order Derivative Wavelets of Spline Smoothing Functions FIR filters for constructing 1D second order derivative wavelets of spline smooth ing functions of any order can also be used to perform a discrete suboctave wavelet transform. H(w) is a lowpass filter for a spline smoothing scaling function. G(w) and K(w) are highpass filters for the decomposition and reconstruction of a wavelet trans form. Smoothing functions 0(w) scaling functions cp (w) and wavelet functions ,J; (w) and y (w) are presented below. The dilation and translation of the mother wavelet for a second order derivative of a spline smoothing function lead to corresponding wavelet frame representations. FIR filter H ( w) is defined as H(w) (A.15) where mis a positive integer. FIR filter G(w) can be defined by (A.16) where c is a selected constant and % is a modulus operator. Using the less constrained Equation (4.2), we can derive FIR filter K(w) as K(w) = 1IH(w)l2 = 1 cos2m(~) = ~ 1:1 cos2j(w). G(w) ci 2 sin 2 (~) c J = O 2 (A.17) The corresponding smoothing function analysis wavelet scaling and synthesis functions become (for second order derivative wavelet case) 0(w) [ (w)lm + 2 sm 4 16 (~) (A.18) J; (w) C )2 sm 4 [ (w) lm+2 16 iw (~) (A.19)
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123
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124 O rd er 3 Sp l ine Its First and Second Order D e ri va ti ve Overlay 2.r.,..,,,~ 1.5 0 5 0 0.5 1 1.5 2''''~~'' 1.5 1 0.5 0 0 5 1.5 ( a) Order 4 Spline It s First and Second Order D e ri va ti ve Overlay 1.5,r.,,,,~ 1.5 ''''''~'' 1.5 1 0.5 0 0 5 1.5 (b) Figure A.1. ( a) A c ub e sp lin e fun ct ion and its first a nd second order d er iv at iv e wavelets, a nd (b) the fourth order s plin e with its first and seco nd order d e rivativ es
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APPENDIX B IMPLEMENTATION OF SUBOCTAVE WAVELET TRA SFORMS In this appendix, we us e ps e udo codes to illu strate a fast impl e m e nt at ion of s ub octave wavelet transforms. Frequency r eso lution fin e tunin g i s ba sed on the filter b a nk theory and bandsplitting techniques. We ass um e that M i s a power of 2 a nd filt ers Gm a nd K m a r e co n st ru cte d dynami ca ll y ba se d on bandsplitting techniques illu strate d (see exa mpl e u se d in c r eat ing Figur e 4.3); G 1 (w) = G(w)H 8 (2w)Hs(4w), G 2 (w) = G(w)H s (2 w)Gs(4w), G 3 (w) = G( w)Gs(2w )Gs(4w), G 4 (w) = G(w)Gs(2w)Hs( 4w), K 1 (w) = K( w) H 8 (2w )ifs( 4w), K 2 (w) = K( w) H 8 (2w)K (4w) K 3 (w) = K(w)Ks(2w)K (4w) K 4 (w) = K(w)K 8 (2w)fl (4w). To redu th bound ary ffect we mirrorxt nd a i g n a l r 1m g i gna l f samp l m a 2N p riodi di er t fun t i n. Through m irr r xt n i n th b tw n th 1 f t and right bound ary amp l wi ll n t aff t th r f atur o f int r t, u h a n arby dg 125
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126 As we have mentioned in Chapter 4, a suboctave wavelet transform is a gen eralization of the traditional dyadic wavelet transform. In the implementation of a suboctave wavelet transform, we, in general, use two sets of FIR filters for suboctave decomposition and reconstruction. For an example of 1D suboctave wavelet trans forms filters H( w), G(w), and G(w) are used for dyadic (octave) decomposition and re co nstruction while filters H 5 (w) G 5 (w), and G 5 (w) are for suboctave bandsplitting to finetune the frequency resolution. Therefore a suboctave wavelet can be con structed as ,J; m(w) = ;/; (w) F~(w) F~(w) ... F;:: 9 2 (M)(w) form E {l 2, ... M}, where ,,P (w) is a dyadic wavelet and F:n(w) is either Hs or G 8 In the following proposition we prove that the constructed function ,,J; m(w) is a suboctave wavelet. Propo sition B. 0.1 Assume that th e function ;/; (w) is a dyadic wavelet. Th e formulat ed function ,,J; m(w) = ;/;(w) F~(w) F~(w) ... F;:: 92 (M)(w) form E {l 2, ... M} is indeed the Fouri er transform of a suboctave wavelet, where M is a power of 2 and F:n (w) is defined as { H ,(iw) if the ith bit of m in binary is 0 otherwise. Proof: Here we prove it (for each ,,p m(w), where m E {l 2 ... M}) using the wavelet admissible condition j +oo 0< 00 Since ;/; (w) is a dyadic wavelet, it satisfies the wavelet admissible condition and we have j +oo 0< 00 00.
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127 For a gen ra l suboctave wave l et decomposition and reconstruction using FIR filters designed for a dyadic wave l et in Appendix A.1, similar to th diagram in Figure 4.3, th modulus va lu es of G(w) a nd H( w) with e 1 have t h e fo ll ow in g upper bounds IG(w)I IH(w)I w W W lei ei 2 sin( 2 ) I = l e sin( 2 ) I S e, l ei(m% 2 )1 cosm(~)I = lcosm(~)I S 1 Se. Now l et's see if each 1/Jm(w ) for an M s uboctave decomposition and reconstruction sat i sfies the wave l et adm i ssib l e condition, / +oo 1Jim(w)l2 dw = +oo 1 Ji (w) F~(w)F~(w) ... F~g 2(M) (w)l2 dw 00 lwl 00 lwl 1+ 0000 e2 log 2 (M) 1,., / ,(w) 12 < lw I 'I" dw = e21og2(M) C, p < oo. T h erefore, 1/J m(w) is a wave l et form E {1 2, ... M} and a finite M. B.1 OneDimensional SubOctave Wave l et Transform The following codes describe the major steps of an a l gor i thm for a 1D suboctav wave l et transform For j from 1 to J Begin For m from 1 to M Begin Wf f (n) = SJ 1f (n) G;J 1 End {Form} SJJ(n) = SJ if (n) H 21 1 End { For j }
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1 28 S u b Octave wave l et decompos i t i o n t h ro u g h t h e two l eve l r ec ur i ve r e l a ti o n s in (4.16) a n d (4. 1 7) i s u t ili zed for t h e im p l e m e n tat i o n At l eve l j, Sj if(n) i s decom posed in to { Wf J (n)} 1 s m S M a n d Sj f (n). W h e n p e rformin g di screte co n vo lu t i on ", we u t ili ze t h e m i rro r exte n s i o n a nd p e ri o di zat i o n w h e n t h e operat i o n exceeds t h e bo u ndary o f a s i g n a l. B. 2 O n e Dim e n s i o n a l In ve r se Su b O ctave Wave l et T ra n s form Th e fo ll ow in g a l go ri t h m pr ese n ts t h e m a j o r ste p s o f a p roce dur e fo r a 1D in ve r se s u b octave wave l et t r a n sfo rm T h e s uboctave wave l et r eco n st ru ct i o n t h ro u g h t h e two l eve l r ec ur s i ve re l at i o n in ( 4. 1 8) i s a ppli e d in t h e impl e m e n tat i o n At l eve l j { wr J (n)h s m S M a nd Sj f (n) a r e sy n t h es i zed to r eco n st ru ct sj1 f (n). Th e di sc r ete co n vo lu t i o n in cor p orates t h e m i rro r exte n s i o n a nd pe ri o di zat i o n w h e n i t exceeds t h e s i g n a l b o und a r y F o r p e r fect deco mp os i t i o n a nd r eco n st ru ct i o n fil te r b a nk s t h e 1D in ve r se s uboctave wave l et tra n s for m w ill r ecove r t h e s i g n a l fr o m i ts s u b octave wave l et tra n sfo r m A f ter app l y in g d e n o i s in g a nd e nh a n ce m e n t o p erators in t h e wave l et d o m a in r eco n st ru ct i o n o f t h ese co m b in e d o p erat i o n s l ea d s to a n i mprove me n t For j from J to 1 step 1 Begin sj1f (n) = Sj](n) fl 2 jl For m from 1 to M Begin Sj1](n) = Sj_ 1 J (n) + Wf f (n) K;J 1 End {For m } End { For j }
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129 B.3 TwoDimensional Sub0 tave Wavelet Transform Th p e udo odes b l ow de cr ib e t h major steps of a n a l gor i thm for a 2D uboctav wav l et transform. For j from 1 to J Begin For m from 1 to M Begin W} m f (n x, ny) = Sj if (n x, ny) { G;1} 1 ~} w}'m J( n x ny) = Sj if (n x ny) { ~ G;1}1} End {Form} Sjf (nx, ny) = sj_if (nx ny) { H 2 1 l, H 2 1I} End { For j } SubOctave wave l et d eco mpo s ition u s in g s imil a r two1 v 1 recursive re l at i o n s a ( 4. 16) a nd ( 4 .1 7) a l ong horizontal and vertica l dir ect i o n i s e mpl oyed during t h e impl e m nt at i on of 2D s ubo ctave wave l et transforms At l eve l j Sj if (n x ny) i decomposed into {W} 'm J(n x ny) w}'m J( n x ny) h '.S m '.S M a nd Sjf (n x, ny)Th di crete co n vo luti on i s ca rri ed out u s in g the mirror xt n s ion and periodization fa 2D discrete fun ct ion (image) a l ong the bo un dary of the im age in e ith r horizontal r v tica l dir e tion. App l y in g w} m J(n x, ny) to d ete t hori zo n ta l var i at i on ( g. rti a l dges) w p rform vertical co nvolution w ith t h Kr neck rd l ta fun t i on [30]. B .4 TwoDimensiona l Inv r The fo ll owing cod pr nt th m a j or o tave wav l t tran form. Th ubo tav wa 1 t 2D r n tru ti n ith :t 1 v 1 r ur i v r l at i o n ( i mi l ar t Equati n ( .1 ) ) i app li l in th impl n h i z l
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130 to reconstruct Sj_ 1 J(nx, ny)The resulting discrete convo lu t ion operator u ses mirror extens i on and periodization beyond the boundary of a n im age a long e ith er horizontal or vertica l direction. For j from J to 1 step 1 Begin Sj1f (nx, ny) = Sjf (n x, ny) { fI 2i1, H 21i} For m from 1 to M Begin sj_if (nx, ny) = sj_if (n x, ny) + wf,m f (nx, ny) { K;J1 L2 1l } + w}'m J(nx, ny) { L 2jl, K;J 1 } End {For m } End { For j }
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BIOGRAPHICAL SKETCH Xuli Zong was born in Harbin, the People 's Republic of China, on July 18 1960. He began his undergraduate study at Harbin Institute of Technology (HIT), China, in 1978. He received B.S. degree in computer science and engineering from HIT in August, 1982. From 1982 to 1987 he served on the faculty of the Computer Science and Engineering Department of HIT. In the Fall of 1988 he started to pursue graduate studies at the University of Florida Gainesville, where he earned M.S. degrees in electrical engineering on August 11 1990 and in computer and information sciences on April 30, 1994 He is going to receive his Ph D. degr ee in computer engineering from the University of Florida in 1997. His research inter ests includ e signal/image processing medical imaging computer vision, data visualization and computer graphics. 137
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I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. An~,~~ Associate Professor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it confo s to accept able standards of scholarly presentation and is fully adequate, in scope nd quality, as a dissertation for the degree of Doctor of Philoso hy. (,Lff_ ~/ / Ti, essor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy . /MJ :=J ,{ ====;:;::;::::=======SartaJ K. Sahni _, Professor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philo ophy. Professor of mputer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ~N\~~ "ohnG. Harris Assistant Professor of Electrical and Computer Engineering
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Thi di rtation was ubmitt d to the Graduate Faculty of the College of En g in ring and to the Graduat School and was ac epted as partial fulfillment of th requir ment for the degree of Do ctor of Philosophy. D cemb r 1997 /J 6~ Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook D ean, Graduate School
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