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Image Denoising Using Histogram-Based Noise Estimation

Permanent Link: http://ufdc.ufl.edu/UFE0022735/00001

Material Information

Title: Image Denoising Using Histogram-Based Noise Estimation
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Kim, I-Gil
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: image, noise
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Engr.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Nowadays digital imaging systems such as digital cameras, camcorders, and televisions are widely used. Such images are commonly susceptible to contamination by noise. To remove the noise effectively from the digital images, exact noise level estimation is very important. In this paper, novel noise level estimation methods and noise reduction methods are introduced that improve noise filtering and thus the resulting images.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by I-Gil Kim.
Thesis: Thesis (Engr.)--University of Florida, 2008.
Local: Adviser: Taylor, Fred J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022735:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022735/00001

Material Information

Title: Image Denoising Using Histogram-Based Noise Estimation
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Kim, I-Gil
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: image, noise
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Engr.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Nowadays digital imaging systems such as digital cameras, camcorders, and televisions are widely used. Such images are commonly susceptible to contamination by noise. To remove the noise effectively from the digital images, exact noise level estimation is very important. In this paper, novel noise level estimation methods and noise reduction methods are introduced that improve noise filtering and thus the resulting images.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by I-Gil Kim.
Thesis: Thesis (Engr.)--University of Florida, 2008.
Local: Adviser: Taylor, Fred J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022735:00001


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5b0edbfcbfd6539bac2e487f5ce8f868c32f0457







IMAGE DENOISING USING HISTOGRAM-BASED NOISE ESTIMATION


By

I-GIL KIM
















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
ENGINEER

UNIVERSITY OF FLORIDA

2008


































2008 I-Gil Kim































To my family, especially my wife Jiyoung and little Julia









ACKNOWLEDGMENTS

I am deeply grateful to my mentor Dr. Paul W. Chun from the bottom of my heart. He

always encouraged me and gave me valuable advice. I also offer my gratitude to Dr. Fred J.

Taylor, my supervisory committee chair, for his direction and guidance. I would like to thank

my committee members. Finally, I express my gratitude to my father, Dr. Koojin Kim. He

instilled in me in the great love I have for the natural world and taught me to be curious about

science. Special thanks go to my family and friends for their love, support, and sacrifices that

allowed me to be me throughout my years of study.









TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ..............................................................................................................4

LIST OF TABLES ......... ..... .... ....................................................6

LIST OF A BBREV IA TION S ......... ............. .............................................................9

A B S T R A C T ........................................... ................................................................. 1 1

CHAPTER

1 INTRODUCTION ............... .......................................................... 13

Conventional Noise Level Estimation Methods ............... .............................................13
Conventional N oise R education M ethods.................................................................... ...... 14

2 BACKGROUNG AND RERATED REASEARCH.................................................15

Bosco's N oise Estim ation (2005, IEEE) ........................................ .......................... 15
Amer's Noise Level Estimation (2005, IEEE) .............. ..........................................16
Shin's Noise Level Estimation (2005, IEEE)...................................................................... 17

3 NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION ........................20

Drawbacks of Conventional Noise Level Estimation Methods...........................................20
Proposed Noise Level Estimation Method based on Histogram Compression...................22

4 NOISE REMOVAL FILTERING.......................................................... ...............34

B ilateral N oise R education Filtering .......................................................................... ....... 34
Im prove ent of B lateral Filtering ........................................................................... .... ... 36
Im pulse N oise R em oval .................. ...... ...... ............. ................. ............ 37
Performance Test of the Proposed Noise Reduction Method by PSNR .............. ..............38

5 C O N C L U S IO N .............................................................................. .......................... .. 4 0

APPENDIX: TEST IMAGES FOR PERFORMANCE COMPARISON.............. ................ 41

L IST O F R E F E R E N C E S .................................................................................... .....................43

B IO G R A PH IC A L SK E T C H .............................................................................. .....................45










LIST OF TABLES


Table


6-1 PSNR for test im ages ......... ... .......... ...... .......... .. ........... ......... 39


page









LIST OF FIGURES

Figure page

2-1 Differences computation in homogeneous areas .................................... ............... 15

2-2 Absolute noise histogram .............................................................................. 16

2-3 Fine structure and texture im ages. ............................................. ............................ 17

2 -4 S p ecial m ask s ........................................................................ 17

2-5 Shin's noise estim ation algorithm ......................................................................... ... ... 19

3-1 F our types of im age .................... ... ........ ................................................ .......... ..... .... 2 1

3-2 H om ogeneou s regions............................................................................. .....................22

3-3 C olor digital im age acquisition ............................................................... .....................23

3-4 Histogram compression on luminance of noisy image................................ ..............24

3-5 Relationship between ox and aw................ .....................................27

3-6 Graphical analysis of the histogram compression for four different types of noisy
im ag es .......................................................... .................................. 2 8

3-7 Graphical analysis of the effect of histogram compression............... ......................29

3-8 H om ogeneous im age perform ance test.................................... ........................... ......... 30

3-9 Less-homogeneous image performance test ......................................... ...............31

3-10 Complex Structure image performance test............................................. ...............31

3-11 Non-homogeneous image performance test............................................. ...............32

3-12 Homogeneous and less homogeneous image performance test......................................33

3-13 Complex structure, and non-homogeneous image performance test..............................33

4-1 Pixel selection for filtering based on a ........................................ ......................... 36

4-2 C central pixel outlier detection ......................................... .............................................37

4-3 Impulse noise reduction by using detecting central pixel outlier ....................................38









A-i The Kodak homogeneous and non-homogeneous photo images....................................41

A-2 Complex structure non-homogeneous photo images............. .............. ............... 42









LIST OF ABBREVIATIONS

b,, Non-overlapping image block

B Blue component of RGB color space

B* Homogeneous block

Ek Absolute difference error between true and estimated noise level

f[k] Original signal

A
f[k] Estimated original signal by filtering

g[k] Contaminated signal

G Green component of RGB color space

h[k, ] Local filter

MSE Mean squared error

n[k] Noise signal

Ps Weight of the proposed method

PSNR Peak signal-to-noise ratio (dB)

R Red component of RGB color space

RGB RGB red, green, and blue color space

W White Gaussian noise

W[k, ] Weight of bilateral filter

W [k, ] Space weight of bilateral filter

WR [k, ;] Range weight of bilateral filter

X Original noise-free image

Y Noisy image

Y, Luminance of YUV color space

YUV Luminance and chrominance color space










z




mmin
Z





- ,


Histogram modified noisy image

Standard deviation of intensity for each block bi

Minimum standard deviation

Standard deviation of signal X, Y, and W

Neighboring point









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Engineer

IMAGE DENOISING USING HISTOGRAM-BASED NOISE ESTIMATION

By

I-Gil Kim

August 2008

Chair: Fred J. Taylor
Major: Electrical and Computer Engineering

Exact noise level estimation is very useful in digital image processing. For example, some

noise removal algorithms use a noise level estimation to adjust the aggressiveness of noise

removal. If an estimated noise level is too low, too much noise will remain in the denoised image.

If an estimated noise level is too high, the features of the original images will be removed from

the denoised image. Accurate noise level estimation will produce better results in the restored

image.

Most conventional noise estimation methods are only focused on estimating the noise level

from a noisy image including a lot of homogeneous (flat) regions. So, the typical methods induce

excessive overestimation and underestimation if the given noisy image doesn't have any

homogeneous regions.

I propose a new noise level estimation method that uses histogram modification to find

more exact noise level from the given noisy image. This new noise estimation method uses a

proposed noise reduction filter based on a bilateral filter. It makes use of block-based noise

estimation, in which an input image is assumed to be contaminated by the additive white

Gaussian noise and impulse noise. A noise reduction filtering process is performed by the filter

bilateral-based. Coefficients of the noise filter are selected as functions of the standard deviation









of the Gaussian noise that is estimated from a given noisy image by the proposed method. To

accurately estimate the amount of noise in various types of noisy images, images are categorized

into four different types based on their homogeneity: homogeneous, less homogeneous, complex

structure, and non-homogeneous images. In the proposed method, by using histogram

modification on a noisy image, fluctuation in a non-homogeneous region can be effectively

suppressed without influencing its noise level. This process increases homogeneous regions in a

given noisy image and improves the performance of noise estimation for most noisy images

regardless of their homogeneity.

To show the effectiveness of the proposed method, it is tested on various kinds of noisy

images ranging from homogeneous to non-homogeneous and its performance is compared with

that of three conventional noise estimation methods. The proposed noise estimation and

reduction method can be efficiently used in various image and video-based applications such as

digital cameras and digital television and is superior because of its performance and simplicity.









CHAPTER 1
INTRODUCTION

Nowadays many people use camera phones and low-end digital cameras in their everyday

life. These devices are particularly subject to noise reduction, when images are acquired in low-

lighting conditions. However these devices are often used in poor lighting-conditions due to their

portability. In order to obtain an acceptable picture, sometimes it is necessary to amplify the

image signal taken under low light environments such as indoor scenes. However, when boosting

an already degraded image signal, noise in the image also can be amplified. An effective noise

reduction filter should change its strength according to the noise level in the noisy image. Here,

we need some measure to estimate the noise level in the given noisy image. In general, we can

get noise level information from the noise standard deviation a. Hence many noise filters depend

on a to adaptively change their smoothing effects [1]-[4].

Conventional Noise Level Estimation Methods

Conventional noise level estimation techniques can be categorized into three main classes

[8]: Block-based, filter-based, and adaptive-based methods. Before 1990, filter-based methods

were commonly used. Such methods perform a pre-filtering operation in which the noisy image

is blurred to suppress image structure. A difference image is computed by subtracting the filtered

image from the original one and then noise level is estimated using the difference image, which

is assumed to contain only the noise signal. After 1990, block-based methods started to play a

role in noise level estimation. Briefly, the block-based method partitions the image into a

sequence of blocks. The estimation of standard deviation of a noisy image is carried out by the

properly calculating the weighted noise level obtained by averaging the noise levels of the most

homogeneous blocks. An adaptive method is a hybrid method which uses elements of the filter-

based and the block-based methods. Its performance depends on the first noise level estimation.









Efficient techniques based on the block based approach are described in [5] and [6]. A

comparison of various methods for determining an approximation of the noise level in an image

is given in [7].

Conventional Noise Reduction Methods

A large number of different noise reduction methods have been proposed so far.

Traditional denoising methods can be generalized into two main groups: spatial domain filtering

and transform domain filtering. Spatial domain filtering methods have long been the mainstay of

signal denoising and manipulate the noisy signal in a direct fashion. Conventional linear spatial

filters like Gaussian filters try to suppress noise by smoothing the signal. While this works well

in the situations where signal variation is low, such spatial filters result in undesirable blurring of

the signal in situations where signal variation is high. To overcome these drawbacks, a number

of new spatial filtering methods such as total variation techniques [9], anisotropic filtering

techniques [10], and bilateral filtering techniques [11,12] have been introduced to suppress noise

while preserving signal characteristics in the regions of high signal variation. Bilateral filtering is

a non-linear and non-iterative filtering technique which utilizes both spatial and amplitudinal

distances to preserve signal detail. On the other hand, transform domain filtering methods

transform the noisy signal into the frequency domain and manipulate the frequency coefficients

to suppress signal noise before the signal is transformed back into the spatial domain. These

techniques had been introduced by Wiener filtering [13] and wavelet-based methods [14,15]

I propose novel approaches to noise level estimation and noise reduction that improve

bilateral filtering. The proposed methods are robust in situations where the given noisy image is

characterized by non-homogeneity and provide improved noise suppression.









CHAPTER 2
BACKGROUND AND RERATED RESEARCH

In this section, I introduce three recent noise estimation algorithms, showing good

performance in IEEE journal. Bosco's and Amer's method are block-based approaches, and

Shin's method is adaptive approach. These three conventional methods will be compared with

my proposed method in chapters 3 and 4.

Bosco's Noise Estimation (2005, IEEE)

In most of block-based noise estimation methods, noise level is obtained by averaging the

noise levels of the most homogeneous blocks. In order to overcome inaccuracy of averaging,

Bosco proposed new method using difference histogram approximation. The detail algorithm is

as follows:

1. Tessellate an input noisy image into a number of 3x3(5x5) nonoverlapping image blocks

2. Compute absolute differences between the central pixel and its neighborhood for each blocks
(Figure 2-1)

3. Select a block as a homogeneous block if the 8 differences are very small

4. Compute noise histogram for selected homogeneous blocks from accumulated differences
(Figure 2-2)

5. Obtain noise level(a) by considering 68% of the noise samples. The index on the x-axis of
the noise histogram allowing reaching the 68% of the noise samples












Figure 2-1. Differences computation in homogeneous areas









In Bosco's method, histogram approximation method is exploited instead of averaging

local standard deviations usually used in conventional method for noise level estimation. In some

of case, it provides better result than other conventional methods. The main weakness of this

method is that it takes more time and storage to accumulate differences.

Example: Noise Level Estimation for the Green Sample Color Channel
















i llliei 24 251 7


Figure 2-2. Absolute noise histogram. The index on the x-axis allowed to reach the 68% of the
total samples represents the current noise level estimation.

Amer's Noise Level Estimation (2005, IEEE)

Selection of homogeneous areas in the given noisy image is very important in most of

conventional noise estimation methods. Accuracy of block-based noise level estimation highly

relies on selecting homogeneous blocks. However, many conventional methods were found to

have difficulties finding homogeneous region in noisy image containing fine structures or

textures (Figure 2-3). Amer introduced a new approach to overcome this drawback using special

masks to find homogeneous regions. The following steps describe the proposed method:

1. Tessellate an input noisy image into a number of 3 x3(5 x5) nonoverlapping image blocks

2. Use eight high-pass operators and special masks for covers to stabilize the homogeneity
estimation for each block. (Figure 2-4)








3. Compute summation of the absolute values of all eight quantities from step 2)
4. Select the block as a homogeneous block if the summation is less than threshold
5. Average local standard deviation of the selected homogeneous blocks
Amer's method gives better noise estimation performance for the images which have texture and

fine structure by using special masking if the noise level of noisy image is not too high. The

number of detected homogeneous regions is rapidly decreased when given noise level is high. So

it is hard to find homogeneous areas in this case.


,.~;ieu;; gw~aiaAS'F


Figure 2-3. Fine structure and texture images.


current pixel


Figure 2-4. Special masks
Shin's Noise Level Estimation (2005, IEEE)
Shin's noise estimation algorithm is based on both block-based and filtering-based

approaches. It selects homogeneous blocks by a block-based approach and then filters the


mask 1 mask 2 mask 3 mask 4




mask 5 mask 6 mask 7 mask 8



H~ffl91









selected homogeneous blocks using a filtering-based approach. Block-based methods require a

low computational load whereas filtering-based approaches yield a stable estimate. Fig. 2 shows

the flowchart of Shin's estimation noise estimation method using adaptive Gaussian filtering.

Each step of Shin's method is described as follows.

1. Tessellate an input noisy image into a number of 3 x 16 nonoverlapping image blocks (b,).

2. Compute the standard deviation (a,) of intensity for each block bi and find the minimum
standard deviation (amin).

3. Select homogeneous blocks (B*) whose standard deviations of intensity are close to Umin

4. Compute Gaussian filter coefficients (olst is set to omin)

5. Gaussian filtering on the selected blocks (B*).

6. Compute the standard deviation(U2nd) of the difference image between noisy and filtered
images within selected blocks (B*), which gives the estimated noise standard deviation
(<- =U2nd)

In general, block-based approaches tend to overestimate the noise level in good quality

images and underestimate it in highly noisy images. This underestimation can be compensated

by adaptive Gaussian filtering of Shin's method. However, its performance is highly depend on

the first estimated noise level (ua,t) and execution time is usually longer than other block-based

approaches because of the second filtering-based estimation.











Noisy image I(i.j)


Figure 2-5. Block diagram of Shin's noise estimation algorithm









CHAPTER 3
NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION

In this chapter I will address the drawbacks of conventional methods for noise level

estimation in digital images degraded by noise and compare performances of each method with

the proposed method for four types of noisy image.

Drawbacks of Conventional Noise Level Estimation Methods

Most conventional methods for estimating the noise standard deviation are generally based

on the following steps:

1. Detect homogeneous regions in the noisy image (because fluctuations in flat areas, pixels are
supposed to be due exclusively to noise).

2. Compute the local standard deviation in the detected homogeneous regions.

3. Repeat steps 1) and 2) until the whole image has been processed.

4. Finally estimate the noise level (or standard deviation of noise) by averaging the computed
local standard deviations.

This method has two major drawbacks. First, if there are no homogeneous regions or too

small areas of flat region in a given noisy image, it is hard to detect the homogeneous regions.

An insufficient number of detected homogeneous regions may result in overestimation or

underestimation. This will induce wrong estimation for noise level. Second, the process of

detecting homogeneous regions in a noisy image requires a high number of computations. The

detection process for a non-homogeneous image may be just time wasted work and the noise

estimation based on such detection has a high probability of being wrong. Theses drawbacks

originate from the fact that most conventional noise estimation methods are only focused on

noisy images including a lot of homogeneous regions. In order to make this problem clear, I

have categorized noisy images into four types as following:

1. Homogeneous image: image contains a lot of homogeneous regions.









2. Less homogeneous image: image contains very few homogeneous regions in a limited area.

3. Complex structure image: original noise-free image has very complex structure, so it is not
easy to find homogeneous regions due to local fluctuations.

4. Non-homogeneous image: there are no homogeneous regions in the given noisy image.













A B












C . D

Figure 3-1. Four types of image. A) homogeneous image, B) less homogeneous image, C)
complex structure image, D) non-homogeneous image

Figures 3-1 shows examples of four different types of image based on homogeneity.

There are homogeneous regions in Figure 3-1 A) and B), but it is not easy to find homogeneous

regions in Figure 3-1 C) and D) by using conventional noise estimation methods. Figure 3-2

shows homogeneous regions detected by Bosco[l] method and violet blocks in each image are

the detected areas.



















A B











C D

Figure 3-2. Homogeneous regions. A) homogeneous image, B) less homogeneous image, C)
complex structure image, D) non-homogeneous image

Proposed Noise Level Estimation Method based on Histogram Compression

In this section the proposed noise estimation method is shown to overcome the

drawbacks of conventional noise estimation methods by means of histogram compression on

intensity of image that tries to exploit correlation among R, G, and B components of color image

regardless of the number of homogeneous regions in the given noisy image.

The best way to estimate an exact noise level from a given noisy image is to make the

image homogeneous without influencing its noise deviation. In other words, if we can suppress

deviation in the original noise-free image without changing deviation of noise in a noisy image,

it is easy to find more exact noise levels from the suppressed noisy image.









Nowadays most digital images are in color and each pixel of the image has red (R), green

(G), and blue (B) components. Each RGB component in a noisy color image has a noise and the

noise tends to have Gaussian distribution in a digital image acquisition system.



Optical Axis





Noise


X 0l Y
Color Filter--


X = 0.3Rx R channel: Rx + WR Y = 0.3(Rx + WR)
+ 0.5Gx G channel : Gx + WG + 0.5(Gx + WG)
+ 0.3Bx B channel : Bx + WB + 0.3(Bx + WB)
where OWR = OWG = OWB


Figure 3-3. Color Digital image Acquisition

Figure 3-3 shows the structure of a current digital image acquisition system. Generally

most common noise in digital image acquisition can be modeled by a white Gaussian noise with

the same standard deviation ow. In order to reduce the correlation between the RGB components,

a conversion from RGB to YUV color space as follows:

Y, (Luminance) = 0.3R, + 0.6G, + 0.1B,

U, (Chrominance) = B, Y,

V, (Chrominance2) = R, Y,

However, if component noise is uncorrelated in RGB space, it is correlated in YUV space.

Also the component noise and RGB component are independent to each other, so they are









uncorrelated in RGB space. A linear operation such as histogram modification in YUV space may

affect uncorrected RGB components and correlated component noise. Histogram compression,

one of several histogram modification methods, can be useful to suppress deviation of RGB

component without affecting deviation of component noise.












(a=0.01, b 128)
Y aX+ b

X 0.2Rx + 0.5Gx +0.3Bx 0.0, b 128) Y 0.2Ry + 0.5Gy +0.3By

Figure 3-4. Histogram Compression on Luminance of Noisy Image. The histogram plots the
number of pixels in the image (vertical axis) with a particular brightness value
(horizontal axis)

The noisy image and original noise-free image have luminance correlation among RGB

components in YUV color space.

X 0.2Rx + 0.5Gx + 0.3Bx

Y=X+W (1)

= 0.2Ry+ 0.5Gy+ 0.3By (2)

= 0.2(Rx+ W) + 0.5(Gx+ WG) + 0.3(Bx+ WB) (3)

where Xis original noise-free image and Yis a noisy image. Wis an additive white Gaussian

noise. Because X and Ware independent of each other, variation of Y has the following relation

to variation of X.

2 2 2
aY x + aW (4)

If we do histogram compression on the luminance of noisy image Y

Z = aY-b, (5)









a= Y- Z (-a)Y+ b

Z=Y-a

= 0.2*Rz + 0.5*Gz + 0.3*Bz (6)

= 0.2(Ry- a) + 0.5(G- a) + 0.3(By- a) (7)

where Z is compressed noisy image in YUV. From equation (6) and (7), the R component of Z, R,

becomes

Rz Ry- a

=Rx+ WR-(1-a)Y b

= Rx+ WR-(-a)(X+ W) b

= [Rx(-a)X+b] + [WR-(1-a)(O.2WR+0.5WG+0.3WB)]

If the histogram compression equation (5) has parameters a=0.01 and b=128, the term (1-a) can

be ignored as in the following equation:

Rz z [Rx-X+128] + [0.8WR-O.5WG-O.3WB)]

= P+ W (8)

where P = Rx-X+128 and Wp 0.8WR-0.5WG-0.3WB. Rz can be divided into the RGB component

term P and noise term Wp. From equation (5) and a=0.01, the variance relation between Z and Y

becomes

az2 = O.Olaz2 0 a2 << 2 (9)

In other words, equation (9) means that the variation of Yis much higher than that of Z and also

the deviations of R components of Y and Z have a similar relation.

OR2 ORY2 (10)

-a2 + UP2 << RX2 + UW from equation (2),(3), and (8)

ap2 + 0.64 U TR2+0.25 OUWG2+0.09 UaB2 << U2 + Ow









White Gaussian noise in Figure 3-3 has the same standard deviation (aw = UwR = UwG = UWB), so

equation (10) becomes

ap2 + 0.98ow2 << ORX2 + ow2

The noise term aw2 can be eliminated by approximation

op2 o o2 (11)

The deviation relation of equation (8) is

OR2 = 2 2+ O2 (12)

Equations (11) and (12) show that the standard deviation of the R component is suppressed by

histogram compression on luminance of a noisy image but the standard deviation is not

suppressed. The numbers of homogeneous regions in a histogram-compressed noisy image are

increased with remaining component noise.

In order to see the effect of histogram compression more clearly, graphical analysis by

using relations among ox, ay, and cw in equation (4) is helpful to understand the proposed

method. Figure 3-4 shows the relation and the values of oy on each pixel of noisy image are

distance from the origin by using the Pythagoras relationship between ax and oy. Each blue

point shows the relative locations of local standard deviation ax, yo, and Cw of 3x3 blocks for R

components. Even if ax and ow are unknown in a given noisy image, we can get the values of

local standard deviation ax before the noise level cw is added to original noise free image. If blue

points are close to the vertical axis, the local blocks of the points are homogeneous including

homogeneous regions in the noisy image. There are a lot of homogeneous blocks in Figure 3-4

because the given noisy image is homogeneous. Hence, many blue points can be found near the

vertical axis.










image id=3, window size=5given std =15



.X






S 1:1i 3l X1:1 X X :.i .X -1
3x3 block. ....













Figure 3-5. Relationship between uxand aw. Each blue pixel shows relative locations of local
standard deviation of 3 x 3 block.

However, the result of this graphical analysis may be different for the four types of image

based on homogeneity as categorized in the previous section. In the case of a non-homogeneous
image, most of the blue points masking be apart from the vertical axis. Figure 3-5 shows the
graphical analysis for the for different types of image and the effect of histogram compression












on R components. The blue points in a red box indicate that the local blocks of the points are
homogeneous. In the case of complex structures and non-homogeneous images, there are very














few homogeneous blocks. So it is not easy to find homogeneous regions using conventional
0 1 :I -' l:l 40 50 i ..:0 ... ..






















noise estimation methods. However, after histogram compression, ox can be suppressed without

influencing distribution of o-w in all four types of images. The total number of points which are

included in the red box in a complex structure and non-homogeneous images is increased. In

other words, homogeneous regions are increased in the given noisy images by histogram

compression.













Homogeneous

tu ,--.',... ,


Less homogeneous Complex Structure


Non-homogeneous


A(%



Histogram Histogram Histogram Histogram





CompFigure 3-6. Graphical analysis of the histogram compression for four different types of noisy






images.



deviations before and after histogram compression for original noise-free image X, noisy image Y,

and additive white Gaussian noise W separately. Figure 3-6 shows the results. It is apparent that
can be increased y histogram compression even if there are very few such homogeneous regions
Figure 3-6. Graphical analysis of the histogram compression for four different types of noisy
images.










in the given notherisy image. With the increased number of histogram compression is regions, it becomes easier

deviato more exactly estimate the overam oppression for original noise-free image X, noisy image Y,ise level.
and additive white Gaussian noise W separately. Figure 3-6 shows the results. It is apparent that

the local standard deviation of the original noise-free image oa can be suppressed by histogram

compression. Histogram compression doesn't affect the local standard deviation of white

Gaussian noise.

From two graphical analyses in Figure 3-5 and 3-6, we can see that homogeneous regions

can be increased by histogram compression even if there are very few such homogeneous regions

in the given noisy image. With the increased number of homogenous regions, it becomes easier

to more exactly estimate the overall noise level.
















x x




Figure 3-7. Effect of histogram Hstogram compression.

Ir Chiompression n lmin e of noisy im e.
W W'




-r 'ar i I .

Ir
Y Y


Figure 3-7. Effect of histogram compression.

The proposed noise level estimation method is based on the following steps:

1. Perform histogram compression on luminance of noisy image.

2. Detect homogeneous regions in the image gotten from the step (1): homogeneous blocks can
be selected by the local standard deviation which is less than a threshold

3. Estimate noise level from the detected homogeneous regions: estimated noise level is
calculated by averaging the local standard deviation of detected homogeneous blocks or
finding the most frequent local standard deviation by using histogram approximation.

Performance of the proposed noise level estimation method was tested by comparing the

results with those obtained using other three noise estimation methods: Bosco[l], Amer[2],

and Shin[3], introduced in chapter 2.













15 ------------------------------------------------------- T10 ----------------------------------






S10 15 A 50 15 B
Given standard deviation Grven standard deviation
image id33. lndow sze5 .blk number=15606
15 n









0 15 15 C
Given standard deviation

Figure 3-8. Performance Test for Homogeneous Image A) Bosco's method, B) Amer's method,
C) Shin's method, D) Proposed method

Figure 3-7 illustrates the performance comparison for the homogeneous image shown in

Figure 3-1. Values on the horizontal axis represent actual given noise levels (standard deviations)

and those on the vertical axis represent estimated noise levels (standard deviations). Red line is

an ideal case of noise estimation in which the estimated amount of noise is equal to the amount

of noise actually added. Blue points represent actual estimated noise levels obtained by using

each NLE method. Conventional noise level estimation methods show good performance

because there are a lot of homogeneous regions. However, underestimation appears at high noise

levels. The proposed method shows better performance at both low and high noise levels.

For the less-homogeneous image test shown in Figure 3-8, overestimation at low noise

levels and underestimation at high noise levels using the conventional methods are induced.

However, the proposed method still shows good performance.














image id=13, window size=5, threshold =20 ,blk number =15606


10






10 5 10 5 A 12 15



Given standard deviation Given standard deviation
0 -- --------------j--:---- --------------- -----------









image id=I3, window size=5,blk number =15606 n m= -. m, :
20~~ ------------------------i------''* -------- ----------------------------------






S5 10 1 5 10 15
Given standard deviation A Given standard deviation B

image id=13, window size-5 blk number =1706
20
i .......--...----...-..-..---- -- ...--. ...---.. -.....
16 ----------------- --- -- ------- -- .--"-------




10



2 -------- ------------- -----------------
-S .-----~"----.-- ------ ------- -- -- ---I-- ---- -

S5 10 15 CL 1 .
Given standard deviation C D : 1



Figure 3-9. Performance Test for Less-homogeneous Image. A) Bosco's method, B) Amer's

method, C) Shin's method, D) Proposed method
image id=60, window s oe=5 sig max =20 blk number =9975 image id=6O, window size=5, threshold =25 ,blk number --9975
2 0 -- - - - -.--- - - -- --.. .. .. .. .. .

10 I--------------------------------7 ----------------------------------------------












0 5 10 15 5 10 1
Given standard deviation A Given standard deviation B


image id=0, window size5 ,blk number =9975 ,=,-0 ,, :.=-


25----------------------1--| ----------------- ---------------------------------------



S-.--.--.----.-----.----------------------- .





-------------------- ... ----------------------


0 5 10 15 0 1, D
Gven standard deviation 0-v standard d i at 1 i



Figure 3-10. Performance Test for Complex Structure Image. A) Bosco's method, B) Amer's
method, C) Shin's method, D) Proposed method
method, C) Shin s method, D) Proposed method










image id=30, window size=5, threshold =25 blk number =15741


S----------------------- 1 --------------- --------
--- -- -- i -- -- -- -t---" -- -- -- -- -- -- -- -- -- -
o4 ------------... --- --------- ---------------- -- ----------- ---------------

SA B
Given standard deviation Given standard deviation

image td=30, window size=5 ,blk number =15741 image id=30, window size=5



................ co -........---- -------
25 -----------------------------j ---------------------------------------------------

u252

.. .C D
S5 10 1






Given standard deviation : 5 ie s Ia I d ,, ai n
Figure 3-11. Performance Test for Non-homogeneous Image. A) Bosco's method, B) Amer'






method, C) Shin's method, D) Proposed method

The proposed method proves particularly strong when applied to a complex structure or


non-homogeneous image as seen in Figure 3-9 and 3-10. In these cases, some of the


conventional methods show less acceptable performance due to excessive overestimation and


underestimation. Neither is a problem in using the proposed method.


To evaluate the performance of the algorithm, the estimation Ek= O--O-e error is first


calculated. Ek is the difference between the true and the estimated noise level. The average and
0 5 10























the standard deviation of the estimation error are then computed from all the measures. In Figure


3-11, the performance comparison is the actual test result for 24 Kodak homogeneous and less


homogeneous photo images in Append Figure 3A-) which are used as test images in image

processing. The left plot A) in Figure 3-11 reveals that the estimation error using the proposed

processing. The left plot A) in Figure 3-11 reveals that the estimation error using the proposed












method is lower than that of other NLE methods for all noise levels. Interestingly, in the right


plot B), the standard deviation of the error using the proposed method is significantly less.



45 1 l 8
arner amer /
4 -..-.-.-.-.-.-.-.-.-.- .-.-.-.-. bosco ----- 16 ----------------...... -------------- ...... bosco ----
proposed /- proposed
35 ----------------------------- 4 ---------------------- ------- ------ sogang --


S12
--- ---.---------------- ----...---- ----------------- 4------------ ---- ---

4 --- ---" --- -- -- -- -- -- -- -- -- -

D 5 -----4----------- ------- -------------- ---- -- -2



05 10 1 5 10 15
Given standard devation A Gien standard devation B


Figure 3-12. Performance Test for Homogeneous and Less homogeneous Image. A) Mean of
absolute error of difference between actual noise and estimated noise level, B)
Standard deviation of error


bosco
20 i---------------------------- -----^=^
2 proposed
0sogang

15 ------------------------ ------------- -------- -----------------------
15


10 ---------------- ---------------- ----------------

10(


SG5 10 1
Given standard deviation


Figure 3-13. Performance Test for Complex structure, and Non-homogeneous Images. A) Mean
of absolute error of difference between actual noise and estimated noise level, B)
Standard deviation of error


Figure 3-12 shows the performance comparison for complex structure and non-


homogeneous images in Appendix A (Figure A-2) which can not be successfully tested in


conventional noise estimation methods. The proposed method shows significantly better noise


level estimation for less homogeneous, complex structure, and non-homogeneous images, all of


which remain problematic in the area of current image processing.









CHAPTER 4
NOISE REMOVAL FILTERING

The proposed noise level estimation method described in chapter 4 has been inserted in a

noise reduction system for white Gaussian noise. Basically its performance depends on the

estimation of standard deviation a for a noisy image. In order to remove noise based on the

estimated noise level using the proposed method, a bilateral noise reduction filter is used because

it is a non-linear filtering technique which utilizes both spatial and amplitudinal distances to

better preserve signal detail.

Bilateral Noise Reduction Filtering

Consider a 2-D signal fthat has been degraded by a white Gaussian noise n. The

contaminated signal g can be expressed as follows:

g[k] f[k] + n[k]

where k=(x,y) The goal of noise reduction from a noisy image is to suppress noise n and extract

original noise free imageffrom noisy image g. In spatial filtering techniques, an estimate of f is

obtained by applying a local filter h to g.

f[k] = h[k, ] x g[k]

In a conventional linear spatial filtering method, the local filter is defined based on spatial

distances between the particular point in the signal at center pixel location (x,y) and its

neighboring points. In the case of Gaussian filtering, the local filter is defined as in the following

equation:

h[k, ] = exp-
4-s

where c represents a neighboring point. Such filters operate under the assumption that an

amplitudinal variation within the neighborhood is small and that the noise signal has large

amplitudinal variations. The noise signal can be suppressed by smoothing the signal over the









local neighborhood. The problem of this assumption is that important signal detail is also

characterized by large amplitudinal variation. Therefore, such filters may induce an undesirable

blurring of signal detail. A simple and effective solution for this problem is to use bilateral

filtering, which is firstly introduced by Tomasi et al. [11] and developed from the Bayesian

approach by Elad [12].

In bilateral filtering, a local filter is defined based on a combination of the spatial distances

and the amplitudinal distances between a center point at (x, y) and its neighboring points. This

can be formulated as a product of two local filters. One is an enforcing spatial locality and the

other is an enforcing amplitudinal locality. In the Gaussian case, the bilateral filter can be

defined by the following equations:

A7 N
f[k]= N W[k,] g[k -]
NW[k, ;]/ -N
;=-N
W[k, = W, [k, ] WR[k, ]


2
W [k, ]= exp -2
2"

W [Y[k]-Y[k -;]2]
WR[k,] = exp -2

The main advantage of defining the filter in this manner is that it allows for non-linear

filtering to enforce both spatial and amplitudinal locality at the same time. The estimated

amplitude at a particular point is influenced by neighboring points if the neighboring points have

similar amplitudes which are more than those with distant different amplitudes.

This can reduce smoothing across signal regions which have large but consistent

amplitudinal variations, thus it is better to preserve such signal detail. Furthermore, the









normalization term of the above formulation helps the bilateral filter smooth away small

amplitudinal differences associated with the noise in smooth regions.

Improvement of Bilateral Filtering

It was advantageous to improve the bilateral noise removal filtering to obtain a more

robust detection of the outliers. The proposed method uses the estimated local standard deviation

a. The bilateral filter averages only pixels that are similar to the central pixel in the filter mask.

For example, assume that the mask is a 5x5 window including 9 valid pixels as seen in Figure 4-

1. In order to improve the performance of bilateral filtering, two slightly different central pixels

can be considered instead of P. Those are the o-biased center P-a and P+a. An interval whose

width is directly proportional to a is used to select the pixels that can be averaged safely. Only

similar pixels are selected for filtering by the interval centered on P because it maximizes the

number of selected pixels.





,. ,. ,.


P-u P P+a
Figure 4-1. Pixel Selection for Filtering Based on a

Pixels are successively averaged using different weights depending on their spatial locality

and locality. The final filtered pixel is obtained by the following equation:


ZW [k, ;]x Px
h[k,;]= N

=-N
where W[k, represents the weight associated to thk,e -th pixel.
-N N

where W[k, gf] represents the weight associated to the -th pixel.


JL JL JL JL X J









Impulse Noise Removal

If the central pixel P is an outlier (defected by impulse noise), unfortunately there are no

pixels similar to it. In this case the o-biased centers of P in the previous section are not useful to

improve bilteral filtering because there will be no pixels which are included by the corresponding

intervals.


outlier
Min Max








P-o P P+a

Figure 4-2. Central Pixel Outlier Detection

Unfortunately, an outlier can be located in the center pixel of the filter mask in Figure 4-2.

To overcome this problem, we need to consider the minimum (Min) and the maximum (Max)

value which is contained in the neighbor pixels of the central pixel. These values are useful to

determine whether the central element in the mask is correct or affected by impulse noise. We

can determine that a center pixel may be an outlier if P-o is greater than the maximum value in

the neighbor pixels. After the center pixel is classified as defective, it should be replaced by a

weighted average of the remaining neighbor pixels.

Figure 4-3 shows the result of proposed impulse noise reduction and bilateral noise

reduction in a noisy image including impulse noise and white Gaussian noise. Impulse noise is

eliminated by using the proposed method with reduction of white Gaussian noise in the cropped

and magnified part of a noisy image.














Impulse
noise
reduction







Figure 4-3. Impulse Noise Reduction by using Detecting Central Pixel Outlier

Performance Test of the Proposed Noise Reduction Method by PSNR

PSNR is a measurement of the similarity between two images. In PSNR higher numbers are

better. If PSNR is higher than 30dB, it is hard to distinguish between the two images with the

human eye. General PSNR formula is the following:

I m-1 n-1 2
MSE = --1 Y I(i, j) K(i, j
mn 0o j=

MAX MAX
PSNR = 10 *log,0(-A 2 = 20 *logo( )
MSE VMSE
where I and K are images compared to each other to show their similarity.

The proposed method was applied to four different types of noisy image with different

characteristics. Each test image is contaminated by white Gaussian noise with standard deviation

of 5 and impulse noise. The PSNR of the restored image was measured for the proposed method

as well as conventional noise estimation methods and general bilateral filtering method. A

summary of the results is shown in Table 6-1. The proposed method achieves good PSNR gains

over other methods for all of the test images. Furthermore, the proposed method achieves better

PSNR gains over less homogeneous, complex structure, and non-homogeneous noisy images.









Table 6-1. PSNR for test images
Method PSNR (DB)
BOSCO AMER SHIN PROPOSED
Image type
Homogeneous 34.5325 34.5667 34.5790 34.5790
Less homogeneous 28.9874 28.9983 28.9283 29.2625
Complex-structure 29.2839 29.2047 29.1903 30.2863
Non-homogeneous 27.5907 27.5440 27.5327 28.0638









CHAPTER 5
CONCLUSION

The proposed noise level estimation and noise reduction method is valid for estimating the

noise level in images affected by additive white Gaussian noise and impulse noise. Conventional

noise estimation methods perform well only when applied to a homogeneous image. However,

the proposed method can estimate more exact noise level from homogeneous to non-

homogeneous images and also shows good performance in noise reduction. It is particularly

strong in estimating the noise level in complex structure and non-homogeneous image.

The proposed noise estimation method requires fewer computational resources than

conventional methods because there is no special process required to detect homogeneous

regions in a non-homogeneous noisy image. Its parallelism structure would speed up

performance on H/W implementation.





APPENDIX A
TEST IMAGES FOR PERFORMANCE COMPARISON


mmm nmm
r kodir1OT kodim02 kodim03 kodim04 kodim05 kodim07


kodimOB kodim09 kodiml0 kodim11 kodiml3 kodiml4


kodiml5 kodiml6 kodimi? kodim06 kodiml2 kodiml8

I .EmmE
kodiml9 kodim20 kodim21 kodim22 kodim23 kodim24
Figure A-1. 24 Kodak Homogeneous and Non-homogeneous Photo Images


























































































Figure A-2. Complex Structure Non-homogeneous Photo Images


;r
J
'F.
r


L









LIST OF REFERENCES


[1] J.Brailean, R.Kleihorst, S.Efstratiadis, A. Katsaggelos, R. Lagendijk, "Noise Reduction Filter
for Dynamic Image Sequences: A Review", Proceedings of the IEEE, Vol.83, No.9, Sept. 1995

[2] S. Battiato, A.Bosco, M. Mancuso, G. Spampinato, "Adaptive Temporal Filtering for CFA
Video Sequences", In Proceedings of IEEE ACIVS'02 Advanced Concepts for Intelligent Vision
Systems 2002, pp. 19-24, Ghent University, Belgium, September 2002

[3] A. Bosco, K. Findlater, S. Battiato, A. Castorina A Noise Reduction Filter for Full-Frame
Imaging Devices" IEEE Transactions on Consumer Electronics Vol. 49, Issue 3, August
2003

[4] A. Bosco, K. Findlater, S. Battiato, A. Castorina, "A Temporal Noise Reduction Filter
Based on Full-Frame Data Image Sensors" in Proceedings of IEEE ICCE 2003

[5] A. Amer, A. Mitiche, and E. Dubois, "Reliable and Fast Structure-Oriented Video Noise
Estimation," in Proc. IEEE Int. Conf. Image Processing, Montreal, Quebec, Canada, Sep. 2002

[6] G. Messina, A. Bosco, A. Bruna, G. Spampinato, Fast Method for Noise Level Estimation
and Integrated Noise Reduction, IEEE Transactions on Consumer Electronics, vol. 51, No. 3, pp.
1028 1033, August, 2005.

[7] D.-H. Shin, R.-H Park, S. Yang, J.-H. Jung, Block-Based Noise Estimation Using Adaptive
Gaussian Filtering, in IEEE Transactions on Consumer Electronics, Vol. 51, No. 1, February,
2005

[8] S. I. Olsen, "Estimation of Noise in Images: An Evaluation," Graphical Models and Image
Process., vol. 55, pp. 319-323, July 1993

[9] L. Rudin, S. Osher, Total variation based image restoration with free local constraints, in:
Proceedings of the IEEE ICIP,
vol. 1, 1994, pp. 31-35.

[10] S. Greenberg, D. Kogan, Improved structure-adaptive anisotropic filter, Pattern Recognition
Lett. 27 (1) (2006) 59-65.

[11] C. Tomasi, R.Manduchi, Bilateral filtering for gray and color images, in: Proceedings of the
ICCV, 1998, pp. 836-846.

[12] M. Elad, On the origin of the bilateral filter and ways to improve it, IEEE Trans. Image
Process. 11 (10) (2002) 1141-1151.

[13] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, Wiley,
New York, 1949.

[14] J. Portilla, V. Strela, M. Wainwright, E. Simoncelli, Image denoising using scale mixtures









of Gaussians in the wavelet domain, IEEE Trans. Image Process. 12 (11) (2003) 1338-1351.

[15] Q. Li, C. He, Application of wavelet threshold to image denoising, in: Proceedings of the
ICICIC, vol. 2, 2006, pp. 693-696.









BIOGRAPHICAL SKETCH

I-Gil Kim obtained his B.S. degree in the Department of Electronics Engineering from

Hong Ik University, Korea, in 1999. After graduation from the university, he moved to the

United States to pursue his graduate studies. He received his M.S degree in Electrical

Engineering from University of Southern California in 2002. He served as a software engineer at

the digital media R&D center in Samsung, Korea, from 2003 to 2005. He was admitted to the

University of Florida in the fall of 2005 and has worked on numerous projects in the fields of

image processing and pattern recognition in the department of electrical and computer

engineering while completing his engineering degree.





PAGE 1

1 IMAGE DENOISING USING HISTOGRAM-BAS ED NOISE ESTIMATION By I-GIL KIM A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORI DA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF ENGINEER UNIVERSITY OF FLORIDA 2008

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2 2008 I-Gil Kim

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3 To my family, especially my wife Jiyoung and little Julia

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4 ACKNOWLEDGMENTS I am deeply grateful to my mentor Dr. Paul W. Chun from the bottom of my heart. He always encouraged me and gave me valuable advi ce. I also offer my gratitude to Dr. Fred J. Taylor, my supervisory committee chair, for hi s direction and guidance. I would like to thank my committee members. Finally, I express my gr atitude to my father, Dr. Koojin Kim. He instilled in me in the great love I have for th e natural world and taught me to be curious about science. Special thanks go to my family and fr iends for their love, support, and sacrifices that allowed me to be me throughout my years of study.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........6 LIST OF ABBREVIATIONS..........................................................................................................9 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 Conventional Noise Level Estimation Methods.....................................................................13 Conventional Noise Reduction Methods................................................................................14 2 BACKGROUNG AND RERATED REASEARCH..............................................................15 Boscos Noise Estimation (2005, IEEE)................................................................................15 Amers Noise Level Estimation (2005, IEEE).......................................................................16 Shins Noise Level Estimation (2005, IEEE).........................................................................17 3 NOISE LEVEL ESTIMATION USI NG HISTOGRAM COMPRESSION..........................20 Drawbacks of Conventional Nois e Level Estimation Methods..............................................20 Proposed Noise Level Estimation Met hod based on Histogram Compression......................22 4 NOISE REMOVAL FILTERING..........................................................................................34 Bilateral Noise Reduction Filtering........................................................................................34 Improvement of Bilateral Filtering.........................................................................................36 Impulse Noise Removal..........................................................................................................37 Performance Test of the Propos ed Noise Reduction Method by PSNR .................................38 5 CONCLUSION..................................................................................................................... ..40 APPENDIX: TEST IMAGES FOR PERFORMANCE COMPARISON.....................................41 LIST OF REFERENCES............................................................................................................. ..43 BIOGRAPHICAL SKETCH.........................................................................................................45

PAGE 6

6 LIST OF TABLES Table page 6-1 PSNR for test images....................................................................................................... ..39

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7 LIST OF FIGURES Figure page 2-1 Differences computation in homogeneous areas...............................................................15 2-2 Absolute noise histogram...................................................................................................16 2-3 Fine structure and texture images......................................................................................17 2-4 Special masks.............................................................................................................. .......17 2-5 Shins noise estimation algorithm......................................................................................19 3-1 Four types of image........................................................................................................ ...21 3-2 Homogene ous regions........................................................................................................22 3-3 Color digital image acquisition..........................................................................................23 3-4 Histogram compression on lu minance of noisy image.....................................................24 3-5 Relationship between X and W.........................................................................................27 3-6 Graphical analysis of th e histogram compression for four different types of noisy images......................................................................................................................... .......28 3-7 Graphical analysis of the e ffect of histogram compression...............................................29 3-8 Homogeneous image performance test..............................................................................30 3-9 Less-homogeneous im age performance test......................................................................31 3-10 Complex Structure im age performance test.......................................................................31 3-11 Non-homogeneous im age performance test.......................................................................32 3-12 Homogeneous and less homoge neous image performance test.........................................33 3-13 Complex structure, and non-hom ogeneous image performance test.................................33 4-1 Pixel selection for filtering based on ..............................................................................36 4-2 Central pixel outlier detection............................................................................................37 4-3 Impulse noise reduction by using detecting central pixel outlier......................................38

PAGE 8

8 A-1 The Kodak homogeneous and non-homogeneous photo images.......................................41 A-2 Complex structure non-homogeneous photo images.........................................................42

PAGE 9

9 LIST OF ABBREVIATIONS bij Non-overlapping image block B Blue component of RGB color space B Homogeneous block Ek Absolute difference error between true and estimated noise level f[k] Original signal ] k [ f Estimated original signal by filtering g[k] Contaminated signal G Green component of RGB color space ] k [ h Local filter MSE Mean squared error n[k] Noise signal Ps Weight of the proposed method PSNR Peak signal-to-noise ratio (dB) R Red component of RGB color space RGB RGB red, green, and blue color space W White Gaussian noise ] k [ W Weight of bilateral filter ] k [ Ws Space weight of bilateral filter ] k [ WR Range weight of bilateral filter X Original noise-free image Y Noisy image Yi Luminance of YUV color space YUV Luminance and chrominance color space

PAGE 10

10 Z Histogram modified noisy image ij Standard deviation of intensity for each block bij min Minimum standard deviation W Y X, Standard deviation of signal X, Y, and W Neighboring point

PAGE 11

11 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Engineer IMAGE DENOISING USING HISTOGR AM-BASED NOISE ESTIMATION By I-Gil Kim August 2008 Chair: Fred J. Taylor Major: Electrical and Computer Engineering Exact noise level estimation is very useful in digital image processing. For example, some noise removal algorithms use a noise level esti mation to adjust the aggressiveness of noise removal. If an estimated noise level is too low, too much noise will remain in the denoised image. If an estimated noise level is too high, the featur es of the original images will be removed from the denoised image. Accurate noise level estimati on will produce better results in the restored image. Most conventional noise estimation methods ar e only focused on estimating the noise level from a noisy image including a lot of homogeneous (flat) regions. So, th e typical methods induce excessive overestimation and underestimation if the given noisy image doesnt have any homogeneous regions. I propose a new noise level estimation method that uses histogram modification to find more exact noise level from the given noisy im age. This new noise estimation method uses a proposed noise reduction filter ba sed on a bilateral filter. It ma kes use of block-based noise estimation, in which an input image is assumed to be contaminated by the additive white Gaussian noise and impulse noise. A noise reduct ion filtering process is performed by the filter bilateral-based. Coefficients of the noise filter ar e selected as functions of the standard deviation

PAGE 12

12 of the Gaussian noise that is estimated from a given noisy image by the proposed method. To accurately estimate the amount of noise in various types of noisy images, images are categorized into four different types based on their hom ogeneity: homogeneous, less homogeneous, complex structure, and non-homogeneous images. In the proposed method, by using histogram modification on a noisy image, fluctuation in a non-homogeneous region can be effectively suppressed without influencing its noise level. Th is process increases ho mogeneous regions in a given noisy image and improves the performance of noise estimation for most noisy images regardless of their homogeneity. To show the effectiveness of the proposed method, it is tested on va rious kinds of noisy images ranging from homogeneous to non-homogeneous and its performance is compared with that of three conventional noise estimation methods. The proposed noise estimation and reduction method can be efficiently used in variou s image and video-based applications such as digital cameras and digital television and is supe rior because of its performance and simplicity.

PAGE 13

13 CHAPTER 1 INTRODUCTION Nowadays many people use camera phones and lo w-end digital cameras in their everyday life. These devices are particularly subject to noise reduction, when images are acquired in lowlighting conditions. However these devices are often used in poor lighting-conditions due to their portability. In order to obtain an acceptable picture, sometimes it is necessary to amplify the image signal taken under low light environments su ch as indoor scenes. However, when boosting an already degraded image signal, noise in the image also can be amplified. An effective noise reduction filter should change its strength accordi ng to the noise level in the noisy image. Here, we need some measure to estimate the noise leve l in the given noisy image. In general, we can get noise level information from the noise standard deviation Hence many noise filters depend on to adaptively change their smoothing effects [1]-[4]. Conventional Noise Level Estimation Methods Conventional noise level estima tion techniques can be categoriz ed into three main classes [8]: Block-based, filter-based, and adaptivebased methods. Before 1990, filter-based methods were commonly used. Such methods perform a prefiltering operation in which the noisy image is blurred to suppress image struct ure. A difference image is comput ed by subtractin g the filtered image from the original one and then noise leve l is estimated using the difference image, which is assumed to contain only the noise signal. After 1990, block-based met hods started to play a role in noise level estimation. Briefly, the bl ock-based method partitions the image into a sequence of blocks. The estimation of standard deviation of a noisy image is carried out by the properly calculating the weighted no ise level obtained by averaging the noise levels of the most homogeneous blocks. An adaptive method is a hybr id method which uses elements of the filterbased and the block-based methods Its performance depends on the first noise level estimation.

PAGE 14

14 Efficient techniques based on the block based approach are describe d in [5] and [6]. A comparison of various methods for determining an approximation of the noise level in an image is given in [7]. Conventional Noise Reduction Methods A large number of different noise reducti on methods have been proposed so far. Traditional denoising methods can be generalized into two main groups: spatial domain filtering and transform domain filtering. Spatial domain filte ring methods have long been the mainstay of signal denoising and manipulate the noisy signal in a direct fashi on. Conventional linear spatial filters like Gaussian filters try to suppress noi se by smoothing the signal. While this works well in the situations where signal variation is low, su ch spatial filters result in undesirable blurring of the signal in situations where signal variation is high. To overc ome these drawbacks, a number of new spatial filtering methods such as tota l variation techniques [9], anisotropic filtering techniques [10], and bilateral f iltering techniques [11,12] have been introduced to suppress noise while preserving signal characterist ics in the regions of high signal variation. Bilate ral filtering is a non-linear and non-iterative filtering techniqu e which utilizes both spatial and amplitudinal distances to preserve signal detail. On th e other hand, transform domain filtering methods transform the noisy signal into the frequency do main and manipulate the frequency coefficients to suppress signal noise before the signal is transformed back into the spatial domain. These techniques had been introduced by Wiener f iltering [13] and wavelet-based methods [14,15] I propose novel approaches to noise level estimation and noise reduction that improve bilateral filtering. The proposed methods are robust in situations where the given noisy image is characterized by non-homogeneity and pr ovide improved noise suppression.

PAGE 15

15 CHAPTER 2 BACKGROUNG AND RERATED REASEARCH In this section, I introduce three recent noise estima tion algorithms, showing good performance in IEEE journal. Boscos and Am ers method are blockbased approaches, and Shins method is adaptive approach. These thr ee conventional methods will be compared with my proposed method in chapters 3 and 4. Boscos Noise Estimation (2005, IEEE) In most of block-based noise estimation met hods, noise level is obtai ned by averaging the noise levels of the most homogeneous blocks. In order to overcome inaccuracy of averaging, Bosco proposed new method using difference hist ogram approximation. The detail algorithm is as follows: 1. Tessellate an input noisy image into a numbe r of 3(5) nonoverlapping image blocks 2. Compute absolute differences between the centr al pixel and its neighborhood for each blocks (Figure 2-1) 3. Select a block as a homogeneous block if the 8 differences are very small 4. Compute noise histogram for selected homoge neous blocks from accumulated differences (Figure 2-2) 5. Obtain noise level( ) by considering 68% of the noise sa mples. The index on the x-axis of the noise histogram allowing reaching the 68% of the noise samples Figure 2-1. Differences com putation in homogeneous areas

PAGE 16

16 In Boscos method, histogram approximation me thod is exploited instead of averaging local standard deviations usually used in conventional method for noise level estimation. In some of case, it provides bette r result than other conv entional methods. The main weakness of this method is that it takes more time an d storage to accumulate differences. Figure 2-2. Absolute noise histogram. The index on the x-axis allowed to reach the 68% of the total samples represents the current noise level estimation. Amers Noise Level Estimation (2005, IEEE) Selection of homogeneous areas in the given noisy image is very important in most of conventional noise estimation methods. Accuracy of block-based noise level estimation highly relies on selecting homogeneous blocks. Howeve r, many conventional methods were found to have difficulties finding homoge neous region in noisy image c ontaining fine structures or textures (Figure 2-3). Amer introduced a new appr oach to overcome this drawback using special masks to find homogeneous regions. The follo wing steps describe the proposed method: 1. Tessellate an input noisy image into a number of 3 3(5 5) nonoverlapping image blocks 2. Use eight high-pass operators and special mask s for corners to stabilize the homogeneity estimation for each block. (Figure 2-4)

PAGE 17

17 3. Compute summation of the absolute values of all eight quantities from step 2) 4. Select the block as a homogeneous block if the summation is less than threshold 5. Average local standard deviation of the selected homogeneous blocks Amers method gives better noise estimation perfor mance for the images which have texture and fine structure by using special masking if the noi se level of noisy image is not too high. The number of detected homogeneous re gions is rapidly decreased when given noise level is high. So it is hard to find homogeneous areas in this case. Figure 2-3. Fine struct ure and texture images. Figure 2-4. Special masks Shins Noise Level Estimation (2005, IEEE) Shins noise estimation algorithm is based on both block-based and filtering-based approaches. It selects homogeneous blocks by a block-based approach and then filters the

PAGE 18

18 selected homogeneous blocks using a filteringbased approach. Block-based methods require a low computational load whereas filtering-based approaches yield a stable estimate. Fig. 2 shows the flowchart of Shins estima tion noise estimation method using adaptive Gaussian filtering. Each step of Shins method is described as follows. 1. Tessellate an input noisy image into a number of 3 nonoverlapping image blocks (bij). 2. Compute the standard deviation (ij) of intensity for each block bij and find the minimum standard deviation (min). 3. Select homogeneous blocks (B*) whose standard deviations of intensity are close to min 4. Compute Gaussian filt er coefficients (1st is set to min) 5. Gaussian filtering on th e selected blocks (B*). 6. Compute the standard deviation(2nd) of the difference image between noisy and filtered images within selected blocks (B*), which gives the estimated noise standard deviation (n =2nd) In general, block-based approaches tend to overestimate the nois e level in good quality images and underestimate it in highly noisy im ages. This underestimation can be compensated by adaptive Gaussian filtering of Shins method. However, its performance is highly depend on the first estimated noise level (1st) and execution time is usually longer than other block-based approaches because of the sec ond filtering-based estimation.

PAGE 19

19 Figure 2-5. Block diagram of Sh ins noise estimation algorithm

PAGE 20

20 CHAPTER 3 NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION In this chapter I will address the drawb acks of conventional methods for noise level estimation in digital images degraded by noise and compare performances of each method with the proposed method for four types of noisy image. Drawbacks of Conventional Noise Level Estimation Methods Most conventional methods for estimating the noi se standard deviation are generally based on the following steps: 1. Detect homogeneous regions in the noisy image (becau se fluctuations in flat areas, pixels are supposed to be due exclusively to noise). 2. Compute the local standard deviation in the detected homogeneous regions. 3. Repeat steps 1) and 2) until the whole image has been processed. 4. Finally estimate the noise level (or standard deviation of noise) by averaging the computed local standard deviations. This method has two major drawbacks. First, if there are no homogeneous regions or too small areas of flat region in a given noisy imag e, it is hard to detect the homogeneous regions. An insufficient number of detected homogeneous regions may result in overestimation or underestimation. This will induce wrong estimation for noise level. Seco nd, the process of detecting homogeneous regions in a noisy image requires a high number of computations. The detection process for a non-homogeneous image ma y be just time wasted work and the noise estimation based on such detection has a high pr obability of being wrong. Theses drawbacks originate from the fact that most conventi onal noise estimation methods are only focused on noisy images including a lot of homogeneous regions. In order to make this problem clear, I have categorized noisy images into four types as following: 1. Homogeneous image: image contains a lot of homogeneous regions.

PAGE 21

21 2. Less homogeneous image: image contains very few homogeneous regions in a limited area. 3. Complex structure image: original noise-free im age has very complex st ructure, so it is not easy to find homogeneous regions due to local fluctuations. 4. Non-homogeneous image: there are no homoge neous regions in the given noisy image. A B C D Figure 3-1. Four types of imag e. A) homogeneous image, B) less homogeneous image, C) complex structure image, D) non-homogeneous image Figures 3-1 shows examples of four diffe rent types of image based on homogeneity. There are homogeneous regions in Figure 3-1 A) and B), but it is not easy to find homogeneous regions in Figure 3-1 C) and D) by using conventional noise estimation methods. Figure 3-2 shows homogeneous regions detected by Bosco[1] method and violet blocks in each image are the detected areas.

PAGE 22

22 A B C D Figure 3-2. Homogeneous regions A) homogeneous image, B) less homogeneous image, C) complex structure image, D) non-homogeneous image Proposed Noise Level Estimation Met hod based on Histogram Compression In this section the propos ed noise estimation method is shown to overcome the drawbacks of conventional noise estimation methods by means of histogram compression on intensity of image that tries to exploit correlation among R, G, and B components of color image regardless of the number of homogeneous regions in the given noisy image. The best way to estimate an exact noise leve l from a given noisy image is to make the image homogeneous without influencing its noise deviation. In other words, if we can suppress deviation in the original noise-f ree image without changing deviati on of noise in a noisy image, it is easy to find more exact noise leve ls from the suppressed noisy image.

PAGE 23

23 Nowadays most digital images are in co lor and each pixel of the image has red (R), green (G), and blue (B) components. Each RGB component in a noisy color image has a noise and the noise tends to have Gaussian distribution in a digital image acquisition system. Figure 3-3. Color Digital image Acquisition Figure 3-3 shows the structure of a current digital image acquisition system. Generally most common noise in digital im age acquisition can be modeled by a white Gaussian noise with the same standard deviation W. In order to reduce th e correlation between the RGB components, a conversion from RGB to YUV color space as follows: Yi (Luminance) = 0.3Ri + 0.6Gi + 0.1Bi Ui (Chrominance1) = Bi Yi Vi (Chrominance2) = Ri Yi However, if component noise is uncorrelated in RGB space, it is correlated in YUV space. Also the component noise and RGB component are independent to each other, so they are Noise R channel : RX + WR G channel : GX + WG B channel : BX + WB where WR = WG = WB X Y Y = 0.3(RX + WR) + 0.5(GX + WG) + 0.3(BX + WB) X = 0.3RX + 0.5GX + 0.3BX

PAGE 24

24 uncorrelated in RGB space. A linear operation such as histogram modification in YUV space may affect uncorrected RGB components and correlated com ponent noise. Histogram compression, one of several histogram modification methods can be useful to suppress deviation of RGB component without affecting de viation of component noise. Figure 3-4. Histogram Compressi on on Luminance of Noisy Imag e. The histogram plots the number of pixels in the image (vertical axis) with a particular brightness value (horizontal axis) The noisy image and original noise-free image have luminance correlation among RGB components in YUV color space. X = 0.2RX + 0.5GX + 0.3BX Y = X + W (1) = 0.2RY+0.5GY+0.3BY (2) = 0.2(RX+WR) + 0.5(GX+WG) + 0.3(BX+WB) (3) where X is original noise-free image and Y is a noisy image. W is an additive white Gaussian noise. Because X and W are independent of each other, variation of Y has the following relation to variation of X. Y 2 = X 2 + W 2 (4) If we do histogram compression on the luminance of noisy image Y Z = aY-b, (5) Y = 0.2RY + 0.5GY +0.3BYX = 0.2R X + 0.5G X +0.3B X Y = aX + b (a=0.01, b = 128)

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25 = Y Z = (1-a)Y + b Z = Y = 0.2*RZ + 0.5*GZ + 0.3*BZ (6) = 0.2(RY) + 0.5(GY) + 0.3(BY) (7) where Z is compressed noisy image in YUV. From equation (6) and (7), the R component of Z, Rz becomes RZ = RY= RX+WR-(1-a)Y b = RX+WR-(1-a)(X+W) b = [RX(1-a)X+b] + [WR-(1-a)(0.2WR+0.5WG+0.3WB)] If the histogram compression equation (5) has parameters a=0.01 and b=128, the term (1-a) can be ignored as in the following equation: RZ [RX-X+128] + [0.8WR-0.5WG-0.3WB)] = P + Wp (8) where P = RX-X+128 and Wp = 0.8WR-0.5WG-0.3WB. Rz can be divided into the RGB component term P and noise term Wp. From equation (5) and a=0.01, the variance relation between Z and Y becomes z 2 = 0.01 zY 2 z 2 << Y 2 (9) In other words, equation (9) means that the variation of Y is much higher than that of Z and also the deviations of R components of Y and Z have a similar relation. RZ 2 << RY 2 (10) P 2 + WP 2 << RX 2 + W 2 from equation (2),(3), and (8) P 2 + 0.64 WR 2+0.25 WG 2+0.09 WB 2 << RX 2 + W 2

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26 White Gaussian noise in Figure 3-3 has the same standard deviation (W = WR = WG = WB), so equation (10) becomes P 2 + 0.98 W 2 << RX 2 + W 2 The noise term W 2 can be eliminated by approximation P 2 << RX 2 (11) The deviation relation of equation (8) is RZ 2 = P 2 + W 2 (12) Equations (11) and (12) show that the standard deviation of the R component is suppressed by histogram compression on luminance of a noisy image but the standard deviation is not suppressed. The numbers of homogeneous region s in a histogram-compressed noisy image are increased with remaining component noise. In order to see the effect of histogram compression more cl early, graphical analysis by using relations among X, Y, and W in equation (4) is helpful to understand the proposed method. Figure 3-4 shows the relation and the values of Y on each pixel of noisy image are distance from the origin by using th e Pythagoras relationship between X and Y Each blue point shows the relative locations of local standard deviation X, Y, and W of 3x3 blocks for R components. Even if X and W are unknown in a given noisy imag e, we can get the values of local standard deviation X before the noise level W is added to original noise free image. If blue points are close to the vertical axis, the loca l blocks of the points are homogeneous including homogeneous regions in the noisy image. There are a lot of ho mogeneous blocks in Figure 3-4 because the given noisy image is homogeneous. He nce, many blue points can be found near the vertical axis.

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27 Figure 3-5. Relationship between X and W Each blue pixel shows relative locations of local standard deviation of 3 x 3 block. However, the result of this graphical analysis may be differe nt for the four types of image based on homogeneity as categorized in the previ ous section. In the case of a non-homogeneous image, most of the blue points may be apart from the vertical axis. Figure 3-5 shows the graphical analysis for the four different types of image and the effect of histogram compression on R components. The blue points in a red box indi cate that the local blocks of the points are homogeneous. In the case of complex structures and non-homogeneous images, there are very few homogeneous blocks. So it is not easy to find homogeneous regi ons using conventional noise estimation methods. However, after histogram compression, X can be suppressed without influencing distribution of W in all four types of images. Th e total number of points which are included in the red box in a complex structure and non-homogeneous images is increased. In other words, homogeneous regions are increas ed in the given noisy images by histogram compression. x w Y 3x3 block masking for getting local Y

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28 Figure 3-6. Graphical analysis of the histogram compression for four different types of noisy images. Another way to show the effect of histogr am compression is to compare local standard deviations before and after histogram compression for original noise-free image X, noisy image Y, and additive white Gaussian noise W separately. Figure 3-6 shows the results. It is apparent that the local standard deviation of the original noise-free image X can be suppressed by histogram compression. Histogram compression doesnt affect the local standard deviation of white Gaussian noise. From two graphical analyses in Figure 35 and 3-6, we can see that homogeneous regions can be increased by histogram compression even if there are very few such homogeneous regions in the given noisy image. With the increased number of homoge nous regions, it becomes easier to more exactly estimate the overall noise level. Histogram Compression Suppressed Suppressed Suppressed Suppressed Histogram Compression Histogram Compression Histogram CompressionHomogeneous Less homogeneous Complex Structure Non-homogeneous

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29 Figure 3-7. Effect of histogram compression. The proposed noise level estimation me thod is based on the following steps: 1. Perform histogram compression on luminance of noisy image. 2. Detect homogeneous regions in the image gotten from the step (1): homogeneous blocks can be selected by the local standard devi ation which is less than a threshold 3. Estimate noise level from the detected homoge neous regions: estimated noise level is calculated by averaging the local standard de viation of detected homogeneous blocks or finding the most frequent local standard deviation by using histogram approximation. Performance of the proposed noise level estim ation method was tested by comparing the results with those obtained using other three noise estimation methods: Bosco[1], Amer[2], and Shin[3], introduced in chapter 2. W W X X Y Y Histogram CompressionX W Y X W Y

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30 A B C D Figure 3-8. Performance Test for Homogeneous Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method Figure 3-7 illustrates the performance co mparison for the homogeneous image shown in Figure 3-1. Values on the horizontal axis represent actual given noise levels (standard deviations) and those on the vertical axis represent estimate d noise levels (standard deviations). Red line is an ideal case of noise estimation in which the es timated amount of noise is equal to the amount of noise actually added. Blue poi nts represent actual estimated noise levels obtained by using each NLE method. Conventional noise level estimation methods show good performance because there are a lot of homogeneous regions. However, underestimation appears at high noise levels. The proposed method shows better perfor mance at both low and high noise levels. For the less-homogeneous image test shown in Figure 3-8, overestimation at low noise levels and underestimation at high noise leve ls using the conventional methods are induced. However, the proposed method still shows good performance.

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31 A B C D Figure 3-9. Performance Test for Less-homoge neous Image. A) Boscos method, B) Amers method, C) Shins method, D) Proposed method A B C D Figure 3-10. Performance Test for Complex Structure Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method

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32 A B C D Figure 3-11. Performance Test for Non-homogeneous Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method The proposed method proves particularly strong when applied to a complex structure or non-homogeneous image as seen in Figure 3-9 and 3-10. In these cases, some of the conventional methods show less acceptable perf ormance due to excessive overestimation and underestimation. Neither is a proble m in using the proposed method. To evaluate the performance of the algorithm, the estimation Ek=|n-e| error is first calculated. Ek is the difference between the true and th e estimated noise level. The average and the standard deviation of the es timation error are then computed from all the measures. In Figure 3-11, the performance comparison is the actual test result for 24 Kodak homogeneous and less homogeneous photo images in Appendix A ( Figure A-1) which are used as test images in image processing. The left plot A) in Figure 3-11 reveals that the es timation error using the proposed

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33 method is lower than that of other NLE methods for all noise levels. Interestingly, in the right plot B), the standard deviati on of the error using the proposed method is significantly less. A B Figure 3-12. Performance Test for Homogeneous and Less hom ogeneous Image. A) Mean of absolute error of difference between actua l noise and estimated noise level, B) Standard deviation of error A B Figure 3-13. Performance Test for Complex structure, and Nonhomogeneous Images. A) Mean of absolute error of difference between act ual noise and estimated noise level, B) Standard deviation of error Figure 3-12 shows the performance comp arison for complex structure and nonhomogeneous images in Appendix A ( Figure A-2) which can not be successfully tested in conventional noise estimation methods. The prop osed method shows significantly better noise level estimation for less homogeneous, complex st ructure, and non-homogeneous images, all of which remain problematic in the area of current image processing.

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34 CHAPTER 4 NOISE REMOVAL FILTERING The proposed noise level estimation method descri bed in chapter 4 has been inserted in a noise reduction system for white Gaussian noi se. Basically its performance depends on the estimation of standard deviation for a noisy image. In order to remove noise based on the estimated noise level using the proposed method, a b ilateral noise reduction filter is used because it is a non-linear filtering technique which uti lizes both spatial and amplitudinal distances to better preserve signal detail. Bilateral Noise Reduction Filtering Consider a 2-D signal f that has been degraded by a white Gaussian noise n. The contaminated signal g can be expressed as follows: g[k] = f[k] + n[k] where k=(x,y) The goal of noise reduction from a noisy image is to suppress noise n and extract original noise free image f from noisy image g. In spatial filtering techni ques, an estimate of f is obtained by applying a lo cal filter h to g. f[k] = h[k, ] g[k] In a conventional linear spatial filtering method, the local filter is defined based on spatial distances between the particular point in the signal at center pixel location (x,y) and its neighboring points. In the case of Gaussian filtering, the local filte r is defined as in the following equation: where represents a neighboring point. Such filter s operate under the assumption that an amplitudinal variation within the neighborhood is small and that the noise signal has large amplitudinal variations. The noise signal can be suppressed by smoothi ng the signal over the 2 s 22 exp ] k [ h

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35 local neighborhood. The problem of this assumpti on is that important si gnal detail is also characterized by large amplitudina l variation. Therefore, such f ilters may induce an undesirable blurring of signal detail. A simple and effective solution for this problem is to use bilateral filtering, which is firstly introduced by Tomasi et al. [11] and devel oped from the Bayesian approach by Elad [12]. In bilateral filtering, a local fi lter is defined based on a combin ation of the spatial distances and the amplitudinal distances between a center point at (x, y) and its neighboring points. This can be formulated as a product of two local filters One is an enforcing sp atial locality and the other is an enforcing amplitudinal locality. In th e Gaussian case, the bilateral filter can be defined by the following equations: ] k [ W ] k [ W ] k [ WR S The main advantage of defining the filter in this manner is that it allows for non-linear filtering to enforce both spatial and amplitudinal locality at the same time. The estimated amplitude at a particular point is influenced by neighboring points if the neighboring points have similar amplitudes which are more than t hose with distant different amplitudes. This can reduce smoothing across signal regi ons which have large but consistent amplitudinal variations, thus it is better to pr eserve such signal detail. Furthermore, the N -N N -N] k [ g ] k [ W ] k [ W 1 ] k [ f 2 R 2 R2 ] ] k [ Y ] k [ Y [ exp ] k [ W 2 s 2 s2 exp ] k [ W

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36 normalization term of the above formulation he lps the bilateral filter smooth away small amplitudinal differences associated with the noise in smooth regions. Improvement of Bilateral Filtering It was advantageous to improve the bilatera l noise removal filtering to obtain a more robust detection of the outliers. The proposed method uses the esti mated local standard deviation The bilateral filter averages onl y pixels that are similar to the central pixel in the filter mask. For example, assume that the mask is a 5x5 window including 9 valid pixels as seen in Figure 41. In order to improve the performance of bilatera l filtering, two slightly different central pixels can be considered instead of P. Those are the -biased center Pand P+ An interval whose width is directly proportional to is used to select the pixels th at can be averaged safely. Only similar pixels are selected for filtering by the interval centered on P because it maximizes the number of selected pixels. Figure 4-1. Pixel Selection for Filtering Based on Pixels are successively averaged using differe nt weights depending on th eir spatial locality and locality. The final filtered pixel is obtained by the following equation: N N N N] k [ W P ] k [ W ] k [ h where W[k,] represents the weight associated to the -th pixel. P P P+

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37 Impulse Noise Removal If the central pixel P is an outlier (defected by impulse noise), unfortunately there are no pixels similar to it. In this case the -biased centers of P in the previous secti on are not useful to improve bilteral filtering because there will be no pixels which are included by the corresponding intervals. Figure 4-2. Central Pixel Outlier Detection Unfortunately, an outlier can be located in th e center pixel of the filter mask in Figure 4-2. To overcome this problem, we need to consider the minimum (Min) and the maximum (Max) value which is contained in the ne ighbor pixels of the central pixel. These va lues are useful to determine whether the central elem ent in the mask is correct or affected by impulse noise. We can determine that a center pixel may be an outlier if Pis greater than the maximum value in the neighbor pixels. After the center pixel is classified as defec tive, it should be replaced by a weighted average of the remaining neighbor pixels. Figure 4-3 shows the result of proposed impul se noise reduction and bilateral noise reduction in a noisy image including impulse noise and white Gaussian noise. Impulse noise is eliminated by using the proposed method with re duction of white Gaussian noise in the cropped and magnified part of a noisy image. outlier Min Max PP P+

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38 Figure 4-3. Impulse Noise Reduction by using Detecting Central Pixel Outlier Performance Test of the Proposed Noise Reduction Method by PSNR PSNR is a measurement of the similarity between two images. In PSNR higher numbers are better. If PSNR is higher than 30dB, it is hard to distinguish between the two images with the human eye. General PSNR formula is the following: where I and K are images compared to each other to show their similarity. The proposed method was applied to four diffe rent types of noisy image with different characteristics. Each test image is contaminated by white Gaussian noise with standard deviation of 5 and impulse noise. The PSNR of the restored image was m easured for the proposed method as well as conventional noise estimation methods and genera l bilateral filtering method. A summary of the results is shown in Table 6-1. The proposed method achieves good PSNR gains over other methods for all of the test images. Furthermore, the proposed method achieves better PSNR gains over less homogeneous, complex st ructure, and non-homogeneous noisy images. Impulse noise reduction 2 1 m 0 i 1 n 0 jj) K(i, j) I(i, mn 1 MSE ) MSE MAX ( log 20 ) MSE MAX ( log 10 PSNRI 10 2 I 10

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39 Table 6-1. PSNR for test images PSNR (DB) Method Image type BOSCO AMER SHIN PROPOSED Homogeneous 34.5325 34.5667 34.5790 34.5790 Less homogeneous 28.9874 28.9983 28.9283 29.2625 Complex-structure 29.2839 29.2047 29.1903 30.2863 Non-homogeneous 27.5907 27.5440 27.5327 28.0638

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40 CHAPTER 5 CONCLUSION The proposed noise level estimation and noise reduction method is valid for estimating the noise level in images affected by additive white Gaussian noise and impulse noise. Conventional noise estimation methods perform well only when applied to a homogeneous image. However, the proposed method can estimate more exac t noise level from homogeneous to nonhomogeneous images and also shows good performa nce in noise reduction. It is particularly strong in estimating the noise level in co mplex structure and non-homogeneous image. The proposed noise estimation method require s fewer computational resources than conventional methods because there is nospecial process required to detect homogeneous regions in a non-homogeneous noisy image. Its parallelism structure would speed up performance on H/W implementation.

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41 APPENDIX A TEST IMAGES FOR PERFORMANCE COMPARISON Figure A-1. 24 Kodak Homogeneous and Non-homogeneous Photo Images

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42 Figure A-2. Complex Structure Non-homogeneous Photo Images

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43 LIST OF REFERENCES [1] J.Brailean, R.Kleihorst, S.Efst ratiadis, A. Katsaggelos, R. Lagendijk, Noise Reduction Filter for Dynamic Image Sequences: A Review, Proceedings of the IEEE, Vol.83, No.9, Sept. 1995 [2] S. Battiato, A.Bosco, M. Mancuso, G. Spampi nato, Adaptive Temporal Filtering for CFA Video Sequences, In Proceedings of IEEE ACIVS 02 Advanced Concepts for Intelligent Vision Systems 2002, pp. 19-24, Ghent Univer sity, Belgium, September 2002 [3] A. Bosco, K. Findlater, S. Battiato, A. Castorina A Noise Reduction Filter for Full-Frame Imaging Devices" IEEE Transactions on Cons umer Electronics Vol. 49, Issue 3, August 2003 [4] A. Bosco, K. Findlater, S. Battiato, A. Castorina, "A Temporal Noise Reduction Filter Based on Full-Frame Data Image Sensors" in Proceedings of IEEE ICCE 2003 [5] A. Amer, A. Mitiche, and E. Dubois, Relia ble and Fast StructureOriented Video Noise Estimation, in Proc. IEEE Int. Conf. Image Processing, Montreal, Quebec, Canada, Sep. 2002 [6] G. Messina, A. Bosco, A. Bruna, G. Spampinato, Fast Method for Noise Level Estimation and Integrated Noise Reduction, IEEE Transactions on Consumer Electronics, vol. 51, No. 3, pp. 1028 1033, August, 2005. [7] D.-H. Shin, R.-H Park, S. Yang, J.-H. J ung, Block-Based Noise Estimation Using Adaptive Gaussian Filtering, in IEEE Transactions on C onsumer Electronics, Vol. 51, No. 1, February, 2005 [8] S. I. Olsen, Estimation of Noise in Imag es: An Evaluation, Graphical Models and Image Process., vol. 55, pp. 319-323, July 1993 [9] L. Rudin, S. Osher, Total va riation based image restoration with free local constraints, in: Proceedings of the IEEE ICIP, vol. 1, 1994, pp. 31. [10] S. Greenberg, D. Kogan, Improved structure-adaptive anisotropic filter, Pattern Recognition Lett. 27 (1) (2006) 59. [11] C. Tomasi, R.Manduchi, Bila teral filtering for gray and colo r images, in: Proceedings of the ICCV, 1998, pp. 836. [12] M. Elad, On the origin of the bilateral filter and ways to improve it, IEEE Trans. Image Process. 11 (10) (2002) 1141. [13] N. Wiener, Extrapolation, Interpolation, and Smoothing of St ationary Time Series, Wiley, New York, 1949. [14] J. Portilla, V. Strela, M. Wainwright, E. Simoncelli, Image denoising using scale mixtures

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44 of Gaussians in the wavelet domain, IEEE Tr ans. Image Process. 12 (11) (2003) 1338. [15] Q. Li, C. He, Application of wavelet threshold to image denoising, in: Proceedings of the ICICIC, vol. 2, 2006, pp. 693.

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45 BIOGRAPHICAL SKETCH I-Gil Kim obtained his B.S. degree in the Depa rtment of Electronics Engineering from Hong Ik University, Korea, in 1999. After gra duation from the university, he moved to the United States to pursue his graduate studies He received his M.S degree in Electrical Engineering from University of Southern Californi a in 2002. He served as a software engineer at the digital media R&D center in Samsung, Korea, from 2003 to 2005. He was admitted to the University of Florida in the fall of 2005 and has worked on numerous projects in the fields of image processing and pattern r ecognition in the department of electrical and computer engineering while completing his engineering degree.