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Title: An Adaptive approach for texture segmentation by multi-channel wavelet frames
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Title: An Adaptive approach for texture segmentation by multi-channel wavelet frames
Series Title: Department of Computer and Information Science and Engineering Technical Report ; 93-025
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Language: English
Creator: Laine, Andrew
Fian, Jian
Publisher: Department of Computer and Information Sciences, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: August, 1993
Copyright Date: 1993
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An Adaptive Approach
for Texture Segmentation
by Multi-channel Wavelet Frames1


Andrew Laine and Jian Fan
Center for Computer Vision and Visualization

August, 1993

TR-93-025


1This work was supported by the National Science Foundation under Grant No. IRI-911 1.. ,'















ABSTRACT


We introduce an adaptive approach for texture feature extraction based on multi-channel
wavelet frames and two-dimensional envelope detection. Representations obtained from both
standard wavelets and wavelet packets are evaluated for reliable texture segmentation. Algo-
rithms for envelope detection based on edge detection and the Hilbert transform are presented.
A,.llll.'.* filters are selected for each technique based on performance evaluation. A K-means
clustering algorithm was used to test the performance of each representation feature set.
Experimental results for both natural textures and si;iit,, t.' textures are shown.















1 Introduction

In the field of computer vision, texture segmentation has been investigated by many re-
searchers using a diversity of approaches. In general, each method consists of two phases:
feature extraction and segmentation. Features for texture representation are of crucial im-
portance for accomplishing segmentation[l]. Previous approaches for representing texture
features can be divided into two categories [2, 3]:
1. Statistical approaches: co-occurrence matrixes [4], second-order statistics [5], numbers
of local extrema [6], local linear transformations [7], autoregressive moving average
(ARMA) [8], and Markov random fields [9] [10] [11][12].

2. Spatial/spatial-f I, ', ii approaches: including features obtained by computing Fourier
transform domain energy [13], local orientation and frequency [14], spatial energy [15],
Wigner distribution [3], multi-channel Gabor filters [16] [17][18] [19][20], wavelet pack-
ets [21], and wavelets [22].
In this paper, we describe a multi-channel approach, closely tied to wavelet multires-
olution analysis. The existence of a multi-channel filtering model is evident in studies of
the human visual system [23]. In fact, pychovisual experiments -i.-. -1 the presence of a
"biological spectrum analy_ i within the human visual system [24].
In an early paper by Bovik [17], complex Gabor filters were used, where texture features
were computed by envelope and phase information extracted from two quadratic compo-
nents of each channel output. In a related paper, A.K.Jain [19] chose real Gabor filters
for analysis. Each filtered channel (output) was subjected to a nonlinear transformation,
)(t) = tanh(ct). An average absolute derivation was then computed in terms of overlapping
windows. Although Gabor filters are easy to design and have desirable properties including
orientation selectivity and filter bandwidth, they are computationally inefficient. In addi-
tion, a large number of channels are required in order to appr ....' .,/, lil cover the complete
frequency plane.
Recent developments in wavelet theory provide a promising alternative through multi-
channel filter banks that have several potential advantages over Gabor filters: (1) Wavelet
filters cover 1, 11, il the complete frequency domain. (2) Fast algorithms are readily available
to facilitate computation. (3) Wavelet based feature extraction may be more efficiently
incooperated into an integrated computer vision system, consisting of compression, edge
detection, and other traditional image processing operations. There have been several recent
studies reporting the success of applying wavelet theory to texture analysis. A.Laine et al [21]
used wavelet packet signatures for texture classification and achieved perfect classification
for 550 samples obtained from 25 distinct natural textures. In addition, M.Unser [22] has
reported promising results using wavelets for both classification and segmentation of textures.
In this paper, we introduce an adaptive method of feature extraction that relies on a
wavelet multi-channel analysis and an envelope detection algorithm. We shall show how to















H2 H2


H, H-






Figure 1: Structure of a discrete wavelet transform (decomposition stages 0, 1, 2 only).

incooperate zero-crossing detection into envelope estimation to provide an adaptive method
of feature extraction while preserving the accuracy of texture boundaries. The remainder of
this paper is organized as follows. Section 2 briefly reviews the framework of discrete wavelet
transforms and the discrete wavelet packet transform, including corresponding variations of
wavelet frames. Section 3 describes our multi-channel feature extraction scheme and method
of filter selection. Section 4 discusses the performance of three segmentation algorithms and
compares experiment results. Finally, Section 5 provides a brief summary and conclusion.


2 Multi-channel wavelet analysis

The general structure and computational framework of the discrete wavelet transform (DWT)
are similar to those found in subband coding systems. The main difference lies in filter design,
where wavelet filters are required to be regular [25]. In this study, we considered two methods
of multiscale analysis:

Discrete wavelet transform (DWT) [25][27] 7-'], which corresponds to an octave-band
filter bank.

Discrete wavelet packet transform (DWPT) [29][30], which corresponds to a general
tree-structured filter bank.

The computational structures for each method (decomposition stages only) are shown
in Figure 1 and Figure 2, respectively. In general, the DWPT imposes a more rigorous
constraint on a wavelet, that is, the wavelet should be orthonormal. However, the DWPT
allows for more flexibility by providing an adaptive basis. For a discussion on their continuous
counterparts, please see [25] [22] [30].
Unfortunately both decompositions are not translation invariant (desirable for accom-
plishing texture analysis [22]). A possible solution is achieved by using an overcomplete
wavelet decomposition called a discrete wavelet frame (DWF) [22]. In this case, a DWF or




















H, HK2


H, H 2'


G1 H,2)


H 2


G, H2


H; ^2



=G2 J-2




HG2 J:
C, Hf2



HG2 J-

H2 -(


Figure 2: Structure of a discrete wavelet packet transform (decomposition stages 0, 1, 2 only).


DWPF scheme is similar to its DWT and DWPT except that no down sampling occurs be-
tween levels. Figure 3 shows a general DWPF as a binary tree for a three level decomposition.


In each case above, the filters Hi(w) and Gi(w) at level I were generated as described in
[25] [26] :


Ii (w)

Qi(w)





G0 k'O


H1i k=1


G2 H2 G2 H2 G2 H2 G2 H2


Figure 3: Tree structure for wavelet packet frames and associated indexes.


Ho(21w)
Go(21w).


G1 k=0


Ho k=1


Gi k=2


Hi k=3


Level 0



Level 1


Level 2














Let S'(w) be the Fourier transform of the input signal at channel k for level 1, then

2t1 () = G()S(w) (3)
Si+1 (w) = H (w)Si(). (4)

From the filter bank point of view, this is equivalent to a filter bank with channel filters
{F(w)|10 < k < 21 1}, where Fk(w) is defined recursively by the formula

FO(w) = Go(w), Ff(w) = Ho(w), (5)
F1+' () = G+,(w)FI(w) = Go(2+1lw)F (w), (6)
F+ (w) = H+,(w)Fw () = Ho(2+l2)F(w). (7)

For 2-D images, we simply use a tensor product extension for which the channel filters
are written as
F/jww) F/(w)Fw). (8)


3 Analytic filter selection and feature extraction

Recent results on texture classification have shown that different filters can have considerable
impact on system performance [21, 22]. Below, we provide some justifications and discuss
considerations on filter selection:

Symmetry. For texture segmentation, accuracy in texture boundary detection is
crucially important for reliable performance. In this application, filters with symmetry
or antisymmetry are clearly favored. Such filters have a linear phase response. The
delay (shift) caused by such phase factors is predictable. Alternatively, filters with
non-linear phase may introduce complicated distortions. Moreover, symmetric or anti-
symmetric filters are also useful in alleviating boundary effects by simple methods of
mirror extension.

Frequency characteristics. For wavelet packet or recursive filter bank decomposi-
tions, Quadrature Mirror Filters (QMF) are preferred because each filter generated
within the filter bank will not exhibit a multi-lobe property. Unfortunately, a com-
pactly supported QMF cannot be symmetric or anti-symmetric [31]. For multi-channel
decompositions using a QMF, all filters are generated from a prototype filter that sat-
isfies the property:

IHo(W)|2 + IHo(w + 7r)2 = 1 (9)
Ho(0) = 1, Ho(7) =0.















Edge detection performance. Two of our proposed feature extraction schemes is
based on edge detection. We considered two approaches to detect edges: identifying the
zero crossings and local maxima. J.Canny's paper [37] provides valuable insights into
the desired properties of a filter used for edge detection. The basic requirement is that a
detection scheme should exhibit good performance under noisy conditions. To achieve
this, a signal is subjected to low-pass smoothing before a derivative operation (for edge
detection by local maxima) or a second derivative operation (for edge detection by zero
crossing). The two operations cascaded are equivalent to a multiplication operation in
the frequency domain. Note that since -- -w, -- i 2, the filters corresponding to
the two approaches can be written as:

F(w) = wL(w), for local maxima detection (10)

and
F(w) = W2L(w), for zero crossing detection (11)
where L(w) is the frequency response of a low-pass filter and thus should have a non-
zero value at = 0. In other words, for local maxima detection, F(w) should have an
order 1 zero at = 0, or F(w) should have an order 2 zero at = 0 if designed for
optimal zero crossing detection. We used this criterion to examine candidate filters.
Since the zero-crossing detector has a higher order of zero at = 0, it exhibits sharper
out-band attenuation, and thus better frequency separation.

Uncertainty factor. In order to discriminate small frequency differences among
channels, we would like each channel (band) to have a flat in-band frequency response
and sharp off-band attenuation. But, such a filter may have a longer support in time
domain, and thus offer less time domain separation. Such contradictory construction
requirements are well demonstrated by the Heisenberg uncertainty principle. Please
see Wilson et al [34] [35] for a further discussion on how the selection of frequency
localization can effect the performance of several image processing applications.

In this investigation, we selected Lemarie-Battle wavelets, which are symmetric and
quadrature mirror filters (QMF). The high-pass filter Go(w) is obtained by frequency shifting
of Ho(w) by

go(n) = (-l)tho(n), or, (12)
Go( ) = Ho(w + 7).

In this way, both low-pass and high-pass filters have zero-phase, and thus no space (time)
shifting will exist after processing (analysis). This construction corresponds to a non-
orthogonal wavelet transform, and only retains perfect decomposition-reconstruction without
downsampling between levels (loose frames). In this application, however, the loss of orthog-
onality is unimportant, as the transform coefficients are not used directly as features. Rather




























(a) (b)


Figure 4: Cover of the frequency line using a Lemarie filter of order 1 and (a) DWF (b)
DWPF.

they are processed by a nonlinear operation and therefore do not comprise an orthogonal
feature set.
For algorithms involving edge detection, we selected an order 1 Lemarie filter, and applied
zero-crossing detection on each channel output. The frequency response for such a filter can
be explicitly written as:


H(w) = co' 2 + cos3)
2 1 t+ 2 cOS2 sW

G(w) = H(w + w) = sin2 ) ( 2 cos (14)

and satisfies the order requirement at w = 0. Figure 4 shows the Lemarie order 1 filter used
in both DWF and DWPF methods of analysis. Figure 5 shows the order 1 scaling function
and associated Lemarie wavelet.
Feature extraction is a crucial step in accomplishing reliable segmentation. A 'good'
feature (representation) should be consistent among the pixels within the same class, while
as disparate as possible between classes for reliable classification. This means that it should
reflect some global view while keeping some discrimination capability at the pixel level.
Therefore, the problem in general sense, is one of time(spatial)-frequency analysis, and a
natural application for methods of wavelet analysis.
Adaptivity means that a feature extraction scheme should be able to accommodate a
large variety of diverse inputs. It should obviously exclude any fixed windowing scheme,
since any such method is hardly adaptive. We next present an efficient representation by
extending the concepts of a channel envelope to construct a feature set for reliable texture




























(a) (b)


Figure 5: Lemarie L1: (a) Scaling function (b) Wavelet

segmentation.
In an earlier study, envelopes obtained from channel outputs were used by A.Bovik [17]
for texture analysis and segmentation in the setting of Gabor multi-channel filtering. The 2-
D complex Gabor filters used by Bovik approximated a pair of conjugate filters. In addition,
the approach of squaring followed by iterative low-pass filtering used by Unser [22] can also
be interpreted as envelope detection. However, the method of iterative low-pass filtering is
not adaptive, and will blur boundaries easily.
We investigated the following envelope-based feature extraction schemes for real wavelet-
based multi-channel methods of analysis. For sake of clarity, we first present each algorithm
for the 1-D case.

1. Envelope estimation by zero crossing. Clearly, edge-based representations are
adaptive. In this method, the maximum value between two adjacent zero-crossings is
found, and assigned to all points within the interval. The technique is described below
in pseudo-code:
Envelope_ZC(x[1: N])
begin
i:= 1;
while ( i < N )
begin
k := i+1;
max = fabs (x[i]);
Search by advancing index k while keep the maximum value
encountered until next zero-crossing point is found;






















0 100 200 300 400 500
1 |-----------------------------------------------

I0 100 200 300 400 500
0




0 100 200 300 400 500


0 100 200 300 400 500


0 100 200 300 400 500

01
0 100 200 300 400 500


Figure 6: Envelope estimation via zero crossings using an L1 filter.
(solid line: envelope; dot line: channel output)
Row 1: input signal; Row 2-4: wavelet coefficients for levels 1 to 3;
Row 5: DC coefficients, level 3.


Assign value max to all indexes from i to k;
Update i := k;
end
end Envelope_ZC;

Figure 6 shows an example using the L1 filter described earlier for a three-level DWF
decomposition. Obviously, accuracy of the zero crossing detection is critical for this
approach to be successful. This motivated the use of a QMF filter with reliable per-
formance in the detection of zero crossing features. Computationally, the algorithm is
very simple and efficient.

2. Envelope estimation by local extrema. This approach is similar to the method
of zero-crossing detection. Local extrema points are first identified. The value of the
left local extrema point is assigned to all points in the interval between two adjacent
extrema.

3. Envelope detection by Hilbert transform. For a narrowband bandpass signal, its
envelope can be computed by a corresponding analytical signal. For a signal x(t), the
analytic signal is defined by:
x(t) = x(t) + jx(t) (15)














where x(t) is the Hilbert transform of x(t):

x(t) = dI. (16)
7 J-oo t T
The frequency characteristic of the Hilbert transform is expressed by:

H(w) >=0 (17)
j I otherwise.

Therefore, the Fourier transform of the analytical signal x(t) is:

X( 2X(w) >=0
X(w) { > (18)
0 otherwise.

Although the frequency characteristic corresponding to a noncausal system cannot be
exactly realized in practice, for a DWT implemented by FFT, the analytic signal can
be approximately computed by setting the FFT values of the filter within the negative
frequency range to zero. The envelope of the original signal x(t) is then simply the
modulus of the analytic signal x(t):

Env[x(t)] = x(t) (19)

Figure 7 shows a example using such an approach. For implementations using convo-
lution, it is necessary to design an approximate FIR Hilbert transformer. There are
many DSP software packages available to assist in this design. But, for a short length
FIR Hilbert transformer, the frequency characteristic at near 0 and 7 will not be sat-
isfactory, and the computational cost will be more than the method of zero-crossing's
detection.
We now extend the above algorithms above for the analysis of two-dimensional image
signals. A two-dimensional analytic signal can be obtained by setting the mirror half plane
to zero. That is, for a 2-D signal f(x, y), the Fourier transform of the analytic signal f(x, y)
is:

S2F(Wx,, ) Wx >= 0
F(,) { F 0 otherwise
or,
2 F,, 2F(W,) y >=0
F(,) { 0 otherwise. (20)

For our 2-D filters, the equivalent complex quadrature filters exhibit the frequency response
shown in Figure 8. This property means the envelope of a 2-D signal can be computed using
the 1-D algorithms described earlier:
































0 100 200 300 400 500







0 100 200 300 400 500
10




J h77.7777777
0 s 1. : x ^ . ..*; :. .: . I I
0 100 200 300 400 500
2


0 100 200 300 400 500


5
0 100 200 300 400 500



0 100 200 300 400 500


Figure 7: Envelope detection via Hilbert transform using an L1 filter.


fy fy


fy fy


Figure 8: Frequency response of equivalent complex quadrature filters (level 1). The diagonal
shadowed areas identify the zeroed half planes. (a) wv, (b) wh, (c) wd, (d) dc components.



























(a) (b)


Figure 9: Envelope representations for horizontal component at level 2. (a) Original
256 x 256, 8-bit texture image. (b) Envelope coefficients obtained from zero-crossing based
algorithm. (c) Envelope coefficients obtained from the Hilbert transform based algorithm.

Edge-based algorithms. The idea is to apply the 1-D algorithm to each row or each
column depending on which vector is processed by highpass filtering.
Envelope_2D_ZC(wh, wv, wd)
{ wh, wv, wd are all [ : N][1 : N] array }
{ wh: horizontal components obtained by highpass filtering on columns and lowpass filtering
on rows. wv: vertical component obtained by lowpass filtering on columns and highpass
filtering on rows. wd: diagonal component obtained by highpass filtering on columns and
highpass filtering on rows. }
begin
For wh, apply Envelope_ZC column-wise;
For wv, apply Envelope_ZC row-wise;
For wd, apply Envelope_ZC column-wise;
end Envelope_2D_ZC;

Hilbert-transform. In this case, the procedure is similar to the edge-based method pre-
sented above, where the 1-D Hilbert-transform replaces the Envelope_ZC algorithm.

Figure 9 displays the envelopes extracted by the two methods above. The results obtained
by the Envelope_2D_ZC algorithm kept the boundary sharp and the in-class region smooth.
The 1-D Hilbert transformer was an FIR (63 tag) filter designed using MONARCH DSP
software for a convolution based implementation of the DWF.














At the end of the feature extraction process, we constructed feature vectors for every
pixel (point) according to the following definitions:

Vit {{1, ,,/ }I
{, = { ,f7j o
where the ;,- represents the envelope value at pixel (i,j) for the horizontal component
at level 1 of a DWF decomposition, and sk,, denotes the envelope value of pixel (i,j) for
the kth component at level L 1 in a DWPF tree.


4 Experimental results for multi-channel texture seg-
mentation

Segmentation algorithms accept as input a set of features and output a consistent labeling
for each pixel. Fundamentally, this can be considered a multi-dimensional data clustering
problem. As pointed out by Haralick [39], at present no general algorithms exist for this
problem. Clustering algorithms that have been previously used for texture segmentation can
be divided into two categories: supervised segmentation and unsupervised segmentation.
For practical applications, unsupervised segmentation is desirable and easy to test. It
is particularly useful in those cases where testing samples are difficult to prepare (making
supervised segmentation infeasible). Since our present work emphasized feature extraction
(representation), we used a traditional K-MEANS clustering algorithm [40] [41] [42]. An
overview of the algorithm is sketched below:

Algorithm K-MEANS(x[1 : N, 1: M],NC)
x[1 : N, 1 : M]: array of structure containing vector and label fields.
NC: number of classes.
begin
begin (Initialization)
Scan the representation matrix x in raster order, and assign a label to
every pixel in x by modulo NC.
Compute the class center {Ck 0 vector for each class k.
end
repeat
Rescan the representation matrix x, and assign pixel (i,j) to the
class k if the Euclidean distance between the pixel and the class
















Table 1: Boundary accuracy for multi-channel segmentation.


Test image Maximum ABE Average ABE f
T1 4.0 1.4 1.0
T2 9.0 2.7 2.1
T4 4.0 1.6 1.1
T5 7.0 1.5 1.6

ABE: Absolute Boundary Error (in pixels).


center Ck is closest.
Update the class centers {Ck} by recomputing their mean vectors.
until no change in labeling occurs.
end K-MEANS;

This simple algorithm labels each pixel independently and did no take into account the
high correlation between neighboring pixels. Clearly, a more sophisticated algorithm should
incorporate some neighborhood constraint into the segmentation process, such as relaxation
labeling. For simplicity, we used median filtering in our preliminary experiments to simulate
the benefit of a local constraint. In particular, we applied a 9 x 9 median filtering to each
initial segmentation.
To test our segmentation algorithm, we carried out experiments using two distinct families
of texture samples:

Natural textures. We used textures obtained from the Brodatz album [43]. Each
testing sample was histogram equalized so that a segmentation result based only on
first order statistics was not possible. Experimental results are shown in Figures 10,
11, 12.

Synthetic textures. We also tested the performance of our algorithm on several
synthetic images of texture. Figure 13 shows a segmentation result on a Gaussian
low-pass texture image [1], and Figure 14 shows a segmentation result on a filtered
impulse noise (FIN) texture image [1]

The accuracy of our segmentation results are summarized in Table 1. In general, we
found that envelope representations obtained from DWPF analysis outperformed similar
representations computed from standard DWF. Figure 15 compares segmentations obtained
from DWF and DWPF analysis alone without benefit of a local constraint, for test image
T4.












































(c) (d)

Figure 10: Multi-channel segmentation result No.l: (a) Test image T1 (256 x 256, 8-bit) con-
sists of D68, wood grain and D17, herringborn weave. (b) Initial segmentation, 6 level DWF.
(c) Boundary obtained from Figure (d) overlayed on original image T1. (d) Segmentation
after 9 x 9 median filtering.

















































(c) (d)

Figure 11: Multi-channel segmentation result No.2: (a) Test image T2 (256 x 256, 8-bit) con-
sists of D24, pressed calf leather and D29, beach sand. (b) Initial segmentation, 6 level DWF.
(c) Boundary obtained from Figure (d) overlayed on original image T2. (d) Segmentation
after 9 x 9 median filtering.


If











































(c) (d)

Figure 12: Multi-channel segmentation result No.3: (a) Test image T3 (512 x 512, 8-bit)
consists of D24, D29, D68 and D17. (b) Initial segmentation, 5 level DWF. (c) Boundary
obtained from Figure (d) overlayed on original image T3. (d) Segmentation after 9 x 9
median filtering.





























(a) (b)











(c) (d)

Figure 13: Multi-channel segmentation result No.4: (a) Test image T4 (256 x 256, 8-bit):
Gaussian LP, left: isotropic Fc = 0, S, = 60; right: non-isotropic Fc = 0, S, = 60, 0o =
0, Bo = 0.175. (b) Initial segmentation, 4 level DWPF. (c) Boundary obtained from Figure
(d) overlayed on original image T4. (d) Segmentation after 9 x 9 median filtering.





























(a) (b)











(c) (d)

Figure 14: Multi-channel segmentation result No.5: (a) Test image T5 (256 x 256, 8-bit):
Filtered impulse noise, left: non-isotropic T = 0.15, S, = 1.0, Sy = 1.5; right: non-isotropic
T = 0.15, S, = 2.0, Sy = 1.0. (b) Initial segmentation, 4 level DWPF. (c) Boundary obtained
from Figure (d) overlayed on original image T5. (d) Segmentation after 9 x 9 median filtering.






























Figure 15: Segmentation results obtained from DWPF and DWF representations, without
local constraint: (a) Initial 4 level DWPF segmentation of image T4. (b) Initial 4 level DWF
segmentation of image T4.


5 Conclusions

In this paper, we present a multi-channel texture segmentation scheme based on wavelet rep-
resentations. The efficacy of several variations including the discrete wavelet frame (DWF)
and discrete wavelet packet frame (DWPF) were investigated and compared. In general,
we found that envelope representations obtained from DWPF analysis outperformed similar
representations computed from DWF.
We describe a method of extracting multi-channel features for texture representation. Our
approach to feature extraction is adaptive in that no windowing is used. Several approaches
for one-dimensional envelope detection were presented and extended for two-dimensional
analysis. The performance of each algorithm for texture representation was evaluated. We
observed that the algorithm based on zero-crossing detection performed better than the
Hilbert transform algorithm. Moreover, the zero-crossing based algorithm naturally incor-
porated edge detection into feature extraction, and was also computationally efficient. Based
on the performance of zero-crossing detection and the need for accurate of texture bound-
aries, we chose a symmetric quadrature mirror filter for zero-crossing detection.
The focus of this paper was on feature extraction for texture segmentation. In our prelim-
inary experiments, we used a K-means clustering algorithm with a rather coarse initialization
method. Nevertheless, our experimental results exhibited reliable performance for several
distinct texture types including macro-texture and micro-textures. In the near further we
plan to develop a signal-adaptive algorithm using DWPF analysis, and try to fully utilize
multiresolution representations to increase reliability and boundary accuracy.















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