An Adaptive Approach for Texture Segmentation by Multi ... - CiteSeerX

An Adaptive Approach for Texture Segmentation by Multi-channel Wavelet Frames

1

Andrew Laine and Jian Fan

Center for Computer Vision and Visualization August, 1993 TR-93-025

1 This work was supported by the National Science Foundation under Grant No. IRI-9111375.

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. Algorithms for envelope detection based on edge detection and the Hilbert transform are presented. Analytic lters 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 synthetic textures are shown.

1

1 Introduction In the eld of computer vision, texture segmentation has been investigated by many researchers using a diversity of approaches. In general, each method consists of two phases: feature extraction and segmentation. Features for texture representation are of crucial importance for accomplishing segmentation[1]. 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 elds [9][10][11][12]. 2. Spatial/spatial-frequency 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 lters [16] [17][18] [19][20], wavelet packets [21], and wavelets [22]. In this paper, we describe a multi-channel approach, closely tied to wavelet multiresolution analysis. The existence of a multi-channel ltering model is evident in studies of the human visual system [23]. In fact, pychovisual experiments suggest the presence of a \biological spectrum analyzer" within the human visual system [24]. In an early paper by Bovik [17], complex Gabor lters were used, where texture features were computed by envelope and phase information extracted from two quadratic components of each channel output. In a related paper, A.K.Jain [19] chose real Gabor lters for analysis. Each ltered channel (output) was subjected to a nonlinear transformation, (t) = tanh(t). An average absolute derivation was then computed in terms of overlapping windows. Although Gabor lters are easy to design and have desirable properties including orientation selectivity and lter bandwidth, they are computationally inecient. In addition, a large number of channels are required in order to approximately cover the complete frequency plane. Recent developments in wavelet theory provide a promising alternative through multichannel lter banks that have several potential advantages over Gabor lters: (1) Wavelet lters cover exactly the complete frequency domain. (2) Fast algorithms are readily available to facilitate computation. (3) Wavelet based feature extraction may be more eciently 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 classi cation and achieved perfect classi cation for 550 samples obtained from 25 distinct natural textures. In addition, M.Unser [22] has reported promising results using wavelets for both classi cation 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 2

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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 brie y 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 lter 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 dierence lies in lter design, where wavelet lters are required to be regular [25]. In this study, we considered two methods of multiscale analysis:  Discrete wavelet transform (DWT) [25][27] [28], which corresponds to an octave-band lter bank.  Discrete wavelet packet transform (DWPT) [29][30], which corresponds to a general tree-structured lter 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 exibility 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 accomplishing 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 3

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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 between levels. Figure 3 shows a general DWPF as a binary tree for a three level decomposition. In each case above, the lters H (!) and G (!) at level l were generated as described in [25] [26] : l

l

H (! ) = H0 (2 ! ) G (! ) = G0 (2 ! ):

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Figure 3: Tree structure for wavelet packet frames and associated indexes. 4

Let S (!) be the Fourier transform of the input signal at channel k for level l, then l

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S2+1 (! ) = G (! )S (! ) S2+1+1 (! ) = H (! )S (! ): l

(3) (4)

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From the lter bank point of view, this is equivalent to a lter bank with channel lters

fF (!)j0  k  2 ? 1g, where F (!) is de ned recursively by the formula l

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F00(! ) = G0 (! ); F10(! ) = H0 (! ); F2+1 (! ) = G +1 (! )F (! ) = G0 (2 +1 ! )F (! ); F2+1+1(! ) = H +1 (! )F (! ) = H0 (2 +1 ! )F (! ): l

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For 2-D images, we simply use a tensor product extension for which the channel lters are written as F  (! ; ! ) = F (! )F (! ): (8) l

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3 Analytic lter selection and feature extraction Recent results on texture classi cation have shown that dierent lters can have considerable impact on system performance [21, 22]. Below, we provide some justi cations and discuss considerations on lter selection:  Symmetry. For texture segmentation, accuracy in texture boundary detection is crucially important for reliable performance. In this application, lters with symmetry or antisymmetry are clearly favored. Such lters have a linear phase response. The delay (shift) caused by such phase factors is predictable. Alternatively, lters with non-linear phase may introduce complicated distortions. Moreover, symmetric or antisymmetric lters are also useful in alleviating boundary eects by simple methods of mirror extension.  Frequency characteritics. For wavelet packet or recursive lter bank decompositions, Quadrature Mirror Filters (QMF) are preferred because each lter generated within the lter bank will not exhibit a multi-lobe property. Unfortunately, a compactly supported QMF cannot be symmetric or anti-symmetric [31]. For multi-channel decompositions using a QMF, all lters are generated from a prototype lter that satis es the property:

jH (!)j + jH (! + )j = 1 0

2

0

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H0 (0) = 1; H0 ( ) = 0:

5

(9)

 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 lter 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 2 the frequency domain. Note that since ! !, 2 ! ! , the lters corresponding to the two approaches can be written as: F (! ) = !L(! ); for local maxima detection (10) and F (! ) = ! L(! ); for zero crossing detection (11) where L(!) is the frequency response of a low-pass lter and thus should have a nonzero value at ! = 0. In other words, for local maxima detection, F (!) should have an order 1 zero at ! = 0, or F (!) should have an order 2 zero at ! = 0 if designed for optimal zero crossing detection. We used this criterion to examine candidate lters. 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 dierences among channels, we would like each channel (band) to have a at in-band frequency response and sharp o-band attenuation. But, such a lter may have a longer support in time domain, and thus oer 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 eect the performance of several image processing applications. In this investigation, we selected Lemarie-Battle wavelets, which are symmetric and quadrature mirror lters (QMF). The high-pass lter G (!) is obtained by frequency shifting of H (!) by g (n) = (?1) h (n); or; (12) G (! ) = H (! +  ): In this way, both low-pass and high-pass lters have zero-phase, and thus no space (time) shifting will exist after processing (analysis). This construction corresponds to a nonorthogonal wavelet transform, and only retains perfect decomposition-reconstruction without downsampling between levels (loose frames). In this application, however, the loss of orthogonality is unimportant, as the transform coecients are not used directly as features. Rather d

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(a) (b) Figure 4: Cover of the frequency line using a Lemarie lter 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 lter, and applied zero-crossing detection on each channel output. The frequency response for such a lter can be explicitly written as: s

2 + cos ! (13) 2 1 + 2 cos ! ! 2 ? cos ! G(! ) = H (! +  ) = sin (14) 2 1 + 2 cos ! and satis es the order requirement at ! = 0. Figure 4 shows the Lemarie order 1 lter 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 classi cation. This means that it should re ect 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 xed windowing scheme, since any such method is hardly adaptive. We next present an ecient representation by extending the concepts of a channel envelope to construct a feature set for reliable texture H (! ) = cos2



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(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 ltering. The 2D complex Gabor lters used by Bovik approximated a pair of conjugate lters. In addition, the approach of squaring followed by iterative low-pass ltering used by Unser [22] can also be interpreted as envelope detection. However, the method of iterative low-pass ltering is not adaptive, and will blur boundaries easily. We investigated the following envelope-based feature extraction schemes for real waveletbased multi-channel methods of analysis. For sake of clearity, we rst 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( [1 : ]) begin

x

N

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;

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Figure 6: Envelope estimation via zero crossings using an L1 lter. (solid line: envelope; dot line: channel output) Row 1: input signal; Row 2-4: wavelet coecients for levels 1 to 3; Row 5: DC coecients, 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 lter 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 lter with reliable performance in the detection of zero crossing features. Computationally, the algorithm is very simple and ecient. 2. Envelope estimation by local extrema. This approach is similar to the method of zero-crossing detection. Local extrema points are rst identi ed. 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 de ned by: x~(t) = x(t) + j x(t) (15) b

9

where x(t) is the Hilbert transform of x(t): 1 1 x() d: x(t) =

(16)

The frequency characteristic of the Hilbert transform is expressed by: ?j ; ! >= 0 H (! ) =

(17)

Therefore, the Fourier transform of the analytical signal x~(t) is: >= 0 X~ (! ) = 20X (! ) ;; !otherwise:

(18)

b

Z

 ?1 t ? 

b

(

j

; otherwise:

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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 lter 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)] = jx~(t)j : (19) Figure 7 shows a example using such an approach. For implementations using convolution, 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  will not be satisfactory, 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: 2F (! ; ! ) ; ! >= 0 F~ (! ; ! ) = 0 ; otherwise or, 2F (! ; ! ) ; ! >= 0 F~ (! ; ! ) = (20) 0 ; otherwise: (

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For our 2-D lters, the equivalent complex quadrature lters 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: 10

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

fy

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

11

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

Envelope 2D ZC(

)

wh; wv; wd

f wh; wv; wd are all [1 : N ][1 : N ] array g f wh: horizontal components obtained by highpass ltering on columns and lowpass ltering

on rows. wv: vertical component obtained by lowpass ltering on columns and highpass ltering on rows. wd: diagonal component obtained by highpass ltering on columns and highpass ltering on rows. g

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 presented 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) lter designed using MONARCH DSP software for a convolution based implementation of the DWF. 12

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

i;j

V

i;j





i;j

i;j



~ g j   ? ; d~ ? g; for DWF ~ ; wv ~ ; wd = ff wh = f s~ ?   L?1? g; for DWPF

l

L 1 0 k;i;j



k

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(2

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~ represents the envelope value at pixel (i; j ) for the horizontal component where the wh at level l of a DWF decomposition, and s~ ? denotes the envelope value of pixel (i; j ) for the kth component at level L ? 1 in a DWPF tree. l



L

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4 Experimental results for multi-channel texture segmentation 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 dicult 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( [1 : 1 : ], x

N;

M

NC

)

x[1 : N; 1 : M ]: array of structure containing vector and label elds. 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 fC~ j0  ?1 g by calculating the mean vector for each class k. k

k

end repeat

NC

Rescan the representation matrix x, and assign pixel (i; j ) to the class k if the Euclidean distance between the pixel and the class

13

Table 1: Boundary accuracy for multi-channel segmentation. Test image Maximum ABE Average ABE  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 C~ is closest. Update the class centers fC~ g 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 ltering in our preliminary experiments to simulate the bene t of a local constraint. In particular, we applied a 9  9 median ltering to each initial segmentation. To test our segmentation algorithm, we carried out experiments using two distinct families of texture samples: k

k

 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 rst 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 ltered 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 bene t of a local constraint, for test image T4. 14

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Figure 10: Multi-channel segmentation result No.1: (a) Test image T1 (256  256, 8-bit) consists 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  9 median ltering.

15

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(b)

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Figure 11: Multi-channel segmentation result No.2: (a) Test image T2 (256  256, 8-bit) consists 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  9 median ltering.

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Figure 12: Multi-channel segmentation result No.3: (a) Test image T3 (512  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  9 median ltering.

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Figure 13: Multi-channel segmentation result No.4: (a) Test image T4 (256  256, 8-bit): Gaussian LP, left: isotropic F = 0; S = 60; right: non-isotropic F = 0; S = 60;  = 0; B = 0:175. (b) Initial segmentation, 4 level DWPF. (c) Boundary obtained from Figure (d) overlayed on original image T4. (d) Segmentation after 9  9 median ltering. c

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Figure 14: Multi-channel segmentation result No.5: (a) Test image T5 (256  256, 8-bit): Filtered impulse noise, left: non-isotropic T = 0:15; S = 1:0; S = 1:5; right: non-isotropic T = 0:15; S = 2:0; S = 1:0. (b) Initial segmentation, 4 level DWPF. (c) Boundary obtained from Figure (d) overlayed on original image T5. (d) Segmentation after 99 median ltering. x

x

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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 representations. The ecacy 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-channelfeatures 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 incorporated edge detection into feature extraction, and was also computationally ecient. Based on the performance of zero-crossing detection and the need for accurate of texture boundaries, we chose a symmetric quadrature mirror lter for zero-crossing detection. The focus of this paper was on feature extraction for texture segmentation. In our preliminary 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 futher we plan to develop a signal-adaptive algorithm using DWPF analysis, and try to fully utilize multiresolution representations to increase reliability and boundary accuracy. 20

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