Supervised Texture Segmentation by Maximising ... - Semantic Scholar

Computer Science Department of The University of Auckland CITR at Tamaki Campus (http://www.citr.auckland.ac.nz)

CITR-TR-99

September 2001

Supervised Texture Segmentation by Maximising Conditional Likelihood Georgy Gimel'farb1

Abstract Supervised segmentation of piecewise-homogeneous image textures using a modified conditional Gibbs model with multiple pairwise pixel interactions is considered. The modification takes into account that interregion interactions are usually different for the training sample and test images. Parameters of the model learned from a given training sample include a characteristic pixel neighbourhood specifying the interaction structure and Gibbs potentials giving quantitative strengths of the pixelwise and pairwise interactions. The segmentation is performed by approaching the maximum conditional likelihood of the desired region map provided that the training and test textures have similar conditional signal statistics for the chosen pixel neighbourhood. Experiments show that such approach is more efficient for regular textures described by different characteristic long-range interactions than for stochastic textures with overlapping close-range neighbourhoods.

1

Center for Image Technology and Robotics Tamaki Campus, The University of Auckland, Auckland, New Zealand. [email protected]

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Supervised Texture Segmentation by Maximising Conditional Likelihood Georgy Gimel'farb Centre for Image Technology and Robotics Department of Computer Science Tamaki Campus, University of Auckland Private Bag 92019, Auckland 1, New Zealand [email protected]

WWW home page: http://www.cs.auckland.ac.nz/~georgy

Abstract. Supervised segmentation of piecewise-homogeneous image

textures using a modi ed conditional Gibbs model with multiple pairwise pixel interactions is considered. The modi cation takes into account that inter-region interactions are usually dierent for the training sample and test images. Parameters of the model learned from a given training sample include a characteristic pixel neighbourhood specifying the interaction structure and Gibbs potentials giving quantitative strengths of the pixelwise and pairwise interactions. The segmentation is performed by approaching the maximum conditional likelihood of the desired region map provided that the training and test textures have similar conditional signal statistics for the chosen pixel neighbourhood. Experiments show that such approach is more ecient for regular textures described by different characteristic long-range interactions than for stochastic textures with overlapping close-range neighbourhoods.

1 Introduction Supervised segmentation is intended to partition a spatially inhomogeneous texture into regions of homogeneous textures after learning descriptions of these latter. Generally, neither textures nor homogeneity have universal formal definitions, so that each statement of the segmentation problem introduces some particular texture descriptions and criteria of homogeneity. For last two decades, one of most popular approaches is to model textures as samples of a discrete Markov random eld on an arithmetic lattice with a joint Gibbs probability distribution of signals (grey levels, or colours) in the pixels [4{ 6, 10, 11]. The Gibbs model relates the joint distribution that globally describes the images to a geometric structure and quantitative strengths of local pixel interactions. Typically only pixelwise and pairwise pixel interactions are taken into account, and the interaction structure is speci ed by a spatially invariant subset of pixels interacting with each individual pixel. The subset forms the pixel neighbourhood. The interaction strengths are speci ed by Gibbs potential functions that depend on signals in the pixel or in the interacting pixel pair.

In this case the texture homogeneity is de ned in terms of spatial invariance of certain conditional probability distributions, and a homogeneous texture has its speci c spatially invariant interaction structure and potentials. The supervised segmentation has to estimate these model parameters from a given training sample, that is, from a piecewise-homogeneous image with the known map of homogeneous regions or from a set of separate single-region homogeneous textures. Then the parameters are used for taking the optimal statistical decision about the region map of a test image that combines the same homogeneous textures. This approach assumes that the parameters estimated from the training sample are typical for all the images to be segmented. This paper considers the supervised segmentation using the conditional Gibbs model of piecewise-homogeneous textures proposed in [7, 8]. Here, the model is modi ed more fully re ect the inter-region relations that usually are quite dierent in the training and test cases. Previously, the two-pass (initial and nal) segmentation had been introduced to implicitly take account of the dierences between the training and test inter-region signal statistics. The modi ed model involves more natural inter-region interactions so that the segmentation can now be achieved in a single pass. The segmentation process approximates the maximum conditional likelihood of a desired region map providing that the image to be segmented and the training sample have the same or closely similar pixelwise and characteristic pairwise interactions in terms of particular conditional signal statistics. We investigate how precise such a segmentation is for dierent texture types, in particular, for stochastic and regular textures eciently described by the Gibbs models of homogeneous textures [8, 9]. The paper is organised as follows. Section 2 describes in brief the modi ed conditional Gibbs model of piecewise-homogeneous textures and shows how the Controllable Simulated Annealing (CSA) introduced in [7, 8] can be used for maximising the conditional likelihood of the desired region map. Section 3 presents and discusses results of the supervised segmentation of typical piecewisehomogeneous stochastic and regular textures. The concluding remarks are given in Section 4.

2 Gibbs model of piecewise-homogeneous textures 2.1 Basic notation Let R = [(m; n) : m = 1; : : : ; M ; n = 1; : : : ; N ] be a nite aritmetic lattice with M
N pixels. Let Q = f0; : : : ; Q , 1g be a nite set of grey levels. Let K = f1; : : : ; K g be a nite set of region labels. Let g = [gi : i 2 R; gi 2 Q] and l = [li : i 2 R; li 2 K] denote a piecewise-homogeneous digital greyscale texture and its region map, respectively, so that each pixel i = (m; n) 2 R is represented by its grey level gi and region label li . The spatially invariant geometric structure of pairwise pixel interactions over the lattice is speci ed by a pixel neighbourhood A. The neighbourhood points

up a subset of pixels (neighbours) f(i + a) : a 2 A; i + a 2 Rg having each a pairwise interaction with the pixel i 2 R. Each oset a = (; ) 2 A de nes a family of interacting pixel pairs, or cliques of the neighbourhood graph [2]. A quantitative strength of pixel interactions in the clique family Ca = f(i; j ) : i; j 2 R; i , j = ag is given by a Gibbs potential function of grey levels and region labels. The conditional Gibbs model in [7, 8] speci es the joint probability distribution of the region maps for a given greyscale image in terms of the characteristic neighbourhood A and the potential V = [Vp ; Va : a 2 A]. Generally, the potential of pixelwise interactions Vpix = [Vp (kjq) : k 2 K; q 2 Q] depends on a grey value q and region label k in a pixel. The potential of pairwise interactions Va = [Va (k; k0jq; q0) : (k; k0) 2 K2; (q; q0) 2 Q2] depends on region label and grey level co-occurrences (gi ; li ; gj ; lj ) in a clique (i; j ) 2 Ca . We assume for simplicity that the potential of pairwise interactions depends only on the grey level dierence d = gi , gj ; d 2 D = f,Q +1; : : :; 0; : : : ; Q , 1g. The inter-region grey level dierences in the cliques of the same family are quite arbitrary for the various region maps of a piecewise-homogeneous texture. Therefore, the potentials should depend actually only on the region label coincidences  = (li , lj ) 2 f0; 1g so that Va (k; k0 jq; q0 )  Va; (kjq , q0 ) where  = 0 for the inter-region and  = 1 for the intra-region pixel interactions. Obviously, the intra-region potentials Va;1 (kjd) depend both on k and d. But the inter-region potentials Va;0 (kjd) actually describe only the region map model and should be independent of region labels and grey level dierences: Va;0 (kjd)  Va;0 . In the original model [7, 8] both the intra- and inter-region potentials depend on k and d because the inter-region statistics is assumed to be similar for the training and test images. But in most cases this assumption does not hold. For the xed neighbourhood A and potentials V, the modi ed conditional Gibbs model of region maps l, given a greyscale texture g, is as follows: !

X Pr(ljg; V; A) = Z 1 exp Ep (ljg; Vp ) + Ea (ljg; Va ) g ;V ;A a2A

(1)

where Zg;V;A is the normalising factor, Ep (ljg; Vp ) is the total energy of the pixelwise interactions:

Ep (ljg; Vp ) =

P

i2R

Vp (li jgi ) = jRj

P

q 2Q

Fp (qjg)

P

k 2K

!

Vp (kjq)Fp (kjq; l; g)

(2)

and Ea (ljg; Va ) is the total energy of the pairwise pixel interactions over the clique family Ca :

Ea (ljg; Va ) =

P

i;j )2C a

(

Va;(li ,lj ) (li jgi , gj )

= jRja Va;0 Fa;0 (l) +

P

d2D

Fa (djg)

P

k 2K

(3)

Va;1 (kjd)Fa;1 (kjd; l; g)

Here, Fp (qjg) and Fp (kjq; l; g) denote the relative frequency of the grey level q in the image g and of the the region label k in the region map l over the grey level q in the image g, respectively, and Fa (djg), Fa;0 (l), and Fa;1 (kjd; l; g) denote the relative frequency of the grey level dierence d over the clique family Ca in the image g, of the inter-region label coincidences in the region map l, and of the intra-region label coincidences for the region k in the region map l over the grey level dierence d in the clique family Ca for the image g, respectively. The factor a = jjCRajj gives the relative size of the clique family Ca .

2.2 Learning the model parameters As shown in [8], the potentials for a given training pair (l ; g ) have a simple rst approximation of the maximum likelihood estimate. Because of the uni ed inter-region potential values, this approximation is now as follows:

8k 2 K; q 2 Q; d 2 D V (kjq) =  F (qjg ) (F (kjq; l; g ) , ) Va; =  a (Fa; (l ) ,  ) Va; (kjd) =  a Fa (djg ) (Fa; (kjd; l ; g ) ,  ) [0] p

[0]

[0] 0

[0]

[0] 1

[0]

0

0

1

(4)

1

where  = jK1 j , 0 = 1 , , and 1 = 2 are the marginal probabilities of region labels and their coincidences, respectively, for the independent random eld (IRF) with the equiprobable region labels. The initial scaling factor [0] in Eq. (4) is computed as P

ep (l ; g ) + (ea;0(l ) + ea;1 (l ; g )) [0]  = e (l ; g ) + Pa2A(e (l ) + e (l ; g ) ) p a;0 0 a;1 1 a2A

(5)

where ep (l ; g ), ea;0 (l ), and ea;1 (l ; g ) are the normalised rst approximations of the Gibbs energies of pixelwise and pairwise pixel interactions:

ep(l ; g ) =

P

q2Q

Fp2 (qjg )

P

Fp (kjq; l ; g ) (Fp (kjq; l ; g ) , )

k 2k ea;0 (l) = 2a Fa;0 (l ) (Fa;0 (l ) , 0 ) P P ea;1 (l; g ) = 2a Fa2 (djg ) Fa;1 (kjd; l ; g ) (Fa;1 (kjd; l ; g ) , 1 ) d2D k2K

(6)

and = (1 , ) and  =  (1 , );  = 0; 1, are the variances of the region label frequencies for the IRF. As in [7, 8], most characteristic interaction structure is recovered in this paper by choosing the clique families with the top values of the total energy of pairwise pixel interactions: ea (l ; g ) = ea;0(l ) + ea;1 (l ; g )

2.3 Supervised segmentation When the model of Eq. (1) is used for segmenting piecewise-homogeneous textures, we assume that the rst-order and second-order statistics of the images g to be segmented and the desired region maps l in Eqs. (2) and (3) are similar to those of the training pair (l ; g ). This assumption is crucial because the likelihood maximisation algorithm we use for segmentation actually tries to minimise a (probabilistic) distance between the above statistics for the training pair and segmented image [8]. Because the model of Eq. (1) belongs to the exponential family of distributions, the log-likelihood function L(Vjl ; g ) = log Pr(l jg ; V; A) with a xed neighbourhood A is unimodal [1] with respect to the potential V. The maximum is approached by the stochastic-approximation-based Controllable Simulated Annealing (CSA) [8] that starts from an arbitrary initial region map l[0], e.g., from a sample of the IRF, with the initial Gibbs potentials of Eq. (4). At each step t, the potentials are modi ed as to approach the conditional pixelwise and pairwise statistics for the training sample (l ; g ) with the like statistics for the current simulated pair (l[t] ; g ). The CSA is easily adapted for approaching the maximum of an arbitrary log-likelihood L(Vjl; g), providing that the pair (l; g) has the same conditional statistics as the training pair. The only change with respect to the maximisation of L(Vjl ; g ) is that the training image g is replaced with the test image g for generating the successive maps l[t] at each step t. Using the signal statistics for the training sample (l ; g ) and for the test image g with the region map l[t] generated by stochastic relaxation with the current potential V[t] , the resulting process modi es the potential values as follows: 8k 2 K; q 2 Q; d 2 D,; a 2 A
[t+1] Vp (kjq) = Vp[t] (kjq) + t Fp (qjg) Fp (kjq; l ; g ) , Fp (kjq; l[t] ; g) ,
(7) Va;[t0+1] = Va;[t1] + t a Fa;0 (l ) , Fa;0 (l[t]) ,
Va;[t1+1] (kjd) = Va;[t1] (kjd) + t a Fa (djg) Fa;1 (kjd; l ; g ) , Fa;1 (kjd; l[t] ; g)

3 Experimental results 3.1 Segmenting an arbitrary training sample The best results of the above approach should be expected for segmenting just the training image. For instance, the training sample in Figure 1,a{b, contains ve arbitrary chosen but spatially homogeneous natural and arti cial textures. In this case the segmentation results in the region maps having from 17.12% to 0.0% of errors when the model of Eq. (1) contains, respectively, from one to six clique families with the top total energies (Figure 1,c{h, and Table 1). In these experiments the search window for choosing the characteristic interaction structures has 60 clique families with the short-range osets (j j  5; jj 

a

b

c

d

e

f

g

h

Fig. 1. Training ve-region texture (a), its ideal region map (b), and segmentation maps obtained with 1 { 6 clique families (c { h), respectively.

5). All the segmentation maps are obtained after 300 steps of the CSA with the scaling factor in Eq. (7) that is changing as follows: [t] = [0] 1 + 0:1001
t The same number of steps and the same schedule for changing [t] is used in all the experiments below.

Table 1. Characteristic clique families included successively into the model of Eq. (1), their Gibbs energies, and the relative segmentation errors " in Figure 1,c{h. A

1 2 3 4 5 6 a = (;  ) [1,0] [-1,1] [0,1] [-2,2] [-3,3] [-4,4] ea (l ; g ) 478.4 440.3 427.7 407.4 400.4 395.4 ", % 17.12 0.54 0.15 0.27 0.0 0.0 j

j

3.2 Segmenting collages of stochastic and regular textures

To investigate the above segmentation in more detail, we use two types of homogeneous textures, namely, stochastic and regular textures that can be eciently simulated by the Gibbs models with multiple pairwise pixel interactions [8, 9]. Figures 2 and 3 present collages of stochastic textures D4, D9, D29, and D57 and regular textures D1, D6, D34, and D101 from [3].

a

b

c

d

e

f

g

h

i

j

k

l

Fig. 2. Four-region training (a,e) and test collages (b{d,f{h) of stochastic textures D4, D9, D29, D57 (a{d) and regular textures D1, D6, D34, D101 (e{h) with their ideal region maps (i{l).

The former four textures have the characteristic short-range interactions whereas the latter ones have mostly the characteristic long-range interactions. In both cases three textures from each group (the stochastic textures D4, D9, and D29 and the regular textures D1, D6, and D34) possess similar statistics of the close-range pairwise pixel interactions (in terms of the relative frequency distributions of grey level dierences). Below we use the collages in Figures 2,a and 2,e with the same region map in Figure 2,i as the training samples for each group of the textures. The search window for choosing the characteristic interaction structure contains 3240 clique families with the short- and long-range osets (j j  40; jj  40).

Piecewise-homogeneous stochastic textures. In this case, relative errors

of segmenting the training image using four, eight, or 16 characteristic clique families with the top-rank Gibbs energies are, respectively, 21.36%, 16.44%, and 13.88%. Here, the ranking of the clique families by their Gibbs energies results in only the close-range interaction structures. The corresponding 16 clique families in terms of their osets a = (; ) are shown in Table 2. The relative errors of segmenting the training sample are slowly decreasing with the neighbourhood size (for instance, 10.15% for jAj = 36). Segmentation errors for the test images in Figure 2,b{d and 3,a{d depend in a similar way

a

b

c

d

e

f

g

h

i

j

k

l

Fig. 3. Four-region test collages of stochastic textures D04, D09, D29, D57 (a{d) and regular textures D1, D6, D34, D101 (e{h) with their ideal region maps (i{l).

on the neighbourhood size so that most of the experiments below are conducted with the same xed characteristic neighbourhood of size jAj = 16. Segmentation of the test images yields larger error rates (23.39{32.43%) caused mostly by misclassi ed parts of the textures D4 and D9 (Figure 4). The main reason is that the stochastic textures D4, D9, and D29 have very similar signal statistics over the chosen characteristic short-range neighbourhood. Thus the individual regions produced by segmentation (Figures 6 and 7) dier from the ideal maps although they are quantitatively, in terms of the chi-square distances between the training and test statistics, and even visually quite homogeneous. Table 3 demonstrates how the segmentation separates the individual textures.

Table 2. Characteristic clique families with the top Gibbs energies ea(l ; g ) selected for the training four-region collage of

stochastic textures in Figures 2,a and 2,i. a = (;  ) ea (  ;  )

(1,0) (0,1) (1,1) (-1,1) (2,0) (2,1) (0,2) (-2,1)

l g 413.1 330.2 277.2 262.9 240.6 209.7 202.1 198.5

a = (;  ) ea (  ;  )

(1,2) (3,0) (-1,2) (3,1) (2,2) (0,3) (-3,1) (-2,2)

l g 189.8 182.3 182.1 169.7 168.4 167.1 162.4 159.9

a: 13.88%

b: 23.39%

c: 25.48%

d: 28.30%

a: 26.45%

b: 28.86%

c: 24.62%

d: 32.43%

Fig. 4. Segmentation of the collages of stochastic textures using the neighbourhood

of size 16; the top and the bottom region maps a{d are obtained for the collages in Figures 2,a{d, and 3,a{d, respectively. Black regions under each map indicate the segmentation errors.

Table 3. Segmentation results (in %) for the training and test collages of stochastic textures (the rows correspond to the ideal regions, and the columns show how many pixels of each ideal region are actually assigned to a particular texture).

Training collage Test collages D4 D9 D29 D57 D4 D9 D29 D57 D4 75.4 15.9 8.5 2.0 47.3{73.0 20.3{44.4 5.5{21.0 0.0{5.3 D9 11.6 82.6 4.2 1.6 6.3{31.7 58.3{73.8 4.9{30.4 0.4{27.2 D29 9.2 2.6 87.4 0.8 4.4{25.6 3.4{9.2 64.5{89.6 0.0{2.1 D57 0.4 0.2 0.3 99.0 0.1{1.0 0.0{9.4 0.0{2.7 89.2{99.6

D4

D9

D29

D57

a

b

c

d

Fig. 5. Homogeneous regions found by segmenting the training and test collages of stochastic textures (the top row of the maps a{d in Figure 4). D4 D9 D29

D57

a

b

Fig. 6. Homogeneous regions found by segmenting the test collages of stochastic textures (the bottom row of the maps in Figure 4,a,b).

D4

D9

D29

D57

c

d

Fig. 7. Homogeneous regions found by segmenting the test collages of stochastic textures (the bottom row of the maps in Figure 4,c,d).

These experiments demonstrate the basic diculty in segmenting stochastic textures by taking account of characteristic conditional pairwise statistics. The close-range characteristic neighbourhoods selected by ranking the total Gibbs energies for the clique families may not be adequate for separating these textures because the conditional signal statistics similar to the training ones can be obtained for regions that dier much from the ideal ones.

Piecewise-homogeneous regular textures In this case we can expect more ecient segmentation using the conditional model in Eq. (1) because simulations of these textures involve usually characteristic long-range interactions [9]. Actually, the relative error of segmenting the training collage in Figure 2,e is 2.67% with the same neighbourhood size of 16 although once again the top-rank total energies correspond to only the close-range interactions (Table 4).

Table 4. Characteristic clique families with the top Gibbs energies ea(l ; g ) selected for the training four-region collage of

regular textures in Figures 2,e and 2,i. a = (;  ) ea (  ;  )

(1,0) (0,1) (1,1) (-1,1) (2,0) (0,2) (2,1) (1,2)

l g 613.7 549.2 360.8 353.5 325.6 323.2 189.8 189.3

a = (;  ) ea (  ;  )

(0,2) (-2,1) (3,0) (0,3) (2,2) (3,1) (-2,2) (-3,1)

l g 183.3 182.1 173.0 159.9 66.8 64.5 60.1 59.0

Figure 8 shows results of segmenting the training collage and test collages of regular textures in Figures 2,e{h and 3,e{h, respectively, using the characteristic

neighbourhood of size 16. The relative errors for the test collages are 1.66{ 14.09%. The test collages in Figure 3,e{h result in less precise segmentation because of many small subregions in these textures that eect the collected conditional statistics. The individual homogeneous regions found by segmentation are shown in Figures 9 { 11, and Table 5 demonstrates the separation of these textures. Most of the errors are caused by the textures D6 and D34 with very similar uniform backgrounds resulting in close similarity between their conditional statistics of short-range grey level dierences.

e: 2.67%

f: 5.80%

g: 5.60%

h: 1.66%

e: 9.86%

f: 9.41%

g: 9.41%

h: 14.09%

Fig. 8. Segmentation of the training and test collages of regular textures using the

neighbourhood of size 16: the top and bottom region maps e{h are obtained for the collages in Figures 2,e{h, and 3,e{h, respectively. Black regions under each map indicate the segmentation errors.

Table 5. Segmentation results (in %) for the training and test collages of regular textures (the rows correspond to the ideal regions, and the columns show how many pixels of each ideal region are actually assigned to a particular texture).

Training collage Test collages D1 D6 D34 D101 D1 D6 D34 D101 D1 96.0 4.0 0.0 0.0 81.9{99.9 0.1{17.7 0.0{0.8 0.0{7.7 D6 1.6 97.0 1.4 0.0 1.0{16.5 75.7{97.6 0.0{17.9 0.0{5.0 D34 0.8 2.4 96.4 0.4 0.0{4.2 1.0{16.7 78.5{99.0 0.0{2.1 D101 0.1 0.0 0.0 99.9 0.0{0.4 0.0{0.3 0.0{1.6 97.7{99.9

The desired distinctions between the close-range conditional statistics may exist also for certain spatially inhomogeneous textures. For example, the collages of regular textures in Figure 12,a,c contain both the homogeneous regular textures D20, D55, D77 and the weakly inhomogeneous texture D36 from [3]. The error rate of segmenting the training image with the characteristic short-range neighbourhood of size 16 is 28.63%. When the size is extended to 36 then the error rates of segmenting the training and test collages are 5.15% and 16.71%, respectively. The resulting region maps for the neighbourhood of size 36 are shown in Figure 12,b,d. The main errors in this case are due to assigning small border parts of the textures D20 and D55 to the texture D77 having similar statistics of the close-range grey level dierences. D1

D6

D34

D101

e

f

Fig. 9. Homogeneous regions found by segmenting the training and test collages of regular textures (the top row of the maps in Figure 8,e,f).

D1

D6

D34

D101

g

h

Fig. 10. Homogeneous regions found by segmenting the test collages of regular textures (the top row of the maps in Figure 8,g,h). D1 D6

D34

D101

e

f

g

h

Fig. 11. Homogeneous regions found by segmenting the test collages of regular textures (the bottom row of the maps e{h in Figure 8).

a

b

c

d

b

D20

D36

D55

D77

d

Fig. 12. Region maps (b) and (d) and the corresponding homogeneous regions obtained by segmenting the training (a) and test (c) collages of regular textures D20, D36, D55, and D77, respectively.

4 Concluding remarks These and other our experiments show that the supervised segmentation by approaching the maximum conditional likelihood is ecient for textures with dierent characteristic interaction structures. But the overlapping close-range structures and similar pairwise signal statistics of individual homogeneous textures may result in a segmentation map that diers considerably from the ideal one although both the maps possess conditional signal statistics similar to the training ones and have visually homogeneous textured regions. The modi ed conditional Gibbs model allows to accelerate segmentation comparing to the previous two-stage scheme [7, 8] by obviating the need for the initial stage. This latter sets to zero the inter-region potentials in order to roughly approximate the desired homogeneous regions even though their inter-region signal statistics are dierent in the test and training samples. Then the initial (and usually quite \noisy") region map is re ned at the nal stage by using both the intra- and inter-region potentials. The one-stage process of Eq. (7) forms the nal region map starting directly from a sample of the IRF. Simultaneously the accuracy of segmentation is slightly improved in that the modi ed model yields smaller dierences between the error rates for the training and test images. For instance, the two-stage segmentation

in [8] results in the error rates of 2.6% and 29.0{35.2% for the training and test collages of the textures D3{D4{D5{D9 from [3], respectively, or 1.9% and 15.3{33.0% for the like collages of the textures D23{D24{D29{D34, and so forth. The one-stage segmentation based on the modi ed Gibbs model produces more predictable results for the test and training samples like 13.9% vs. 23.4{32.4% for the stochastic textures with very similar close-range pairwise signal statistics or 2.7% vs. 1.7{14.1% for the regular textures, respectively. Our experiments show that the choice of most characteristic pixel neighbourhoods should be based not only on partial Gibbs energies of pairwise pixel interactions but also on the accuracy of segmentation. If the top-rank Gibbs energies correspond mostly to the close-range neighbourhoods then these latter can be ecient only for segmenting the textures with suciently dierent close-range pairwise signal statistics of the homogeneous regions.

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