OPTICAL REVIEW Vol. 3, No. 3 (1996) 184-191

Stochastic Segmentation of Severely Degraded Images Using Gibbs Random Fields Dong-Woo KIM, Gee-Hyuk LEE and Soo-Yong KIM

Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Republic of Korea

(Received January 8, 1996; Accepted April 1, 1996)

This paper deals with segmentation of noisy images using Gibbs random field (GRF) with an emphasis on modeling of the region process. For noisy image segmentation using the multi-level logistic (MLL) model with the second-order neighborhood system, which is commonly used in image processing, the segmentation performance is degraded significantly in case of low signal to noise ratio. By comparison with the Ising model that explains the magnetic properties of ferromagnetic material, it is evident that the characteristics of the region process modeled using the MLL model with the second-order neighborhood system are different in nature from the expected characteristics of a region. To solve this problem we added the term of the magnetic energy associated with the magnetic field of a spin system (or image) to the energy function of GRF. Using the modified model for the region process, the result of image segmentation was improved and did not depend on the cooling schedule in simulated annealing.

Key words : Gibbs random field, multi-level logistic model, Ising model, ferromagnetism, simulated annealing

1. Introduction

Since the 1980's there has been much research on sto- chastic image modeling and processing using statistical techniques as computational power has improved. 1-~) Modeling images using stochastic models such as simulta- neous auto-regressive (SAR), Markov random field (MRF), and Gibbs random field (GRF), various problems in image processing including image restoration' 6) and segmenta- tion,2,7 9) and texture modelingZ,3,10 13) and segmenta- tion2,7,14 17) become problems of statistical estimation. Many studies were reported, especially on GRF which is mathematically more tractable than MRF. Various classes of GRF as image models were developed including the multi-level logistic (MLL) model, 2) autobinomial modeV 'H) and many others for different forms of energy function. These models are regarded as generalizations of the Ising model to multi-valued random fields and can be classified into region models and texture models according to the signs of model parameters.

A summary of this study is as follows. Firstly, we describe statistical behaviors of the Ising model and some difficulties in using the MLL model with the second-order neighborhood system which is commonly used in image processing. For image segmentation using this MLL model segmentation performance degrades significantly as signal to noise ratio (SNR) is decreased below 1.0. As images degrade severely, the function which is to be optimized in the statistical estimation procedure becomes more complicated. In this case satisfactory results cannot be obtained using dynamic programming (DP) or deter- ministic relaxation (DR), and it is effective to use simu- lated annealing (SA). Even with the latter, segmentation performance degrades significantly as the cooling schedule slows. All these difficulties are due to the fact that the MLL model as a region model does not have a regional feature.

Secondly, to solve the above problem we treated an image as a spin system consisting of magnetic dipoles placed at the sites of a rectangular lattice, as in the Ising model. Then we added an additional energy term repre- senting the magnetic energy associated with the magnetic field of a spin system (or image) into the energy function of GRF as well as the term of interaction energy between neighboring spins (or pixe!s ). The approximate form of the magnetic energy term used in this paper can be reduced to that of an interaction energy term among higher-order neighborhoods. In this respect a satisfactory region model cannot be obtained with the second-order neighborhood system. The advantages of the modified model can be summarized as follows. We were able to control the size of clusters in realizations of image from the region model by changing the weighting factor between the interaction energy and the magnetic energy. Segmentation perfor- mance was improved with the modified model, and the improvement was more outstanding with poorer images of lower SNR. Segmentation results did not depend on cooling schedules in the SA optimization procedure.

In the modified model, we presented the magnetic energy formula confined to a binary image case, that is, magnetic dipoles (or pixels) have only two values, spin up and down or +1 and -1 . In further study we will try to generalize this model to cover multi-level images.

This paper is organized as follows. In Sect. 2, we sum- marize some background material on techniques of statisti- cal image processing including the definitions of MRF and GRF. In Sect. 3, we outline some dimculties in using the MLL model with the second-order neighborhood system. Using the analogy with ferromagnetic materials, we show that the proper regional feature can be obtained by adding a magnetic energy term into the energy function of GRF. In Sect. 4, we report some experiments on some test images. Finally, some concluding remarks are given in Sect. 5.

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2. Image Model and Segmentation Algorithm Using GRF

In this section, we summarize some statistical models and methods used in this paper such as the definitions of MRF and GRF, MLL model, noisy image model, maxi- m u m a posteriori (MAP) segmentation algorithm, and Gibbs sampler. The detailed treatments of the above models and methods can be found in Geman and Geman, 1) Derin and his colleagues, 2'9) and Besag. 1s,19) 2.1 Gibbs Random Fields

In statistical image processing, images are modeled using two-dimensional (2-D) random fields (r.f.) and a particular image is considered to be a realization from the random fields. We consider a finite rectangular lattice L defined as

L={(i , j ) : 0-< i -<Nx-1, 0_<j~<N2-1} (1)

and a r.f. X={Xi~} over L. X consists of N1N2 random variable (r.v.) X~.

Given a lattice L, a neighborhood system

~= { ~?i:( i , j ) ~ L,~7o c _ L} (2)

is defined as the collection of subsets of L such that

1) (i,j)~r]ij, 2) ( i , j ) ~ l k t ~ ( k , l ) ~ j .

The subset of L, r/is, is the neighborhood of pixel (i,j). The neighborhood systems which are commonly used in image processing are shown in Fig. l(a); These systems have a structural hierarchy. For the first-order neighborhood system 71 which has the smallest range, rf consists of the four nearest neighbors of pixel (i j ) and for the second- order neighborhood system r/2, r/g~ 2 consists of the eight nearest neighbors.

For a given lattice-neighborhood system pair (L,r/) MRF is defined with the following local characteristics of condi- tional probabilities:

P(Xij=xdXk~=x~t,(k,1)~:(ij))

= P(Xi~= xi~lX~l= xkl,( k,l) C ~i~) (3)

A r.f. X is MRF if and only if for all the r.v. Xij in X, the conditional probability of X~ conditioned by all the r.v. Xk~ except X~j is equal to the conditional probability of Xo conditioned by only the neighbors of (i,j).

A clique c for a given (L, r/) is defined as a subset of L such that

1) c consists of a single pixel, or

2) ( i , j ) ~ c , ( k , l ) ~ c ~ ( i , j ) ~ k t .

The clique types of the first-order and the second-order neighborhood systems are shown in Fig. l(b) and (c), respectively.

For a given pair (L,r/) a r.f. X is a Gibbs r.f. if and only if its joint distribution has the form of

P ( X = x ) l e x p [ - f l U ( x ) ] (4)

where U(X)=c~ c Vc(x), energy function (5)

5 4 3 4 5

4 2 1 2 4

3 1 (i,j) 1 3

4 2 1 2 4

5 4 3 4 5

(a)

(b)

(ij)

(c) Fig. 1. Neighborhood systems and cliques: (a) A hierarchically ordered sequence of neighborhood systems that are commonly used in image modeling; (b) First-order neighborhood system and associ- ated clique types; (c) Second-order neighborhood system and associ- ated clique types.

V~(x), potential associated with clique c C, the collection of all cliques in L

Z = ZxeX p [ - f lU(x) ] , partition function. (6)

V~(x) is a function which depends only on the values of x in clique c. The above GRF image model has been applied successfully to various problems of image processing such as image restoration and segmentation, texture modeling, classifica- tion and segmentation.

In the next section, we review the MLL model which is commonly used as an image model, and the Ising model which is used in statistical mechanics to describe the magnetic properties of ferromagnetic material. 2.2 Multi-Level Logistic Model

This model was originally proposed by Derin and his

186 OPTICAL REVIEW Vol. 3, No. 3 (1996) D.-W. RIM et at

colleagues 2'") and has been used in the studies of image and texture segmentation. For a given r.f. X over a lattice L, a particular lattice is defined by a neighborhood system V and clique potentials Vc(x). The MLL model is defined as GRF having the following clique potential. X~j is a random variable taking values in an M-valued set Q={q~,q2 . . . . . qM }, and we assign a parameter to each clique type except the single pixel type as shown in Fig. 2, where M corre- sponds to the number of gray-levels. The clique potential is defined as

V~(x)= { - ~ k ifotherwiseall Xi~ in ck are equal (7)

The clique potential of the single pixel type is defined as

V~(x)=ak if x~j=qk (8)

The MLL model can be considered a generalization of the Ising model where binary r.f. are extended into M-valued random fields. The Ising model TM was used to explain the interaction between molecules in ferromag- netic material before the use of GRF in image processing; in this model, the random variables representing the spin states of molecules can have only two values: up and down. The energy function is defined as

N EI{s,}=-- ~. t~spsq--H~=is, (9)

<P,q>

where sp represents the spin state of molecule p ( p = 1,2,..., N) and takes values in {1,-1}. <p,q> representing the nearest neighbor molecule pairs are the same as pair cliques of the first-order neighborhood system. When there is no external magnetic field (H=0) and the interac- tion coefficients are constant (e~ = r the energy function becomes

El{Sp}:----~ ~, S[~q (10) <P,q>

The MLL model becomes the same as the Ising model if its neighborhood system and model parameters are de- aided as follows:

1) the first-order neighborhood system, r/L 2) X~ are binary r.v. taking values in {1,-1}.

4) the potentials for single pixel cliques V~(x)=0. In this paper, we confine our discussion to a binary case.

However, the necessity of a new magnetic energy term and the resulting performance improvement are expected to be valid in the case of multi-level images as well. The interaction energy used in this paper is as follows:

UI {X}------ ~1 ~, X/~q-- e 2 ~, X ~ q (11) <p,q>l <p,q>2

The second-order neighborhood system, ~12 was used, and

Fig. 2. Assigning model parameters to clique types in the MLL model.

only the clique potentials for pair cliques were nonzero. The clique types (p,q>i are shown in Fig. 3. Incidentally, the autobinomial model 3''''19) is another widely used image model. In a binary image, the above three models all become the same. 2.3 M A P Segmentation Algorithm

Noisy images are modeled by the sum of two r.f.

Y=F(X)+ W where F is a simple function from {1,-1} to {g+,g_}, gray-levels of image, i.e., F(1)=g+ and F(-1)=g_. We representing noise is an i.i.d. Gaussian r.f. with the mean, 0 and the variance, o "2. The problem of image segmenta- tion is changed into that of statistical estimation. Gener- ally, MAP segmentation is used, which is to find ~ that maximizes the a posteriori distribution P(X=x[Y=y) with respect to x for a given observation y. Using Bayes' rule

P(X=x] Y=y)= P(Y=y]X=x)P(X=x) (13) P(Y=y)

Since P(Y=y) is independent of x in the optimization procedure, the problem becomes maximization of the numerator of the right-hand side of Eq. (13). Because of the exponential form of the probability distributions, tak- ing logarithm simplifies the expression as follows:

]n P( Y- y[X=x)+ln P(X=x) N1N2 2 1 2

- 2 l n ( 2 r c o - ) - ~ c ( y ~ j - g x , j ) - l n Z - f l U l ( x )

: l o - (14)

where/0 is a constant and can be ignored in the maximiza- tion procedure.

The number of all possible images is enormous and, in this particular case, is 2 ~lN=. In the optimization of P ( X = x[ Y = y ) with respect to x, various methods such as DP, 2'9) DR, 9) and SA l'n'lT) can be used. In this study we used SA. 2.4 Realization f rom R.F.

Realization of particular images from a given r.f. can be done by Gibbs sampleP ) or Metropolis algorithm. TM Gibbs sampler is the iteration algorithm of the following two steps:

1) A pixel in a r.f. is chosen randomly or in a deterministic manner.

2) The value of the selected pixel is changed according to the conditional probability which is conditioned by the values of the neighboring pixels.

<p,q>~ <p,q>2 <p,q>3

<p,q>4 <p,q>5

Fig. 3. Pair clique types (p,q)' for z/1 to V 2.

OPTICAL REVIEW Vol. 3, No. 3 (1996)

It was proved that an output of Gibbs sampler satisfies the probability distribution of the r.f. regardless of the initial state of an input image if each pixel in the r.f. is chosen an infinite number of times. 1~ Some examples of realization generated using Gibbs sampler are presented in the next section.

3. Some Difficulties with the MLL Model and its Solu- tion

W e present some images realized f rom the MLL model in Fig. 4. Using the model parameters given in the figure caption, each image was generated by Gibbs sampler after 50 iterations. 2~ The images can be classified into two groups: region model (Fig. 4(a) and (b)) and texture model (Fig. 4(c) and (d)). For the region model all the parameters associated with clique potentials are positive, and for the texture model one or more model parameters are negative. Two compare the convergence properties of the two models at the realization procedure, we present in Figs. 5 and 6 the changes in images generated by Gibbs sampler after varying the number of iterations. While the texture model reaches equilibrium state only after tens of itera- tions, the region model is relatively slow to reach this state and, what is worse, when it is achieved all regions are merged into a single region. According to the theorem on Gibbs sampler, an image generated by Gibbs sampler satisfies a given distribution only after infinite number of iterations. Therefore, the typical region images shown in Fig. 4(a) and (b), which consist of several clusters, are misleading realizations f rom the MLL region model.

D..W. KIM et aL 187

Comparison with the Ising model clarifies this difficulty in modeling region process with the MLL model. The Ising model represents ferromagnet ism with ~>0, and

Fig. 4. Some image realizations from the MLL model with ~/z using Gibbs sampler. 50 iterations: (a) #k=l.0, M (# of grey-level)=2; (b) #~=0Z,, M=2; (e) fla=A=/~=l.0, fl,=-l.0, M=4; (d) fll=/~=l.0,/~= #,=-0.5, M=4.

Fig. 5. A texture realization from the MLL model with ;7 z using Gibbs sampler: (a) 0 iteration: (b) 5 iterations; (c) 50 iterations; (d) 500 iterations.

Fig. 6. A scene(region) realization from the MLL model with r/2 using Gibbs sampler: (a) 10 iteration; (b) 50 iterations; (c) 250 iterations; (d) 1000 iterations.

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Fig. 7. Image realizations at various temperatures. r (a) fl=0.4; (b) fl=0.3; (c) fl=0.25; (d) fl=0.2; (e) fl-0.15; (f) fl=0.1.

antiferromagnetism with e <0 in Eq. (10). These phenom- ena correspond to the region and texture models of the MLL model, respectively. It is well known that there is a critical temperature, To, for ferromagnetism, 2~ so that a specimen becomes all spin-up or all spin-down state below T~, and becomes a totally disordered state above T~. An abrupt phase transition occurs between the two state. Figure 7 shows realizations from the following probability distribution at various values of fl(=l/kT):

P(X=x) lexp[-flU~(x)] (15)

With a sufficiently high value of fl, it is in an all spin-up or all spin-down state. As fl is lowered or the temperature rises, spin transitions start to take place and result in a spot noise. The density of spots increases until fl is 0.2. An abrupt phase transition occurs when fl is between 0.2 and 0.15. The realization becomes totally disordered when fl is below 0.15. No realizations whether above or below T~ are appropriate for the region process; it can therefore be said that the MLL model is inappropriate for the region process model.

D.-W. KIM et aL

This problem is known as undesirable phase transition in image processing literature and has been addressed by many authors, 3''1'22'23~ but has not yet been solved satisfacto- rily. In this paper we offer an approach to the problem using the analogy between an image and a magnetic spin system, or a ferromagnetic specimen whose properties have been successfully described by the Ising spin system model, the origin of all random field image models. In nature, ferromagnetic specimens are not magnetized with all spins in one direction even if the temperature is below To. If a whole specimen is magnetized in one direction (single domain structure), each magnetic field from all the magnetic dipoles (or pixels) contributes constructively to the total extemal magnetic field with the others. This causes a strong magnetic field around the specimen, and the magnetic energy from this external magnetic field makes the single domain structure energetically unfavor- able. Therefore, a ferromagnetic material tends to break up into domains (multi-domain structure). Each domain is fully magnetized in accord with the results of Ising model simulation, but the various domains can be randomly oriented and thus present an unmagnetized appearance from the macroscopic point of view. In other words, a mutli-domain specimen does not generate a large external magnetic field and an associated large amount of magnetic energy. This results in lower total energy consisting of interaction and magnetic energy. 24~

We regarded an image as a spin system consisting of magnetic dipole moments placed at a rectangular lattice and introduced the magnetic energy from the magnetic field as well as the interaction energy between neighboring dipoles (or pixels). This enabled an image to have multi- domain structure consisting of clusters of controllable size.

If there are N magnetic moments r~p with their loca- tions ?p (p=1,2 . . . . . N), the magnetic energy of the system is

UM=--~-!gp.r~ (16) #

where B# is the magnetic field produced at the position of ~p by all the moments except r~p.

gp-- ,J#-- 5-!, [ - _m~3+ 3(r~q- F~)~ ] (17) ~l~ q~PL rpq rpq 5 ]

Supposing that a lattice is placed in the plane z=0, rnqx= mqy=O, and that rnqz=xq=l or - 1 as shown in Fig. 8, r~q. ~ becomes zero. Therefore,

Bpx=Bw=O, Bpz=-cqZ.~ xq (18) 7.pq 3

UM (x)=- ~Bp~rnp~

: c ~ p xp=cq (19) rpq3 "

As is shown in Eq. (19), the interactions between one pixel and all the others have to be calculated to consider the magnetic energy. This leads to disappearance of the advan- tage of random field models, that is, only the interactions

OPTICAL REVIEW Vol. 3, No. 3 (1996)

between neighboring pixels have to be considered; and the computing time drastically increases with the size of neighborhood. However, the amount of influence of neighboring pixels in the magnetic energy term decreases significantly as the distance between two interacting pixels increases because it is inversely proportional to the cube of the distance. We thus considered only the influence of fifth-order neighborhood in calculating the magnetic energy. The influence of pixels in sixth-order neighbor- hood is less than 4% that of nearest neighbor pixels. If we reuse ( i j ) indexing and combine all the constants into a weighting factor w, the magnetic energy UM(X) and the

/\z

Fig. 8.

Y

N magnetic moments over a finite rectangular lattice.

D.-W. KIMetat 189

total energy function U(x) becomes:

UM(X)= Z Z X i ~ (2O) (i.j) (h,l)~ ~,,~ ~ / ( i - - k y+( j - - l y a,

U ( X ) -- UI (~7C) ~- W U M (X) . (21)

The size of clusters in the image model can be controlled by changing the weighting factor w between U~(x) and UM(X). Some example realization from U(x) are shown in Fig. 9.

4. Experiments and Discussion

We investigated the effect of the addition of the mag- netic energy term into the energy function on the perfor- mance in segmentation procedure. In a binary image SNR is often defined as

SNR= Ig§ (22) (7

where g§ and g_ are two gray-levels and o" is the standard deviation of noise. When an image is severely degraded with SNR below 1.0 the function l n P ( Y = y [ X = x ) + In P ( X = x ) to be optimized becomes complicated with a lot of local minima, so the optimization is hard using DP or DR. In this paper SA was used in the optimization procedure. For cooling schedule, we use the following geometric decreasing sequence:

T , = T~. T,_I, n=1,2 . . . . (2.3)

In Figs. 10, 11, and 12 we present the segmentation results for SNR=I .0 , 0.6667, and 0.5, respectively. At SNR=I .0 the performances of the original M LL model and the modified model are comparable. However, as SNR is

Fig. 9. Some image realizations from the modified model: (a) w= 0.15; (b) w-0.154; (e) w-0.157; (d) w-0.16.

Fig. 10. Segmentation of test image with SNR=I.0. g§ g_= 120, o'=20, w-0.157; (a) original; (b) noisy; (c) segmented using the MLL model with 7f, e-7.2%; (d) segmented using the modified model, e=7.8%.

190 OPTICAL REVIEW Vol. 3, No. 3 (1996)

lowered, the performance of the modified model is more outs tanding than the original. The error rate of the seg- menta t ion results for three SNR's are given in Table 1. The error rate is defined as

D.-W. KIM et al.

e = ~ X l O 0 ( % ) (24)

where Ne is the number of mis-segmented pixels and Nt is the number of total pixels in an image.

W h e n SA is used in a complicated optimization prob- lem, it is c o m m o n to determine a cooling schedule experi- mental ly, by trial and error. F igure 13 shows the results of segmentat ion using the two models and varying the cool- ing parameter, T~. For the MLL model, Ta should be chosen with great care to obtain an acceptable segmenta- t ion result. The performance significantly degrades when

Table 1. Segmentation results using the MLL model with r/2 and the modified model with 3 different SNRs (1.0, 0.6667, 0.5).

e (%) SNR

MLL model with r/z The modified model

1.0 7.2 7.8 0.6667 26.2 15.8 0.5 35.0 22.3

Fig. 11. Segmentation of test image wkh SNR=0.6667. g§ g_=120, o'=30, w=0.157: (a) original; (b) noisy; (c) segmented using the M L L model with rf, e=26.2%; (d)segmented using the modified model, e=15.8%.

Fig. 12. Segmentation of test image with SNR=0.5. g+-130, g_= 120, o'-20, w=0.157: (a) original; (b) noisy; (c) segmented using the MLL model with r/2, e-35.0%; (d) segmented using the modified model, e-22.3%.

Fig. 13. Segmentation of test image with various cooling speeds Td: (a) original; (b) noisy, SNR-0.6667; (c) (f) MLL model with z/2, (c) Td=0.97, e--27.6%, (d) Td=0.98, e=26.2%, (e) Tr e-30.8%, (f) Td=0.995, e--34.1%; (g)-(j) the modified model, (g) Ta-0.995, e 22.0%, (h) Td=0.997, e=18.1%, (i) T~-0.998, e-15.8%, (j) Ta-0.999, e=15.4%.

OPTICAL REVIEW Vol. 3, No. 3 (1996)

Table 2. Segmentation results using the MLL model with 7/2 and the modified model at various cooling speeds.

MLL model with ~/2 The modified model

Ta e (%) Ta e (%) 0.97 27.6 0.995 22.0 0.98 26.2 0.997 18.1 0.99 30.8 0.998 15.8 0.995 34.1 0.999 15.4

Td is higher or lower than the chosen value. I t is essential that one knows the proper cooling parameter, Td, before- hand. Wi th the modified model, on the other hand, we have more freedom in choosing the parameter. The perfor- mance is acceptable once Ta is chosen above a certain value. The error rates of the segmentat ion results are shown with varying Td for both models at Table 2. As Td approaches 1.0, the amoun t of computat ion great ly increases, so we may choose a proper Td by compromis ing between the performance wanted and the computat ional load. The approximate expression of magnetic energy used in this paper can be rewri t ten as the pair clique potential form with the fifth-order ne ighborhood system r] s.

UM (X)= Cl ~, XI~q-~C 2 ~, Xp,~q~-...JuC 5 ~-a (25) <p,q>l <p,q>2 <p,q>S "~I~q

The types of pair cliques <p,q>i are shown in Fig. 3. In this respect, one is not expected to obtain a satisfactory image mode] with the second-order neighborhood system alone, but has to use the third or higher-order neighborhood system as well.

5. Conclusions

We have pointed out some difficulties in using GRF as an image model and have described a modified model to improve the performance in image segmentat ion. Using the MLL model with the second-order ne ighborhood system, the region process is difficult to model while the texture process is rather easy. Earlier work on the model- ing seemed to give successful results, but fur ther investiga- t ion showed the realizations were not in the equi l ibr ium state hut in the intermediate of the iterative methods such as Gibbs sampler. This intermediate state has nothing to do with realization in a t rue sense. To get around this problem we introduced a new energy te rm representing the external magnet ic energy into the total energy func- t ion which originally contained only the interact ion energy. Wi th this new term, we were able to control the size of the clusters in the realization by adjusting the weight ing factor between the two energy terms.

D.-W. KIM et al. 191

Several exper iments with both models revealed the superior performance of the modified model, especially with low SNR and poor image quality. The performance was also less dependent on the critical choice of the cooling parameter compared with the original model.

Acknowledgments

This research was partially supported by a Korea Science and Engineering Foundation Grant and KAIST Basic Research Fund.

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