Hidden Markov Random Field and FRAME Modelling for TCA-Image Analysis

Max-Planck-Institut für demografische Forschung Max Planck Institute for Demographic Research Konrad-Zuse-Strasse 1 · D-18057 Rostock · GERMANY Tel +49 (0) 3 81 20 81 - 0; Fax +49 (0) 3 81 20 81 - 202; http://www.demogr.mpg.de

MPIDR WORKING PAPER WP 2005-032 NOVEMBER 2005

Hidden Markov random field and FRAME modelling for TCA-image analysis

Katy Streso ([email protected]) Francesco Lagona ([email protected])

This working paper has been approved for release by: Jutta Gampe ([email protected]) Head of the Office of Statistics and Information Services.

Working papers of the Max Planck Institute for Demographic Research receive only limited review. Views or opinions expressed in working papers are attributable to the authors and do not necessarily reflect those of the Institute. HIDDEN MARKOV RANDOM FIELD AND FRAME MODELLING FOR TCA-IMAGE ANALYSIS

Katy Streso Francesco Lagona StatLab, Ofﬁce of Statistics and Information Services StatLab, Ofﬁce of Statistics and Information Services Max Planck Institute for Demographic Research Max Planck Institute for Demographic Research Konrad - Zuse - Strasse 1 Konrad - Zuse - Strasse 1 18057 Rostock, Germany 18057 Rostock, Germany [email protected] [email protected]

ABSTRACT nal saw cuts and artifacts (for example on the right). The Tooth Cementum Annulation (TCA) is an age estimation marked rectangle delimits the area that is used for the ap- method carried out on thin cross sections of the root of plication in Section 5. human teeth. Age is computed by adding the tooth erup- Paleodemographers at the Max Planck Institute for tion age to the count of annual incremental lines that are Demographic Research use large databases of images like called tooth rings and appear in the cementum band. Algo- the one depicted in Figure 1 to identify mortality profiles rithms to denoise and segment the digital image of the tooth of past human populations. Hence algorithms are needed section are considered a crucial step towards computer- to denoise and segment these images automatically. assisted TCA. The approach pursued in this paper relies Standard methods like singular value decomposi- on modelling the images as hidden Markov random fields, tion, Fourier transform and regression smoothing mea- where gray values are assumed to be pixelwise condition- sure texture features and are for this reason not flexible ally independent and normally distributed, given a hidden enough to fulfill the above task. In the course of this random field of labels. These unknown labels have to be paper, TCA-images are therefore described by a statisti- estimated to segment the image. To account for long-range cal model, specifically a Hidden Markov Random Field dependence among the observed values and for periodicity (HMRF) model. Section 2 introduces these models and the in the placement of tooth rings, the Gibbsian label distri- distribution of the hidden field is specified by a FRAME bution is specified by a potential function that incorporates model ([2]). With this Markov random field (MRF), macro-features of the TCA-image (a FRAME model). Es- macro-features of TCA-images such as long-range auto- timation of the model parameters is carried out by an EM- correlation among observed gray values and periodic place- algorithm that exploits the mean field approximation of the ment of tooth rings can be modelled. Section 3 describes label distribution. Segmentation is based on the predictive the estimation of the model parameters via an EM algo- distribution of the labels given the observed gray values. rithm that exploits the mean field approximation of the hidden field distribution. Section 4 specifies the FRAME KEY WORDS model for the application to TCA-images and describes the EM, FRAME, Gibbs distribution, (hidden) Markov random Gibbs sampler that is used to simulate from this prior dis- field, mean field approximation, TCA tribution. The sensible results of fitting the hidden FRAME model to real images like the one depicted in Figure 1 by 1 Introduction using the EM algorithm are discussed in Section 5.

Tooth Cementum Annulation ([1]) is an age estimation 2 The Hidden FRAME Model method based on annual incremental appositions in the ce-

mentum of mammalian teeth. A 90-110 Ñ thick cross HMRF modelling allows us to address both denoising and

section, polished or unpolished, is photographed with a Le- segmentation by means of a labelling problem ([3]). To

= f½;:::;ÆÅg

ica DC350F camera system under bright-ﬁeld and 200 or illustrate, let Ë be the set of pixels form-

¢ Å 400 times magniﬁcation. TCA-images are then 8 or 16 bit ing a rectangular lattice of size Æ . In the course of

gray scale pictures of size 1030x1300 or 1016x1300 pixels. this paper a pixel will interchangeably be denoted by i,or

Ü; Ý µ The dark parts of the annual lines, often called tooth rings, ´ when the two dimensions of the lattice need to be

are empirically 1 to 3 Ñ thick and result roughly in 5 to emphasized. The observed image is represented by an ar-

Y Y ¾ Ê

20 pixel thin lines under 400 times magniﬁcation. ray , where i is the gray value observed at pixel

i Y i

Figure 1 displays a typical good quality TCA-image . The value i is assumed to be drawn from the th con- Y

of the unpolished section extracted from a person aged 41. tinuous random variable i , belonging to the random ﬁeld

Y =´Y ;:::;Y µ ÆÅ

It is expected to ﬁnd 34 horizontal tooth rings in the marked ½ . Analogously, we deﬁne the array

¾G = f¼; ½;:::;Gg cementum band. Additionally, the image contains diago- of labels i that need to be estimated µ i ; j i g i Y ´ f ¾Ë Y i bÓÖ Óf µ ´ ½ È Òeigh Å j µ ¢ µ Æ j ehv hsnaGusa otfunction cost Gaussian a TCA-images chosen For have application. we the on depend both of choice fseiyn egbrodsrcue oeprecisely, More structure. define let’s neighborhood a specifying of and hr h parameters the where where mixture the to according modelled is rt admvariable random crete tec ie n suethat assume and pixel each at h admfield random the h MRF The X µ= ÆÅ ¾Ëj ¾G Y £ µ ´ j ¥a the ¥ ; Gg i f f ´ h lattice the ntestigo MF h on itiuinof distribution joint the HMRF, a of setting the In µ ¾ j egbrn relationship neighboring iue1 yia noihdTAiaeo odqaiy(S0066fo h C aaaeo h P DR) MPI the of database TCA the from (IS-0000666 quality good of TCA-image unpolished typical A 1. Figure µ= È i ;:::; ´ µ= Y g neighborhood Y ½ j i ´ ´ ¾ ´ ¾ È g cementum band f µ f i scle otfnto reiso est.The density. emission or function cost called is Æ ; stedsrbto faMro admfield random Markov a of distribution the is g Ë

a oe pta eednisb means by dependencies spatial model may i £=´£ f Y hti nirﬂxv n symmetric; and anti-reﬂexive is that ´ µ = i

i ´ e i £ Æ enda the as defined , Ô ¾ fpixel of i sabnr eainhpon relationship binary a as ; ssmldfo h dis- the from sampled is i i . ¾ steset the as ¾ i hcodnt of coordinate th r unknown. are µ µ i and ´ Æ j i ´ Y È : = ¡ i ¼ ËÒ ¾Ëg i > j j i ne hsstig h admfield random the setting, this Under novsoefilter one involves pc otenihoho system neighborhood the to spect ovligtelblimage label by modelled the efficiently convolving is image model, the FRAME about the knowledge In prior [5]. and [4] [2], in developed mainly for stands which FRAME, F called is TCA-images for lized h le responses filter The vlaigteefitrrsoss nissmls version, distribution simplest Gibbs a its is In distribution FRAME the responses. filter these evaluating where oawd ait fee ag cl textures. scale large even of applied variety be wide can a it to and theory filtering and modelling analysis: HMRF texture of areas important hereby two combines model elegantly FRAME hidden The 4). (Section application aaercfamily parametric iews yteptnilfunction potential the by pixelwise µ µ i ilters, 1. 2. ´

the ¥ ´ Æ È È ; hoods h pcfi omo h R oe htsalb uti- be shall that model MRF the of form specific The ½ f µ] Z i R = egbrodsystem neighborhood µ´ stenraiigcntn.Teeeg function energy The constant. normalizing the is µ no Fields andom µ= i £ Æ Ì µ´ ´ F [´ È £ (positivity) Ì Ì Ì F F F ¾Ë È i ´ e hti nw pt h parameter the to up known is that n h function the and Z a nd M Æ ihsial lesadby and filters suitable with to aximum stesto l neighbor- all of set the as (Markovianity). Æ tpixels at Å ffrall for if £ ¢ h hieo the of choice The . Æ sa is E sdie ythe by driven is toyadwas and ntropy ¾G i MRF r evaluated are ihre- with (1) Ì . 3 Parameter Estimation and Segmentation MCMC algorithm ([7]). This would require to generate a

Markov chain at each pixel which is not feasible. The alter- Ì

In order to estimate and , the maximum likelihood es- native approach we suggest is based on the approximation

timates (MLE) and can in principle be found by maxi-

È ´µ È j :

´iµ

mizing the likelihood function Æ (6)

i¾Ë

X Y

Ä´ ;ÌjY µ= È ´jÌ µ f ´ Y j ; µ : i

i (2)

¢Å

Æ In this paper, the conﬁguration is chosen according to the

i¾Ë

¾G ~

theory of mean ﬁeld approximation theory ([8]) where is

However, this maximization is intractable because of the set to the expected values of the label image:

Æ ¢Å

size of the label space G .

= E [ ] j ¾ Æ ´iµ: j The EM algorithm is a widely used technique to solve j for all

this kind of problem. The algorithm depends on the predic- É

È ´ jE [ ]µ

´iµ

tive probability that is usually computed via MCMC. In our The product Æ is then a valid probabil-

i¾Ë

application this is again not feasible because of the size of ity distribution and minimizes the Kullback-Leibler diver-

´µ TCA-images. We suggest to use mean ﬁeld approximation gence to the true prior distribution È among all prod- to make the EM tractable. ucts of this kind. The E-step of the EM algorithm hereby

To illustrate, let us recall that the EM algorithm starts changes to

´¼µ

Ì h

with preliminary estimates and of the parameters i

´Ø ½µ

;Ì ÐÓg È ´; Y j ;ÌµjY; and , and then proceeds iteratively by alternating two E

steps. In the E-step of the Ø-th iteration the conditional ex-

pectation of the complete log-likelihood, with respect to the X

ÐÓg È = g j ; ;Ì f ´Y j ; ;Ìµ

i i i

Æ ´iµ

unknown labels

g =¼

i¾Ë

h i

´Ø ½µ

E ÐÓg È ´Y; j ;ÌµjY; ;Ì

¡È = g jY ; ; ;Ì

i i

Æ ´iµ

X X

´Ø ½µ

= È jY; ;Ì ÐÓg È ´; Y j ;Ìµd

= ÐÓg È = g j ;Ì ·ÐÓgf ´Y j = g; µ

i i i

Æ ´iµ

Æ ¢Å

¾G

g =¼

i¾Ë

´Ø ½µ

f Y j = g; È = g j ;Ì

i i i

Æ ´iµ Ì

is calculated, where and are the estimates

from the previous iteration. The M-step of the EM algo- G

´Ø ½µ

È = g j ;Ì f Y j = g;

i i i

´iµ

rithm maximizes this expectation to update and : Æ

g =¼

h i

´Øµ ´Ø ½µ

;Ì = aÖg ÑaÜ E ÐÓg È ´Y; j ;ÌµjY; ;Ì

The parameter estimates can therefore be updated by the

f ;Ì g

Æ íµ íµ Equations (3) to (5) and replacing Æ therein by , Since each iteration is guaranteed to increase the (incom- that are computed iteratively. Our EM algorithm then takes plete) log-likelihood (2) under mild assumptions, the EM the following form: algorithm will converge to a local maximum ([6]). EM algorithm using MFA for fitting a hidden FRAME model In the case of a Gaussian random field, the EM algorithm reduces to the three updating formulas ([6])

1. input TCA-image Y

È Initialization

´Ø ½µ

´¼µ

Y È = g jY; ; ;Ì

i i

Æ ´iµ

2. initialize label conﬁguration by thresholding

i¾Ë

´Øµ

(3) 3. initialize parameters

´Ø ½µ

È = g jY; ; ;Ì

´iµ

Æ Updating

i¾Ë

Ø =½ : Ø

4. for ÑaÜ

´Øµ

update label image by

´Øµ

´Ø ½µ

hi =

Y È = g jY; ; ;Ì

g i

i 5.

Æ ´iµ

i¾Ë

´Øµ

6. for each site i (randomly permuted)

´Ø ½µ

g =¼ : G

Ø ½µ

´ 7. for

È = g jY; ; ;Ì

Æ ´iµ

´Ø ½µ ¾Ë

i 8. calculate the conditional probability

(4) ¾

´Ø ½µ

g ´Ø ½µ ´Ø ½µ

f Y j = g; ; » e

i i

X X

´Ø ½µ

´Ø ½µ Øµ

´ 9. approximate the prior energy and probability

È = g jY; ; ;Ì Ì = aÖg ÑaÜ

Æ ´iµ

´Ø ½µ

fÌ g

g =¼

i¾Ë

F £ ´j µ Í = g j ;Ì

´Ø ½µ

Æ ´iµ

j ¾C ´iµ

¡ ÐÓg È = g j ;Ì :

´iµ

Æ (5)

´Ø ½µ

Í =g j ;Ì

´ µ

Æ ´iµ

´Ø ½µ

È = g j ;Ì

Æ ´iµ

È = g jY; ; ;Ì

´Ø ½µ

´iµ

The conditional probabilities Æ are

Í =g j ;Ì

´ µ

Æ ´iµ

not available in closed form and could be evaluated by an e

g =¼ 0.004 0 −0.004 10 10 0 0 −10 x −10 y

Figure 2. 3-D surface and image of a Gaborcosine function Figure 3. A typical image of size 128x128 pixels simulated

=½6 « =¼

with Ì and by the Gibbs sampler using the FRAME model (1) with the

= j:j ﬁlter displayed in Figure 2, and 8 gray levels.

10. calculate the posterior probability

¼ ¼

´Ø ½µ ´Ø ½µ

Ü = Ü cÓ× « · Ý ×iÒ «; Ý = Ü ×iÒ « · Ý cÓ× « Ö =4

= g jY ; ; ;Ì

È with ,

i i

Æ ´iµ

Ø ½µ ´Ø ½µ ´Ø ½µ

´ being the aspect ratio and being a normalizing factor. The

» f Y j = g; ; È = g j ;Ì

i i i

´iµ Æ Gaborcosine function above is an elongated Gaussian bell 11. calculate the expected label

multiplied by a cosine wave, where parameter Ì changes

´Ø ½µ ´Ø ½µ

the wavelength and « determines the orientation of the co-

g ¡ È = g jY ; ; ;Ì

i i

Æ ´iµ =¼

g sine wave. For example, Figure 2 shows the Gaborcosine

h i =

Ì = ½6 « = ¼ Ü; Ý ¾ [ ½¿; ½¿]

Ø ½µ

´ function for , and . This

´Ø ½µ

È = g jY ; ; ;Ì

i i

Æ ´iµ =¼

g ﬁlter can capture waves or lines of width 16 and orientation

´Øµ

= hi

12. set 0¡.

= ¼

update parameters In the application for TCA-images we ﬁx «

=¼ :G

13. for g which is the main direction of tooth rings. In order to

14. update g according to Equation (3) cover the range of possible tooth ring widths we chose

´Øµ

¾ f¾; 4; 6; 8; ½¼; ½¾; ½4; ½6; ½8g

Ì . We remark that our

15. update g according to Equation (4) approach is different to that in [2], because we are inter- 16. update Ì according to Equation (5) ested in reconstructing tooth rings, that resemble only one feature of interest. We do not want to synthesize percep-

The initialization of and the sequential updating of tional equivalent images, including noise. Besides simpli-

the true labels were chosen according to the recommenda- fying the FRAME model to incorporate only one ﬁlter (one

Ø tions in [8]. The number of iterations ÑaÜ was chosen feature), the potential function that evaluates this ﬁlter according to the last gain in the likelihood response is assumed to be known and chosen to be the sim-

plest among the upright curves, namely the absolute value

X X

= j:j

´Ø ½µ ´Ø ½µ

´Ø ½µ

Ä´ jY µ= È = g jY ; ; ;Ì :

i i

´iµ

Æ Figure 3 displays a typical image drawn from the

g =¼ ¾Ë

i FRAME model using the Gaborcosine ﬁlter with param-

=½6 « =¼

Segmentation can ﬁnally be carried out by exploiting eters Ì and and the absolute valued potential

function. This image comes very close to the ideal TCA-

= g jY ; ; ; Ì È

i i

´iµ Æ by means of thresholding. image that one could have in mind about parallel running tooth rings. Orientation and width of these lines are deter- 4 Application mined by both parameters of the Gaborcosine ﬁlter.

The image in Figure 3 was generated by Gibbs sam- F

This Section is devoted to specifying the ﬁlter family Ì pling. (See e.g. [3].) The single site Gibbs sampler for

´¼µ

Ø = ¼

and the potential function that we have used for TCA- example initializes at time and then updates

´Ø·½µ

image analysis and to describe a simulation algorithm for each pixel by repeatedly sampling a candidate from

Ø·½µ ´Øµ

generating a typical image from this model. ´

the full conditional È . The transition prob-

i i

Filtering theory is well recognized in texture analy- ËÒ

´Øµ ´Ø ½µ

´ j µ

sis at least since [9]. Marˇcelja ([10]) has shown that two- abilities È are then guaranteed to converge to

´µ dimensional Gabor functions closely conform to the recep- the stationary distribution È . tive ﬁeld proﬁles of simple cells in the striate cortex. We remark that the Gibbs sampler can be applied

in the present case because the FRAME model is a MRF F

We deﬁne the ﬁlter «;Ì on the basis of the real valued, even-symmetric Gabor function: model. This can be proven by the application of the

Hammersley-Clifford theorem ([11]).

¼¾ ¼¾

ÖÜ ·Ý

µ ´

¾ When choosing a random initial image and a random

¾Ì

GcÓ× ´Ü; Ý µ=c ¡ e Ü ; cÓ×

Ì;« (7) updating order of the pixels, the Gibbs sampler consists of Ì Figure 4. The mean ﬁeld approximation of the cementum band of TCA-image 1

the following steps for the FRAME model:

Gibbs sampling algorithm for the FRAME model

´¼µ F

1. input initial white noise image and ﬁlter

´¼µ

£ 2. precompute the ﬁlter response F 3. repeat sufﬁciently often

Figure 5. The black rings from the mean ﬁeld approxima-

¡ Å Æ ¡ Å = jË j

4. repeat Æ times ( size of the image)

Ü; Ý µ

5. randomly select site ´ tion of part of TCA-image 1 overlayed onto the original

¼ ¼

Ü ;Ý µ 6=´Ü; Ý µ

6. for all ´

´Ø·½µ ´Øµ

¼ ¼ ¼

7. set ¼

´Ü ;Ý µ ´Ü ;Ý µ

´Ø·½µ

8. for each gray value g of label

Ü;Ý µ

´ the TCA-image in Figure 1. The MRF model is speciﬁed

¼ ¼

Ü ;Ý µ ¾fÆ ´Ü; Ý µ; ´Ü; Ý µg

9. for all ´

½ by the FRAME model (1). The parameters ¼ , and a

10. calculate the new ﬁlter responses ¾

common variance as well as the ﬁlter parameter are

´Ø·½µ ¼ ¼ ´Øµ ¼ ¼

£ ´Ü ;Ý µ= F £ ´Ü ;Ý µ

F estimated by an EM algorithm as stated in Section 2. The

label image is obtained from the mean ﬁeld at the last

´Ø·½µ ´Øµ

¼ ¼

·F ´Ü Ü ;Ý Ý µ

Ü;Ý µ ´Ü;Ý µ

´ iteration.

´Ø·½µ

11. for each gray value g of label Figure 4 shows the predictive probability

´Ü;Ý µ

~ ^

´Ø·½µ

È ´ j ; ; Ì µ

´iµ

Æ of the pixels in the cementum band of

= g ½

12. set with (conditional)¼ probability

´Ü;Ý µ

´Ø·½µ ¼ ¼

A @

£ ´Ü ;Ý µ

F Figure 1, where and are the estimates of the last itera-

j´ µ j

¼ ¼

´Ü ;Ý µ¾Æ ´Ü;Ý µ

e ^ = ¾8444

´Ø·½µ ´Øµ

tion. The parameter estimates are the means ¼ ,

½ ¼

È = g j =

¾ 7

´Ü;Ý µ Æ ´Ü;Ý µ

^ =¾87½4 ^ =5:5 ¡ ½¼ È

½ , the common variance and the

´Ø·½µ

¼ ¼

A @

´Ü ;Ý µ F £

µ j j´

¼ ¼

´Ü ;Ý µ¾Æ ´Ü;Ý µ =½4

ring width Ì .

e =¼

g For illustration purposes a smaller part (the one

´Ø·½µ

13. update the ﬁlter response F

marked in Figure 1) of this mean ﬁeld is thresholded ( i

~ ^

È ´ j ; ; Ì µ < ¼:5 =½ ¼

i i

´iµ if Æ and otherwise). The

If the computer precision is not enough to calculate the

middle lines of the black rings are then superimposed onto

´Øµ ´Ø·½µ

= g j

conditional probability È in step 12, the original image. The reader can count approximately 32

Æ ´Ü;Ý µ ´Ü;Ý µ one can easily insert a nourishing one. tooth rings in Figure 5. From the known age we expect To detect convergence, the Gelman-Rubin multi- 34 tooth rings in the image presented in Figure 1. This is quite a good estimate that is typically conﬁrmed by a dozen variate convergence statistic Ê ([12]) is used on every

(20x20)th pixel of the image. The Gibbs sampler stops it- additional TCA-images.

< ½:¾

erating when Ê . The algorithm above needs about

´jF j¡ÆÅ ¡ G ¡ Ë µ jF j Ç operations where is the area covered 6 Conclusion by the ﬁlter and Ë is the number of sweeps of the Gibbs sampler. For segmentation of TCA-images we set up a hidden Markov random ﬁeld model and exploited the EM-

algorithm. This procedure requires the approximation of ¡

5 Results

È = g jY; ; ;Ì

´iµ the posterior probabilities Æ and

The aim of analysis of TCA-images in this paper is to un- the ﬁnal segmentation . The Gibbs sampler proved to be

= f¼; ½g cover the black and white labelling (G ) in order infeasible in both cases except for small images. For exam- to be able to estimate the number of tooth rings. For this ple the simulation of the predictive distribution in Figure 3 purpose a Gaussian hidden Markov random field is fitted to took about 55 hours on a PC and programmed in Matlab. We therefore chose to use the mean field approximation [6] Geoffrey J. McLachlan and Thriyambakam Krishnan. to estimate the posterior probabilities and thresholded the The EM Algorithm and Extensions. Wiley Series in mean field of the last iteration for the final segmentation. Probability and Statistics. John Wiley & Sons, New This compound estimation procedure took all together 10 York, 1997. hours and gave reasonable results. Despite of the good overall age estimate, the reader [7] Yongyue Zhang, Michael Brady, and Stephen Smith. can see in Figure 5 that some rings are not well met and that Segmentation of Brain MR Images Through a Hidden bifurcations occur in the label image (Figure 4). This is due Markov Random Field Model and the Expectation- to two reasons. On the one side the reconstruction of the Maximization Algorithm. IEEE Transactions on TCA-image is heavily influenced by the shape of the single Medical Imaging, 20(1):45Ð57, January 2001. filter we estimate. The hidden FRAME model in this form [8] Gilles Celeux, Florence Forbes, and Nathalie Peyrard. can hereby only take into account strong local changes of EM procedures using mean field-like approximations tooth rings. In order to overcome this global property of for Markov model-based image segmentation. Pat- the FRAME model, one would need to select location de- tern Recognition, 36(1):131Ð144, January 2003.

pendent ﬁlters, i.e. estimating the ﬁlter parameter Ì at each

pixel i. On the other side, we assumed that the orientation [9] Anil K. Jain and Farshid Farrokhnia. Unsupervised of tooth rings is mainly horizontal. By estimating not only Texture Segmentation Using Gabor Filters. Pattern

the ring width Ì from the bank of ﬁlters, but also the ori- Recognition, 24(12):1167Ð1186, 1991.

entation «, one could overcome this limitation and would therefore avoid the bifurcations, that now mainly occur in [10] S. Marˇcelja. Mathematical description of the re-

areas where tooth rings have another orientation. sponses of simple cortical cells. Journal of the Op- ¾

¾ tical Society of America, 70(11):297Ð300, November

Additionally different variance parameters ½ ¼ 1980. might also change results and therefore such heteroscedas- ticity assumption should to be tested. The mean field ap- [11] Julian Besag. Spatial Interaction and the Statisti- proximation is not the only possible one for the approxi- cal Analysis of Lattice Systems. Journal of the mation (6). Celeux, Forbes and Peyrard ([8]) also mention Royal Statistical Society. Series B (Methodological), mode field approximation and simulated field approxima- 36(2):192Ð236, 1974. tion that should be tested for quality and speed in the case of TCA-image analysis. Moreover, a larger number of ex- [12] Stephen P. Brooks and Andrew Gelman. General periments on images of different quality need to be imple- Methods for Monitoring Convergence of Iterative mented in order to test the accuracy of the procedure. Simulations. Journal of Computational and Graph- ical Statistics, 7(4):434Ð455, 1998. References

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[3] Stan Z. Li. Markov Random Field Modeling in Image Analysis. Computer Science Workbench. Springer, Tokyo [et al.], 2001.

[4] Song Chun Zhu, Yingnian Wu, and David B. Mum- ford. Filters, Random Fields and Maximum Entropy (FRAME): Towards a Uniﬁed Theory for Texture Modeling. International Journal of Computer Vision Archive, 27(2):107 Ð 126, March 1998.

[5] Song Chun Zhu and David B. Mumford. Prior Learn- ing and Gibbs Reaction-Diffusion. IEEE Transac- tions on Pattern Analysis and Machine Intelligence, 19(11):1236Ð1250, November 1997.