hidden markov random field and frame modelling for tca...

Hidden Markov Random Field and FRAME

modelling for TCA image analysis

Katy Streso and Francesco Lagona

StatLab, Office of Statistics and Information Services, Max Planck Institute forDemographic Research, Konrad - Zuse - Strasse 1, 18057 Rostock, [email protected] and [email protected]

Summary. Tooth Cementum Annulation (TCA) is an age estimation method car-ried out on thin cross sections of the root of the human tooth. Age is computed byadding the tooth eruption age to the count of annual incremental lines which arecalled tooth rings and appear in the cementum band. Algorithms to denoise andsegment the digital image of the tooth section are considered a crucial step towardscomputer-assisted TCA. The approach in this paper relies on modelling the imagesas hidden Markov random fields, where gray values are assumed to be pixelwiseconditionally independent and normally distributed, given a hidden random field oflabels. These unknown labels have to be estimated to segment the image. To ac-count for long-range dependence among the observed values and for periodicity inthe placement of tooth rings, the Gibbsian label distribution is specified by a poten-tial function that incorporates macro-features of the TCA image (a FRAME model).An estimation of the model parameters is made by an EM algorithm exploiting themean field approximation of the label distribution. Segmentation is based on thepredictive distribution of the labels given the observed gray values.

Key words: EM, FRAME, Gibbs distribution, (hidden) Markov random field,mean field approximation, TCA

1 Introduction

Tooth Cementum Annulation [WGV03] is an age estimation method based on annualincremental appositions in the cementum of mammalian teeth. A 90-110 µm thickcross section, polished or unpolished, is photographed using a Leica DC350F camerasystem with bright-field and 200 or 400 times magnification. TCA images are then8 or 16 bit gray scale pictures of 1030x1300 or 1016x1300 pixels in size. The darkparts of the annual lines, often called tooth rings, are empirically 1 to 3 µm thickand roughly result in thin lines of 5 to 20 pixel with 400 times magnification.

Figure 1 displays a typical TCA image of intermediate quality of the unpolishedsection extracted from a person aged 41. It is expected to find 34 horizontal toothrings in the marked cementum band. Additionally, the image contains diagonal sawcuts and artifacts (for example, on the right). The marked rectangle as well as thewhole cementum band indicate areas used for an application in Section 4.

1226 Katy Streso and Francesco Lagona

cem

entu

m b

and

Fig. 1. An unpolished TCA image (IS-0000666 from the TCA database of the MPI)

Paleodemographers at the Max Planck Institute for Demographic Research uselarge databases of images as depicted in Figure 1 to identify mortality profiles ofpast human populations. Computer-aided evaluation of TCA images is consideredan important step towards defining a protocol for TCA-based age estimation.

In the course of this paper, TCA images are denoised and segmented by a HiddenMarkov Random Field (HMRF) model, were the distribution of the hidden field isspecified by a FRAME prior [ZWM97]. This is a Markov Random Field (MRF) thatmodels macro-features of TCA images such as long-range auto-correlation amongobserved gray values and periodic placement of tooth rings. The HMRF is introducedSection 2. Section 3 describes the estimation of the model via an EM algorithm thatexploits the mean field approximation of the hidden field distribution. In Section4 we specify the FRAME model for the application to TCA images and presentsensible results of fitting the hidden FRAME model to TCA images.

Hidden MRF and FRAME modelling for TCA image analysis 1227

2 The Hidden FRAME Model

HMRF modelling allows us to address both denoising and segmentation by meansof a labelling problem [Li01]. To illustrate, let S = {1, . . . , NM} be the set of pixelsforming a rectangular lattice of size N ×M . In the course of this paper, a pixel willinterchangeably be denoted by i, or (x, y) when the two dimensions of the latticeneed to be emphasized. The observed image is represented by array Y , where Yi ∈ R

is the gray value observed at pixel i. Value Yi is assumed to be drawn from the ithcontinuous random variable Yi, belonging to the random field Y = (Y1, . . . ,YNM ).Analogously, we define array λ of labels λi ∈ G = {0, 1, . . . , G} that need to beestimated at each pixel and assume that λi is sampled from the discrete randomvariable Λi, defined as the ith coordinate of the random field Λ = (Λ1, . . . , ΛNM ).

In the HMRF setting, the joint distribution of Y is modelled according to themixture

f(Y ) =X

λ∈GN×M

�P (λ)

Yi∈S

f(Yi|λi)�, (1)

with known number of components, where P (λ) is the distribution of a Markovrandom field and f(Y |λ) is called cost function. The choice of both depends on theapplication. For TCA images, we have chosen a Gaussian cost function

f(Yi|λi, θ) =1√

2πσλi

e−

(Yi−µλi)2

2σ2λi ,

where parameters θ = {µg , σ2g |g ∈ G} are unknown. MRF P (λ) may model spatial

dependencies by means of specifying a neighborhood structure. More precisely, letus define

• a neighboring relationship as a binary relationship on lattice S that is anti-reflexive and symmetric;

• neighborhood N(i) of pixel i as the set N(i) = {j ∈ S|j neighbor of i} and• neighborhood system N as the set of all neighborhoods N = {N(i)|i ∈ S}.

Under this setting, random field Λ is a MRF with respect to neighborhood systemN , if for all λ ∈ GN×M

1. P (λ) > 0 (positivity),2. P

�λi|λS\i

�= P (λi|λN(i)) (Markovianity).

The specific form of the MRF model that will be utilized for TCA images is calledFRAME, which stands for Filters, Random Fields and Maximum Entropy and wasmainly developed in [ZWM97], [ZWM98] and [ZM97]. In the FRAME model, priorknowledge of images is efficiently modelled by convolving label image λ with suitablefilters and by evaluating the filter responses. In its simplest version, the FRAMEdistribution is a Gibbs distribution

P (λ) =1

Ze

Pi∈S

φ[(FT ∗λ)(i)]

, (2)

where Z is the normalizing constant. The energy function involves one filter FT thatis known up to parameter T . The filter responses (FT ∗ λ)(i) to λ at pixels i are


evaluated pixelwise by the potential function φ. The choice of the parametric fam-ily FT and function φ is driven by the application (Section 4). The hidden FRAMEmodel thus elegantly combines two important areas of texture analysis: HMRF mod-elling and filtering theory, and it can be applied to a wide variety of even large scaletextures.

3 Parameter Estimation and Segmentation

In order to estimate θ and T , the maximum likelihood estimates (MLE) θ and T canin principle be found by maximizing the likelihood function L(θ, T |Y ) as presentedin Equation (1). However, this maximization is intractable because of the size oflabel space GN×M . The EM algorithm is a widely used technique to solve this kindof problem. The algorithm depends on the predictive probability usually computedvia MCMC. In our application this is computationally too expensive because of thesize of neighborhoods and the size of images that are considered for TCA application.We suggest to use mean field approximation (MFA) to make the EM tractable.

Let us recall that the EM algorithm starts with preliminary estimates θ(0) and

T (0) of the parameters θ and T , and then proceeds iteratively by alternating twosteps. In the E-step of the t-th iteration, the conditional expectation of the completelog-likelihood, with respect to the unknown labels λ, say

Ehlog P (Y, λ|θ, T )|Y, θ

(t−1), T

(t−1)i=

Zλ∈GN×M

P�λ|Y, θ

(t−1), T

(t−1)�

log P (λ, Y |θ, T )dλ

is calculated, where θ(t−1) and T (t−1) are the estimates from the previous iter-

ation. The M-step of the EM algorithm maximizes this expectation to update θ

and T . Since each iteration is guaranteed to increase the (incomplete) log-likelihoodL(θ, T |Y ) under mild assumptions, the EM algorithm will converge to a local maxi-mum [MK97]. With a Gaussian random field, the EM algorithm reduces to the threeupdating formulas [MK97]

µ(t)g =

Pi∈S

YiP�λi = g|Yi, λN(i), θ

(t−1), T (t−1)�P

i∈S

P�λi = g|Yi, λN(i), θ

(t−1), T (t−1)� ,�

σ(t)g

�2

=

Pi∈S

�Yi − µ

(t)g

�2


(t−1), T (t−1)�P

i∈S


(t−1), T (t−1)� ,

T(t)=argmax

{T}

Xi∈S

GXg=0

log P�λi = g|λN(i), T

�P�λi = g|Yi, λN(i), θ

(t−1), T

(t−1)�

(3)

The conditional probabilities P�λi = g|Y,λN(i), θ, T

�are not available in closed

form and could be evaluated by an MCMC algorithm [ZBS01]. This would requireto generate a Markov chain at each pixel, which is too time consuming for large scale


textures modelled by large neighborhoods. The alternative approach we suggest isbased on the approximation

P (λ) ≈Yi∈S

P�λi|λN(i)

�. (4)

In this paper, the configuration λ is chosen according to the theory of mean fieldapproximation [CFP03], where λ is set to the expected values of the label image:λj = E[λj ] for all j ∈ N(i). The parameter estimates can therefore be updated byEquations (3) and replacing λN(i) therein by λN(i), which are computed iteratively.Our EM algorithm then takes the following form:

EM algorithm using MFA for fitting a hidden FRAME model

1. input TCA image Y

2. initialize label configuration λ(0) by thresholding and initialize parameters3. for t = 1 : tmax

4. 〈λ〉 = λ(t−1) (update label image)5. for each site i (randomly permuted)6. for g = 0 : G7. calculate the cond. prob. f

�Yi|λi = g, µ(t−1), σ(t−1)

�according to Equ.

(??)

8. approximate the prior prob. P�λi = g|λN(i), T

(t−1)�

≈

e

Pj∈N(i)

φ

��F

T (t−1) ∗λ

�(j)

�GP

g=0e

Pj∈N(i)

φ

��F

T (t−1) ∗λ

�(j)

�9. calculate the posterior probability


(t−1), T

(t−1)�

=f�Yi|λi = g, θ(t−1)

�P�λi = g|λN(i), T

(t−1)�

GPg=0

f�Yi|λi = g, θ(t−1)

�P�λi = g|λN(i), T (t−1)

�10. calculate the expected label 〈λi〉 =

GPg=0

g·P

�λi=g|Yi,λN(i),θ

(t−1),T (t−1)

�GP

g=0P

�λi=g|Yi,λN(i),θ

(t−1),T (t−1)

�11. set λ(t) = 〈λ〉12. for g = 0 : G (update parameters)

13. update µg and�σ

(t)g

�2

according to Equations (3)

14. update T according to Equation (3)

The initialization of λ and the sequential updating of the labels were chosenaccording to the recommendations in [CFP03]. The number of iterations tmax wasselected such that the absolute relative change in the log-likelihood

L(θ(t−1), T

(t−1)|Y ) =Xi∈S

log

GXg=0


(t−1), T

(t−1)�

. (5)

over the last five iterations is less than 10−6. Segmentation can finally be carried

out by exploiting P�λi = g|Yi, λN(i), θ, T

�by means of thresholding.


4 Application

This Section is devoted to specifying the ingredients of the FRAME model that wehave used for TCA image analysis: the filter family FT and the potential functionφ. A simulated image that was generated by the Gibbs sampler will illustrate thismodel. Finally results from fitting a Gaussian hidden FRAME model to TCA imagesare shown.

Filtering theory is well recognized in texture analysis at least since [JF91].Marcelja [Ma80] has shown that two-dimensional Gabor functions closely conformto the receptive field profiles of simple cells in the striate cortex. We define filtersFT,α on the basis of real valued, even-symmetric Gabor functions:

GcosT,α(x, y) = c · e−(rx′2+y′2)

2T2 cos

�2π

Tx′

�, (6)

with x′ = x cos α+y sin α, y′ = −x sin α+y cos α, r = 4 being the aspect ratio and c

being a normalizing factor. The Gaborcosine function above is an elongated Gaussianbell, multiplied by a cosine wave, where parameter T changes the wavelength andα determines the orientation of the cosine wave. For example, Figure 2 (a and b)shows the Gaborcosine function for T = 16, α = 0 and x, y ∈ [−13, 13]. This filtercan capture waves or lines of width 16 and orientation 0◦.

(a) (b) (c)

Fig. 2. 3-D surface (a) and image (b) of a Gaborcosine function with T = 16 andα = 0, and a typical image of 128x128 pixels in size simulated by the Gibbs samplerusing the FRAME model (2) with this filter, φ = |.| and 8 gray levels (c).

In the application for TCA images, we fix α = 0, which is the main directionof tooth rings. In order to cover the range of possible tooth ring widths, we choseT ∈ {2, 4, 6, 8, 10, 12, 14, 16, 18}. We remark that our approach is different to thatin [ZWM97], because we are interested in reconstructing tooth rings that resembleonly one feature of interest. We do not want to synthesize perceptional equivalentimages, including noise. Besides simplifying the FRAME model to incorporate onlyone filter (one feature), the potential function φ that evaluates the filter responseis assumed to be known and chosen to be the simplest among the upright curves,namely the absolute value φ = |.|.


Figure 2 (c) displays a typical image drawn from the FRAME model using theGaborcosine filter with parameters T = 16 and α = 0 and the absolute valuedpotential function. This image comes very close to the ideal TCA image that onecould have in mind about parallel running tooth rings. The orientation and width ofthese lines are determined by both parameters of the Gaborcosine filter. The imagein Figure 2 (c) was generated by single Gibbs sampling. (See [SL05].)

To uncover the black and white labelling (G = {0, 1}) of the TCA image inFigure 1, a Gaussian hidden Markov random field was fitted. The MRF model wasspecified by the FRAME model (2). Parameters µ0, µ1 and a common variance σ2

as well as the filter parameter T were estimated by an EM algorithm, as stated inSection 2. Label image λ was obtained from the mean field at the last iteration.Figure 3 (a) shows the predictive probability P (λi|λN(i), θ, T ) of the pixels in the

cementum band of Figure 1 (including the artifact on the right), where θ and T

are the estimates of the last iteration. The parameter estimates are the means µ0 =29026, µ1 = 29052, the common variance σ2 = 4.8 · 107, and the ring width T = 14.For illustration purposes, a smaller part (marked in Figures 1 and 3 (a)) of thismean field is thresholded (λi = 0 if P (λi|λN(i), θ, T ) < 0.5 and λi = 1 otherwise).The middle lines of the black rings are then superimposed on the original image (seeFigure 3 (b)).

(a) (b)

Fig. 3. The mean field approximation of the cementum band of TCA image 1 (a)and its black rings overlayed onto part of the original (b)

The median number of dark rings in label image 3 (a) is 35. From the known age,the expected number of tooth rings (the known age minus average tooth eruptionage) is 33.61 in the image presented in Figure 1.

Additional experiments with TCA images of mixed quality gave the followingresults:

These results are competitive to the manually counted number of rings.


Image number of rings Image number of rings Image number of ringsIS-000 expected estimated IS-000 exp. estim. IS-000 exp. estim.

0231 40.94 46 0688 35.44 34 1547 38.19 280592 60.39 64 1157 34.39 30 1692 34.39 340682 35.44 33 1225 40.94 46

5 Conclusion

TCA images can be efficiently modelled by a HMRF and were evaluated by theestimation of the predictive label distribution, via an EM algorithm using MFA.Despite of the good overall age estimate, visual inspection of Figure 3 shows twodrawbacks for TCA application: tooth rings are not well detected and bifurcationsoccur in the label image. This is due to the rigidity of the hidden FRAME model thatuses one filter, and hereby accounts only for strong local changes of tooth rings. Toadd flexibility one could model non-homogeneity by estimating location dependentfilters, or one could additionally estimate the filter orientation α. But estimating thefilter parameter T at each pixel gave unstable and biased results. Experiments for thesecond approach as well as standard error calculations are in progress. Additionallyone should test mode field and simulated field approximation [CFP03] for qualityand speed instead of approximating (4) by mean field approximation.

References

[WGV03] Wittwer-Backofen, U., Gampe, J., Vaupel, J.W.: Tooth Cementum Annu-lation for Age Estimation: Results from a Large Known-Age Validation Study.American Journal of Physical Anthropology, 123(2), 119–129 (2003)

[ZWM97] Zhu, S.C., Wu, Y.N., Mumford, D.B.: Minimax Entropy Principle and ItsApplication to Texture Modeling. Neural Computation, 9(8), 1627–1660 (1997)

[Li01] Li, S.Z.: Markov Random Field Modeling in Image Analysis. Springer, Tokyo[et al.] (2001)

[ZWM98] Zhu, S.C., Wu, Y., Mumford, D.B.: Filters, Random Fields and MaximumEntropy (FRAME): Towards a Unified Theory for Texture Modeling. Interna-tional Journal of Computer Vision Archive, 27(2), 107–126 (1998)

[ZM97] Zhu, S.C., Mumford, D.B.: Prior Learning and Gibbs Reaction-Diffusion.IEEE Trans. PAMI, 19(11), 1236–1250 (1997)

[MK97] McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. JohnWiley & Sons, New York (1997)

[ZBS01] Zhang, Y., Brady, M., Smith, S.: Segmentation of Brain MR ImagesThrough a Hidden Markov Random Field Model and the Expectation-Maximization Algorithm. IEEE Trans. on Medical Imaging, 20(1), 45–57 (2001)

[CFP03] Celeux, G., Forbes, F., Peyrard, N.: EM procedures using mean field-likeapproximations for Markov model-based image segmentation. Pattern Recogni-tion, 36(1), 131–144 (2003)

[JF91] Jain, A.K., Farrokhnia, F.: Unsupervised Texture Segmentation Using GaborFilters. Pattern Recognition, 24(12), 1167–1186 (1991)

[Ma80] Marcelja, S.: Mathematical description of the responses of simple corticalcells. Journal of the Optical Society of America, 70(11), 297–300 (1980)


[SL05] Streso, K., Lagona, F.: Hidden Markov Random Field and FRAME Mod-elling for TCA Image Analysis. Proceedings of the Third IASTED InternationalConference on Signal Processing, Pattern Recognition, and Applications, 310 –315 (2006)

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