Improving 2D mesh image segmentation with Markovian Random Fields

Alex J. Cuadros-Vargas, Leandro C. Gerhardinger, Mario de CastroJoao Batista Neto and Luis Gustavo Nonato

Instituto de Ciencias Matematicas e de Computacao - USPCP 668, Sao Carlos, SP, Brazil 13560-970

alexj,leandro,mcastro,jbatista,gnonato@{icmc.usp.br}

Abstract

Traditional mesh segmentation methods normally oper-ate on geometrical models with no image information. Onthe other hand, 2D image-based mesh generation and seg-mentation counterparts, such as Imesh [6] perform the taskby following a set of well defined rules derived from the ge-ometry of the triangles, but with no statistical informationof the mesh elements.

This paper presents a novel segmentation method thatcombines the original Imesh image-based segmentation ap-proach with Markovian Random Field (MRF) models. Ittakes an image as input, generate a mesh of triangles and,by treating the mesh as a Markovian field, produces qualityunsupervised segmentation.

The results have demonstrated that the method not onlyprovides better segmentation than that of original Imesh,but is also capable of producing MRF-like segmentationoutput for certain types of images, with considerable cutin processing times.

1 Introduction

Image segmentation by Markovian Random Field(MRF) models is an expanding research field. Traditionally,the attractiveness of the MRF approach to segmentation isits ability to correctly model images with texture patternsthat exhibit random behaviour, providing good segmenta-tion [13, 9]. Moreover, the process is accomplished in anunsupervised fashion and is also capable of representingmany stochastic problem by modelling complex featureswith a relatively reduced number of parameters.

Applications of MRF to segmentation may range fromremote sensing [23] to medicine [22] and are commonlyapplied to 2D data. However, some results can be foundin the 3D domain. Chistopher [4] has developed a MRFsegmentation method for inherent 3D medical data, such asComputed Tomography and Magnetic Resonance Imaging.

In the MRF approach to segmentation, an image is rep-resented by the capture of local features, determining forevery pixel in the image its grey level dependence with re-lation to its neighbours’ grey level. This dependence is ex-pressed in terms of a conditional probability distribution,computed over the intensity of each pixel. Any decom-position that possess neighbourhood relationships of suchfashion and well established segmentation properties can besegmented by MRF.

A typical example of such decomposition are imagesrepresented by meshes, as neighbourhood relationships caneasily be defined for them. Furthermore, meshes can bebuilt so as to incorporate some specific features of the im-age, turning segmentation an easier and faster task. Anotheraspect is that images represented by meshes possess a re-duced number of cells (when compared with that of pixel orvoxel representations), demanding a lower computationaleffort in the segmentation process.

Traditionally, mesh segmentation methods operate ongeometrical models with no image information attached[25, 14, 24]. Such methods aim to find structures of interestin the mesh suitable for computer vision applications.

In spite of good properties of mesh representation, theuse of MRF in such decomposition has never been explored.Another important fact not usually considered is that meshrepresentation provides, as well as the traditional informa-tion required by MRF, additional geometrical informationthat can empower the MRF segmentation output. In fact,instead of solely looking at the pixel grey level, improvedsegmentation could be attained if MRF were combined withfeatures provided by meshes such as area, edge length, cellaspect ratio and so on.

This paper presents a novel unsupervised 2D segmen-tation method that combines both MRF and meshes. Themethod extends the concept of MRF segmentation, tradi-tionally focused on pixel local features and its neighbour-hood, to a broader framework in which features extractedfrom meshes are also considered in the computation of thesegmented image. The results have shown a significant im-

provement in the quality of the final segmentation whencompared with that accomplished with a set of well definedmesh segmentation rules alone, the original Imesh. Also,the new method exhibits much lower segmentation times incomparison with traditional MRF segmentation methods.

This paper is organised as follows. Section 2 describesthe MRF image segmentation method. The concepts andthe decomposition process for the mesh generation are de-scribed is section 3. The combination of both MRF andmesh is introduced in section 4 and some results are pre-sented in section 5. Conclusions are finally found in section6.

2 Markovian Random Field Model to seg-mentation

Many objects in real images are characterised by textureof random or non deterministic behaviour. For such casesa probabilistic approach to segmentation can be desirable.Many works on stochastic models for images have been de-veloped [12, 7, 13, 5] and successfully applied in differentsituations [18, 1, 2, 21].

For such models, objects in a digital image are seen as agroup of pixels enclosed in a given area. The representationof each object can be done by labelling them and construct-ing a label field. Generally, stochastic segmentation algo-rithms define such labels as random variables, grouped intoa so called 2D random field.

Many random field models exist [19] such as Gibbs,Ising, Potts, Pickard and Markov. The latter, is very ad-equate for modeling texture (especially micro–texture) forthat it defines a probability function for the field by meansof purely local features, ie, neighbouring features.

Unsupervised segmentation by stochastic models be-longs to the family of incomplete data problems. Hence,the segmentation task is, in the end, a problem of statisticaloptimization of a given criterion. In this model the imageto be segmented is theobserved data; the results sought,normally named class map or label field, is considered themissing dataand; the set ofmodel parametersis an item tobe computed. The implementation of such approach com-prises a two-fold process: a) parameter estimation and b)the statistical optimization itself.

The parameter estimation process can be accomplishedby some iterative algorithms: EM (Expectation Maximiza-tion), SEM (Stochastic Estimation-Maximization) and ICE(Iterative Conditional Estimation). For the statistical opti-mization, two processes are known: MAP (Maximizationa Posterior) [7, 15] and MPM (Maximizer of the PosteriorMarginals) [16, 5], which is more appropriate to image seg-mentation than MAP [16]. While the latter assigns the samecost for every incorrect segmentation over the entire image,the former assigns a cost based on the number of misclassi-

fied pixels. This work has adopted the EM/MPM approachas proposed by Comer [5]. The approach will be brieflydescribed in the next section.

2.1 The EM/MPM segmentation

In the Markovian EM/MPM image segmentationmethod, the observed data is the observed image, given byY = Y1, ...,YN and the label fieldX = X1, ...,XN is the miss-ing data. X andY are 2D random fields of a rectangulargrid S with N variables. According to Bayes’ Theorem,the posterior probability pX|Y is proportional tothe prior probability pX times a likelihoodfunction fY|X . Hence,

pX|Y(x|y,θ) ∝ pX(x)× fY|X(y|x,θ), (1)

where θ is the parameter array for the model andy =y1, ...,yn andx = x1, ...,xn are sample realizations ofY andX, respectively. Eachxs can be assigned the value of a labelk = 1, ...,L, with L being the number of labels previouslydefined. On the other hand,ys are grey levels in the range0, ...,255.

The segmentation methods seeks to find an estimate forthe label fieldx through the MPM optimization method,whose minimization criterion is the expected number ofmislabelled pixels. As an unsupervised segmentationmethod, the EM algorithm is then employed to estimate theparametersθ of the model.

In the original formulation proposed by Comer and Delp[5] for image segmentation, the probabilityfYs|X , ∀s ∈{1, ...,N} is considered a normal distribution that only de-pends onXs. Hence,

fY|X(y|x,θ) =N

∏s=1

fYs|Xs,θ(ys|xs,θ)

=N

∏s=1

1√2πσ2

xs

exp

(− (ys−µxs)

2

2σ2xs

).

(2)The array of parametersθ are the means and variance

of each label, ie,θ = [µ1,σ21 , ...,µL,σ

2L ]. The distribution

pX of the label field is aMarkovian Random Field[19, 8] that, according to Hammersley-Clifford’s theorem[3, 12], has a conditional probability mass function given asa Gibbs distribution

pX(x) =1Z

exp(−U(x)) , (3)

whereZ is a normalisation factor andU(x) is anenergyfunction in the form

U(x) = ∑c∈C

Vc(x). (4)

Vc(x) is a function that depends on thecliques and thefamily {Vc : c∈C} is calledpotential . The set of pixelsC⊆ S is a clique in the label field if, for any pixelsr,s∈C,thens∈ Gr . That is, they are all neighbours. A collectionG = {Gs⊆ S,s∈ S} is a neighbouring system forS if, forany pixel ins∈ S, thens 6∈ Gs ands∈ Gr ⇐⇒ r ∈ Gs,∀r ∈S. The elements ofGs are then the neighbour pixels of apixel s. Figure 1 shows a pixel and its 4– and 8–connectedneighbourhood, with its respective cliques.

(a) (b)

(c) (d)

Figure 1. ( a) 4-connected neighbourhood and(b) its corresponding cliques. ( c) 8-connectedneighbourhood and ( d) its correspondingcliques.

For image segmentation purposes, an adequate energyfunction is given by

U(x) = ∑{r,s}∈C

β t(xr ,xs)+ ∑{r}∈C

γxr , (5)

where

t(xr ,xs) ={

0, if xr = xs,1, if xr 6= xs.

(6)

Notice that only the double cliques (two pixels) andsingle (one pixel) are taken into account in this model.The parameterβ , called Spatial InteractionParameter , defines a weight for the penalty functiont(xr ,xs) of a double clique. This function aims to preventneighbour pixel from having different labels. The set of pa-rametersγk,k = 1, ...,L, which acts upon the single cliques,can be considered as the occurrence costs of each labelk.Since this is a unsupervised segmentation, the parametersγk are, generally, assumed to be 0,∀k.

3 Mesh Generation

This section shows the process of mesh generation.Some basic definitions and terminologies are introduced,followed by the description of the mesh computation froman image.

3.1 Basic Concepts

Let S be a set of points inR2. A triangulation (mesh)of S is a two-dimensional simplicial complexM whose ver-tices are the points ofS, and anyk-simplex ofM, k = 0,1,is contained in at least a 2-simplex (triangle) ofM. If theunion of all simplices inM makes up the convex hull ofSand the circumcircle of each triangle inM does not containin its interior any point ofS thenM is calledDelaunay tri-angulation[10].

A planar graphis a graphG with vertices inR2 whereeach edge is a straight-line segment with ends inG and ife1 ande2 are two edges ofG, e1∩ e2 is either empty or avertex of G. Given a planar graphG, a mesh conformedby G is a triangulationM where each vertex ofG is in Mand if e is an edge inG then|e|= |e1∪e2∪·· ·∪ek|, whereei ∈ M, i = 1, . . . ,k are edges ofM and | · | represents theunderling space, i.e., each edge ofG can appear subdividedas a set of edges inM.

Let Sbe a set of points andM be a mesh (triangulation)of S, if M = M1∪M2∪·· ·∪Mk, where eachMi is a triangu-lation andMi ∩M j , i 6= j is either empty or a planar graphthen{M1,M2, · · · ,Mk} is said ak-partitioningof M in sub-meshes Mi , i = 1, . . . ,k.

An m×n imageis a functionI : [0, . . . ,m]× [0, . . . ,n]→R+ that assigns to each pointp∈ [0, . . . ,m]× [0, . . . ,n]⊂Z2

a non-negative scalarI(p). The pair(p, I(p)) is called pixel.

3.2 Image Mesh Decomposition

Image mesh decomposition concerns to the process ofgenerating a mesh (triangulation) that fits features containedin the image. In our context, triangles should not be allowedto cross different regions of the image, that is, each triangleshould be spatially contained in only one region of the im-age.

Let T be the set of triangles of a Delaunay triangu-lar meshM whose vertices are points of an imageI andE : T → R+ be a function that associates an error measureto each triangle inT. In fact, functionE measures how gooda triangle is regarding a specific property, that is,E enablesto decide whether or not a triangle must belong to the trian-gulation.

Different strategies to define the functionE have beenpresented in the literature, but usually such approaches relyon evaluatingE by traversing all pixels inside a triangle

t so as to decide, based on some characteristic of the im-age, whether or nott is an appropriated triangle. In general,whenE indicates thatt is a bad triangle, the triangulation isupdated by inserting new points withint, thus eliminating itfrom M [11].

Although widely employed, the bad triangles removalstrategy described above presents two main drawbacks.Traversing all pixels within a triangle may demand a highcomputational cost. Each time the triangulation is updatedall the new triangles must be scanned in order to evaluateE.Another concern is the insertion of new points in the trian-gulation. If not handled properly, this operation can result inan accumulation of points around already existing vertices.

To avoid the problems described above, we adopt anstrategy based on the medians of the triangles to define thefunctionE. By traversing medians only, one can reduce thecomputational effort while being effective in detecting tri-angles that go across different regions of the image.

Let h1,h2,h3 be the three medians of a trianglet ∈ M.Consider the sets of pointsPh j = {p ∈ h j | E (p) ≥ cE }whereE is an edge detection operator andcE is an userdefined scalar. Therefore,Ph j is the set of points where theedges of the image intersect the medianh j .

Let αi , i = 1,2,3 be the baricentric coordinates of a pointpk ∈ Ph j andA(pk) = min{αi} be a function that associatesto eachpk its smallest baricentric coordinate. LetDM(pk)be the square distance betweenpk and its closest vertexin M, that is,DM(pk) = min

v j∈M{d2(pk,v j)}, whered(·, ·) is

the Euclidean distance.ph j denotes the point ofPh j whereDM(pk) is maximal. From these definitions, the error func-tion E can be stated as follows:

E(t) = max{A(ph j )}, j = 1,2,3 (7)

Figure 2(b) gives a geometric interpretation of the func-tion E. The baricentric coordinates of a pointpk is relatedwith the areas of the triangles formed bypk and the verticesof the triangle that containspk. Therefore,A(ph j ) measureshow much the area of a trianglet is enclosed within a re-gion of the image. A small value ofA(ph j ) indicates thath j

intersects an edge of the image close to the boundary of thetriangle. Thus, values ofE(t) close to zero indicate thatt iswell fitted within a region in the image. Hence, a triangletis considered unsuitable ifE(t) > cE, where 0≤ cE ≤ 1 isan user defined scalar.

Unsuitable triangles are eliminated by inserting, in theDelaunay triangulation, the pointph j such thatE(t) =A(ph j ). Sinceph j are points chosen to be as far as possiblefrom the vertices ofM, the problem of dense accumulationof points around existing vertices is reduced.

After the mesh generation process, segmentation can beperformed by combining image attributes with topologicaland geometric information contained in the mesh so as to

a) b)

Figure 2. Error measure in a triangle. a) De-tection of region borders, b) Geometric inter-pretation of the function E.

group cells into sub-meshes.

4 Mesh + MRF segmentation

This section shows how the traditional MRF segmenta-tion model over neighbouring pixels can be extended to 2Dmesh structures.

The greatest different between mesh structures and im-ages is in the geometry of the elements handled by both.Image pixels have the same shape, geometry and numberof neighbours. 2D meshes, on the other hand, contain tri-angles of varied sizes, symmetry and a neighbourhood withdifferent number of elements. Moreover, each neighbouringelement may exhibit a wide–range adjacency degree. Fig-ures 3(a) and 3(b) illustrate a general neighbouring schemefor a 2D mesh of a set of triangles and its respective cliques.

(a) (b)

Figure 3. ( a) Neighbourhood of a 2D meshand (b) its corresponding cliques.

To handles mesh triangles, the original EM/MPM mustbe modified. The MPM estimation algorithm for the labelfield X is extended so as to assign a weight, whose value de-pends on the planar geometry shared by the triangle and itsneighbours, to every neighbouring element of every trianglein the mesh. Equation (6) is then rewritten as:

t(xr ,xs) =

{0, if xr = xs,as,r

ps, if xr 6= xs,

(8)

whereas,r is the length of the edge shared by the trianglessandr, whereasps is the perimeter ofs.

The parameter estimation algorithm EM is also altered.For an image, the intensity of each pixel has the same im-portance (weight) in the process. Mesh triangles, however,due to their varied geometry may possess different weights.The estimates for the set of parametersθ, as described byComer and Delp [5], becomes:

µk(p) =

N

∑s=1

ysP(Xs = k|Y = y,θ(p−1))×as

N

∑s=1

P(Xs = k|Y = y,θ(p−1))×as

(9)

and

σ2k (p) =

N

∑s=1

(ys−µk(p))2P(Xs = k|Y = y,θ(p−1))×as

N

∑s=1

P(Xs = k|Y = y,θ(p−1))

(10)and as is the area of the triangles, used as a weight forthe computation of variance and mean values of grey levelof each labelk. This proposal is equivalent to dividing, inequation 2, the varianceσ2

xsby the area of the triangle.

5 Results

To best illustrate the benefits of the proposed method weprovide results from MRF segmentation alone (as describedin section 2 and the Imesh method [6], a image based meshgeneration and segmentation technique with no MRF-likeneighbourhood relationship (as described in section 3).

We then confront both of them with the results obtainedfrom the proposed method, for different values ofβ (Eq.5). This parameter governs the influence of the geome-try over the Mesh+MRF segmentation, as stated in Eq. 8.Whenβ = 0, the geometry information from Eq. 8 is notconsidered. In this paper, we refer to this segmentation asMesh+MRF with no geometry information. Whenβ > 0,not only the neighbouring relationship of the MRF model,but also geometry information is added to the segmentationprocess. The higher theβ parameter, the more geometry in-formation the model conveys. This segmentation is referredto asMesh+MRF with geometry information .

The behaviour of the Mesh+MRF segmentation methodfor increasing values ofβ is illustrated in Figure 4. Figures

a) b)

c) d)

e) f)

Figure 4. a) Original CT slice of a mon-key head; b) Mesh Generated from (a); c)Mesh+MRF with no geometry information(β = 0). Mesh+MRF segmentation with geom-etry information: d) β = 1; e) β = 2; f) β = 3

4(a) and 4(b) are the original CT slice of a monkey head andits image-generated mesh, respectively. Figures 4(c), 4(d),4(e) and 4(f) are the 2–class segmentation results forβ = 0,1, 2 and 3, respectively. Whenβ = 0, no geometry informa-tion from the triangles is considered and the segmentationis poor. As the value ofβ increases the perimeter and thelength of the edge shared by neighbouring triangles becomemore relevant, yielding a smoother and more accurate seg-mentation.

Figure 5 illustrates the segmentation process (3 classesconsidered) for a CT scan slice of a foot. Figures 5(a) and5(b) are, respectively, the original image and the mesh gen-erated by the Imesh technique. Segmentation for the MRFmethod alone and Imesh (with no MRF model added) aregiven in 5(c) and 5(d), respectively. Images 5(e) and 5(f)are the results obtained with the Mesh+MRF segmentationwithout and with geometry information, respectively.

Notice how the segmentation from Mesh+MRF with ge-ometry information is significantly more consistent than

a) b)

c) d)

e) f)

Figure 5. a) Original MRI image of a foot; b)Mesh Generated from (a); c) Segmentation byMRF alone; d) Mesh segmentation alone; e)Mesh+MRF segmentation with no geometryinformation; f) Mesh+MRF segmentation withgeometry information ( β = 3)

that computed from mesh segmentation without the MRFextension. Although not as good as the Markovian segmen-tation alone (Figure 5.c), due to the highly textural natureof the foot image, the result in 5(f) is an evidence of how animage-based mesh segmentation method can benefit fromincorporating features originally designed for texture imagesegmentation such as MRF.

The final experiment of Figure 6 shows how the newmethod can contribute to shorten the processing time whencompared with MRF segmentation alone. Image 6(a). isa high resolution (140x570) satellite image of the Superiorlake on the Canadian border with the USA. Images 6(b) and6(c) are, respectively, the segmentation results (2 classes)for the MRF alone and the Mesh+MRF method with ge-ometry information, respectively. Although very similar inquality, the segmentation by the the new method is approx-imately 10x faster than that by MRF (see table 1).

a)

b)

c)

Figure 6. a) Original satellite image of a lake;b) Segmentation by MRF alone; c)Mesh+MRFsegmentation with geometry information

6 Conclusions

This paper introduces a new method that combines MRFwith Mesh to achieve image segmentation. Unlike tradi-tional mesh segmentation techniques, which operate on ge-ometrical models, our method takes an image as input, gen-erate a mesh of triangles and, by treating this mesh as aMarkovian field, produces quality unsupervised segmenta-tion.

From the experiments we can devise two major contri-butions from this work. The first is the improvement inthe quality of the segmentation when compared with thesegmentation criteria so far available in the original Imeshmethod [6]. Such criteria did not take into account thestatistical information now provided by the addition of theMRF functionality. The experiment of Figure 5 illustratessuch behaviour. For image with some textural informationand a considerable amount of noise, such that of the foot,the proposed method yields a more homogeneous segmen-tation than traditional Imesh, for example. However, asexpected, for such kind of images the MRF segmentation

a)

b)

Figure 7. Meshes generated from fig. 6.a. a)Mesh employed in the segmentation of 6.c; b)Better quality mesh;

Image MRF Mesh+MRFMesh Gen. Segment Total

Monkey(256x256) 10 1.125 0.468 1.593Foot(256x256) 12 1.263 1.375 2.638Lake(1140x570) 90 6.64 2.343 8.983

Table 1. Segmentation times (in seconds) forthe MRF and Mesh+MRF methods

method alone gives a better result. On the other hand, asdemonstrates the experiment of Figure 6, the Mesh+MRFmethod can still provide comparable results with those ofMRF alone for images with some textural information andirregular edges, provided the edges of the object sought arebetter delineated.

The second major contribution is a significant reductionin the segmentation times, especially when compared withMRF segmentation alone. As opposed to traditional MRFsegmentation methods that operate on pixels, our approachdeals with a mesh of triangles. Since the number of trian-gles of the mesh is much smaller than that of pixels in thecorresponding image, the resulting segmentation is muchfaster. The values in table 1 give some evidences of thisbehaviour. Despite being a two-stage process (mesh gen-eration and segmentation itself) the overall segmentation

times for the Mesh+MRF method are much smaller thanthose achieved with MRF method alone. The difference inprocessing times becomes more evident, for data of largerdimensions such as the satellite image of the lake. Reportedsegmentation time have been computed on a Xeon 3.2 GHzsystem, with 2GB RAM, running Windows.

Finally, it should be pointed out the ability of the methodto incorporate quality criteria to the mesh elements [20, 17],while keeping the edges obtained during the segmentationprocess. Figure 7(a) shows the original mesh over the seg-mented image of the Lake 6(a). Figure 7(b) depicts a refinedmesh derived from 7(a), with all edges preserved. A mini-mum angle value of 30o has been used as a quality criterionin this example.

AcknowledgementsWe wish to acknowledge the Brazilian financial agencies

CNPq (proc. 307268/2003-9, FAPESP (proc. 04/02810-0, 03/02815-0 and 02/05243-4) and CAPES involved in fi-nancing the project that generated these results.

References

[1] A. Balan. Tecnicas de segmentacao de imagens aereas paracontagem de populacao de aves. Master’s thesis, ICMC,Universidade de Sao Paulo, Sao Carlos, SP, Brasil, Fevereiro2003.

[2] A. Balan, A. Traina, C. Traina, and P. Azevedo. Fractal anal-ysis of image textures for indexing and retrieval by content.pages 581–586. 18th IEEE Symposium on Computer-BasedMedical Systems, June 2005.

[3] J. Besag. Spatial interaction and the statistical analysis oflattice systems. Journal of the Royal Statistical Society,36(2):192–236, 1974.

[4] L. Christopher.Bayesian segmentation of three dimensionalimages using the EM/MPM algorithm. PhD thesis, PurdueUniversity, 2003.

[5] M. Comer and E. Delp. The em/mpm algorithm for seg-mentation of textured images: Analysis and further exper-imental results. IEEE Transactions on Image Processing,9(10):1731–1744, October 2000.

[6] A. Cuadros-Vargas, L. Nonato, M. R., and E. T. Imesh: Animage based quality mesh generation technique. InBrazilianSymposium on Computer Graphics and Image Processing,18 (SIBGRAPI). IEEE Computer Society, 9–12 Oct. 2005.

[7] H. Derin and H. Elliott. Modeling and segmentation of noisyand textured images using gibbs random fields.IEEE Trans-actions on Pattern Analysis Machine Intelligence, 9(1):39–55, 1987.

[8] E. Dougherty.Random Processes for Image and Signal Pro-cessing. IEEE Series on Imaging Science and Engineering.SPIE/IEEE, 1998.

[9] R. Fjortoft, W. Pieczynski, and Y. Delignon. Generalizedmixture estimation and unsupervised classification based onhidden markov chains and hidden markov fields.Scandina-vian Conference on Image Analysis, 2001.

[10] S. Fortune. Voronoi diagrams and delaunay triangulation.In Computing in Euclidean Geometry, Wor1d Scientific Pub.Co., pages 193–233, 1994.

[11] M. Garland and P. Heckbert. Fast polygonal approximationof terrains and height fields. Technical Report CMU-CS-95-181, Carnegie Mellon University, 1995.

[12] S. Geman and D. Geman. Stochastic relaxation, gibbs dis-tributions, and the bayesian restoration of images.IEEETransactions on Pattern Analysis and Machine Intelligence,6(6):721–741, 1984.

[13] N. Giordana and W. Pieczynski. Estimation of generalizedmultisensor hidden markov chains and unsupervised imagesegmentation.IEEE Transaction on Pattern Analysis andMachine Intelligence, 19(5):465–475, 1997.

[14] L. Guillaume, D. Florent, and B. Atilla. Curvature tensorbased triangle mesh segmentation with boundary rectifica-tion. In Computer Graphics International. IEEE ComputerSociety, 16-19 Jun 2004.

[15] S. Lakshmanan and H. Derin. Simultaneous parameter esti-mation and segmentation of gibbs random fields using sim-ulated annealing.IEEE Transactions on Pattern AnalysisMachine Intelligence, 11(8):799–813, 1989.

[16] J. Marroquin, S. Mitter, and T. Poggio. Probabilistc solutionof ill-posed problems in computacional vision.The Ameri-can Statistician, 82:76–89, March 1987.

[17] G. Miller, E. Steven, and N. Walkington. When and whyRuppert’s algorithm works. InProceedings of the 12th In-ternational Meshing Roundtable, pages 91–102. Sandia Na-tional Laboratory, September 2003.

[18] J. Rajapakse, J. Giedd, and J. Rapoport. Statistical approachto segmentation of single-channel cerebral mr images.IEEETransactions on Medical Imaging, 16(2), April 1997.

[19] A. Rosholm.Statistical Methods for Segmentation and Clas-sification of Images. PhD thesis, Technical University ofDenmark, 1997.

[20] J. Ruppert. A delaunay refinement algorithm for quality2-dimensional mesh generation.J. Algorithms, 18(3):548–585, 1995.

[21] W. Schwartz and H. Pedrini. Segmentacao de imagens deterrenos baseada na associacao de caracterısticas de texturascom dependencia espacial modelada por campo aleatorio demarkov. Goiania, 2005. XII Simposio Brasileiro de Senso-riamento Remoto.

[22] H. Sheshadri and A. Kandaswamy. Detection of breast can-cer by mammogram image segmentation.Journal of CancerResearch and Therapeutics, 1(4):232–234, 2005.

[23] C. Yonggfeng and S. Hong. An unsupervised segmentationmethod mased on mpm for sar images.IEEE Geoscienceand Remote Sensing Letters, 2(1):55–60, 2005.

[24] H. Zhang and R. Liu. Mesh segmentation via recursive andvisually salient spectral cuts. InVision, Modeling, and Visu-alization. Erlangen, 16-18 Nov 2005.

[25] Y. Zhidong, S. Kumar, and C.-C. Kuo. Mesh segmentationschemes for error resilient coding of 3-d graphic models.IEEE Trans. on Circuits and Systems for Video Technology,15(1):138–144, 2005.