+ All Categories
Home > Documents > Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen...

Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen...

Date post: 24-Jun-2020
Category:
Upload: others
View: 17 times
Download: 0 times
Share this document with a friend
7
General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from orbit.dtu.dk on: Jul 05, 2020 Markov Random Fields on Triangle Meshes Andersen, Vedrana; Aanæs, Henrik; Bærentzen, Jakob Andreas; Nielsen, Mads Published in: 18th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision Publication date: 2010 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Andersen, V., Aanæs, H., Bærentzen, J. A., & Nielsen, M. (2010). Markov Random Fields on Triangle Meshes. In V. Scala (Ed.), 18th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision: Comunication Papers Proceedings (pp. 265-270)
Transcript
Page 1: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

You may not further distribute the material or use it for any profit-making activity or commercial gain

You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Jul 05, 2020

Markov Random Fields on Triangle Meshes

Andersen, Vedrana; Aanæs, Henrik; Bærentzen, Jakob Andreas; Nielsen, Mads

Published in:18th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision

Publication date:2010

Document VersionEarly version, also known as pre-print

Link back to DTU Orbit

Citation (APA):Andersen, V., Aanæs, H., Bærentzen, J. A., & Nielsen, M. (2010). Markov Random Fields on Triangle Meshes.In V. Scala (Ed.), 18th International Conference in Central Europe on Computer Graphics, Visualization andComputer Vision: Comunication Papers Proceedings (pp. 265-270)

Page 2: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

Markov Random Fields on Triangle Meshes

Vedrana AndersenDTU Informatics

R. Petersens Plads2800 Kgs. Lyngby

[email protected]

Henrik AanæsDTU Informatics

R. Petersens Plads2800 Kgs. Lyngby

[email protected]

Andreas BærentzenDTU Informatics

R. Petersens Plads2800 Kgs. Lyngby

[email protected]

Mads NielsenDIKU

Universitetsparken 12100 Copenhagen

[email protected]

ABSTRACTIn this paper we propose a novel anisotropic smoothing scheme based on Markov Random Fields (MRF). Ourscheme is formulated as two coupled processes. A vertex process is used to smooth the mesh by displacing thevertices according to a MRF smoothness prior, while an independent edge process labels mesh edges accordingto a feature detecting prior. Since we should not smooth across a sharp feature, we use edge labels to control thevertex process. In a Bayesian framework, MRF priors are combined with the likelihood function related to themesh formation method. The output of our algorithm is a piecewise smooth mesh with explicit labelling of edgesbelonging to the sharp features.

Keywords: Mesh, smoothing, Markov Random Fields.

1 INTRODUCTIONMarkov Random Fields (MRF) have been used ex-tensively for solving Image Analysis problems at alllevels. The local property of MRF makes them veryconvenient for modeling dependencies of image pix-els, and the MRF-Gibbs equivalence theorem providesa joint probability in a simple form, making MRF the-ory useful for statistical Image Analysis. While someexamples are mentioned below, MRF have rarely beenused for mesh processing. One reason could be thatMRF are usually defined on regular grids, but this isby no means required.

In this paper we demonstrate that feature preservingmesh smoothing may conveniently be cast in termsof MRF theory. Using this theory we can explic-itly model our knowledge of properties of the surface(prior knowledge, e.g. how smooth the surface shouldbe, which sharp features should it contain) and ourknowledge of the noise (likelihood, e.g. how far do webelieve the measured position of a vertex is likely tobe from the true position). The central element of theMRF formulation is that we use Bayes rule to expressthe probability of any mesh configuration by defining

Permission to make digital or hard copies of all or part ofthis work for personal or classroom use is granted with-out fee provided that copies are not made or distributed forprofit or commercial advantage and that copies bear this no-tice and the full citation on the first page. To copy otherwise,or republish, to post on servers or to redistribute to lists, re-quires prior specific permission and/or a fee.

its of prior and likelihood independently. This divisionof responsibilities often turns out to be a benefit.

For instance, a big advantage of the MRF formulationis that we can use the likelihood to keep the mesh fairlyclose to the input, avoiding the shrinkage associatedwith many other schemes. Unlike [Hildebrandt andPolthier, 2007] we do not obtain a hard constraint, butmeshes far from the input can be made arbitrarily un-likely by choosing an appropriate likelihood function.

We investigate the use of MRF for formulating pri-ors on 3D surfaces in a number of different ways.The smoothness prior encodes the belief that a smoothsurface (according to some fairness criterion) is moreprobable than a noisy surface. In particular, we showhow we can use one MRF to perform explicit labellingof edges according to how sharp they are, and anotherMRF to find optimal vertex positions according to thesmoothness prior. Using our edge labelling from thefirst MRF to control the vertex smoothing, we are ableto recapture very subtle sharp features on the noisymesh.

2 RELATED WORK

Mesh-smoothing algorithms have a long history in thefield of geometry processing since the early work of[Taubin, 1995], which demonstrated the connectionbetween various explicit linear methods using the socalled umbrella operator and low pass filtering. In[Desbrun et al., 1999] a discrete Laplace Beltramioperator was introduced and the connection betweensmoothing and mean curvature flow was explained.Both techniques are efficient, but fail to distinguish

Page 3: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

between the noise and the features of the underlyingobject.

To address this problem, anisotropic diffusion [Des-brun et al., 2000] and diffusion smoothing of the nor-mal field [Tasdizen et al., 2002] were proposed. Theresults are impressive, but the computation complex-ity puts a limit on the size of the model. More ef-ficient methods were also developed, such as non-iterative feature-preserving smoothing [Jones et al.,2003] based on robust statistics, and an adaptation ofbilateral filtering to surface meshes [Fleishman et al.,2003].

Another feature preserving smoothing method, fuzzyvector median smoothing [Shen and Barner, 2004], isa two-step smoothing procedure. In the first step facenormals are smoothed using a robust method whichemploys distance to median normal as smoothingweight. In the next step vertex positions are updatedaccordingly. More recently, in [Diebel et al., 2006]a Bayesian approach was proposed. This methoduses a smoothness prior and the conjugate gradientmethod for optimization. It is feature-preserving, butwithout an explicit feature detection scheme. Similarto [Diebel et al., 2006], we use a Bayesian approach,but unlike that method we obtain feature preservationby explicitly detecting the set of chosen features. Ourmethod is also more flexible, allowing us to use avariety of priors and likelihood potentials.

The method for recovering feature edges proposed in[Attene et al., 2005] is based on the dual process ofsharpening and straightening feature edges. Vertex-based feature detection using an extension of the fun-damental quadric is utilized in a smoothing method de-scribed by [Jiao and Alexander, 2005].

Comprehensive study on the use of MRF theory forsolving Image Analysis problems can be found inbooks [Li, 2001; Winkler, 2003]. MRF theory isconvenient for addressing the problem of piecewisesmooth structures. In [Geman and Geman, 1984] afoundation for the use of MRF in Image Analysisproblems is presented in an algorithm for restorationof piecewise smooth images, where gray-level processand line processes are used. Another application ofMRF for problems involving reconstruction of piece-wise smooth structures is [Diebel and Thrun, 2005],where high-resolution range-sensing images are re-constructed using weights obtained from a regular im-age.

There are some previous examples of using MRF the-ory to 3D meshes, but the applications are somewhatdifferent. In [Willis et al., 2004] MRF are used inthe context of surface sculpting with the deforma-tion of the surface controlled by MRF potentials mod-

elling elasticity and plasticity. MRF was also used formesh analysis and segmentation in [Lavoué and Wolf,2008].

Our work investigates the possibility of formulatingsurface priors in terms of MRF, and using those pri-ors for reconstructing the surface from the noisy date.Unlike most other mesh smoothing algorithms, our ap-proach does not only preserve sharp ridge features, butalso explicitly detects the ridges.

The method described here is not automatic and re-quires an estimation of a considerable set of param-eters. However, this allows a great control over theperformance of the priors.

3 MESH SMOOTHING USING MRFMarkov Random Fields is a powerful framework forexpressing statistical models originating in computa-tional physics, and it has proven highly successful inImage Analysis [Li, 2001; Winkler, 2003]. A MRFis, essentially, a set of sites with associated labels andedges connecting every site to its neighbors. The la-bels are the values which we wish to assign (e.g. pixelcolor, vertex position or edge label), and it is a centralidea in MRF theory that the label at a given site mustonly depend on the labels of its neighbors. This frame-work lends itself well to mesh based surfaces, wherethe neighborhood of a vertex can be naturally definedvia its connecting edges.

Apart from a well developed mathematical frameworkone of the main advantages of MRF is that its Marko-vianity (local property) makes is quite clear what theobjective function is and what a MRF based algorithmaims at achieving. Exponential distributions are oftenused, and the joint probability distribution function ofgiven configuration f (e.g. combined vertex location)is given by

P( f ) ∝ e−∑U( f ) ,

where the U( f ) can be seen as energy terms or poten-tials defined on neighborhoods. In order to find themost likely configuration f , we need to obtain

minf

∑U( f ) . (1)

In our proposed framework, we wish to smooth a givenmesh. Some of the U( f ) in (1) are thus data (likeli-hood) terms penalizing the displacement of the ver-tices in the smoothed mesh relative to the originalmesh. Other terms would be prior terms which expresshow likely a surface is a priori, i.e. without makingreference to how far removed it is from the data.

3.1 LikelihoodWe want the output of the smoothing to relate to theinput mesh, which has an underlying true surface cor-

Page 4: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

v v1

v2

v3

v4

Figure 1: Left: A neighborhood structure for thesmoothness prior. The neighbors of the vertex v aremarked red. When we move vertex v, we only needto look at its neighboring vertices to calculate thechange in the joint smoothness potential. Right:A collection of 4 vertices, expressing two adjacent

faces.

rupted by the noise of the data-acquisition device. As-suming isotropic and Gaussian measurement noise wechoose quadratic function for the likelihood energy

UL(v) = α‖v0−v‖2

where v0 and v denote the initial and the current po-sition of the vertex v. The constant α is used as theweight determining how much faith one has in thedata.

There is always a possibility of plugging in a differ-ent likelihood function in our model, e.g. a volumepreserving likelihood function or likelihood utilizingsome specific knowledge about data acquisition pro-cess.

3.2 Smoothing PotentialAlongside the data term we also have some a pri-ori terms expressing our assumptions about howa smoothed mesh should look. Firstly, we have asmoothing potential, which is basically a penaltyfunction, ρ , based on the difference between thenormals of adjacent faces, see Figure 1

Us (v1,v2,v3,v4) = ρ(n123−n243) , (2)

where n123 and n243 are the normals of the two adja-cent faces. The suitable MRF neighborhood for aboveformulation is defined as follows: two different ver-tices are neighbors if they belong to the adjacent faces.In this smoothing scheme the label of each mesh ver-tex is its spatial position, which is adjusted to mini-mize the chosen energy function.

The choice of the smoothness potential can greatlyinfluence the feature preserving property of thesmoothing. On the one side, there is a over-smoothingquadratic potential developed by [Szeliski andTonnesen, 1992]

ρ(x) = ‖x‖2 ,

e e1 e2

θ12

Figure 2: Left: A neighborhood structure for theedge support prior. The neighbors of the edge e aremarked red. The neighboring edges support eachother if they lie along the same line. Right: A pair ofedges. The support for the edges e1 and e2 depends

on the size of the angle θ12.

on the other side, there is a feature preserving squareroot potential developed by [Diebel et al., 2006]

ρ(x) = ‖x‖ .

In our case, feature preservation will be handled by theexplicit edge labelling, which allows us to use the ag-gressive quadratic potential for smooth regions, with-out thinking about its feature preservation properties.

3.3 Edge LabellingIn many mesh smoothing tasks the presence of clearridge features in the result is part of our a priori ex-pectation. This is included in our MRF model wherewe, as an integral part of the smoothing process, labelmesh edges as being ridge edges or not. Edge labelε is a number from the interval [0,1] which indicateshow probable it is that the given edge is a part of asharp ridge feature. Those labels will later be used tointroduce discontinuities in the smoothing process.

Edge labelling is in itself based on a MRF model con-sisting of two terms, edge sharpness term UE1, and theneighborhood support term UE2.

The larger the dihedral angle φe, of a mesh edge is, themore probable it is that the edge lies along the surfaceridge. The first term is thus given by

UE1(e) = (φ0−φe)ε , (3)

where φ0 is a ridge sharpness threshold, and ε is thelabel assigned to the edge e.

The second term of the edge labelling is the neighbor-hood support, i.e. the presence of other ridge edgesalong the same ridge line. We assign a support energyto all pairs of edges, see Figure 2. A measure of paral-lelism between the edges is used in the formulation ofthe support potential

UE2(e1,e2) =−cos(θ12)ε1ε2 , (4)

where θ12 is the angle between the edges e1 and e2,and ε1 and ε2 are the labels assigned to e1 and e2. Fea-

Page 5: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

ture edges lying on a straight line will have a max-imum support, the orthogonal edges do not supporteach other, and feature edges meeting at a sharp an-gle are discouraged.

There are additional constrains one can use to defineridge edges, like e.g. dihedral angle changing slowlyalong the ridge line, or the expectation that the ridgeedge itself is smooth.

3.4 The Coupled ModelThe smoothing potential and the edge labellingare coupled in a feature preserving scheme, whichsmoothes the mesh, but not over the edges labelledas sharp. This is obtained by using edge labels asweights for the smoothing potential, which is now, forthe setting as in Figure 1

Us (v1,v2,v3,v4) = (1− ε23)ρ(n123−n243) .

The edges labelled as sharp with will not contributeto the smoothness potential, and the smoothed surfacewill be allowed to form a ridge along those edges.

In total, we are minimizing the sum of three terms:the likelihood term, (weighted) smoothing potential,and the edge labelling potential, which in turn consistsof the edge sharpness term and neighborhood supportterm.

3.5 OptimizationAt present we use the Metropolis sampler [Winkler,2003] with simulated annealing for the optimization,i.e. computing a solution to (1). This is a some-what cumbersome but flexible method, allowing forwidespread experimentation with different objectivefunctions. The clear advantage of this approach is thatwe do not make any assumptions about the potentials.

The Metropolis sampler is a random sampling algo-rithm, which generates a sequence of configurationsfrom a probability distribution using a Monte Carloprocedure. The sampling scheme consists of randomlychoosing a new label for a single site, and replacingthe old label with the probability which is controlledby the current temperature. For an initially high tem-perature, the new configuration can be accepted evenif it has a smaller probability that the old one. This al-lows the algorithm to leave local energy minima. Thetemperature then gradually decreases and the systemconverges.

In our case, a new label is either a new vertex posi-tion (randomly sampled in the vicinity of the presentposition), or a new edge label for the ridge detection.Instead of optimizing simultaneously over all definedpotentials, we have in each iteration of the optimiza-tion process first detected the feature edges (consider-

Figure 3: Smoothing fandisk model using our fea-ture preserving method with explicit edge labelling.Left: Fandisk model corrupted with the Gaussiannoise. Edges are initially labelled based only on thesharpness of the dihedral angle. Right: The result-ing smooth mesh and the resulting edge labelling.

ing vertex positions to be fixed), and than displaced thevertices (considering edge labels to be fixed).

More specialized and efficient algorithms have beendeveloped for many kind of MRF problems e.g. viafiltering, belief propagation and graph cuts (in case ofdiscrete labels). After showing that MRF is a good for-mulation of the mesh smoothing problem, the searchfor faster optimization method is part of our ongoingwork. A conjugate gradient method would probablyprovide sufficiently good results in a more efficientway.

4 RESULTSThe results of our experiments prove the feasibilityand versatility of using MRF on triangular meshes.Explicit edge labelling when smoothing models withsharp ridge features is shown it the Figure 3. In an ini-tial noisy mesh it is impossible to detect feature edgesbased only on the local information. However, ouralgorithm converges to a configuration where all theridges get correctly labelled and even the subtle fea-ture edges get detected. Correct edge labelling allowsus to choose aggressive smoothing prior and obtain re-sults superior to using only a single feature preservingprior, as demonstrated in the Figure 4. Note that, un-like the fuzzy vector median smoothing (which is gen-erally very successful in preserving edges and smoothregions), our method detects and preserves a subtleridge in the front of the model, and is partly preservinga disappearing ridge close to models back. The mostother smoothing methods will either miss those subtleridges, or will not remove the low frequency noise.

5 DISCUSSIONThere are many alternative ways of using MRF on tri-angle meshes. Instead of labelling vertices with spa-

Page 6: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

Figure 4: Smoothing fandisk model using the dif-ferent feature preserving methods. Top row: Origi-nal model and the model corrupted with the Gaus-sian noise. The two subtle ridges are circled in theoriginal model. Middle row: Results of fuzzy vec-tor median smoothing and MRF smoothing usingonly the feature preserving square root potential.Bottom row: Results of MRF smoothing using thequadratic potential and the explicit edge labelling.

Note the preserved subtle ridges.

tial positions, vertex labels can also be used to classifyvertices into smooth segments. Furthermore, vertexlabels could be used to detect features, classifying thevertices into those that are a part of the smooth surface,those that are on the ridge and vertices that are a cor-ner, in a manner similar to [Lavoué and Wolf, 2008].MRF can also be defined on mesh faces, either for seg-mentation or aligning face normals.

Having enough prior knowledge of the problem athand, one can tailor the surface potentials to obtain thedesired result. By including the curvature information

Figure 5: Obtaining curvature clamping by pro-viding curvature information to edge detection pro-cess. Left: Initial mesh. Right: The result of clamp-ing the curvature to discourage the concave sharp

ridges.

in the edge labelling process we can detect only certainridges, while skipping the others, obtaining curvatureclamping behavior mentioned in [Botsch et al., 2008]and being the focus of the recent article [Eigensatz etal., 2008], see Figure 5. Extending the size of the ver-tex neighborhood it is possible to formulate the priorfor piecewise quadratic surfaces and also model theridge behavior more precisely.

To demonstrate the great flexibility and versatility ofthe MRF formulation we include another example ofmesh smoothing. Inspired by a two-step smoothingmethod [Shen and Barner, 2004], we used MRF toobtain the smooth normal field, which is then usedfor reconstructing vertex positions. Now we have themesh faces as the sites of the MRF, with the MRFlabels being the normal direction of the faces. Thevertex update step is taken directly from [Shen andBarner, 2004], which in turn uses a method developedby [Taubin, 2001] where the system of equations getssolved in a least squares sense to obtain the vertex po-sitions update.

One of the important differences between the vertexbased smoothing and face based smoothing is the pos-sibility to preform smoothing of the normals withoutchanging the geometry of the mesh, which makes thisapproach more effective. The disadvantage is that it isnot so straightforward to include displacement-basedlikelihood function. The results of using this methodcan be seen on the Figure 6.

REFERENCES[Attene et al., 2005] Marco Attene, Bianca Falci-

dieno, Jarek Rossignac, and Michela Spagnuolo.Sharpen & bend: Recovering curved sharp edgesin triangle meshes produced by feature-insensitivesampling. IEEE Trans. on Visualization and Comp.Graph., 11(2):181–192, 2005.

[Botsch et al., 2008] Mario Botsch, Mark Pauly, LeifKobbelt, Pierre Alliez, Bruno Lévy, StephanBischoff, and Christian Rössl. Geometric model-

Page 7: Markov Random Fields on Triangle Meshes · Markov Random Fields on Triangle Meshes Vedrana Andersen DTU Informatics R. Petersens Plads 2800 Kgs. Lyngby Denmark va@imm.dtu.dk Henrik

Figure 6: Smoothing a noisy cube using the faceand the edge processes. Left: A synthetic cube cor-rupted with Gaussian noise with the initial normalfield and the initial edge labelling. Right: The re-sulting mesh, with the smooth normal field and the

resulting edge labelling.

ing based on polygonal meshes. Eurographics 2008Full-Day Tutorial, 2008.

[Desbrun et al., 1999] Mathieu Desbrun, MarkMeyer, Peter Schröder, and Alan H. Barr. Implicitfairing of irregular meshes using diffusion and cur-vature flow. In SIGGRAPH ’99: Proc. of the 26thAnnual Conf. on Comp. Graph. and InteractiveTechniques, pages 317–324, 1999.

[Desbrun et al., 2000] Mathieu Desbrun, MarkMeyer, Peter Schröder, and Alan H. Barr.Anisotropic feature-preserving denoising of heightfields and images. In Proc. of Graphics Interface,pages 145–152, 2000.

[Diebel and Thrun, 2005] James R. Diebel and Sebas-tian Thrun. An application of Markov random fieldsto range sensing. In Proc. of Conf. on Neural Infor-mation Processing Systems, 2005.

[Diebel et al., 2006] James Richard Diebel, SebastianThrun, and Michael Brünig. A Bayesian methodfor probable surface reconstruction and decimation.ACM Trans. on Graphics, 25, 2006.

[Eigensatz et al., 2008] Michael Eigensatz,Robert Walker Sumner, and Mark Pauly. Curvature-domain shape processing. Comp. Graph. Forum,27(2):241–250, 2008.

[Fleishman et al., 2003] Shachar Fleishman, IddoDrori, and Daniel Cohen-Or. Bilateral meshdenoising. In SIGGRAPH ’03: ACM SIGGRAPH2003 Papers, pages 950–953, 2003.

[Geman and Geman, 1984] Stuart Geman and Don-ald Geman. Stochastic relaxation, Gibbs distri-butions, and the Bayesian restoration of images.IEEE Trans. on Pattern Analysis and Machine In-telligence, 6(332):721–741, 1984.

[Hildebrandt and Polthier, 2007] Klaus Hildebrandtand Konrad Polthier. Constraint-based fairing ofsurface meshes. In SGP ’07: Proc. of the 5thEurographics Symp. on Geometry Processing,

pages 203–212, 2007.[Jiao and Alexander, 2005] Xiangmin Jiao and

Phillip J. Alexander. Parallel feature-preservingmesh smoothing. In Int. Conf. on ComputationalScience and Its Applications (4), pages 1180–1189,2005.

[Jones et al., 2003] Thouis R. Jones, Frédo Durand,and Mathieu Desbrun. Non-iterative, feature-preserving mesh smoothing. In SIGGRAPH ’03:ACM SIGGRAPH 2003 Papers, pages 943–949,2003.

[Lavoué and Wolf, 2008] Guillaume Lavoué andChristian Wolf. Markov Random Fields forImproving 3D Mesh Analysis and Segmentation.In Eurographics 2008 Workshop on 3D ObjectRetrieval, 2008.

[Li, 2001] Stan Z. Li. Markov Random Field Mod-eling in Image Analysis. Springer Verlag, Tokyo,second edition, 2001.

[Shen and Barner, 2004] Yuzhong Shen and Ken-neth E. Barner. Fuzzy vector median-based sur-face smoothing. IEEE Trans. on Visualization andComp. Graph., 10(3):252–265, 2004.

[Szeliski and Tonnesen, 1992] Richard Szeliski andDavid Tonnesen. Surface modeling with orientedparticle systems. In SIGGRAPH ’92: Proc. of the19th Annual Conf. on Comp. Graph. and Interac-tive Techniques, pages 185–194, 1992.

[Tasdizen et al., 2002] Tolga Tasdizen, RossWhitaker, Paul Burchard, and Stanley Osher.Geometric surface smoothing via anisotropicdiffusion of normals. In VIS ’02: Proc. of the Conf.on Visualization 2002, pages 125–132, 2002.

[Taubin, 1995] Gabriel Taubin. A signal processingapproach to fair surface design. In SIGGRAPH ’95:Proc. of the 22nd Annual Conf. on Comp. Graph.and Interactive Techniques, pages 351–358, 1995.

[Taubin, 2001] Gabriel Taubin. Ibm research report:Linear anisotropic mesh filtering. Technical ReportRC22213, IBM Research Division T.J. Watson Re-search Center, 2001.

[Willis et al., 2004] Andrew Willis, Jasper Speicher,and David B. Cooper. Surface sculpting withstochastic deformable 3d surfaces. In ICPR ’04:Proc. of the 17th Int. Conf. on Pattern Recognition,volume 2, pages 249–252, 2004.

[Winkler, 2003] Gerhard Winkler. Image Analysis,Random Fields and Markov Chain Monte CarloMethods: A Mathematical Introduction. SpringerVerlag, 2003.


Recommended