Surface from Scattered PointsA Brief Survey of Recent Developments

Oliver SchallMax-Planck-Institut fur Informatik

schall@mpi-sb.mpg.dehttp://www.mpi-sb.mpg.de/˜schall

Marie SamozinoINRIA - Sophia Antipolis

Marie.Samozino@sophia.inria.frhttp://www-sop.inria.fr/geometrica/team/Marie.Samozino

AbstractThe paper delivers a brief overview of recentdevelopments in the field of surface reconstruc-tion from scattered point data. The focus is oncomputational geometry methods, implicit surfaceinterpolation techniques, and shape learningapproaches.

Keywords: surface reconstruction, scattereddata

1 IntroductionIn the mid-1980s the problem of surface recon-struction from scattered data was probably first ad-dressed by Boissonnat [17]. Further attraction waslead to surface reconstruction from point clouddata by the research of Hoppe et al. [39]. Sincethen surface reconstruction became an active re-search field with two main surface reconstructionapproaches.

One important research direction are Delaunay-based methods [2, 9, 11, 13, 16, 24, 27, 31] whichusually involve the computation of a Delaunaycomplex or of dual structures using the scattereddata. The surface is usually reconstructed by ex-traction from the previously computed Delaunaycomplex. Detailed surveys on computational ge-ometry methods for surface reconstruction can befound in [22, 26]. A short survey is presentedin [23].

The second important research direction are vol-umetric methods [14, 20, 56, 66]. Those approx-

imate the scattered data by a 3D function. Thesurface can then be explicitly reconstructed by ex-tracting it as the zero-level set of the computedfunction. A survey can be found in [53]. Re-cent research trends include Moving Least Squares(MLS) surfaces [1, 4, 5, 12] defined implicitly bythe MLS projection operator.

Furthermore, different methods like zipperingmeshes obtained from range images [67] and sta-tistical learning [41, 42, 73] have been developed.

Recent research in surface reconstruction fromscattered data is also focused on reconstructionfrom scattered polygonal data [43, 64]. As pointcloud data acquired from real world objects us-ing for instance range scanners is always corruptedwith noise, the reconstruction from noisy pointdata was recently addressed in [21, 28, 47, 58] aswell as the processing of uncertain data in [12,59].Furthermore, point cloud data can be undersam-pled in regions of low visibility for the scanningdevice. Therefore, the problem of surface re-construction from incomplete data is addressedin [25, 43, 63, 68].

A very comprehensive, yet preliminary, surveyof surface reconstruction techniques can befound in [3]. We contribute to [3] by testingseveral surface reconstruction software tools andrevealing their strengths and weaknesses. We alsodemonstrate that surface reconstruction resultscan be drastically improved if an appropriatepreliminary filtering method [62] is applied tooriginal scattered data.

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The rest of this report is organized as follows. InSection 2 Delaunay-based methods are considered.Section 3 gives an overview of approaches com-puting an implicit surface. Learning-based meth-ods are reviewed in Section 4. Finally, a filteringand denoising method is described in Section 5.

2 Delaunay-based Methods

In this section we are presenting an overviewof Delaunay-based surface reconstruction ap-proaches from scattered data. After reviewingideas and properties of some approved methodswe will present algorithms for the reconstructionof noisy point data being part of the most recentresearch.

Delaunay-based surface reconstruction ap-proaches were mainly developed in the field ofcomputational geometry. These methods arebased on the idea that computing a surface fromscattered point data P demands the exploration ofthe neighborhood of all samples in all directions tofind possible neighbors for every sample point. Asthis is accomplished by the Delaunay triangulationit is predestinated for surface reconstruction. Tofind the final surface some methods computea subset of the Delaunay triangulation. Thosealgorithms are usually referred to as restrictedDelaunay-based methods. Two famous algorithmsfrom this group are the Crust and the Coconealgorithm which will be described in the followingsection.

2.1 Crust & Cocone

The Crust algorithm for surface reconstructionwas designed by Amenta and Bern [7] from theCrust algorithm for curve reconstruction [8]. Thealgorithm works as follows. Assuming we aregiven a point set P sampling an unknown smoothsurface S the Crust algorithm first computes theVoronoi diagram V (P ) of P . Then it determinesthe set of poles Q of the Voronoi diagram.

Let Vp be the Voronoi cell of the sample pointp in the Voronoi diagram. Then the positive polep+ is defined as the farthest Voronoi vertex of Vp

from p. The pole vector np is defined as the vec-tor pointing from the sample point p to p+. Thepole vector is a good approximation for the sur-face normal at p. The negative pole p− is definedas the Voronoi vertex of Vp farthest from p in theopposite direction of the positive pole meaning thevector from p to p− and the pole vector np enclosean angle of more than π

2(see left image of Figure

2). The poles provide a good approximation forthe medial axis of the surface sampled by P .

Figure 1: 2D example of the Delaunay triangu-lation D(P ∪ Q) of the sample pointsP (black) and their Voronoi poles Q(white). The Crust is indicated by solidlines.

After this the Crust algorithm computes the De-launay triangulation D(P ∪Q). To extract the can-didate triangles of the surface that should be re-constructed from the computed Delaunay triangu-lation, the algorithm discards all triangles havingnot all vertices in P . The idea behind choosingthe surface as a restriction of the Delaunay com-plex D(P ∪ Q) is that all triangles crossing themedial axis of the sampled surface S are removed(see Figure 1). The candidate triangles form ingeneral not the resulting surface but they containa surface that is homeomorphic to S if the sam-pling P is dense enough. Therefore, a manifoldextraction stage is applied to find the final surfacecalled the Crust.

The Crust algorithm was one of the first surfacereconstruction methods that could provide guar-antees. Amenta and Bern showed using a resultof Edelsbrunner and Shah [32] that if P is an ε-sample for ε ≤ 0.06 then it is guaranteed that thecandidate triangles contain the restricted Delaunaytriangulation DS(P ) which is the Delaunay trian-gulation D(P ) restricted to the sampled surface S.Although it is impossible to detect the triangles ofDS(P ) in the set of candidate triangles withoutknowing S this information is used in the mani-fold extraction step to compute an approximationof S.

The Crust algorithm demands the computationof two Delaunay triangulations being D(P ) andD(P ∪ Q) determining the runtime and memorycomplexity being O(||P ∪ Q||2).

The Cocone algorithm was developed byAmenta, Choi, Dey and Leekha [10] from theCrust algorithm. It computes similar to the Crustalgorithm a set of candidate triangles containingthe restricted Delaunay triangulation DS(P ) if thegiven sampling is dense enough. This is done by

2

p−

p+

SS p

p

np

Vp

Figure 2: Left: Voronoi cell Vp of sample p inter-secting surface S. The positive pole p+,the pole vector np and the negative polep− are illustrated. Right: Cocone illus-tration. Adapted from [26].

first computing the Voronoi diagram V (P ) to findthe pole vectors for every sample point. Then a co-cone Cp for every sample point p is defined as thecomplement of the double cone with the apex at p,the pole vector np as axis and an opening angle of3π4

. The cocone Cp is clipped inside the Voronoicell Vp. As the pole vector np approximates thesurface normal at p the cocone determines a neigh-borhood around the tangent plane at p (see rightimage of Figure 2). After this the algorithm detectsall Voronoi edges in Vp being intersected by the co-cone Cp for every point p. The dual Delaunay tri-angles then build the candidate triangle set. Afterthe whole candidate triangle set is determined themanifold extraction is performed to find the result-ing surface. The cocone has the same guaranteeslike the Crust algorithm. It is proven that the re-stricted Delaunay triangulation DS(P ) is part ofthe candidate triangle set if P is an ε-sample withε ≤ 0.06.

The Cocone algorithm demands in contrastto the Crust algorithm only the computation ofthe Delaunay triangulation D(P ). Thus theruntime and memory complexity is reduced toO(||D(P )||2).

Important extensions of the presented methodsare algorithms that produce watertight surfaces.These methods belong usually to the class of in-side/outside labeling algorithms. They first com-pute a Delaunay triangulation of the scattered data.Then the resulting tetrahedra are labeled inside oroutside depending whether the tetrahedron is in-side the solid bounded by the scattered data or out-side. After all tetrahedra are labeled the resulting

Figure 3: Power Crust (top) and Tight Cocone(bottom) reconstructions of the Dragonusing samples from the VRIP files takenfrom the Stanford Scanning Repository.

surface can be extracted by retaining only trianglesthat are shared by one inside and one outside tetra-hedron. As there is no remaining path from theinside of the bounded solid to the outside not pass-ing a retained triangle the resulting mesh is wa-tertight. Two important methods generating wa-tertight reconstructions are Power Crust developedby Amenta, Choi and Kolluri [11] and Tight Co-cone of Dey and Goswami [27]. Reconstructionsof both algorithms are illustrated in Figure 3.

2.2 Recent DevelopmentsThe presented computational geometry methodsare supported by rigorous mathematical results andprovide guarantees under the presented conditions.Unfortunately, the precondition that P is an ε-sampling with small ε does not often hold in prac-tice. Therefore, computational geometry methodsface difficulties while dealing with noisy data andundersampling (see top row of Figure 6). To over-come these problems recent research is focused onthis area.

Dey and Goswami [28] present the Robust Co-cone algorithm which applies an observation used

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in the Power Crust algorithm. It was observedthat the polar balls (balls surrounding the polesand touching the nearest samples) approximate thesolid bounded by the sampled surface. If the ballsof adjacent tetrahedra intersect deeply both tetra-hedra belong to the same component being eitherinside or outside. On the other hand if balls in-tersect shallowly they belong to different compo-nents. Dey and Goswami showed that this mightnot be true for noisy regions. They also show thatthe polar balls in these regions have a small radiusand can thus be detected. Robust Cocone only pre-serves the samples on the outer Delaunay balls andreconstructs a watertight surface only using them.

Kolluri et al. [47] introduce the Eigen Crustalgorithm. The method computes a watertightsurface reconstruction from noisy scattered datawith outliers using spectral graph partitioning. Tocompute the surface the idea is again to labeleach tetrahedron of the Delaunay triangulation (orequivalently each Voronoi vertex) inside or out-side. The Eigen Crust algorithm labels the Voronoivertices in two steps. The first stage constructs apole graph, whose nodes represent the poles. Theedges of the pole graph are weighted according tothe likelihood that pairs of tetrahedra are on thesame side of the surface. This weighted graph isrepresented by the pole matrix. Finally, the small-est eigenvalue of the pole matrix is computed to di-vide the graph in two subgraphs containing insideand outside poles. To label also tetrahedra that arenot duals of poles the Eigen Crust algorithm buildsa second graph from the Voronoi vertices that arenot poles. As the labeling of the non-pole verticesis more ambiguous the algorithm classifies them inorder to produce a smooth surface with low genus.

3 Implicit Surface Interpolation

Given a set of points X = xi ⊂ IR3 scat-tered over a surface, the main idea behind most ofthe implicit surface interpolation techniques con-sists of building a function y = f(x) whose zerolevel set Z(f) = x : f(x) = 0 approxi-mates / interpolates X . Usually y = f(x) is con-structed as a composition (weighted sum) of sim-ple primitives.

3.1 Radial Basis Functions (RBF)

Radial Basis Function (RBF) techniques are nowstandard tools for geometric data analysis [33,53] in pattern recognition [44], statistical learn-ing [36], and neural networks [37]. Properties ofRBFs are widely studied in mathematical litera-ture [18, 30, 40, 71]. (See also references therein.)

Given a scattered point dataset, we interpolateor approximate it by the zero level-set of a com-posite function f : IR3 → IR defined as a linearcombination of relatively simple primitives

f(x) =m∑

i=1

αiΦ(x, ci) (1)

where Φ(., ci) : IR3 → IR are functions centeredat ci and αi are the unknown weights [35].

We want to constrain the solution to be stableto translation and rotation of the point set. Thefunctions Φ are thus given by:

Φ(x, ci) = φ(‖x − ci‖), (2)

where ‖.‖ denotes the Euclidian distance and φ :IR+ → IR.

Relations between the RBF and varia-tional approaches to scattered data interpola-tion / approximation are analyzed in [30, 37].

Reconstruction using Radial Basis Functionsgives a smooth implicit interpolating or approxi-mating surface, since both the implicit solution andits zero level set have the same continuity proper-ties as the ones of the basis functions Φ.

Let F = fi be a set of n values of a function fat some scattered distinct points X = xi ∈ IR3.We want to find a function f : IR3 → IR such that∀i = j, . . . , n

f(xj) =

m∑

i=1

αiφ(‖xj − ci‖) = fj (3)

The reconstruction problem thus boils down to de-termining the vector α = α1, . . . , αn by solv-ing a linear system of equations given by the con-straints (3). Since all constraints are located on thesurface, all fis are valued zero. In order to avoidthe trivial solution α =

−→0 , we add interior and

exterior constraints where the function is non zeroand assign them the values −d and d, respectively.We compute the weights α = [αi] using (3), anddenoting [φ(‖xi−xj‖)] = AX,Φ, we have to solvethe following linear system:

AX,Φ · α = F. (4)

In order to obtain a unique solution the matrixAX,Φ is required to be invertible, at least onthe subspace of ~α vectors where the solution issearched. A common solution is to use the sub-space such as:

∀α ∈ IRn

n∑

i=1

αip(xi) = 0 ∀p ∈ IPq

(5)

4

where IPq is the set of polynomials of order up toq. With this condition, the functions φ are condi-tionally positive definite [18]. Furthermore, to re-cover the right number of variables and unknowns,a polynomial p ∈ IPq will be added to (1)

f(x) =

m∑

i=1

αiΦ(x, ci) + p(x) (6)

Some conventional radial basis functions are:φ : IR+ → IR

biharmonic RBF φ(r) = r with a linear polyno-mial

pseudo-cubic RBF φ(r) = r3 with a linear poly-nomial

triharmonic RBF φ(r) = r3 with a quadraticpolynomial

thin plate RBF φ(r) = r2 log(r) with a linearpolynomial

All functions listed above have an unboundedsupport. The corresponding equations lead to adense linear system, therefore recovering a solu-tion is tractable only for small data sets. To over-come this problem Morse et al. [55] use Gaus-sian as Compactly Supported RBFs to obtain asparse interpolation matrix. More generally, a lotof other Compactly Supported RBFs (CSRBF) canbe used for reconstruction as proposed in [69, 72]but they are not well-suited for reconstruction fromincomplete data. To handle large and incom-plete data sets two strategies have been proposed.One approach uses polyharmonic RBF (i.e. non-compactly supported functions) [19], reduces thenumber of centers by a greedy selection procedureand performs fast evaluation using the so-calledFast Multipole Method (FMM). Another approachconsists of using locally supported functions [65],where the partition of unity is used for blending,and the function support is computed locally forall centers, as described in [58]. A multiresolutionversion of this approach has been proposed in [57].

We can notice that radially symmetric functionsare not suited for piecewise smooth surface recon-struction. Dinh et al. [29] have presented a methodusing anisotropic basis functions to overcome thisissue.

3.2 Partition of Unity (PU)”Divide and conquer” is the main idea behind thePartition of Unity approach. The main idea con-sists of breaking the domain into smaller subdo-mains where the problem can be solved locally.The data is first approximated on each subdomain

separately, and the local solutions are blent to-gether using a weighted sum of local subdomainapproximations. The weights are smooth functionsand sum up to one everywhere on the domain.

Tobor et al. [65] combine the Partition of Unitymethod and the radial basis functions. Ohtake etal. [56] use weighted sums of different kinds ofpiecewise quadratic functions in order to capturethe local shape of the surface. This way implicitsurfaces from very large scattered point sets canbe reconstructed.

Consider a global bounded domain Ω in an Eu-clidian space. Divide Ω into M mildly overlap-ping subdomains Ωii=1,...,M with Ω ⊆ ∪iΩi.”Mildly overlapping” herein means that the in-tersection of two incident subdomains contains atmost one data point. Together with this coveringwe construct a Partition of Unity, i.e. a familyof non-negative continuous compactly supportedfunctions wii=1,...,M such that Supp(wi) ⊆ Ωi

and∑M

i=1wi = 1 everywhere. Let X = xi ∈

IR3 be a set of n points on the surface. For eachcell Ωi, a set Xi = x ∈ X/x ∈ Ωi is built, andthe surface is approximated on each subdomain bya local approximant fi. The global function is thendefined as a combination of the local functions as:

f(x) =

n∑

i=1

wi(x)fi(x). (7)

The condition∑M

i=1wi = 1 can be obtained from

any other set of smooth functions Wi by a normal-ization process:

wi(x) =Wi(x)∑n

j=1Wj(x)

. (8)

The weighting function Wi determines the con-tinuity of the global reconstruction function f . Wecan generate these functions using local geometryof the corresponding cell (distance function, centerand radius of cell, etc.).

For domain decomposition, in general, spacesubdivision based trees are proposed (binary trees,octrees, etc.). The cells can have different shapessuch as as axis-aligned bounding boxes, balls oraxis-aligned ellipsoids. The degree of subdivisioncan be adapted to both local sample density anddesired smoothness. The choice of the local fittingmethod is another degree of freedom: Radial BasisFunctions are used in [65, 70], while quadrics areused in [56]. Furthermore, one can adapt the fit-ting strategy for each cell according to the numberof points and distribution of associated normals.Reconstructing sharp features is this way possibleas described in [56].

An interesting combination of the RBF and PUapproaches was recently proposed in [58] where a

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Figure 4: RBF+PU reconstruction of the StanfordHappy Buddha model from its originalregistered scans.

partition of unity is used to obtain an initial roughapproximation of given scattered data and thenRBFs are used to refine the PU approximation. Anexample of such PU+RBF reconstruction is shownin Figure 4.

3.3 Moving Least Squares (MLS)Moving-least squares (MLS) surfaces have beenintroduced by Levin [48, 49]. The work of Alexaet al. [4, 5] first used MLS surfaces in point-basedgraphics. MLS surfaces are defined by a projec-tion operator which projects points from a tubularneighborhood onto the surface. The neighborhoodis usually defined by a union of balls centered atthe input points pi. The traditional projection op-erator is computational expensive because of anassociated non-linear optimization problem. A de-tailed survey of point-based techniques includinga short description of the traditional projection op-erator can be found in [45].

Adamson and Alexa [1] propose a simpler pro-jection technique for the definition of an im-plicit surface from point cloud data. They itera-tively project a point x onto a plane defined by aweighted average of neighboring points

a(x) =

∑j Φ(||x − pj ||)pj∑

j Φ(||x − pj ||)

and the normal n(x). If normals nj for inputpoints are given n(x) can be computed by aver-

aging the input normals

n(x) =

∑j Φ(||x − pj ||)nj

||∑

j Φ(||x − pj ||)nj ||,

otherwise n(x) is determined by the normal of theweighted least-squares fitting plane of the pointspj . After convergence the point x reaches a posi-tion on the MLS surface.

Recent work of Amenta and Kil [12] analysesthe stability of the original projection operator forpoints that are not sufficiently close to the MLSsurface. Furthermore, they present an explicit defi-nition of MLS surfaces in terms of critical points ofthe energy function EMLS along lines determinedby a vector field.

4 Learning-based MethodsLearning-based methods used in surface recon-struction have their origin in the work of Kohonen[46] where Self-Organizing Maps (SOMs) wereintroduced. Later Fritzke [34] defined a special-ized class of SOMs called the Growing Cell Struc-tures (GCSs). Both neural networks have beenused for surface reconstruction. Barhak and Fis-cher [15] use SOMs for grid fitting. GCSs are ap-plied by Hoffmann and Varady [38] to free-fromsurface reconstruction. Yu [73] uses SOMs for sur-face reconstruction from scattered data. In this re-port we will focus on recent publications by Ivris-simtzis et al. [41, 42] using GCSs for the recon-struction of point cloud data.

The Growing Cell Structure presented by Ivris-simtzis et al. is an incrementally expanding neu-ral network with triangle mesh connectivity (neu-ral mesh) growing by splitting and removing ver-tices depending on their activity. The idea of thealgorithm is to randomly sample the given inputpoint set P and to use these samples as a trainingset for the initial neural mesh which is usually cho-sen as a tetrahedron. By processing samples theneural mesh M learns the geometry and the topol-ogy of the surface represented by the input pointset. Geometry learning is accomplished by ran-domly choosing a sample s ∈ P and moving thenearest vertex v ∈ M towards s. After each up-date the 1-ring neighborhood of v is smoothed. Toimprove the connectivity and to allow growing ofM the algorithm can introduce new vertices to theneural mesh by splitting the most active vertex aswell as removing the least active vertex. The activ-ity of all vertices is measured by a signal counter.The counter of the nearest vertex v to the samples is increased by one to assign v a higher activ-ity value. The signal counter of all other verticesis multiplied with a positive number α < 1 to re-

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Figure 5: Neural mesh reconstruction (500K ver-tices) of the Atlas model with samplesobtained from VRIP files in the DigitalMichelangelo Archive [50].

move gradually the effect of old activity. Topol-ogy learning is achieved by triangle removal andboundary merging to allow the adaptation of Mto boundaries and handles. As the growing of theneural mesh depends on the sampling density of Plarge triangles occur in regions of M approximat-ing parts of the surface with a low sampling den-sity. When the sampling density becomes insignif-icant such that the area of the corresponding trian-gles is over a defined threshold, they are removedand a boundary is created. If two boundaries arenear to each other measured with an approxima-tion of the Hausdorff distance the two boundariesare merged to create a handle.

The latest works of Ivrissimtzis et al. focus onthe improvement of topological learning. One ideaof [42] is to create an ensemble of neural meshes.An ensemble is a set of neural meshes with a com-bined output which has a smaller statistical errorthan any of its elements. To create a combinedneural mesh from meshes with different connec-tivity a volumetric method is used.

Due to the sampling-based reconstruction thepresented algorithm is able to process large data(see Figure 5). Furthermore, learning-based meth-ods are well-suited for the reconstruction of noisydata. One drawback of the reviewed method is the

reconstruction time complexity depending on thenumber of vertices of the neural mesh.

5 Filtering and DenoisingThe performance of surface reconstruction ap-proaches is usually reduced if the input data isnoisy. One approach to overcome this problem isto filter and denoise the input before it is used forsurface reconstruction [52, 54, 62].

Schall et al. [62] propose a method for deal-ing with uncertain and noisy surface scattereddata. Given a set of noisy input points P =p1, . . . ,pN the method computes for every in-put point pi a local uncertainty function measur-ing the contribution of pi to the smooth surfaceS which is supposed to be reconstructed. The lo-cal functions are composed to a global uncertaintyfunction whose local minima are determined. Thecomputed minima are approximation centers of thesampled surface S and form a sparse set of pointswhich are used as vertices of a mesh. The meshcan be reconstructed using an arbitrary meshingtechnique. The presented method robustly pro-duces accurate smooth meshes with minimal topo-logical noise. The top images of Figure 6 showreconstructions of the Stanford Bunny model fromits original scans using Power Crust (left) andTight Cocone (right). The bottom row of Figure 6demonstrates reconstruction results created usingthe same Power Crust and Tight Cocone methodsfrom the approximation centers generated accord-ing to [62].

6 Discussion and ConclusionIn this survey we presented several surface recon-struction methods. These methods are mainly ei-ther Delaunay-based and use a subset of a com-puted Delaunay complex to reconstruct the surfaceor represent the surface implicitly by the zero-levelset of a defined function. Other techniques arelearning-based or filter and denoise the input data.

The area of surface reconstruction is still a fieldwith many open problems and research directions.Recent research trends focus on reconstruction ofscattered polygonal data and noisy point clouddata. Furthermore, other methods avoid surface re-construction but visualize the surface representedby the point cloud data directly by for instance ray-tracing Point Set Surfaces or using surface splat-ting techniques.

One of the recent approaches to processing scat-tered 3D point data consists of using point-basedrendering primitives, as first suggested by Levoyand Whitted [51]. Efficient implementations of the

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Figure 6: Reconstructions of the Stanford Bunnyusing Power Crust (left) and Tight Co-cone (right) directly from the range scans(top row) and from the approximationcenters (bottom row).

approach were presented in [60, 61] and in manysubsequent works (see Figure 7). See the recentsurveys [6, 45].

AcknowledgementsWe would like to thank Tamal Dey, Nina Amentaand Szymon Rusinkiewicz for making their soft-ware available. Furthermore, we thank YutakaOhtake for providing us his software and IoannisIvrissimtzis for images from his papers. We aregrateful to the anonymous reviewers of this paperfor their constructive comments and useful sug-gestions. The Atlas and St. Matthew datasets arecourtesy of the Digital Michelangelo Project. TheBunny, Dragon and Buddha datasets are courtesyof the Stanford 3D scanning repository. This re-search was supported in part by the European FP6NoE grant 506766 (AIM@SHAPE).

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