spectral surface reconstruction from noisy point cloudsjrs/papers/eigencrust.pdf · spectral...

Eurographics Symposium on Geometry Processing (2004)R. Scopigno, D. Zorin, (Editors)

Spectral Surface Reconstruction from Noisy Point Clouds

Ravikrishna Kolluri Jonathan Richard Shewchuk James F. O’Brien

University of California, Berkeley

Abstract: We introduce a noise-resistant algorithm for reconstructing a watertight surface from point cloud data.It forms a Delaunay tetrahedralization, then uses a variant of spectral graph partitioning to decide whether eachtetrahedron is inside or outside the original object. The reconstructed surface triangulation is the set of triangularfaces where inside and outside tetrahedra meet. Because the spectral partitioner makes local decisions based ona global view of the model, it can ignore outliers, patch holes and undersampled regions, and surmount ambiguitydue to measurement errors. Our algorithm can optionally produce a manifold surface. We present empiricalevidence that our implementation is substantially more robust than several closely related surface reconstructionprograms.

Categories and Subject Descriptors(according to ACM CCS): I.3.5 [Computing Methodologies]: Computer Graph-ics Computational Geometry and Object Modeling

1. Introduction

Laser range finders record the geometry of real-world three-dimensional objects in the form of point coordinates sam-pled from their surfaces, plus auxiliary information suchas the laser’s position. Surface reconstruction algorithmsrecreate geometric models from these data, expressed ina form more useful for applications such as rendering orsimulation—for example, as a surface triangulation or assplines. Laser range finders are imperfect devices that in-variably introduce at least two kinds of errors into the datathey record:measurement errors(random or systematic) inthe point coordinates, andoutliers, which are spurious pointsfar from the true surface. Furthermore, objects often haveregions that are not accessible to scanning and so remainundersampled or unsampled. The point cloud in Figure1(bottom center) suffers from all the above.

When these problems are severe, data arise for which noalgorithm can construct an accurate, consistent, watertightmodel of an object’s surface solely by examining local re-gions of a point cloud independently. A successful algorithmmust take a global view.

Figure 1: A watertight manifold surface triangulation re-constructed by our eigencrust algorithm; a photograph ofthe source object; the point cloud input to the algorithm,with 4,000 artificial random outliers; and the sorted com-ponents of the two eigenvectors used for the reconstruction.2,008,414 input points; 12,926,063 tetrahedra; 3,605,096output triangles; genus 14 (source object has genus 1); 437minutes reconstruction time, including 13.5 minutes to tetra-hedralize the point cloud, and 157 minutes and 265 minutesto compute the first and second eigenvectors, respectively.

c© The Eurographics Association 2004.

Kolluri, Shewchuk, and O’Brien / Spectral Surface Reconstruction

Our innovation is to introduce the techniques of spectralpartitioning and normalized cuts into surface reconstruction.These techniques are used heavily for tasks such as imagesegmentation and parallel sparse matrix arithmetic, wherepartitioning decisions based on a global view of an imageor a matrix can outperform local optimization algorithms.Although the global optimization step makes our algorithmslower than many competitors, it reconstructs many modelsthat other methods cannot.

The general technique we use to produce a surface froma point cloud is well known: compute the Delaunay tetrahe-dralization of the points, then label each tetrahedroninsideor outside. (Recent advances in Delaunay software make itpossible to tetrahedralize sets of tens of millions of points.)The output is a triangulated surface, composed of every tri-angular face where aninside tetrahedron meets anoutsidetetrahedron. This procedure guarantees that the output sur-face iswatertight—it bounds a volume, and there is no routefrom the inside to the outside of the volume that does notpass through the surface. Watertight surfaces are importantfor applications such as rapid prototyping and volume meshgeneration for finite element methods.

Our contribution is an algorithm for labeling the tetrahe-dra that is substantially more robust than previous methods.We call the surface it produces theeigencrust. The eigen-crust algorithm creates a graph that represents the tetrahedra.A spectral partitioner slices it into two subgraphs, aninsidesubgraph and anoutside subgraph. Because the spectral par-titioner has a global view of the point set, it is effective atidentifying the triangular faces that are most likely to lie atthe interface between an object and the space around it.

2. Related work

There has been much work on reconstructing surfaces frompoint clouds. The idea of labeling each Delaunay tetrahe-dron inside or outside, then extracting a surface using thelabels, appears in an early paper of Boissonnat [Boi84],and is also harnessed in the Tight Cocone algorithm of Deyand Goswami [DG03] and in the Powercrust algorithm ofAmenta, Choi, and Kolluri [ACK01] (with cells of a powerdiagram replacing tetrahedra). In this paper, we compareimplementations of the Tight Cocone and Powercrust algo-rithms with our eigencrust software.

A recent advance is the Robust Cocone algorithm of Deyand Goswami [DG04], a Delaunay-based reconstruction al-gorithm that is provably robust against small coordinate er-rors. Their algorithm is not robust against undersamplingor outliers, and in fact is easily defeated by undersampling.We observe that a variant of our spectral algorithm could beused to help the Robust Cocone algorithm to label tetrahe-dra when the sample set is not dense enough for the RobustCocone’s original labeling algorithm to succeed. Unfortu-nately, we did not have sufficient time to obtain a RobustCocone implementation for comparison in this paper.

Another branch of surface reconstruction algorithms arethose that define a function over space whose zero set is asurface, which can be triangulated by techniques such asmarching cubes [LC87]. These algorithms vary widely inhow they compute the function. Hoppe, DeRose, Duchamp,McDonald, and Stuetzle [HDD∗92] provide one of the ear-liest algorithms, which locally estimates the signed distancefunction induced by the “true” surface being sampled. Bit-tar, Tsingos, and Gascuel [BTG95] use the medial axis of thepoint set to improve the speed of zero-set methods and theirability to reconstruct topologically complex objects. Cur-less and Levoy [CL96] developed an algorithm that is par-ticularly effective for laser range data comprising billionsof point samples, like the statue of David reconstructed bythe Digital Michelangelo Project [LPC∗00]. A zero-set ap-proach by Carr et al. [CBC∗01] adapts the radial basis func-tion-fitting algorithm of Turk and O’Brien [TO99] to sur-face reconstruction. Ohtake, Belyaev, Alexa, Turk, and Sei-del [OBA∗03] use a partition-of-unity method with a fasthierarchical evaluation scheme to compute surfaces for datasets with over a million points.

Closely related to the zero-set approaches are the level-setalgorithms of Whitaker [Whi98] and Zhao, Osher, and Fed-kiw [ZOF01]. Other important reconstruction algorithms arethe mesh zippering algorithm of Turk and Levoy [TL94],Bernadini et al.’s [BMR∗99] ball-pivoting algorithm, andKobbelt, Botsch, Schwanecke, and Seidel’s [KBSS01] meth-od for extracting surfaces from volumetric data, which in-cludes effective methods of extracting sharp corners that areoften missed by marching cubes.

Different branches of algorithms have different advan-tages. The main advantages of the Delaunay algorithms arethe effortlessness with which they obtain watertight surfaces;the ease with which they adapt the density of the trianglesto match the density of the points (unlike marching cubes),for models whose point density varies greatly from regionto region; and the theoretical apparatus that makes it pos-sible to prove the correctness of some of these reconstruc-tion algorithms on well-sampled smooth surfaces [ABK98,AB99,AK00,ACDL02]. Here, we show that spectral parti-tioning can help make these methods robust for surfaces thatare noisy or not sufficiently well-sampled for the theoreticalguarantees to apply.

Spectral methods for partitioning graphs were introducedby Hall [Hal70] and Fiedler [Fie73] and popularized byPothen, Simon, and Liou [PSL90]. They are used for taskssuch as image segmentation, circuit layout, document clus-tering, and sparse matrix arithmetic on parallel computers.The goal of graph partitioning is to cut a graph into two sub-graphs, each roughly half the size of the original graph, sothat the total weight of the cut edges is small (each edge isassigned a numerical weight). There are many ways to for-mulate the graph partitioning problem, which differ in howthey trade off the weight of the cut against the balance be-tween the two subgraphs. Most graph partitioning formula-tions are NP-hard, so practical partitioning algorithms (in-



cluding spectral methods) are heuristics that try to find anapproximate solution.

One of the most effective formulations of spectral par-titioning is the normalized cutscriterion of Shi and Ma-lik [ SM00], which is particularly effective at trading off sub-graph balance against cut weight. We make a simple modifi-cation to the Shi–Malik algorithm (closely related to a tech-nique of Yu and Shi [YS01]) that greatly improves our sur-face reconstruction algorithm’s speed and the quality of thesurfaces it produces.

3. Spectral Surface Reconstruction

We assume the reader is familiar with the notions of De-launay triangulations and Voronoi diagrams in three dimen-sions, and with the geometric duality that maps each Delau-nay tetrahedron to a Voronoi vertex and each Delaunay faceto a Voronoi edge. See Fortune [For92] for an introduction.

The eigencrust algorithm begins with a setS of samplepoints in space. LetS+ be the setS augmented with eightbounding box vertices, the corners of a large cube that en-closes the sample points. (The width of the cube should bemuch greater than the diameter ofS, so that no sample pointlies near any side of the cube). LetT be the Delaunay tetra-hedralization ofS+. Let Q be the Voronoi diagram ofS+

(and the geometric dual ofT). For each tetrahedront in thetetrahedralizationT, there is a dual vertexv of the VoronoidiagramQ, andv is the center of the sphere that circum-scribest.

The goal is to label each tetrahedron—or equivalently,each Voronoi vertex—inside or outside. The eigencrust al-gorithm labels the Voronoi vertices in two stages. Each stageforms a graph and partitions it.

In the first stage, our algorithm labels a subset of theVoronoi vertices called thepoles, following Amenta andBern [AB99]. We form a graphG, called thepole graph,whose nodes represent the poles. See Figure2 for a two-dimensional example. The edges ofG are assigned numer-ical weights that reflect the likelihood that certain pairs ofpoles are on the same side of the unknown surface that wewish to reconstruct.

The graphG is represented by apole matrix L. We par-tition the poles ofG by finding the eigenvectorx that corre-sponds to the smallest eigenvalue of a generalized eigensys-tem Lx = λDx, and using that eigenvector to cut the graphinto two pieces, theinside andoutside subgraphs. Thus welabel each poleinside or outside.

In the second stage, we form another graphH whosenodes represent the Voronoi vertices that arenot poles, andpartition H to label all the Voronoi vertices (equivalently,the tetrahedra) that were not labeled in the first stage. Thegoal of the second stage is different from the goal of thefirst: the non-poles are somewhat ambiguous—most of themcould arguably be eitherinside or outside—so the partitioner

Figure 2: Top: the Delaunay triangulation (black) andVoronoi diagram (red) of a set of points sampled from aclosed curve. Red Voronoi vertices are poles; white Voronoivertices are not. Note that three-dimensional examples havemany more non-pole Voronoi vertices than two-dimensionalexamples. Center: the negatively weighted edges of the polegraph G (green), before the bounding box triangles are col-lapsed into a single supernode. Yellow triangles are the du-als of poles labeledinside by the first stage of spectral par-titioning. The black triangle does not dualize to a pole; it islabeledinside by the second partitioning stage. Bottom: thepositively weighted edges of G.

tries to assign them labels that produce a relatively smoothsurface with low genus.

Now all the Voronoi vertices have labels, so all the tetra-hedra ofT have labels. The eigencrust is a surface triangula-



tion consisting of every triangular face ofT where aninsidetetrahedron meets anoutside tetrahedron. If the points inSare sampled densely enough from a simple closed surface,then the eigencrust approximates the surface well.

If all the tetrahedra adjoining a sample point are labeledoutside (or all areinside), the point does not appear in thereconstructed surface triangulation. In Section4, we see thatthis effect provides our algorithm with effective and auto-matic outlier removal. No other effort to identify outliers isrequired.

Why are the Voronoi vertices labeled in two separatestages? Because the non-poles are ambiguous, they tend to“glue” the inside and outside tetrahedra together. If they areincluded in the first partitioning stage, the graph partitioneris much less successful at choosing the right labels, and runsmore slowly too. A two-stage procedure produces notablybetter and faster results.

We have chosen the graphs’ edge weights (by trial anderror) so the algorithm tries to emulate the provably correctCocone algorithm of Amenta et al. [ACDL02] when there isneither noise nor outliers, and the sampling requirements ofthe Cocone algorithm are met. Our algorithm usually returnssignificantly different results only under conditions wherethe Cocone algorithm has no guarantee of success.

Our algorithm includes three optional steps. After the firstpartitioning stage, we can identify some tetrahedra that maybe mislabeled due to noise, and remove their labels (so theyare assigned new labels during the second stage). This stepimproves the resilience of our algorithm to measurement er-rors. After the second partitioning stage, we can convert thesurface to a manifold (if it is not one already) by relabelingsomeinside tetrahedraoutside. After the final surface recov-ery step, we can smooth the surface to make it more usefulfor rendering and simulation.

3.1. The Pole Graph

Imagine that we form a graph whose nodes represent the ver-tices of the Voronoi diagramQ (and their dual tetrahedra inT), and whose edges are the edges ofQ (omitting the edgesthat are infinite rays). Suppose we then assign appropriateweights to the edges, and partition the graph intoinside andoutside subgraphs.

Unfortunately, this choice leads to poor results. The maindifficulty is that the Delaunay tetrahedralizationT invari-ably includes flat tetrahedra that lie along the surface weare trying to recover, as Figure3(a) illustrates. These tetra-hedra are not eliminated by sampling a surface extremelyfinely; they are a natural occurrence in Delaunay tetrahe-dralizations. Many of them could be labeledinside or out-side equally well. They cause trouble because they can formstrong links with both the tetrahedra inside an object and thetetrahedra outside an object, and thus prevent a graph parti-tioner from finding an effective cut between the inside andoutside tetrahedra.

vus

c

(a) (b)Figure 3: (a) No matter how finely a surface is sampled,tetrahedra can appear whose circumscribing spheres arecentered on or near the surface being recovered. (b) TheVoronoi cell c of a sample point s. The poles of s—theVoronoi vertices u and v—typically lie on opposite sides ofthe surface being recovered, especially if the cell is long andthin.

To solve this problem, Amenta and Bern [AB99] iden-tify special Voronoi vertices calledpoles. Poles are Voronoivertices that are likely to lie near the medial axis of the sur-face being recovered. The Voronoi vertices whose duals arethe troublesome flat tetrahedra are rarely poles, because theproblem tetrahedra lie near the object surface, not near themedial axis.

Each sample points in Scan have two poles. Letc be theVoronoi cell ofs in Q (i.e. the region of space composed ofall points that are as close or closer tos than to any othersample point inS+). See Figure3(b) for an example. TheVoronoi cell c is a convex polyhedron whose vertices areVoronoi vertices. It is easy to compute the vertices ofc,because they are the centers of the circumscribing spheresof the tetrahedra inT that haves for a vertex.

Let u be the vertex ofc furthest froms; u is called apoleof s. Let v be the vertex ofc furthest froms for which theangle∠usvexceeds 90◦; v is also called a pole ofs. Fig-ure 3(b) illustrates the two poles of a typical sample point.The eight bounding box vertices inS+ are not considered tohave poles. LetV be the set of all the poles of all the samplesin S.

Amenta and Bern show that in the absence of noise, thetetrahedra that are the duals of the poles are likely to extendwell into the interior or exterior of the object whose surfaceis being recovered. The tetrahedra whose duals are not polesoften lie entirely near the surface, as Figure3(a) shows, so itis ambiguous whether they are inside or outside the object.

The eigencrust algorithm identifies the setV of all poles,then constructs a sparsepole graph G= (V,E). The setE ofedges is defined as follows. For each samples with polesuandv, (u,v) is an edge inE. For each pair of sampless, s′

such that(s,s′) is an edge of the Delaunay tetrahedralizationT, let u andv be the poles ofs, and letu′ andv′ be the polesof s′; then the edges(u,u′), (u,v′), (v,u′), and (v,v′) areall edges ofE. Every pole is the dual of a tetrahedron, sotetrahedra that adjoin each other are often linked together inG, whereas tetrahedra that are not close to each other are notlinked.



Cv

Cu

φ

v

u

t

t

v

u

sv

φ

u

C Cv u

(a) (b)Figure 4: (a) Small angles of intersection between circum-scribing spheres may indicate that two tetrahedra are on op-posite sides of the surface being recovered. (b) Large anglesof intersection usually indicate that two tetrahedra are onthe same side of the surface.

We assign edge weights in a heuristic manner based onseveral observations of Amenta and Kolluri [AK00]. If Sis sampled sufficiently densely from a smooth surface, theVoronoi cells are long and thin, and the longest dimension ofeach cell is oriented roughly perpendicular to the surface, asFigure3(b) depicts. Of course, point sets that arise in prac-tice are often not sampled densely enough, but if a sampleshas a long, thin Voronoi cellc, the likelihood is high that itspolesu andv are on opposite sides of the surface. Therefore,we assign the edge(u,v) a negative weight, to indicate thatif one of u or v is labeledinside, the other should probablybe labeledoutside.

Let tu andtv be the tetrahedra inT whose duals areu andv. Let Cu andCv be the circumscribing spheres oftu andtv.The spheresCu andCv intersect at an angleφ. Amenta andKolluri show that ifφ is small, as illustrated in Figure4(a),thenc is quite long and thin, and the likelihood is high thattu andtv lie on opposite sides of the surface. Ifφ is close to180◦, thenc is relatively round, and it is unsafe to concludethattu andtv lie on opposite sides. We assign(u,v) a weightof wu,v =−e4+4cosφ, so thatwu,v is most negative whenφ isclosest to zero.

We assign positive weights to the other edges inE. Theseweights are the glue that hold proximal tetrahedra togetherand ensure thatG is likely to be cut only near the originalsurface, where the glue is weakest. Let(u,v) be an edgeof E that is not assigned a negative weight—thus, there is aDelaunay edge(s,s′) for which u is a pole ofs andv is apole ofs′, but there is no sample points′′ whose poles areuandv. Again, lettu andtv be the tetrahedra that are dual tou andv, and letCu andCv be their circumscribing spheres,which intersect at an angleφ. We assign(u,v) a weight ofwu,v = e4−4cosφ. Amenta and Kolluri show that ifφ is closeto 180◦, as illustrated in Figure4(b), thenu andv are likelyto lie on the same side of the surface, so we use a large,positive edge weight. Ifφ is close to 0◦, we choose a smalledge weight, so thatu andv are not strongly glued together.

It may occur that the spheresCu andCv do not intersect atall, in which case we remove the edge(u,v) from E.

We could partition the graphG directly, but we knowapriori that certain tetrahedra must be labeledoutside, and itis advantageous to fix their labels prior to the partitioningstep. LetO be the set of poles whose dual tetrahedra areknown to be outside the object being reconstructed. We takeadvantage of this information by forming a new graphG′

that is similar toG, but the poles inO are collapsed into asinglesupernode z. If u andu′ are poles inO, and(u,u′) isan edge ofG, the edge is eliminated (not present inG′). If vis a pole inG that is not inO, then in the new graphG′, theedge(z,v) has weightwz,v = ∑u∈O wu,v. Collapsingoutsidepoles into a single supernode makes the spectral partitionerfaster and more accurate.

What poles doesO contain? There are several types oftetrahedra that can be labeledoutside prior to the partitioningstep.

• Any tetrahedron with a vertex of the cubical bounding boxmust beoutside.

• If the point samples were acquired by a laser range finder,the tetrahedra that lie between the laser source and anysample point it recorded must beoutside. Of course, theremay be measurement errors in the positions of the samplepoints and the laser source, so we recommend only label-ing those tetrahedra that the laser penetrated more deeplythan some tolerance depth, multiple times.

• For particularly difficult reconstructions, a user may vi-sually identify specific points in space that are outsidethe object. The tetrahedra containing these points are la-beledoutside. Collapsing just one such tetrahedron intothe outside supernode can change the labeling of manyother tetrahedra, so this is occasionally a practical option.(No example in this paper takes advantage of this possi-bility.)

Optionally, the algorithm may create aninside supernodeas well, with a large negative weight connecting theoutsideandinside supernodes. This is particularly useful for recon-structing one-sided building facades or other sets of samplepoints that do not represent closed volumes. For this pur-pose, the tetrahedra adjoining the front face of the bound-ing box are labeledoutside, and the tetrahedra adjoining theback face areinside.

3.2. Spectral Partitioning

From the modified pole graphG′, we construct apole matrixL. (L is often called theLaplacian matrix, but our use of neg-ative weights makes that name a misnomer.)L is sparse andsymmetric, and has one row and one column for each node ofthe graphG′. For each edge(vi ,v j ) of G′ with weightwvi ,v j ,the pole matrixL has the componentsLi j = L ji = −wvi ,v j .(Positive, “attractive” weights become negative matrix com-ponents, and negative, “repulsive” weights become positivematrix components.) The diagonal components ofL are therow sumsLii = ∑ j 6=i |Li j |. The remaining components ofL



Figure 5: Watertight skeleton surface, and the sorted com-ponents of the eigenvectors computed during the two par-titioning stages. The points are densely sampled from asmooth surface, so the eigenvectors are polarized. 327,323input points; 2,334,597 tetrahedra; 654,596 output trian-gles; genus zero; 12.3 minutes reconstruction time, includ-ing 2.8 minutes for the tetrahedralization, and 5.1 minutesand 4.2 minutes for the eigenvectors.

(the off-diagonal components not represented by an edge ofG′) are zero. IfG′ is connected (which is always true in ourapplication) and includes at least one edge with a negativeweight,L is guaranteed to be positive definite.

The spectral analysis of a Laplacian matrix or pole ma-trix can be intuitively understood by analogy to the vibra-tional behavior of a system of masses and springs. Imaginethat each node ofG′ represents a mass located in space, andthat each edge represents a spring connecting two masses.Positive edge weights imply attractive forces, and negativeedge weights imply repulsive forces. The eigensystem ofL represents the transverse modes of vibration of the mass-and-spring system. The lowest-frequency modes give cluesas to where the graph can be cut most effectively: theinsidemasses are usually found vibrating out of phase with theout-side masses.

We take advantage of this observation by finding theeigenvectorx associated with the smallest eigenvalueλ ofthe generalized eigensystemLx = λDx, whereD is a diago-nal matrix whose diagonal is identical to that ofL. BecauseL is sparse, we compute the eigenvectorx using the iterativeLanczos algorithm [Lan50,PSL90]. Each component of theeigenvectorx corresponds to one column ofL, and thereforeto one node ofG′, to one pole ofQ, and to one tetrahedronof T. (The exception is the component ofx that correspondsto the supernodez.)

Figure5 shows a reconstruction of a skeletal hand and theeigenvectors computed during the two partitioning stages.The components of the eigenvectors are sorted in increasingorder. When our method is applied to noise-free point setsthat are densely sampled from smooth surfaces, we find thatthe eigenvectorx is relatively polarized: most of its compo-nents are clearly negative or clearly positive, with few com-ponents near zero. However, noisy models produce moreambiguous labels—see Figures1, 6, and8. One of the com-ponents ofx corresponds to theoutside supernodez. Sup-pose this component is positive; then the nodes ofG′ whosecomponents are positive are labeledoutside, and the nodeswhose components are negative are labeledinside. (If thecomponent corresponding toz is negative, reverse this label-ing.)

This procedure differs from the usual formulation of nor-malized cuts [SM00] in one critical way. The standard nor-malized cuts algorithm does not use negative weights, so itsLaplacian matrixL is positive indefinite—it has one eigen-value of zero, with an associated eigenvector whose compo-nents are all 1. Therefore, the standard formulation uses theeigenvector associated with the second-smallest eigenvalue(called theFiedler vector) to dictate the partition. Our polegraph has negative weights, our pole matrix is positive def-inite, and we use the eigenvector associated with the small-est eigenvalue to dictate the partition. Because the negativeweights encode information about tetrahedra that are likelyto be on opposite sides of a surface, we find that this formula-tion reconstructs better surfaces, and permits us to calculatethe eigenvector much more quickly than normalized cuts intheir standard form.

The Lanczos algorithm is an iterative solver which typi-cally takes aboutO(

√n) iterations to converge, wheren is

the number of nodes inG′. (L is an n× n matrix.) Theconvergence rate also depends on the distribution of eigen-values of the generalized eigensystem, in a manner that isnot simple to characterize and is not related to the conditionnumber. The most expensive operation in a Lanczos itera-tion is matrix-vector multiplication, which takesO(n) timebecauseL is sparse.

For undersampled and noisy models, theO(n√

n) runningtime is justified. Because spectral partitioning searches fora cut that is “good” from a global point of view, it elegantlypatches regions that are undersampled or not sampled at all(see Figure6). Measurement errors may muddy up the edgeweights, but a good deal of noise must accumulate globallybefore the reconstruction is harmed (see Section4). Thereare many faster surface reconstruction algorithms, and theyare often preferable for clean models. But most algorithmsare fooled by undersampling, outliers, and noise, and manyleave holes in the reconstructed surface or make serious er-rors in deciding how to patch holes.



Figure 6: Left: the eigencrust reconstruction of the Stanford Bunny data (raw, unsmoothed point samples with natural noise butno outliers) patches two unsampled holes in the bottom of the bunny. The eigenvectors are less polarized than the eigenvectors inFigure5, reflecting the labeling ambiguities due to measurement errors. Also illustrated is the bunny after Laplacian smoothing,described in Section3.6. 362,272 input points; 2,283,480 tetrahedra; 679,360 output triangles; genus zero; 19.1 minutesreconstruction time, of which 17.5 minutes is spent computing the eigenvectors. Right: the eigencrust patches a large unsampledregion of a model of a hand.

3.3. Correcting Questionable Poles

An optional step, strongly recommended for noisy models,removes labels whose accuracy is questionable, so that somepoles will be relabeled during the second step. Most poleslie near the medial axis of the original object, and dualize totetrahedra that extend deeply into the object’s interior. How-ever, random measurement errors in the sample point coordi-nates can create spurious poles that are closer to the surfacethan the medial axis. Fortunately, a spurious pole is usuallyeasy to recognize: it dualizes to a small tetrahedron that isentirely near the object surface.

Laser range finders typically sample points on a squaregrid. Using the three-dimensional coordinates of those sam-ples, we compute a grid spacing` equal to the median lengthof the diagonals of the grid squares. Any labeled tetrahe-dron whose longest edge is less than 4` is suspicious, so weremove its label. The grid resolution is typically small com-pared to the object’s features, so poles near the medial axisare unaffected.

We find that this step consistently leads to more accu-rate labeling of noisy models. It is unnecessary for smooth,noise-free models.

3.4. Labeling the Remaining Tetrahedra

The first partitioning stage labels each tetrahedron whosedual Voronoi vertex is a pole, and labels some of the othertetrahedra too (such as those touching the bounding box).Many tetrahedra with more ambiguous identities remain un-labeled. To label them, we construct and partition a secondgraphH. The goal of the second partitioning stage is to la-bel the ambiguous tetrahedra in a manner that produces arelatively smooth surface of low genus.

The graphH has two supernodes, representing all thetetrahedra that were labeledinside andoutside, respectively,during the first stage.H also has one node for each unlabeledtetrahedron. If two unlabeled tetrahedra share a triangularface, they are connected by an edge ofH. If an unlabeledtetrahedron shares a face with a labeled tetrahedron, the for-mer is connected by an edge to one of the supernodes.

We have tried a variety of ways of assigning weights tothe edges ofH. We obtained our best results (surfaces withthe fewest handles) by choosing each weight to be the “as-pect ratio” of the corresponding triangular face, defined asthe face’s longest edge length divided by its shortest edgelength. These weights encourage the use of “nicely shaped”triangles in the final surface, and discourage the appearanceof “skinny” triangles (whose large edge weights resist cut-ting).

H has just one negative edge weight: an edge connect-ing the inside andoutside supernodes, whose weight is thenegation of the sum of all the other edge weights adjoiningthe supernodes. The negative edge ensures that the supern-odes are assigned opposite signs in the eigenvector.

We partitionH as described in Section3.2. A tetrahedronis labeledinside if the corresponding value in the eigenvectorhas the same sign as theinside supernode, and vice versa.

As an alternative to this stage, the first partitioning stagecan label power cells of the poles instead of labeling Delau-nay tetrahedra—in other words, we replace the Powercrust’spole labeling algorithm [ACK01] with our spectral pole la-beling algorithm. In the Powercrust algorithm,everypowercell is the dual of a pole, so there is nothing left to label afterthe first partitioning stage.

Figure 7 shows that the eigencrust is poor at capturingsharp corners, and the Powercrust algorithm is much better,but the hybrid algorithm is even more effective. Spectralpartitioning labels power cells better than the original Pow-ercrust. The hybrid spectral Powercrust algorithm shares thePowercrust’s advantage of recovering sharp corners well, butit also shares its disadvantage of increasing the number ofvertices many-fold. The number of vertices in each model is4,100, 52,078, and 51,069, respectively.

3.5. Constructing Manifolds

An optional heuristic step searches for local topological ir-regularities that prevent the reconstructed surface from beinga manifold, and makes the surface a manifold by relabelingselected tetrahedra frominside to outside.



Eigencrust Powercrust Hybrid

Figure 7: Reconstructions of a mechanical part by threealgorithms. The eigencrust algorithm uses both eigenvec-tors, whereas the hybrid (spectral Powercrust) algorithmuses only the first.

These irregularities come in two types. First, consider anyedgee of the Delaunay tetrahedralizationT. The tetrahedrathat havee for an edge form a ring arounde. If the recon-structed surface is a manifold, there are three possibilities:the tetrahedra in the ring are alloutside, they are allinside,or the ring can be divided into a contiguous strand ofinsidetetrahedra and a contiguous strand ofoutside tetrahedra. Ifthe ring of tetrahedra arounde do not follow any of thesepatterns—if there are two or more contiguous strands ofin-side tetrahedra in the ring—then we fix the irregularities byrelabeling some of the tetrahedra frominside to outside sothat only one contiguous strand ofinside tetrahedra survives.The surviving strand is chosen so that it contains theinsidetetrahedron that was assigned the largest absolute eigenvec-tor component during the second partitioning stage.

The second type of irregularity involves any point samples in S. If the reconstructed surface is a manifold, then thetetrahedra that haves for a vertex are either alloutside, allinside, or divided into one face-connected block ofoutsidetetrahedra and one face-connected block ofinside tetrahedra.A topological irregularity ats may take the form of twoin-side tetrahedra that haves for a vertex, but are not connectedto each other through a path of face-connectedinside tetrahe-dra all havings for a vertex. In this case, theinside tetrahedraadjoinings can be divided into two or more face-connectedcomponents. Only one of theseinside components survives;we relabel the othersoutside. The surviving component isthe one that contains a pole ofs. (In the unlikely case thatthere are two such components, choose one arbitrarily.)

It is also possible to have two or more face-connected

components ofoutside tetrahedra (and just one componentof inside tetrahedra). LetW andX be two of them. We com-pute the shortest face-connected path fromW to X, wherethe length of a path is defined to be the sum of the absoluteeigenvector components of theinside tetrahedra on the path.The tetrahedra on the shortest path are relabeledoutside.

These operations are repeated until no irregularity re-mains. The final surface is guaranteed to be a manifold.One can imagine that for a pathological model this proce-dure might whittle down the object to a few tetrahedra, butin practice it rarely takes an unjustifiably large bite out of anobject.

3.6. Smoothing

Triangulated surfaces extracted from noisy models arebumpy. The final optional step is to use standard Lapla-cian smoothing [Her76] to remove the artifacts created bymeasurement errors in laser range finding, and to make themodel more amenable to rendering and simulation. Lapla-cian smoothing visits each vertex in the triangulation in turn,and moves it to the centroid of its neighboring vertices.We performed five iterations of smoothing on the smoothedbunny and dragon in Figures6 and8.

4. Results

Our implementation uses our own Delaunay tetrahedraliza-tion software, and TRLAN, an implementation of the Lanc-zos algorithm by Kesheng Wu and Horst Simon of the Na-tional Energy Research Scientific Computing Center.

Figure8 illustrates the performance of the spectral algo-rithm on the Stanford Dragon. The raw data exhibit randommeasurement errors and include natural outliers. Spectralreconstruction yields a watertight manifold surface, and re-moves all the outliers.

Figure 9 shows how several algorithms degrade as ran-domly generated outliers are added to the input data. Even1,200 random outliers have no influence on the eigencrustexcept to affect how a hole at the base of the hand is patched.The Tight Cocone and Powercrust algorithms are incapaci-tated by relatively few outliers, although they correctly re-construct models without outliers.

The fourth row of Figure9 shows the degradation of sev-eral algorithms as increasing amounts of random Gaussiannoise are added to the point coordinates of the StanfordBunny, which already includes measurement errors. Theexpression under each reconstruction is the variance of theGaussian distribution used to produce additional noise ineach coordinate, expressed in terms of the grid spacing`defined in Section3.3. The eigencrust remains a genus zeromanifold when the added noise has variance 2`, but beginsto disintegrate as the measurement errors become notablylarger than the resolution of the range data. With added noiseof variance`, the Powercrust algorithm succeeds, but withvariance 2̀ the structure is full of holes. (It is a watertightsurface, but what it bounds is Swiss cheese.) The Tight Co-



Figure 8: Eigencrust reconstruction of the Stanford Dragonmodel from raw data. The point cloud (upper left) has manyoutliers, which are automatically omitted from the spectrallyrecovered surface. 1,769,513 input points; 11,660,147 tetra-hedra; 2,599,114 surface triangles; genus 1; 197 minutesreconstruction time.

cone algorithm can only cope with noise of less than 0.8`variance.

Figure10illustrates the three algorithms on a set of pointsdensely sampled from the smooth splines of the Utah Teapot.The difficulties here are more subtle. The teapot’s spout pen-etrates the body deeply enough to create an ambiguity thatneither the Powercrust nor Tight Cocone algorithm solvescorrectly. The spectral reconstruction algorithm treats theinterior spout sample points as outliers, and correctly omitsthem from the eigencrust.

The spectral algorithm is not infallible. It occasionallycreates unwanted handles—thirteen on the angel. The globaleigenvector computation is slow. Tetrahedron-labeling algo-rithms do not reconstruct sharp corners well—observe howthe teapot adjoins its spout. This problem can sometimesbe overcome by labeling power cells rather than tetrahedra,at the cost of much larger model complexity. Either way,however, spectral surface reconstruction is remarkably ro-bust against noise, outliers, and undersampling.

Acknowledgments

We thank Chen Shen for rendering many of our models,and Nina Amenta and Tamal Dey for providing their sur-face reconstruction programs. This work was supported inpart by the National Science Foundation under Awards ACI-9875170, CMS-9980063, and CCR-0204377, in part by theState of California under MICRO Award 02-055, and in partby generous support from Pixar Animation Studios, IntelCorporation, Sony Computer Entertainment America, andthe Okawa Foundation.

References

[AB99] AMENTA N., BERN M.: Surface Reconstruc-tion by Voronoi Filtering.Discrete and Compu-tational Geometry 22(1999), 481–504.

[ABK98] AMENTA N., BERN M., KAMVYSSELIS M.:A New Voronoi-Based Surface ReconstructionAlgorithm. InComputer Graphics (SIGGRAPH’98 Proceedings)(July 1998), pp. 415–421.

[ACDL02] AMENTA N., CHOI S., DEY T. K., LEEKHA

N.: A Simple Algorithm for HomeomorphicSurface Reconstruction.International Journalof Computational Geometry and Applications12, 1–2 (2002), 125–141.

[ACK01] AMENTA N., CHOI S., KOLLURI R.: ThePower Crust. InProceedings of the Sixth Sym-posium on Solid Modeling(2001), Associationfor Computing Machinery, pp. 249–260.

[AK00] AMENTA N., KOLLURI R.: Accurate and Ef-ficient Unions of Balls. InProceedings of theSixteenth Annual Symposium on ComputationalGeometry(June 2000), Association for Com-puting Machinery, pp. 119–128.

[BMR∗99] BERNADINI F., MITTLEMAN J., RUSHMEIER

H., SILVA C., TAUBIN G.: The Ball-PivotingAlgorithm for Surface Reconstruction.IEEETransactions on Visualization and ComputerGraphics 5, 4 (Oct. 1999), 349–359.

[Boi84] BOISSONNAT J.-D.: Geometric Structuresfor Three-Dimensional Shape Reconstruction.ACM Transactions on Graphics 3(Oct. 1984),266–286.

[BTG95] BITTAR E., TSINGOS N., GASCUEL M.-P.:Automatic Reconstruction of Unstructured 3DData: Combining Medial Axis and Implicit Sur-faces.Computer Graphics Forum 14, 3 (1995),457–468.

[CBC∗01] CARR J. C., BEATSON R. K., CHERRIE J. B.,M ITCHELL T. J., FRIGHT W. R., MCCAL -LUM B. C., EVANS T. R.: Reconstruction andRepresentation of 3D Objects with Radial BasisFunctions. InComputer Graphics (SIGGRAPH2001 Proceedings)(Aug. 2001), pp. 67–76.

[CL96] CURLESS B., LEVOY M.: A VolumetricMethod for Building Complex Models fromRange Images. InComputer Graphics (SIG-GRAPH ’96 Proceedings)(1996), pp. 303–312.

[DG03] DEY T. K., GOSWAMI S.: Tight Cocone: AWater-tight Surface Reconstructor.Journal ofComputing and Information Science in Engi-neering 3, 4 (Dec. 2003), 302–307.

[DG04] DEY T. K., GOSWAMI S.: Provable SurfaceReconstruction from Noisy Samples. InPro-ceedings of the Twentieth Annual Symposiumon Computational Geometry(Brooklyn, NewYork, June 2004), Association for ComputingMachinery.



200 outliers 1,200 outliers 1,800 outliers 6,000 outliers 10,000 outliers

Eigencrust

200 outliers 1,200 outliers 200 outliers 1,200 outliers

C Tight Cocone[DG03]

PowercrustB[ACK01]

Eigencrust,+2` Eigencrust,+3` Eigencrust,+5` Tight Cocone,+0.8` — Powercrust,+2` —

Figure 9: Top row: 25,626 noise-free points plus randomly generated outliers. Second row: eigencrust reconstructions of thismodel maintain their integrity to 1,200 outliers, then begin to degrade. Third row: other Delaunay-based algorithms degrademuch earlier. Fourth row: Stanford Bunny reconstructions from raw data, with natural noise plus added random Gaussiannoise. The amount of random noise is expressed as the variance of the Gaussian noise added to each point coordinate, in termsof the grid spacing̀ defined in Section3.3. The eigencrust maintains its integrity with+2` added noise; all the other bunnyreconstructions here exhibit serious failures.



Eigencrust Tight Cocone Eigencrust Powercrust

Figure 10: Reconstructions from 253,859 points sampled on the Utah Teapot. The spout’s splines penetrate into the body ofthe teapot, causing difficulties for both the Powercrust and Tight Cocone algorithms. A cutaway view shows that Powercrustmislabels asoutside a cluster of power cells where the spout enters the body. The spectral algorithm correctly identifies thesame poles asinside.

[Fie73] FIEDLER M.: Algebraic Connectivity ofGraphs. Czechoslovak Mathematical Journal23, 98 (1973), 298–305.

[For92] FORTUNE S.: Voronoi Diagrams and DelaunayTriangulations. InComputing in Euclidean Ge-ometry, Du D.-Z., Hwang F., (Eds.), vol. 1 ofLecture Notes Series on Computing. World Sci-entific, Singapore, 1992, pp. 193–233.

[Hal70] HALL K. M.: An r-Dimensional QuadraticPlacement Algorithm.Management Science 11,3 (1970), 219–229.

[HDD∗92] HOPPE H., DEROSE T., DUCHAMP T., MC-DONALD J., STUETZLE W.: Surface Recon-struction from Unorganized Points. InCom-puter Graphics (SIGGRAPH ’92 Proceedings)(1992), pp. 71–78.

[Her76] HERMANN L. R.: Laplacian-IsoparametricGrid Generation Scheme.Journal of the En-gineering Mechanics Division of the AmericanSociety of Civil Engineers 102(Oct. 1976),749–756.

[KBSS01] KOBBELT L. P., BOTSCH M., SCHWANECKE

U., SEIDEL H.-P.: Feature-Sensitive Sur-face Extraction from Volume Data. InCom-puter Graphics (SIGGRAPH 2001 Proceedings)(Aug. 2001), pp. 57–66.

[Lan50] LANCZOS C.: An Iteration Method for the So-lution of the Eigenvalue Problem of Linear Dif-ferential and Integral Operators.J. Res. Nat.Bur. Stand. 45(1950), 255–282.

[LC87] LORENSEN W. E., CLINE H. E.: March-ing Cubes: A High Resolution 3D SurfaceConstruction Algorithm. InComputer Graph-ics (SIGGRAPH ’87 Proceedings)(July 1987),pp. 163–170.

[LPC∗00] LEVOY M., PULLI K., CURLESS B.,RUSINKIEWICZ S., KOLLER D., PEREIRA

L., GINZTON M., ANDERSON S., DAVIS J.,

GINSBERG J., SHADE J., FULK D.: The Digi-tal Michelangelo Project: 3D Scanning of LargeStatues. InComputer Graphics (SIGGRAPH2000 Proceedings)(2000), pp. 131–144.

[OBA∗03] OHTAKE Y., BELYAEV A., ALEXA M., TURK

G., SEIDEL H.-P.: Multi-Level Partition ofUnity Implicits. ACM Transactions on Graphics22, 3 (July 2003), 463–470.

[PSL90] POTHEN A., SIMON H. D., LIOU K.-P.: Par-titioning Sparse Matrices with Eigenvectors ofGraphs.SIAM Journal on Matrix Analysis andApplications 11, 3 (July 1990), 430–452.

[SM00] SHI J., MALIK J.: Normalized Cuts and ImageSegmentation. IEEE Transactions on PatternAnalysis and Machine Intelligence 22, 8 (Aug.2000), 888–905.

[TL94] TURK G., LEVOY M.: Zippered Poly-gon Meshes from Range Images. InCom-puter Graphics (SIGGRAPH ’94 Proceedings)(1994), pp. 311–318.

[TO99] TURK G., O’BRIEN J.: Shape TransformationUsing Variational Implicit Functions. InCom-puter Graphics (SIGGRAPH ’99 Proceedings)(1999), pp. 335–342.

[Whi98] WHITAKER R. T.: A Level-Set Approach to 3DReconstruction from Range Data.InternationalJournal of Computer Vision 29, 3 (1998), 203–231.

[YS01] YU S. X., SHI J.: Understanding Popoutthrough Repulsion. InIEEE Conference onComputer Vision and Pattern Recognition(Dec.2001).

[ZOF01] ZHAO H.-K., OSHER S., FEDKIW R.: FastSurface Reconstruction Using the Level SetMethod. InFirst IEEE Workshop on Variationaland Level Set Methods(2001), pp. 194–202.


spectral surface reconstruction from noisy point cloudsjrs/papers/eigencrust.pdf · spectral...

Documents