IEEE TRANSACTIONS ON VISUALIZATION AND ...stoledo/Pubs/TVCG-0047-0504.pdfIEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 0, NO. 0, AUGUST 2004 3 6 spectral basis vectors

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 0, NO. 0, AUGUST 2004 1

Geometry-aware Bases for Shape ApproximationOlga Sorkine, Daniel Cohen-Or,Member, IEEE,Dror Irony, and Sivan Toledo

Abstract— We introduce a new class of shape approx-imation techniques for irregular triangular meshes. Ourmethod approximates the geometry of the mesh using alinear combination of a small number of basis vectors. Thebasis vectors are functions of the mesh connectivity andof the mesh indices of a number ofanchor vertices. Thereis a fundamental difference between the bases generatedby our method and those generated by geometry-obliviousmethods, such as Laplacian-based spectral methods. Inthe latter methods, the basis vectors are functions of theconnectivity alone. The basis vectors of our method, incontrast, are geometry-aware, since they depend on boththe connectivity and on a binary tagging of vertices thatare “geometrically important” in the given mesh (e.g.,extrema). We show that by defining the basis vectors to bethe solutions of certain least-squares problems, the recon-struction problem reduces to solving a single sparse linearleast-squares problem. We also show that this problem canbe solved quickly using a state-of-the-art sparse-matrixfactorization algorithm. We show how to select the anchorvertices to define a compact effective basis from which anapproximated shape can be reconstructed. Furthermore,we develop an incremental update of the factorization ofthe least-squares system. This allows a progressive schemewhere an initial approximation is incrementally refined bya stream of anchor points. We show that the incrementalupdate and solving the factored system are fast enough toallow an on-line refinement of the mesh geometry.

Index Terms— shape approximation, basis, mesh Lapla-cian, linear least-squares

I. I NTRODUCTION

SHAPE approximation is an important problem incomputer graphics and CAGD. Reducing the amount

of data needed to represent a specific shape is oftennecessary for modeling, efficient storage and transmis-sion of 3D models. Irregular triangle meshes are thepredominant means of representing shapes, and in thelast decade there has been a vast amount of work onmesh simplification techniques [1]. These techniques areclosely related, and can be regarded as descendants ofknot removal techniques developed for spline curves andsurfaces [2]. Other approximation techniques, suited for

All the authors are with the School of Computer Science, Tel AvivUniversity, Tel Aviv 69978, Israel.

E-mail: {sorkine|dcor|irony|stoledo}@tau.ac.il

semi-regular connectivity, are based on wavelet repre-sentations or subdivision surfaces [3]–[6].

Mesh simplification techniques aim to approximate agiven shape with as few vertices or triangles as possible,while keeping the error of the approximation, in somegiven metric, lower than a prescribed tolerance. A differ-ent class of approximation techniques retains the originalconnectivity of the given mesh and approximates onlyits geometry [7]–[9]. Karni and Gotsman [7] introducea spectral method where the mesh is approximated byreconstructing its geometry using a linear combinationof a number of basis vectors. The basis is derivedfrom the spectral decomposition of the Laplacian matrixassociated with the mesh connectivity [10]. Chou andMeng [8] encode the geometry of the mesh using vectorquantization of the displacement coordinates. Based onan analysis of the spectral basis of the Laplacian, Sorkineet al. [9] introduce a method where the quantizationis applied to the geometry vector transformed by theLaplacian operator.

Laplacian-based methods are attractive for mesh pro-cessing, since they benefit from the powerful set of toolsfrom linear algebra and signal processing. The eigenvec-tors of the mesh Laplacian matrix can be viewed as anextension of the Fourier transform basis functions for theirregular connectivity case, and the eigenvalues representthe frequencies [7], [11]. The spectral basis is readilydefined on the given irregular mesh and does not requirealtering the input representation. In addition to geometry-compression applications [7], [12], spectral propertieshave been studied for the design of fairing filters andmodeling tools [11], [13], mesh watermarking [14] andspherical parameterization [15].

However, together with their appealing properties, onemust bear in mind that pure Laplacian-based methods aregeometry-oblivious, since the basis vectors are functionsof the connectivity alone. It is possible to use thegeometric Laplace-Beltrami operator (see, e.g., [16]),however, its construction requires heavy use of themesh geometry, which is not practical for compressionapplications. Our newgeometry-awaremethods derivethe basis both from the mesh connectivity and limited ge-ometrical information. The basis vectors in our methodsare centered around selected “geometrically important”anchor vertices. This allows a terse capturing of impor-


Original model, 28747 vertices 35 basis vectors 160 basis vectors 960 basis vectors

Fig. 1. Reconstruction of mesh geometry using geometry-aware bases. A geometry-aware basis function is centered around a certainanchorvertex of the mesh. The locations of the anchors used in reconstruction in the second left figure are marked by red spheres.

tant features of the surface and leads to compact andefficient representation of the mesh geometry. Figure 2illustrates the reconstruction using such basis vectorson a 2D curve example. The bottom row shows themeshes reconstructed using geometry-aware bases. Thelocations of the anchors are marked by small dots. Thereconstructed mesh passes close to the original locationsof the anchor points, which enables good approximationof such features as the tips of the bird’s wings and tail.For comparison, reconstruction of this mesh using ananalogous number of spectral basis vectors misses outthe features. This behavior is evident in large as well asin small scale.

It should be noted that explicit computation of the ba-sis vectors is, generally speaking, too expensive for largemeshes. Geometry representation using the Laplacianeigenbasis [7] requires finding a partial spectral decom-position of a large symmetric matrix. This computationis too expensive to be applied in practice to anything butsmall meshes.

The method that we present here avoids explicitcomputation of the underlying basis. Instead of directlyrepresenting the geometry by the coefficients of thelinear combination of the basis vectors, we reduce thereconstruction problem to solving a sparse linear least-squares system, as explained in Section II. State-of-the-art least-squares solvers make the solution efficient andenable reconstruction of the mesh as a whole.

A. Overview

The proposed geometry-aware representation of ashape is a linear combination ofk basis functions, whichare vectors that assign a real value to each vertex ofthe mesh. The basis functions are an implicit functionof the connectivity of the mesh and of the indices ofk vertices that we callanchors. Each basis function isselected so that it fulfils the following conditions in theleast-squares sense: it attains the value1 at one of theanchors and0 at the other anchors, and it is the smoothestamong all the functions that satisfy these requirements(we also propose a slightly different definition for strictly

interpolatory anchors, but the principle is the same).The smoothness of a function is defined in a discretemanner using the connectivity of the mesh. Specifically,we require that the position of a vertex deviates as littleas possible from the average of its neighbors in the mesh.These definitions result in smooth basis functions thatare easy to combine into an approximation that attainsspecific values at the anchors. Furthermore, a fast sparseleast-squares solver with updating capability allows usto efficiently recover a representation of the shape fromthe coefficients of the linear combination.

A number of recent papers have shown that theconnectivity of the mesh often encodes some useful in-formation about the geometry of the shape that the meshrepresents [17], [18]. Isenburg et al. [17] reconstruct ashape from the connectivity by a non-linear optimizationof a uniform edge-length criterion. In [18] it was shownthat augmenting the connectivity with a few well-placedanchors significantly increases the geometric value of theinformation encapsulated in the connectivity alone. Theleast-squares system that is used to reconstruct the so-called LS-mesh in [18] is essentially the same systemthat arises from our basis vectors. In this paper, we fullyexplore the application of geometry compression, boththeoretically and experimentally. Progressive compres-sion is made possible thanks to the proposed algorithmthat quickly augments the existing representation withnew anchors without fully solving the least-squares re-construction system again. We rigorously analyze theunderlying basis vectors, which provides a theoreticalframework for studying this type of approximation ap-proaches.

The effectiveness of adding anchors with geometricinformation was used earlier in [9] to reduce the low-frequency error caused by quantization of the differentialcoordinates of the mesh. There, a linear least-squaressystem was solved to reconstruct the mesh geometryfrom a quantized differential representation, and thework focused on the analysis of the visual impact ofthe quantization error. In our case, the mesh verticesdo not hold any geometric information – it is entirelyencapsulated in the basis functions.


6 spectral basis vectors 45 spectral basis vectors

6 geometry-aware vectors 45 geometry-aware vectors

Fig. 2. Reconstruction of the swallow curve (simple closed path)using different bases. The top row shows reconstruction using theLaplacian eigenvectors, which are the discrete Fourier basis functionin this case. The bottom row displays reconstruction with geometry-aware basis vectors. The reconstructed mesh is shown in black, whilethe original mesh is tinted in blue. The geometry-aware bases betterapproximate the features of the shape, on large as well as on smallscales.

The main contributions of this paper include efficientalgorithms for producing a geometric approximation ofa shape and for recovering the approximate shape fromthe compact representation. Our algorithms are based onseveral advanced computational linear algebra tools: theability to control the conditioning of the least-squaresproblems that we solve, the ability to solve them quickly,and the ability to quickly add anchors by updating asparse factorization of an augmented Laplacian matrix.We also provide evidence that the new method comparesfavorably with spectral methods, both in terms of com-pression ratios for a given approximation error and interms of running times. The paper explores the theoryof augmenting the connectivity with geometric data, insearch for a better understanding of shapes in generaland approximation of irregular meshes in particular.

II. GEOMETRY-AWARE BASES

Most of the techniques for approximating and encod-ing mesh geometries represent the geometry as a linearcombination of basis functions. In this section we presentthe specific basis functions that we use and explain whythis basis is effective.

A mesh functionis a real vector that assigns a valueto each vertex in the mesh. Abasis functionis simply amesh function, and abasisis a set of basis functions thatspansRn, wheren is the number of vertices in the mesh.The coordinates of the vertices, say thex coordinates,are a mesh function that expresses the location of thevertices inR3 as a linear combination of the functionsof the standard basis, whose functions assign1 to onevertex and0 to all the others. The coordinates can alsobe expressed as a linear combination of other basisfunctions.

The bases that we use, like Laplacian-spectral bases,can be constructed by solving a series of minimizationproblems. This construction is perhaps not the mostnatural one for Laplacian-spectral bases, but it is the mostnatural for our bases. Let us describe this constructionfor the well-known Laplacian-spectral bases first. Thecombinatorial Laplacian of a mesh is then × n sym-metric positive semi-definite matrixL = D −A, whereA = (aij) is the adjacency matrix (aij = 1 if vertices iandj are neighbors andaij = 0 otherwise) andD is thediagonal matrix whoseith entry on the diagonal equalsthe valency (degree) of vertexi.

Given the LaplacianL of the mesh, the first Laplacian-spectral basis functionu1 is the function that minimizes1

‖Lu1‖ subject to‖u1‖ = 1. The next basis functionu2

is the one that minimizes‖Lu2‖ subject to‖u2‖ = 1 andto u2 ⊥ u1. In general,uk minimizes‖Luk‖ subject to‖uk‖ = 1 and touk ⊥ span{u1, . . . ,uk−1}. The func-tions uk are the eigenvectors ofL sorted by the eigen-values. The minimization problems above favor smoothbasis functions, because the transformationx 7→ Lx as-signs to each vertexi the difference betweenxi and theaverage of its neighbors, multiplied by the number ofneighbors. Therefore,u1 is the smoothest vector inRn,the constant vector,u2 is the smoothest mesh functionorthogonal tou1, and so on. The first functionu1 isalways the same, while the shapes of the rest depend onthe topology of the mesh.

A. Relaxed geometry-aware bases

Our basis also solves a series of minimization prob-lems, but they are chosen in a geometry-aware manner.Given a set ofk vertex indices1 ≤ a1, a2, . . . , ak ≤ n,the ith functionvi in our basis minimizes

‖Lvi‖2 +

∑j 6=i

ω2|(vi)aj− 0|2 + ω2|(vi)ai

− 1|2 .

1We use the following notation. Vectors are denoted by uprightbold letters, e.g.,x, and their elements are denoted by italic letters,e.g.,xi. All the vectors in this paper are column-vectors and all thenorms are2-norms.


0 50 100 150 200 250 300-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.25 first vectors out of 5 anchor vectors

0 50 100 150 200 250 300-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.25 first vectors out of 20 anchor vectors

(a) (b)

Fig. 3. Geometry-aware basis functions on a 1D domain. The meshhere is a simple closed path with 274 vertices. Plot (a) displays thefive basis functions corresponding to a set of five anchors; (b) showsthe first five basis functions out of a 20-anchor basis.

The interpretation of this minimization problem is thefollowing. The basis functionvi minimizes the sum oftwo terms. The first term is the non-smoothness invi,and the second is the deviation ofvi from given values inthek mesh locationsa1, . . . , ak, which we callanchors.These values are1 at ai and0 at aj , j 6= i. Therefore,vi

tries simultaneously to be smooth everywhere, to be largeat ai, and to vanish on all the otheraj ’s. The weightωcontrols the impact of the anchors. Our algorithms neveruse basis functions other than the firstk (the number ofanchors), so there is no point in characterizing them.(Formally, all completions of this set ofk functions to abasis ofRn are equivalent for our algorithms.)

Figure 3 shows five geometry-aware basis functionson a mesh consisting of a simple path. On this mesh,the first Laplacian-spectral basis functions are simplylow-frequency sines and cosines. The geometry-awarefunctions are also fairly smooth, but most of their“energy” is concentrated near a single anchor. Basisfunctions for largerk are less smooth, because theanchors get closer to each other, forcing the functionsto attain values near0 and near1 within short intervals.Intuitively, a few geometry-aware functions should allowus to approximate smooth mesh functions whose extremaare at or near the anchors more accurately than a fewgeometry-oblivious Laplacian-spectral functions.

We express approximations of mesh functions using aset of k anchors and the coefficientsc = (c1, . . . , ck)T

of the correspondingk geometry-aware functionsV = (v1, . . . ,vk). Given this representation of the ap-proximation, we reconstruct the approximationx in thestandard basis by solving a single least-squares mini-mization problem,

x = argminx

{‖Lx‖2 +

k∑i=1

ω2|xai− ci|2

}=

=k∑

i=1

civi = V c

The equality follows from the linearity of the minimum-norm solution to least-squares problems. The coefficientmatrix L of this least-squares problem hasn + k rowsandn columns. The significance of this expression is thatit shows that we can reconstructx from V without anyreference to the basis vectorsV . Thus, assuming w.l.o.g.that(a1, a2, . . . , ak) = (1, 2, . . . , k), we reconstructx bysimply finding the vector that minimizes the norm of

L

ω Ik×k | 0

x1

...xn

−

|0|

ω c1...

ω ck

. (1)

SinceL is typically very sparse, this least squares canbe solved very quickly even whenn is large. It shouldbe noted that the reconstruction is not interpolatory atthe given values on the anchors – it only approximatesthem in a least-squares sense.

There are at least three categories of constraints thatwe can apply to the anchors. The method that we pre-sented above charges a quadratic penalty for deviationsof x from x at the anchors. We can use different weightsfor these penalties and for the smoothness penalties.Another option is to use box constraints, which requirethat xai

lies within a box centered aroundxai[19]. The

algorithmic issues in this approach are more complexthan in the other approaches, so we have not pursuedit. The third approach is interpolatory; it requires thatxai

= xai. This is the limiting case of the two other

approaches. We explain this approach next.

B. An interpolatory scheme

We can create slightly different geometry-aware basesby forcing the basis functions to attain prescribed valuesat specific mesh locations. Given a set ofk vertex indices1 ≤ a1, . . . , ak ≤ n, the ith function wi in the basisminimizes‖Lwi‖ subject to(wi)ai

= 1 and (wi)aj= 0

for j 6= i.Given the indices of thek anchors and the coef-

ficients c of the basis functions(w1, . . . ,wk) = W ,the approximationx can be reconstructed as follows.We use the equationxai

= ci to eliminate xaifrom

the system. This effectively deletes columnai and rown + ai from the coefficient matrixL (whereω = 1) andchanges the right-hand side. After all these equations areeliminated, the resulting coefficient matrixL hasn rowsandn− k columns. To reconstruct the unknown valuesxj , j /∈ {ai}, we solve the least-squares problem

minx‖Lx− (−L1:n,{ai}c)‖ .


Combining the minimizer with the known valuesxai

= ci yields the approximation

x =k∑

i=1

ciwi = Wc .

The coefficient matrixL of this least-squares problemis smaller and sparser thanL, so in general it willbe even easier to solve the least-squares problem thatreconstructsx.

The main disadvantage of this basis, compared to therelaxed basis{vi}, is that adding interpolatory anchorsis computationally more expensive than adding least-squares anchors; we explain this issue below, in Sec-tion III. When a large weightω is used for the anchorsin the relaxed scheme, the solution effectively becomesvery close to interpolatory. See Figure 4 that visualizesthe influence of differentω’s.

C. Approximating mesh functions

So far we have seen the basis functions and howto reconstruct an approximation given the indices ofthe anchors and the coefficients of the basis functions.We now turn to the question of how to generate thecoefficientsc = (c1, . . . , ck)T given a mesh functionxand a seta1, . . . , ak of anchors.

Perhaps the best way to definec is by requiring thatthe approximationx = V c or x = Wc of a meshfunction x be as close as possible, in the2-norm, tox. That is, to require thatc minimizes ‖V c − x‖ or‖Wc− x‖ (depending on the basis used). Solving thesesystems is potentially expensive. A naive way to computethese optimalc’s is to computeV or W explicitly,by solving the least-squares problems that define theircolumns, and then to solve the densen×k least-squaresproblem. Note that to reconstructx = V c or x = Wc,we do not use an explicit representation ofV or W .

For interpolatory geometry-aware bases, another nat-ural way to choosec is by settingci = xai

. This ensuresthat x coincides withx at the anchors. The2-norm ofthe errorx − x is likely to be higher than if we definec so as to minimize the error, but now the error isconcentrated away from the anchors. It turns out thatsettingci = xai

works well even for relaxed geometry-aware basesV . Employing large weights (ω → ∞) onthe relaxed anchors effectively makes the relaxed schemeinterpolatory, while maintaining the advantage of theupdating capability (see Section III). In practice, we setω = 10n.

III. T HE PROGRESSIVE SCHEME

One of the best aspects of relaxed geometry-awarebases is that we can quickly improve the approximationas soon as the location of additional anchors becomesknown. This allows a client to display a rough approx-imation as soon as the location of a few anchors isreceived from a server or retrieved from storage.

When the locations of additional anchors becomeknown to the client, it can produce a more accurateapproximation by updating the system with the newinformation. The following system is solved:

Lnewx = (01×n, ω c1, . . . , ω ck, ω ck+1, . . . , ω ck+m)T ,

whereLnew is the updated system matrix comprised ofthe previousL, and additional rows for the new anchorsak+1, . . . , ak+m; ck+1, . . . , ck+m denote the new coef-ficients. The key to utilizing additional anchors is anefficient updating scheme to a sparse factorization ofL.The system (1) can be solved using a sparse Choleskyfactorization of the normal equations,LT L = RT R,whereR is sparse and upper triangular. The factorizationis done once, for an initial set of anchors. Suppose thatwe now add an anchorak+1. This adds a row toL,and addsω2 to the ak+1th diagonal element ofLT L.To reconstruct the new approximation, we need a newCholesky factorization ofLT

newLnew. Fortunately, we canupdate the previous factorization in time proportional tothe number of nonzeros inR. Furthermore, the updatedoes not modify the nonzero structure ofR, only thenumerical values of its entries.

We updateR as follows. We essentially eliminate thesingle nonzero in the new row inL using a series ofGivens rotations that we perform on that row and onrows of R. The first rotation is performed on rowak+1

of R and annihilates theak+1th element in the new row.This, however, introduces nonzeros to several elements inthe new row, elements with column indices greater thanak+1. We then eliminate the nonzero element with thesmallest column index in the new row, say indexi, usinga Givens rotation on rowi of R. The Givens rotationnever modifies the nonzero structure of rows inR,because the next row that we update is always the parentin the elimination tree ofLT L of the previous row. Sincewe updateR using a series of orthogonal transformations(the Givens rotations) and since the addition of aω2 tothe diagonal ofLT L only improves its conditioning, theupdating process is always numerically stable.

Our incremental update method can be viewed asa special case of the general algorithm proposed byDavis and Hager [20]. Due to the specific structure ofthe change inL, the update in our case is particularly


ω = 1.0 ω = 10.0 ω = 100.0 interpolatory reconstruction

ω = 1.0 ω = 10.0 ω = 100.0 interpolatory reconstruction

Fig. 4. The effect of different weights on the relaxed scheme. The first three columns display the reconstruction with the relaxed schemeusing the same set of anchors, with different weights. The rightmost column shows the reconstruction using the interpolatory scheme. Close-up on the ear is shown in the bottom row; the red spheres denote the position of the anchor vertices in the original mesh. As the weight ofthe anchors grows, the reconstruction approaches to being interpolatory.

efficient. More specifically, Davis and Hager show howto update the Cholesky factorR when an arbitrary rowis added or removed fromL. Our algorithm solves aspecial case of this general update/downdate problem:the case of adding a row with a single nonzero. Inthe general case, the nonzero structure ofR mightchange, and so does its elimination tree. These changesrequire a sophisticated algorithm to take care of sparsity.Furthermore, sinceR can fill as a result of an update,the cost of a series of updates can be hard to predict.In contrast, in our case the nonzero structure ofRand the elimination tree do not change, so the pathin the elimination tree from the vertexak+1 to theroot gives the sequence of elimination operations thatmust be performed. This special case is considerablysimpler. Another difference between the algorithm ofDavis and Hager and ours is that we use orthogonalGivens rotations to eliminate the new row inL, whereasthey use nonorthogonal operations. As a consequence,our algorithm performs4 floating-point operations pernonzero in R that is modified, and their algorithmperforms only2 per modified nonzero. Using fast Givensrotation in our algorithm would bring the two algorithmto the same cost per modified nonzero, but due to theinsignificance of the update costs, we did not implementsuch an approach. In short, our algorithm is, essentially,a special case of [20]. But in our case, much of themachinery developed in [20] is not needed.

Since three-dimensional meshes typically have smallvertex separators, and due to the special structure of ourupdates, we can provide a tighter bound on the cost

of an update operation than was given by Davis andHager. They show that the cost of an update operationis proportional to the number of nonzeros inR thatare being modified. The same is true in our algorithm.However, in our case we can argue that under a rea-sonable assumption, the number of modified nonzerosin R is proportional ton, the size of the mesh; in mostcases,n is much smaller than the number of nonzerosin R. Suppose that a mesh can be embedded on thesurface of a body with bounded genus (that is, withoutmany holes). Then the mesh has excluded minors, whichimplies that it has aO(

√n) approximately-balanced

vertex separator [21]. Once separated, the same holdsfor the parts. The separators form a tree, and the path inthe elimination tree from rowak+1 to the root is also apath in this separator tree. The number of nonzeros inRthat is modified is at most the sum of the squared sizesof the separators on this path, which is at most(

c√

n)2 +

(c√

(2/3) n)2

+(c√

(2/3)2 n)2

+ · · · <

< 1.8 c2 n ,

for some constantc that depends on the genus. Note thatfor such meshes, the total number of nonzeros inR isΘ(n log n), so the update only modifies a small fractionof them. In particular, the update is much cheaper thansolving a single least-squares problem with the computedfactor R.

In the graphics literature, updating linear systems ofequations due to changes of boundary conditions wasalso performed by James and Pai [22]. However, theyuse the capacitance-matrix approach, where a change


of rank s in the original system matrix requiresO(s3)operations for update. This type of approach is notefficient when it comes to incremental updates, since theupdate has to be applied to the original factorization ofthe first L. Thus, the cost would be cubic in the totalnumber of added anchors.

In general, updating a sparse factorization with arbi-trary constraints can be both expensive and unstable. Forexample, updating the factorization of the interpolatoryscheme is probably more difficult than updatingR toaccommodate additional relaxed anchors. However, it iseasy to add relaxed anchors to a factorization of aninterpolatory basis. What is important is what kind ofconstraint we add, not how the original factorization wasproduced. Therefore, we only employ relaxed anchorsupdate, which is guaranteed to be stable.

IV. SELECTING ANCHORS

The norm of the approximation error‖x − x‖ isgoverned by two factors: the condition number ofLand the angle betweenx and span{v1, . . . ,vk}. Thefirst factor depends on the location of the anchors inthe topology of the mesh, and is independent of thegeometry of the shape. It is proven thatL is well-conditioned if, loosely speaking, no vertex is too far (interms of mesh edges) from an anchor, i.e., if the anchorsare well-distributed across the mesh graph. Theoreticalbounds on the condition number ofL, as well as apractical algorithm for choosing an initial set of anchorsto conditionL, can be found in [23]. The second factordepends on the interaction between the the geometry ofthe given shape and the basis functionsv1, . . . ,vk.

We use an iterative greedy heuristic to reduce theangle betweenx and span{v1, . . . ,vk}. Given a set ofanchorsa1, . . . , ak−1, we compute an approximationxof the given shape and find the vertex on whichx differsmost fromx. That vertex becomes the next anchor,ak.Since we try to approximate at least three mesh functionsusing the same anchors (the mesh function in three spacedimensions), we actually select the vertex whose spacial3D location in the approximated shape has the largestgeometric distance to its location in the original mesh.

The above incremental selection scheme is well suitedfor progressive transmission of the mesh geometry: theserver sends the anchors to the client in the same orderin which they were chosen by the greedy algorithm.

V. RESULTS

We have tested our shape approximation method onseveral 3D models. We report results only for the relaxedgeometry-aware bases due to lack of efficient updating

capability for the interpolatory scheme, as discussed inSection III. To reconstruct an approximation from aninitial set of anchors, the client needs to compute thesparse factorization ofL (the connectivity is supposed tobe already known) and to solve for the mesh functionsx, y andz. When more anchors become known, the fac-torization is updated and we solve forx, y, z again. Therunning times of these key ingredients are summarizedin Table I. The factorization is the most costly part, andis computed only once; the update and solve times arevery small. We have used the direct solvers provided byTAUCS [24]. All our experiments were carried out on a2.4 GHz Pentium 4 machine.

The compressed representation consists of the indicesof the chosen anchors and the basis coefficients, whichare the locations of the anchor vertices in the originalmodel. The coefficients are uniformly quantized, and allthe data is encoded using an arithmetic encoder. How-ever, since the locations and the indices of the anchorsare scattered across the mesh, entropy-encoding typicallydoes not further reduce the size of the representation.Thus, roughlyk log n bits are needed to represent theindices of thek anchors and3kq bits for the coefficients,where q is the quantization level. Note that since ourscheme favors smooth reconstructions, the approximatedshapes do not suffer from “jaggies” effects that wouldbe caused by quantization ofall the x, y, z coordinates.We usedq between 10 to 12 bits.

The results of approximations using varying numbersof basis vectors are shown in Figures 1 and 6. One canobserve that the main features of the models, such asextruding parts, are captured in the very early stagesof the progressive scheme (i.e. with a small number ofbasis vectors). We have compared our results with themethod of Karni and Gotsman [7]. The spectral basisof the mesh Laplacian [7] is a natural candidate forcomparison with our geometry-aware bases, since bothcompression methods preserve the mesh connectivity,unlike the compression schemes that require semi-regularremeshing [3]–[6]. We have carried out such a compar-ison; however, it is limited to small meshes only. Asdiscussed above, the spectral method requires computinga partial eigendecomposition of the Laplacian, whichis time- and space-consuming. We used MATLAB ’seigs function to find the first several eigenvectors ofsome submeshes of theCamel model (see Figure 5).Computation of the first 1000 eigenvectors of a meshwith 3220 vertices took about 4 minutes on a 2.4 GHzmachine with 2 GB or RAM. The computation usedmore than 1 GB of RAM, and indeed, on a similarmachine with only 1 GB of RAM, the computationtook about 20 minutes due to paging. Computing the


TABLE I

RUNNING TIMES IN SECONDS OF THE DIFFERENT COMPONENTS OF SOLVING THE LINEAR LEAST-SQUARES SYSTEMS. Factor STANDS

FOR THE FACTORIZATION TIME OF THE NORMAL EQUATIONS MATRIX; SolveIS THE TIME OF SOLVING FOR A SINGLE MESH FUNCTION

BY BACK -SUBSTITUTION; Average updateIS THE AVERAGE TIME SPENT ON UPDATING THE FACTORIZATION BY ONE RELAXED ANCHOR

(THE NUMBERS IN PARENTHESES DENOTE THE RANGE OF ANCHOR AMOUNTS OVER WHICH THE AVERAGE WAS COMPUTED). Worst-case

STANDS FOR THE LONGEST UPDATE TIME OBSERVED OVER THE UPDATES OF THE PREVIOUS COLUMN.

Model # vertices Factor Solve Average update (range)Worst-caseCamel hump 1334 0.031 0.002 0.00007 (1–1000) 0.0002Camel mouth 3210 0.101 0.006 0.0002 (1–3000) 0.0003Camel leg 3220 0.121 0.006 0.0002 (1–3000) 0.0004Camel head 11381 0.503 0.029 0.0010 (1–10000) 0.0013Pig 28747 1.558 0.065 0.0020 (1–28000) 0.0032Camel 39074 2.096 0.073 0.0021 (1–39000) 0.0034Feline 49864 2.750 0.110 0.0025 (1–49000) 0.0034Max Planck 100086 7.713 0.240 0.0110 (1–100000) 0.0120Igea 134345 11.826 0.444 0.0200 (1–130000) 0.0215

first 1000 eigenvectors of a mesh representing the entirehead of the camel, with 11,381 vertices, took about21 minutes on the 2 GB RAM machine. On largermeshes, the eigenvector computation simply failed dueto lack of memory. For example, we were not ableto compute more than about 5000 eigenvectors of the11,381-vertex mesh, even on a machine with 2 GB RAM.We note that MATLAB ’s eigs function uses a state-of-the-art sparse eigensolver calledARPACK [25], which isimplemented in Fortran. Thus, this performance is notdue to MATLAB ’s interpreter and nor to a poor choice ofalgorithm; it is essentially the inherent cost of computingeigenvectors.

Figure 5 summarizes the comparison results for thetested small meshes in the form of rate-distortion curves.Typically, up to 10 - 20% of then eigenbasis vectors areneeded for visually lossless reconstruction. As suggestedby Karni and Gotsman [7], we quantized the spectralcoefficients to 14 bits. Stronger quantization leads todistortion of the reconstructed shape even when morethan 50% of the full basis is used, since quantization inthe transformed domain behaves differently than quan-tization in the standard basis. The spectral coefficientswere compressed with an arithmetic encoder. The rate-distortion curves report three error metrics as a functionof the file size of the compressed geometry: the max-norm error, theL2 error measured by the Metro tool [26]and a simple RMS of distance between the mesh vertices.The graphs show that our method does a better job interms of the max-norm metric, which is perhaps not sur-prising because the anchor selection scheme specificallyaims at minimizing this norm. As for theL2 and simpleRMS metrics, the two algorithms perform practically thesame. For theCamelhump mesh, which is fairly smoothand featureless, the geometry-oblivious spectral method

performs only slightly better.It should be mentioned that to alleviate the computa-

tion problem of the spectral basis, Karni and Gotsman [7]propose to partition the mesh into patches, each of smallenough size to make its spectral decomposition feasible.In their subsequent work, Karni and Gotsman [12] usefixed bases, derived from 6-regular connectivity patches.However, partitioning the mesh is prone to visible dis-continuity artifacts along the boundaries between thesubmeshes, similar to the blocking artifact in JPEGencoding. We emphasize that our method is computation-ally efficient while it achieves nearly equal performancein terms of compression ratios.

VI. CONCLUSIONS AND DISCUSSION

We have presented a method to approximate thegeometry of a shape based on its connectivity and anumber of anchor vertices. The “tagging” of the anchors,together with the connectivity, yield a geometry-awarebasis that spans a subspace which is close to the givenshape. The coefficients that approximate the shape inthat subspace are readily given by the spatial location ofthe anchors. Reconstructing the approximated shape onlyrequires the solution of a sparse least-squares problem.The technique is simple and easy to implement given therequired linear algebra building blocks. The complexitiesof the geometry and the connectivity of the irregularmesh are completely hidden by the linear algebra ob-jects, the matrices and the vectors. The efficiency ofthe technique stems from the existence of sophisticatedlinear algebra tools, such as sparse-matrix factorizations,updating techniques, and so on.

There are a number promising directions for futurework. One is the relationship between the triangle countreduction and geometry encoding [27], [28]. The scheme


that we presented is not fully progressive, in the sensethat the mesh has always the full connectivity. It wouldbe desirable to find a way to incorporate geometry-awarebases into progressive meshes [29], [30].

Another direction is to study the relation of our basesto non-uniform B-spline bases. We can view these basesalong three axes: their orthogonality, their supports, andtheir applicability to irregular 3D meshes. In this designspace, our method can be described as non-orthogonal,globally-supported bases for irregular meshes. The po-tential computational difficulties that are normally posedby non-orthogonality and global support are avoided heresince we do not compute the basis vectors explicitly.

Our method can also be viewed as a family of regu-larization methods for a discrete ill-posed problem [31].Any lossy geometry-encoding method tries to describemesh functions with many degrees of freedom usingrelatively little data. Therefore, any such method is bydefinition ill-posed (many shapes have the same com-pressed representation). In our case, the data consists ofthe indices of the anchors and the values that the shapeattains there. To reconstruct a unique shape from thisdata, one must add a side condition. The condition thatwe attach is a smoothness condition, that we impose inthis irregular discrete case using the Laplacian matrix.To improve smoothness even further and/or to make thealgorithmic challenges more manageable, we can relaxthe equality constraints at the anchors and replace themwith either penalties or box constraints. This viewpointwould lead to exactly the same algorithms that we havedeveloped in this paper.

The smoothness side condition that regularizes thereconstruction is clearly unsuitable for models that arenot smooth. Our method is, however, suitable for modelswith localized sharp features, as long as many anchorsare used in the vicinity of the sharp features. The repro-duction of sharp features near anchors can be controlledby the weights of smoothness constraints versus theweights of location constraints.

We believe that this work contributes to a moreprofound understanding of shapes represented by irreg-ular meshes. There is a broad spectrum of techniquesto select a basis for effectively representing geometry,ranging from splines and parametric free-form surfacesto wavelet bases for image encoding. Recently, re-searchers began proposing using over-complete bases.This technique, known as basis pursuit [32], starts witha large and redundant set of basis vectors, and uses anoptimization algorithm to try to find a combination ofvery few basis vectors that well approximate a giveninput vector (shape). This can sometimes lead to verysparse representations, but the costs of generating the

basis vectors and finding a sparse representation areconsiderable. In that context, our method can be seenas a specific over-complete basis, and as a way togenerate a sparse representation without resorting to anoptimization or search algorithm.

ACKNOWLEDGMENT

We thank the editor and the reviewers for theircomments and suggestions on this work. Models arecourtesy of Cyberware, Stanford University and Max-Planck-Institut fur Informatik. This work was supportedin part by grants 572/00 and 8001/02 from the IsraelScience Foundation (founded by the Israel Academy ofSciences and Humanities), by grant 2002261 from theUS-Israeli Binational Science Foundation, by the IsraeliMinistry of Science, by an IBM Faculty PartnershipAward and by the German Israel Foundation (GIF).

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200 300 400 500 600 700 800 900 1000 1100 1200

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Fig. 5. Rate-distortion curves for small parts of theCamelmodel. The graphs display different error measures:L∞ stands formaxi ‖pi − pi‖where pi = (xi, yi, zi); RMS stands for the root-mean-square geometric distance between corresponding vertices in the original andapproximated models;L2 error was measured using the Metro tool. Our experiments show that the geometry-aware approximation methodis very close to the spectral method in its performance. TheL∞ error of our method tends to be smaller, while theL2 error is practicallythe same.

Original model, 39074 vertices 100 basis vectors,e=0.01 600 basis vectors,e=0.0022 1200 basis vectors,e=9.8·10−4 3600 basis vectors,e=2.07·10−4

0.5KB (0.10 bits/vertex) 3.3KB (0.69 bits/vertex) 6.7KB (1.40 bits/vertex) 19.8KB (4.15 bits/vertex)

Original model, 49864 vertices 100 basis vectors,e=0.0098 500 basis vectors,e=0.0034 4000 basis vectors,e=0.0012 9000 basis vectors,e=7.2·10−4


Original model, 100086 vertices 100 basis vectors,e=0.0078 1000 basis vectors,e=0.0027 3000 basis vectors,e=0.0013 10000 basis vectors,e=4.22·10−4


Fig. 6. Reconstruction of several models using an increasing number of geometry-aware basis vectors. The sizes of the encoded geometryfiles are displayed below the models. The lettere denotes theL2 error value. Refer to Table I for the timings.


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Olga Sorkine received the BSc degree inmathematics and computer science from TelAviv University in 2000. Currently, she isa PhD student at the School of ComputerScience at Tel Aviv University. Her researchinterests are in computer graphics and includeshape modeling, mesh processing and approx-imation.

Daniel Cohen-Or is an Associate Professor atthe School of Computer Science at Tel AvivUniversity. He received a BSc in both Mathe-matics and Computer Science (1985), an MScin Computer Science (1986) from Ben-GurionUniversity, and a PhD from the Department ofComputer Science (1991) at State Universityof New York at Stony Brook. His currentresearch interests include rendering, visibility,

shape modeling and image synthesis.

Dror Irony is a PhD student in the School ofComputer Science at Tel Aviv University. Hereceived his BSc in mathematics and computerscience in 1996 and his MSc in computerscience in 2000, both from Tel Aviv Univer-sity. Dror’s Master thesis dealt with a newparallel communication-efficient dense linearsolver and some related theoretic and practicalresults. His research today is focused in stable

direct algorithms for sparse and banded matrices. From 1996 until2000, Dror worked for Motorola Communication Israel.

Sivan Toledo is an associate professor ofComputer Science at Tel Aviv University. Hereceived his BSc and MSc from Tel AvivUniversity, both in 1991. He received his PhDfrom MIT in 1995, and worked as a postdoc-toral associate at the IBM TJ Watson ResearchCenter and at the Xerox Palo Alto ResearchCenter before joining Tel Aviv University in1998.

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