Extraction and remeshing of ellipsoidal representations from mesh data

Patricio D. Simari Karan Singh

Dynamic Graphics ProjectDepartment of Computer Science

University of Toronto

AbstractDense 3D polygon meshes are now a pervasive product

of various modelling and scanning processes that needto be subsequently processed and structured appropri-ately for various applications. In this paper we addressthe restructuring of dense polygon meshes using theirsegmentation based on a number of ellipsoidal regions.We present a simple segmentation algorithm where con-nected components of a mesh are fit to ellipsoidal surfaceregions. The segmentation of a mesh into a small numberof ellipsoidal elements makes for a compact geometricrepresentation and facilitates efficient geometric queriesand transformations. We also contrast and compare twopolygon remeshing techniques based on the ellipsoidalsurfaces and the segmentation boundaries.

Key words: Geometry representations, geometry pro-cessing, compression, ellipsoid, surface segmentation,k-means clustering, polygon remeshing.

1 Introduction

Polygon meshes are currently the most widely used rep-resentation for 3D objects in Computer Graphics. Theirpopularity stems from their simplicity and flexibility toapproximate any geometric shape as well as the ability ofcurrent graphics hardware to process and render a largenumber of polygons efficiently. Approximating a realistic3D model can require up to several million faces, whichstate of the art hardware can render in real-time. Suchdense polygon meshes are typically the result of scan-ning 3D physical data [9] or tessellations of geometricdata with a different mathematical representation. Thesemeshes often have elements of noise, holes, and other ir-regularities. Furthermore, the choice of vertex samplesand their topological connectivity are largely an artifactof the construction process and the shape may benefitfrom geometric restructuring. The geometric processingand storage of dense meshes is also expensive and ap-proaches to mesh decimation, compression and progres-sive restructuring [4, 8] is also a much studied subject.One important observation we make here is that whilesome of these problems can be isolated and solved, all

Figure 1: Ellipsoidal surface regions are ideal for mod-elling curved surfaces often found in organic objects.Left: The original triangulated mesh representing a handusing 5k faces. Centre: A decimated version using180 faces. Right: An ellipsoidal approximation using 18primitives and their boundaries.

data processing potentially modifies the shape suggestedby the original mesh data. It is thus preferable for meshprocessing algorithms to acknowledge the irregularitiesof dense meshes and handle data in its original form.

The main goal of this paper is to explore the poten-tial of approximating the surface of a dense mesh usinga relatively small number of ellipsoidal regions. Such anapproximation allows us to address a number of issuesrelated to mesh processing including geometric queries,transformations, efficient rendering and compact repre-sentation.

1.1 Motivation and background

This paper is motivated by the fact that most organic ob-jects and many manufactured objects have large curvedareas. Industrial designers working in physical media of-ten use characteristically shaped tools like French curves[12], sweeps and steels that have traditionally been con-structed using conic curve and quadric surface sections.

In this paper we consider the ellipsoid as a representa-tive quadric with which to approximate 3D objects. Ellip-soids have certain advantages over planes as an approx-imation primitive. The processing of various commongeometric queries such as normal, curvature, proximityand intersection with a ray for an ellipsoid is analytic andequally efficient for ellipsoids as for planes. For curvedregions not only are far fewer ellipsoids required for ap-proximation but they have the potential of capturing localshape curvature precisely. In areas where the surface ofinterest is flat, a region from an ellipsoid with sufficientlylarge radii can be used. Figure 1 compares the visual re-sults of modelling a hand decimated1 using planar facesand an approximation of superior visual quality using anorder of magnitude fewer ellipsoids.

Ellipsoids, unlike planes, being closed surfaces, arealso capable of providing information about the volumeof closed objects. Our algorithm also allows for el-lipsoidal fitting that spans from being surface oriented,where a number of ellipsoidal surface regions approx-imate the surface as best possible, to volume oriented,where the same number of ellipsoids are used to capturethe overall volume enclosed by an object. In this case, weuse the entire ellipsoid and not just regions of its surface.

The argument for not using primitives constructed withcubic or higher degree polynomials is one of marginalgains in terms of their ability to approximate curved ge-ometry better than quadrics and that evaluation of variousgeometric queries on higher order primitives is not as ef-ficient. One disadvantage of ellipsoids that we recognizeis that they do not capture saddles or regions where theprincipal curvatures are convex and concave. We note,however, that human perception typically segments shapealong such concave boundaries. The inability of the el-lipsoid to capture the local curvature of such a shapefragment automatically favours segmentation along suchboundaries.

Once a shape has been segmented into an ellipsoidalapproximation, it is only necessary to store the ellipsoidsand the segmentation boundaries at which they meet inorder to represent the object. We will present and contrasttwo schemes for remeshing from such a representation.

1We apply GSI’s decimation which uses a variant of the MAPS al-gorithm [8].

Related workAlthough there have been many greedy techniques fordense mesh simplification, Hoppeet al. [5] present anoptimization-oriented approach. They introduce an en-ergy function that models the desire for small meshes aswell as fidelity of representation. By adjusting the weightof these two opposing goals, it is possible to choose howmuch simplification versus fidelity to the original mesh isdesired.

In [7], Katzet al. propose a mesh segmentation schemebased on fuzzy clustering of mesh faces. Their goal wasto obtain a segmentation of connected mesh regions thatrepresented meaningful components. In their approachhowever, once the segmentation is obtained, there is nosimplification of the representation of segments.

Recently, Cohen-Steineret al. [3] have introduced theapplication ofk-means style clustering of mesh faces intoconnected planar regions. Using this segmentation, theypropose a remeshing scheme in which each region is sim-plified into a plane. This results in a compact represen-tation that places the planar elements optimally with re-spect to the original mesh. Curved regions, however, stillrequire many such planes in order to be well approxi-mated.

In [1], Bischoff et al. propose a representation for 3Dmodels based on ellipsoids. However, a relatively largenumber of primitives is used and they retain the entireellipsoidal surface. Their motivation for their variant ofthe ellipsoid decomposition is the robust transmission ofgeometric objects. It represents a coarse approximationwhich is then increasingly refined by the transmission ofthe mesh’s original vertices.

Pentland, in [11], used superquadrics as geometricprimitives for representing 3D objects. The fitting of su-perquadrics to range data has been well studied [13, 6]. Inthese cases however, the motivation was not simply thatof a compact representation, but the visual recognition ofdifferent objects based on this segmentation.

2 Error metric for ellipsoidal surface regions

Let us assume we have a triangular mesh surface regionR which is a set of mesh faces, that we are approximat-ing with an ellipsoidal primitiveP . We wish to knowhow well P approximates said region. In other words,we would like to define an error functionE(R,P ).

We begin by considering the distance between a givenvertexvi and an ellipsoidP . There are several distancemetrics that one may consider when defining the errorfunction. One such function is Euclidean distance. Inthe case of ellipsoids, it is simpler to compute the radialdistance from a point to its surface.

Euclidean distance: We defined Euclidean distance

from a vertexvi to ellipsoidP , as the Euclidean distancefrom vertexvi to its radial projection on ellipsoidP , de-notedΠP (vi).

Eeuc(vi, P ) = ‖vi −ΠP (vi)‖2

The vertexvi will often have associated a vectorni in-dicating the surface normal at said vertex. It may be de-sirable for our ellipsoidal primitive to approximate thesesurface normals as well. Thus, we introduce an angulardistance.

Angular distance: The angular distance between ver-tex vi and ellipsoidP is a measure of the angle betweenthe vertex normal atvi, denotedni, and the ellipsoidalsurface normal at the radial projection ofvi on ellipsoidP , denotednP (vi). We define said angular distance as

Eang(vi, P ) = ‖ni − nP (vi)‖2

Finally, given a surface mesh, it is possible to approx-imate curvature values given for the data. In our case weuse a mean curvature estimate for meshes presented in[10]. As in the case of the vertex normals, it may be de-sirable for the ellipsoidal surfaces to approximate thesevalues as well. For this purpose, we introduce the curva-ture distance metric.

Curvature distance: The curvature distance betweenvertexvi and ellipsoidP measures how well theP fol-lows mesh curvature at said vertex. We define the curva-ture distance betweenvi and ellipsoidP as the magnitudeof the difference between the estimated mean curvature atvi, denotedHi, and the mean curvature on the ellipsoidat the radial projection ofvi, denotedHP (vi).

Ecur(vi, P ) = (Hi −HP (vi))2

An argument against using Gaussian curvature is that itwould always be positive, even when considering the in-ner surface (see negative ellipsoids below.) At a sad-dle point in the data, the Gaussian curvature is negative,while the mean curvature is zero. Our metric models thefact that a “flat” ellipsoid (with mean curvature close tozero at the point of interest by virtue of very large radii)could approximate this small region.

Composite distance:Now, we can compose these er-ror metrics into a single value forvi andP by taking aweighted sum

E(vi, P ) = αEeuc(vi, P )+βEang(vi, P )+γEcur(vi, P )

The values ofα, β, andγ indicate the relative importancethat we place on each of the individual error metrics.

Given a triangular mesh facefj = (v1, v2, v3), we de-fine its distance to the ellipsoidP as follows

E(fj , P ) =∑i

E(vi, P )

Figure 2: Without negative ellipsoids positive curvature iscaptured by one ellipsoid, but the negative curvature re-gion is approximated by the seven remaining. With neg-ative ellipsoids one ellipsoid is sufficient for each region.

Now, given a mesh surface regionR = {f1, . . . , fn}, wedefine the fitting error ofP with respect to regionR asthe sum of errors for each facefi to P .

E(R,P ) =∑i

E(fi, P )

If the mesh is partitioned inton regionsR1, . . . , Rn, eachapproximated by an ellipsoidP1, . . . , Pn, the error of theapproximation is defined as the sum of errors at each re-gion.

E({R1, . . . , Rn}, {P1, . . . , Pn}) =∑i

E(Ri, Pi)

We should note that there can be additions and variationsto these definitions. For example, when calculating theerror for a region, each face’s error can be weighted bythe face’s area, thus indicating that larger faces are moreimportant.

A typical value forα in our case isα = 1/a2 whereais the mesh’s average edge length. This helps normalizethe weight of the Euclidean metric for mesh’s sampled atdifferent resolutions. In the case ofβ andγ, they are notdependent on mesh scale2. We tried different weightsand found settings of1.2 and5 × 10−4, respectively, towork well.

Negative ellipsoidsWhen modelling surfaces using ellipsoidal surface sec-tions we are not able to capture bowl-shaped concavities,i.e. regions of negative curvature, if we consider only theellipsoid’s outer surface. In order to be able to capturethese types of regions while still using the same type ofprimitive we introduce the notion of anegative ellipsoid.

In essence, a negative ellipsoid has the same shape asa regular ellipsoid, except that we consider its inner sur-face when measuring angular distance between normals

2The curvature estimate we use [10] normalizes for varying face ar-eas

and curvature. Thus, we extend the relevant previous def-initions as follows:

Eang(vi, P ) ={‖ni − nP (vi)‖2 if P is pos.‖ni + nP (vi)‖2 if P is neg.

Ecur(vi, P ) ={

(Hi −HP (vi))2, if P is pos.(Hi +HP (vi))2, if P is neg.

In essence, we are simply considering the negative ellip-soid to have its surface normals pointing inward and itsmean curvature to be the compliment of the mean curva-ture defined on the outer surface. Figure 2 illustrates theeffect of negative ellipsoids to surface segmentation.

3 Error metric for ellipsoid volume

We mentioned earlier that ellipsoids could also be usedto represent a region by approximating not only its shape,but also its volume. Here we introduce an error metric tobe used when this is intended.

Evol(R,P ) = αEeuc(R,P ) + δ(V(R)− 43πabc)2

wherea,b, andc are the radii of ofP , V(R) is the ap-proximate volume englobed by mesh region R, andδ isa weight determining the relative importance of approxi-mating volume versus fitting the data points to the ellip-soid’s surface.

A typical weight for δ in our case isδ = 105/V2,whereV is the approximate volume of the entire mesh.The denominator ensures that the volume penalty termis normalized for meshes of different volumes, while thenumerator’s large magnitude shows a preference for ap-proximating each region’s volume over fitting the pointsindividually.

Figure 3 illustrates the relevance of the volume penaltyweightδ. The mesh is that of a semi-cylindrical section.Initially, the fitting of the ellipsoid using the volume met-ric with a highδ value approximates the volume enclosedby the convex hull of the section. As we decrease therelative magnitude ofδ in the error metric, the ellipsoidfitting transitions from approximating volume, to approx-imating surface.

4 Ellipsoidal segmentation

Our segmentation approach is an extension of Lloyd’s al-gorithm. Starting with an initial classification, the algo-rithm alternates between a fitting step, and a classifica-tion step. In the fitting step, we update eachPi with theellipsoid that minimizesE(Ri, Pi). In the classificationstep, the regionsRi are re-computed, assigning each facefj of the mesh to the region that minimizesE(fj , Pi)while also under the constraint that the regions must re-main connected. To this purpose we implement the prior-ity queue region flooding scheme found in [3], inserting

Figure 3: As we decrease the relative magnitude of δ inthe volume fitting error metric, the ellipsoid fitting transi-tions from approximating volume, to approximating sur-face. a. δ = 1010/V2, b. δ = 105/V2, c. δ = 102/V2, d.δ = 0.

faces to the queue based on their error to their associ-ated ellipsoid. We also implement the region teleporta-tion scheme to avoid local minima. The reader is referredto the publication for greater details.

Ellipsoid fitting We parameterize ellipsoid primitivesas tuple of nine scalars, three to represent the ellipsoidscentre, three to indicate the length of each radius, andthree to indicate its alignment. When considering neg-ative ellipsoids in the case of surface segmentation, wealso store if the ellipsoid is positive or negative.

For each mesh regionRi of the current segmentation,we find thePi that minimizesE(Ri, Pi). Given that thesign value is discrete, we first minimize fixing the signto be positive, then fixing it to be negative, and keep theresult with lowest error.

The minimization requires an initial estimate of the pa-rameters. We estimate the ellipsoid’s centre as the geo-metric centroid of the vertices of the current region. Inorder to estimate the alignment and radii, we perform asingular value decomposition of the regions vertices, cen-tring their mean at the origin. The resulting eigenvectorsestimate the regions orientation and the eigenvalues allowus to estimate the radii.

Termination: The algorithm terminates when the de-sired number of iterations have been performed, reportingthe segmentation with least error found. Should the algo-rithm converge to a local minimum, it may continue itssearch for a better segmentation simply by teleporting a

Figure 4: Examples of volume segmentation. Top:Bunny (3K faces), 8 ellipsoids, 50 iterations. Centre:Dinopet (6K faces), 22 ellipsoids, 100 iterations. Bot-tom: Horse (3K faces), 20 ellipsoids, 120 iterations.

region as mentioned above.Figure 4 shows examples of volume segmentation. In

these cases, we set a high volume error weightδ. This en-sures that volume is approximated as closely as possibleand only then is the placement of the ellipsoid modified tofit the surface. For our volume approximationV of eachmesh segment we use the volume of the convex hull.

The top of figure 8 shows the results of surface seg-mentation. Here the actual ellipsoid’s volume is not con-sidered and only the fit to the surface is taken into ac-count. Figures 1 and 5 show extracted ellipsoidal repre-sentations.

4.1 Smoothing segmentation boundariesOnce we have obtained an ellipsoidal segmentation, thesegmentation boundaries are given by edges whose twoadjacent faces have been assigned to different regions.

Figure 5: Venus (5K faces). Top left: original mesh, frontand back views. Right: obtained ellipsoidal representa-tion with smoothed boundaries, 12 ellipsoids, 394 uniqueboundary vertices, 100 iterations. Bottom left: error overiterations. Our reclassification flooding scheme and re-gion teleportation can sometimes increase the error. Af-ter several iterations however, the algorithm stabilizes ona minimum.

Smoothing of these segmentation boundaries may be de-sired. To achieve this, we perform a constrained relax-ation of the boundary vertices. For each boundary pointp, we consider its boundary neighbours as well as itsradial projection to all associated ellipsoids. Then wetake the centroid of these associated pointsq and updatep ← (1 − ε)p + εq, for some smallε. After a numberof iterations, the result is points which lie along the ap-propriate ellipsoidal intersections and tend to be evenlyspaced along the boundary.

Figures 1 and 5 show smoothed segmentation bound-aries in the rendered ellipsoidal representations.

4.2 Impact of different metrics

Each of the components of the surface fitting metric in-crementally allows for a better fit of the ellipsoidal sur-face to the mesh region. This is illustrated in figure 6.

Initially, we start out by using only the Euclidean dis-tance, settingβ andγ to 0. Then we run the segmen-tation for ten iterations and arrive at the segmentation atthe top of figure 6. To the right, we see the original meshcoloured according to segments, and to the left we showthe resulting ellipsoids. Notice that since we are fittingsurfaces, we are only interested in the ellipsoidal surfaceregion near the mesh data. The ellipsoid is free to growas large as needed to fit the mesh segment since only a

Figure 6: Mesh surface segmentations resulting from dif-ferent settings of weight values for the composite dis-tance metric Top: only Euclidian distance is used, settingother weights to zero. Centre: Euclidean and angular dis-tances are used in error metric. Bottom: mean curvatureis incorporated to the error metric along with Euclideanand angular distance.

region of its surface will be kept for the final representa-tion.

Using only Euclidean distance does not suffice for agood surface fit. This is particularly noticeable in the re-gion of the bill.

Next, we incorporate angular distance into the metric.Now, once again, we run the segmentation for ten itera-tions and arrive at the segmentation at the centre of figure6. The resulting segmentation follows the surface muchmore closely. However, we can still see that the fittingis not as good as it could be. The tail is much flatter,and the ellipsoid at the bill is not following closely as itapproaches the head.

Finally, we incorporate mean curvature and use allthree metrics. This provides more information to disam-biguate the ellipsoid fitting and can be appreciated at thebottom of figure 6, in which the ellipsoidal surfaces fol-low the mesh almost perfectly.

5 Remeshing of ellipsoidal representations

Given our ellipsoidal representation of a surface, it maybe desired to reconstruct a mesh for use in applicationsthat will not handle this representation directly. To hiseffect we introduce two remeshing schemes. First, weexplain how to use the boundary regions of an ellipsoidalsurface.

5.1 Using ellipsoidal region boundariesDifferent applications will require that it be determined ifa given point on an ellipsoid lies within its region bound-

aries. For example, if we are casting a ray at an objectrepresented as a union of ellipsoidal regions, it is easyto determine if said ray intersects an ellipsoid. However,it intersects the object only if the intersection point lieswith the region’s boundaries on the ellipsoid. This prob-lem will also arise when we address remeshing below.

If we have a planar polygon, it is easy to determine ifa given point lies inside or outside said polygon. In ourcase, however, the polygon lies on the surface of an el-lipsoid. We reduce the problem of determining if a pointon an ellipsoid is within the boundaries to that of a planarpolygon through the use of stereographic projection.

In our representation, for each ellipsoid we have a setof boundary regions described as a series of cycles ofboundary vertices. For every ellipsoid, there is a trans-formation which maps it to the unit sphere. If we havea point on the ellipsoid’s surface, we simply transformthe point with said transformation along with all bound-aries so the point and boundaries lie on the unit sphere.Now, we take a stereographic projection of the point andboundaries. The original point is within the region of in-terest if and only if the projected point lies within thepolygon determined by the outer boundary’s projectionand outside that of all hole boundaries.

In our case we use the following stereographic pro-jection, which maps point(x, y, z) on the unit sphere to(xp, yp) on theXY plane:

xp = 2x/(1− z); yp = 2y/(1− z)

Note that the sphere’s north pole is mapped to infin-ity. However, since clearly no region will use the entiresphere, we can simply find a rotation that places the re-gion of interest as close to the south pole of the sphere aspossible, while not including the north pole. This rota-tion is then used after transforming to the unit sphere andprior to projecting.

An illustration of the stereographic projection of a re-gion is shown at the top of figure 7. For illustrative pur-poses, the unit sphere is centred at(0, 0, 1) so that theprojection can be better viewed.

5.2 Remeshing with parametric tessellation of sur-faces

One way in which we may obtain a mesh from our el-lipsoid representation is by generating for each regiona parametric tessellation. In our implementation we goabout this as follows.

For each region, we generate points on the unit sphere.This is easily done by generating points in regular in-tervals of azimuth and elevation. Next, we considerthe stereographic projection of these points along withthe relevant boundaries, having mapped these to the unit

Figure 7: Tessellation of an ellipsoidal surface region.Top: Generated points on the unit sphere are stereo-graphically projected along with segmentation bound-aries. Bottom: Points outside of boundaries are discardedand the remaining points are triangulated.

sphere as described above. This is illustrated at the top offigure 7.

We keep the points that respect the boundaries, andthen perform a constrained Delaunay triangulation onprojected points respective to the boundaries. Note that instereographic projections, circles on the sphere’s surfaceare mapped to circles on the plane, so a Delaunay triangu-lation on the projected points corresponds to a Delaunaytriangulation on the sphere’s surface. Then through thetransformation to ellipsoid space, the anisotropic natureof the surface is captured in the triangulation.

To achieve the triangulation, we begin with a triangula-tion of the boundaries, and then iteratively add each pointto the triangulation. For each one, we find the face inwhich it lies, insert it, resulting in the face being split intothree triangles, and then perform edge flipping in orderto re-establish the Delaunay triangulation. Once this isdone, the faces obtained directly translate to faces on thesphere’s surface, which can now be transformed into theellipsoid region. The bottom of figure 7 illustrates theresulting triangulation of the surface region of interest.The centre of figure 8 shows meshes resulting from thisapproach.

This scheme has some drawbacks. The uniform para-metric sampling of the sphere to obtain the points resultsin the poles being much more densely sampled. This isclearly visible at the centre of figure 8. Also, it is hardto determine a priori how many points to use for each re-gion. In our approach we used the same amount of pointsfor each ellipsoid. As a result, vertices are not placedoptimally.

Figure 8: Top: Ellipsoidal surface segmentation. Teddy(5K faces), 8 ellipsoids, 50 iterations, 130 unique bound-ary vertices. Hand (5K faces), 18 ellipsoids, 50 itera-tions, 463 unique boundary vertices. Centre: remeshingby parametric tessellation. Bottom: remeshing by itera-tive vertex addition using same total number of vertices.

5.3 Remeshing by iterative vertex additionIn order to address the drawbacks of the previous ap-proach we propose a different remeshing scheme whichattempts to place vertices where they are most needed.

Initially, we start out with a triangulation of only theregion boundaries as described for the previous scheme.Now we iteratively add vertices as follows. Our candi-dates are the face centres of the current mesh. For eachcandidate we measure its radial distance to the relevantellipsoid and choose the one that is farthest but is also farenough from the points already added according to somethreshold value. This single point is projected onto theellipsoid and then added using the same technique de-scribed for the previous scheme. Note that edge flippingwill be constrained to the relevant ellipsoidal region, thus,boundary edges, which we consider feature edges [2], arenot flipped.

The bottom of figure 8 shows the results for this al-gorithm. We use the same total number of vertices asresulted from our other approach. In the results, it canbe seen that regions of higher curvature are naturally as-

signed more vertices while regions of lower curvature im-ply less vertices, and thus larger faces.

6 Conclusions and future work

We have presented the idea of modelling a 3D object us-ing a relatively small number of ellipsoidal surface re-gions, each represented by an ellipsoid and boundariesindicating where adjacent regions meet. This results in acompact representation which supports efficient geomet-ric queries and transformations.

In order to obtain the segmentation, we introducedmetrics based on Euclidean radial distance, surface nor-mals and surface curvature; all properties which are eas-ily and efficiently computable for an ellipsoidal surface.Based on these metrics we use a simple segmentation al-gorithm based on a variant ofk means clustering.

Given that a triangulated mesh is sometimes desir-able, we also presented two remeshing schemes whichwe compared. We showed that in the second scheme,vertices are placed intelligently, resulting in regions ofhigher curvature being densely sampled, while flatter re-gions are represented by larger faces in the resultingmesh.

It should be noted that for highly dense meshes, the re-sulting segmentation boundaries may include many ver-tices. These boundaries are simply a sequence of pointsand can easily be simplified for a more compact represen-tation.

Also we could add another parameter to the ellipsoidrepresentation in order to introduce bending (allowing forbanana-shaped surfaces.) This could be useful both involume and surface segmentations, in the latter case al-lowing the possibility of modelling saddle regions, withthe trade off of decreasing the efficiency of geometricqueries.

References

[1] S. Bischoff and L. Kobbelt. Ellipsoid decomposi-tion of 3d-models.3DPVT Proceedings, pages 480–488, 2002.

[2] S. Bischoff and L. Kobbelt. Towards robust trans-mission of geometric data.Computers & Graphics,26(5):665–675, 2002.

[3] David Cohen-Steiner, Pierre Alliez, and MathieuDesbrun. Variational shape approximation.ACMTrans. Graph., 23(3):905–914, 2004.

[4] Hugues Hoppe. Progressive meshes.ComputerGraphics, 30(Annual Conference Series):99–108,1996.

[5] Hugues Hoppe, Tony DeRose, Tom Duchamp, JohnMcDonald, and Werner Stuetzle. Mesh optimiza-

tion. In Proceedings of the 20th annual conferenceon Computer graphics and interactive techniques,pages 19–26. ACM Press, 1993.

[6] Ales Jaklic, Ales Leonardis, and Franc Solina.Seg-mentation and recovery of superquadrics. KluwerAcademic Publishers, 2000.

[7] S. Katz and A. Tal. Hierarchical mesh decomposi-tion using fuzzy clustering and cuts.ACM Transac-tions on Graphics, 22(3):954–961, 2003.

[8] Aaron W. F. Lee, Wim Sweldens, Peter Schroder,Lawrence Cowsar, and David Dobkin. Maps: mul-tiresolution adaptive parameterization of surfaces.In SIGGRAPH ’98: Proceedings of the 25th annualconference on Computer graphics and interactivetechniques, pages 95–104. ACM Press, 1998.

[9] Marc Levoy, Kari Pulli, Brian Curless, SzymonRusinkiewicz, David Koller, Lucas Pereira, MattGinzton, Sean Anderson, James Davis, JeremyGinsberg, Jonathan Shade, and Duane Fulk. Thedigital michelangelo project: 3d scanning of largestatues. InProceedings of the 27th annual con-ference on Computer graphics and interactivetechniques, pages 131–144. ACM Press/Addison-Wesley Publishing Co., 2000.

[10] Mark Meyer, Mathieu Desbrun, Peter Schroder, andAlan H. Barr. Discrete differential-geometry opera-tors for triangulated 2-manifolds. In Hans-ChristianHege and Konrad Polthier, editors,Visualizationand Mathematics III, pages 35–57. Springer-Verlag,Heidelberg, 2003.

[11] Alex P. Pentland. Perceptual organization and therepresentation of natural form.Artificial Intelli-gence., 28(3):293–331, 1986.

[12] Karan Singh. Interactive curve design using digi-tal french curves. InProceedings of the 1999 sym-posium on Interactive 3D graphics, pages 23–30.ACM Press, 1999.

[13] F. Solina and R. Bajcsy. Recovery of paramet-ric models from range images: The case for su-perquadrics with global deformations.IEEE Trans.Pattern Anal. Mach. Intell., 12(2):131–147, 1990.