llu´ısros ,kokichisugihara federicothomas · 2 l. ros et al.: trihedral polygonizations 1 2a 2b...

1 Introduction

Towards shaperepresentation usingtrihedral mesh projections

Lluıs Ros1, Kokichi Sugihara2,Federico Thomas1

1 Institut de Robòtica i Informàtica Industrial,CSIC-UPC, Llorens Artigas 4-6, 08028 Barcelona,SpainE-mail: [email protected],[email protected] Department of Mathematical Informatics, Universityof Tokyo. 7-3-1, Hongo, Bunkyo-ku, Tokyo113-8656, JapanE-mail: [email protected]

Published online: ?? ?? 2002c© Springer-Verlag 2002

This paper explores the possibility of ap-proximating a surface by a trihedral polyg-onal mesh plus some triangles at strategicplaces. The presented approximation hasattractive properties. It turns out that theZ-coordinates of the vertices are completelygoverned by the Z-coordinates assigned tofour selected ones. This allows describingthe spatial polygonal mesh with just its 2Dprojection plus the heights of four vertices.As a consequence, these projections essen-tially capture the “spatial meaning” of thegiven surface, in the sense that, whateverspatial interpretations are drawn from them,they all exhibit essentially the same shape.

Key words: Shape representation – Polygo-nization – Trihedral mesh – Reconstructionfrom projections – AND/OR graphs

Correspondence to: L. Ros

A polygonal mesh is a piecewise linear 2-manifoldmade up with planar polygonal patches, glued alongthe edges, and possibly containing holes. A poly-gonization method is an algorithm able to con-struct a polygonal mesh approximating a givensurface. The literature on polygonization methods,mainly on triangulations, is vast (see for exam-ple [3, 18], or [6] for a survey on triangulationsand algorithms to simplify them). In general, themain goal is to obtain meshes that are close tothe surface within a known error, as a way tounderstand and represent the surface shape [12].Other goals have been to increase the speed ofpolygonization and the ability of the polygonizerto satisfy some constraints in the solution (e.g.,one might request the most accurate approxima-tion using a given number of line segments ortriangles).In general, a polygonal mesh cannot be reconstructedfrom its projection onto a plane because infinitelymany meshes generate exactly the same projection.For example, for the triangular mesh projection inFig. 1, there are many different reconstructions, as il-lustrated. The first two seem to have no meaning; but,actually, there is a rather “hidden” meaningful recon-struction of Nefertiti’s face available. Can we obtaina spatial mesh approximating this face so that its pro-jection still keeps its spatial meaning?Certainly, there is a class of meshes whose pro-jections fully determine the spatial shape once theheights of four vertices are given. We call these pro-jections unequivocal because their reconstructionsrepresent essentially the same object. For example,the projection in Fig. 2a unequivocally representsa truncated tetrahedron, as seen in Fig. 2d–f. Observethat it suffices to set the heights of P, Q, T and Rto determine those of S and U, using the fact thatall cofacial vertices must be coplanar and, hence, Smust lie on the face-plane RPQS, and U on SQTU.In general, if all vertices of a polyhedron have ex-actly three incident faces, the heights of a vertexand its three neighbours are sufficient to determinethe heights of the rest, as Sect. 2 explains. One ofour goals is then to approximate any given surfacewith a polygonal mesh yielding unequivocal projec-tions that uniquely identify the spatial shape oncethe heights of four vertices (selected in this way)are given. Section 2 presents the trihedral polygonalmesh, the model we use to this end, and shows howits projections are unequivocal in the sense givenabove.

The Visual Computer (2002) 18:???–???Digital Object Identifier (DOI) 10.1007/s00371-002-0176-9

2 L. Ros et al.: Trihedral polygonizations

1

2a 2b 2c

2d 2e 2f

Fig. 1. Arbitrary reconstructions of this triangulatedprojection have no spatial meaning except for a veryspecific one: it shows Nefertiti’s faceFig. 2a–f. A truncated tetrahedron (a) and three pos-sible reconstructions (d–f). The slightest perturba-tion destroys the correctness of the projection (b), butthis can be avoided adding new triangular faces (c)

Nevertheless, we need to go beyond this goal if thisrepresentation is to be useful. Consider what hap-pens if the (x, y) vertex positions in Fig. 2a areslightly altered (Fig. 2b). The new projection nolonger represents a correct truncated tetrahedron,for to be so, the edges joining the two triangularfaces, when extended, should be concurrent at theapex of the (imaginary) original tetrahedron. Equiv-alently, note that once P, Q, R and T are given,the height of U is overconstrained, for it can becalculated from both the coplanarity of SQTU orthat of RPTU. For generic vertex positions, thetwo values of this height do not necessarily coin-cide, and the only spatial reconstruction that keepscofacial vertices coplanar is a trivial one, withall vertices lying on a single plane [10, 11]. Thismakes the four provided heights inconsistent be-tween each other. In sum, the consistency of the

four heights only holds at very specific positionsof the vertices and inevitable discretization errorswill make this representation useless. This prob-lem is common in computer vision [13] and com-puter graphics [15, 17], and mathematical charac-terizations of generically consistent projections aregiven in [14, 16]. The way we use to make thisrepresentation robust against these errors followsfrom this observation: if the height of a vertex ina projection is overconstrained because the vertexlies on several planes that fix it, we just introducenew triangular faces around it for preventing thisto occur (Fig. 2c). Section 3 gives a fast algorithmto this end, derived from this observation, usingthe so-called T/TT-transformations. Section 4 de-scribes a complementary optimization step that prop-erly places these transformations to minimize thereconstruction errors by reducing the problem to

L. Ros et al.: Trihedral polygonizations 3

a cyclic AND/OR graph search. We finally concludein Sect. 5.

2 Trihedral polygonal meshes

Trihedral meshes, i.e., those where all vertices haveexactly three incident faces, produce unequivocalprojections. Indeed, Fig. 3 shows that in them, af-ter fixing the planes of two adjacent faces, we haveenough data to derive the heights of the remainingvertices. Clearly, the heights of the bold vertices fixthe shadowed face-planes and the heights of othervertices on the face-planes. At this point, any othersurrounding face has three vertices whose heights areknown, and so, its plane can be fixed too. The sameargument can be iteratively applied and the resultis a height propagation reaching all vertices in theprojection.In the schematic representation of this height prop-agation (Fig. 3) every face f receives three incom-ing arrows from the three vertices that fix it. Thederivation of heights for the rest of vertices on fis indicated with outgoing arrows from f . The re-sult is a tree-shaped structure spanning all verticesand faces. In this tree, a path from any of the ini-tial four vertices to any other vertex will be hereafterreferred to as a propagation wave. Note that heightpropagations where a face is fixed from three (al-most) collinear vertices must be avoided. Section 4gives a way to compute propagations eluding thesecollinearities.A trihedral mesh approximating a convex or concavesurface can be readily obtained by distributing a setof random points all over the surface and comput-ing its tangent planes at these points. This leads toa plane arrangement whose upper envelope, if thesurface is convex, or lower envelope, if it is concave,provides a good mesh approximation of the surface.Since the tangent plane orientations are random, anythree of such planes meet in a single point, and hencethe mesh is trihedral.Alternatively, a trihedral mesh approximation ofa piece of concave or convex surface can be ob-tained by starting with a rough mesh approximationand iteratively applying a bevel-cutting [5] and/ora corner-cutting [2] operation to attain the desiredapproximation.The situation becomes more complex when concav-ities, convexities and saddle-like shapes are simulta-neously present, as neither of the previous strategies

overconstrained

Fig. 3. A height propagation starting at four prespecified(bold) vertices. Several vertices can have an overcon-strained height

can be directly applied. A straightforward solutionto find a trihedral mesh approximation of a generalsurface consists of computing the three-dimensionalVoronoi diagram of a set of points sampling thesurface. The algorithm encompasses the followingsteps, where S denotes the input surface to be poly-gonized:1. Generate points on S at random with a density

proportional to the curvature of S.2. Replace each point p produced in step 1 by a pair

of generating points, p′ and p′′, placing them onthe normal to S at p, one in each side of S, atthe same distance from p. This distance must besufficiently small so that the line-segment con-necting p′ and p′′ penetrates the surface exactlyonce.

3. Construct the three-dimensional Voronoi diagraminduced by all generating points. Each such pointwill yield one polyhedral Voronoi cell in the dia-gram, possibly unbounded.

4. For each polygonal face separating two cells, ifthe two cells correspond to points on differentsides of S, select this face as part of the desiredoutput polygonization. Collecting such faces, weget a polygonization with only trihedral or tetra-hedral vertices, with three and four incident faces,respectively.

5. Use one of the local operations shown in the bot-tom row of Fig. 4 to replace every tetrahedralvertex with trihedral ones. Output the resultingpolygonization.


Fig. 4. Top row: a triangulated model of a Venus (left) is symmetrically sampled (center) to obtain a polygonization with onlytrihedral and tetrahedral vertices (right). Center row: zoom into three selected areas. Bottom row: local operations used toremove tetrahedral vertices

We easily see that this algorithm produces a trihe-dral mesh. Since we select faces from the Voronoidiagram of points in general position, the poly-gonization obtained in step four either containstrihedral or tetrahedral vertices. (Any five pointsin general position will not be co-spherical, andhence, the vertices of the Voronoi diagram willhave at most four incident Voronoi cells.) Any

tetrahedral vertex of this polygonization can be fi-nally removed by using one of the two operationsshown in the bottom row of Fig. 4. If there isa plane that separates the vertex from its neigh-bours, we simply truncate the vertex (bottom leftof Fig. 4), otherwise it is a saddle point and we re-place it with four trihedral vertices (bottom rightof Fig. 4).


Fig. 5. a,b T and TT-transformations. c Overhanged and self-intersecting reconstructions induced by T-transformations atlocally non-convex faces. In all figures, newly added edges are shown in thick lines

This strategy has been partially implemented untilstep four, using the Qhull package to compute the re-quired 3D Voronoi diagram [9]. Strictly, Qhull com-putes convex hulls, but it can also be used to de-rive Delaunay triangulations, halfspace intersectionsabout a point, Voronoi diagrams, furthest-site De-launay triangulations, and furthest-site Voronoi dia-grams in 2D, 3D, and higher dimensions. It imple-ments the Quickhull algorithm for computing convexhulls [1] and is able to handle rounding errors inher-ent to the use of floating point arithmetic. The toprow of Fig. 4 shows the results of the strategy whenapplied to a triangulated surface of a Venus sculp-ture. The input sculpture is shown (left) together withthe two layers of generating points produced in step 2(center), and the polygonization generated in step 4(right). The middle row of Fig. 4 shows three en-largements of selected areas. One can come up withother “trihedrization” strategies with possibly bet-ter results, mainly concerning the smoothness of theapproximation, but this does not modify the discus-sions that follow.

3 T and TT-transformations

In a trihedral mesh a projection is overconstrainedbecause any of its vertices lies on three faces, andpotentially up to three propagation waves can deter-mine a height at the same time (Fig. 3). However, asdone in Fig. 2c, this can be avoided by adding trian-gular faces. To this end, we first compute an arbitrary

height propagation spanning all vertices and checkwhich of them receives more than one wave. We thentake one overconstrained vertex v at a time and pre-vent all but one waves from reaching v as follows. Tostop the wave getting v from face f , we apply eitherof these two transformations (Fig. 5a,b):

– A T-transformation, which places a new edgejoining the two neighbouring vertices of v in f ,say vl and vr .

– A TT-transformation, which places a new vertexv′ on f near v and the three new edges (v′, v),(v′, vl) and (v′, vr).

After either transformation, f is split into two orthree faces respectively, so that it cannot constrainthe height of v anymore. Also, the added triangles areinnocuous because all heights can still be determinedfrom the four initial ones.Which transformation is preferred depends on thegeometry of face f around vertex v. If all pointsinside the triangle vlvrv belong to f , we say thatf is locally convex at v. So, for situations wheref is locally convex at v, simplicity prevails andT-transformations are enough (Fig. 5a). When localnon-convexities are present (Fig. 5b), T-transform-ations would yield occluded or partially occludedcrossing edges whose spatial reconstructions haveoverhanged parts or self-intersecting faces (Fig. 5c).Here, TT-transformations are preferred for they canavoid this.An observation complements the strategy. In anoverconstrained vertex v, either two or three in-


a b c

d e f

Fig. 6a–f. A projected dodecahedron (a) together with a height propagation (b) and the T-transformations it yields (c). A pro-truded tetrahedron (d) and two possible corrections: e involving TT-transformations, and f, involving only T-transformations

coming propagation waves arrive. If no more thanone of them comes through a locally non-convexface, then we can always drop the incidence con-straint in this vertex by only using T-transformations:we just leave the eventual “bad” wave to deter-mine the height of v and stop the others withT-transformations. This completes the descriptionof a one-sweep algorithm removing overdetermi-nation. As an example, Fig. 6a–c shows a pro-jected dodecahedron before and after applyingT-transformations.In general, when the approximated surface is uni-formly convex, or uniformly concave, all faces ofthe resulting trihedral polygonal mesh will be lo-cally convex, and hence T-transformations will suf-fice. However, even when local non-convexities ex-ist at the faces, there still might be some heightpropagations where only T-transformations suffice.In Fig. 6e, for example, an algorithm comput-ing an arbitrary propagation can be forced to useTT-transformations, whereas with a proper search,a robust projection is obtained only with T-transform-ations (Fig. 6f). But one certainly finds correctprojections where no propagation solely usingT-transformations can be found [10, Sect. 8.4].CE

a

4 Optimal propagations and cyclicAND/OR graphs

The algorithm in the preceeding section corrects theincidence structure by finding an arbitrary heightpropagation and inserting a T- or a TT-transformationwhenever a vertex height is determined by two ormore faces. However, arbitrary propagations mighttravel along “degenerate paths” where the planes forsome of the faces are determined by three aligned(or almost aligned) vertices. Clearly, these degen-erate propagations must be avoided if we want tominimize the errors during the reconstruction of thespatial shape from the initial set of four heights.The following experiment exemplifies this point. Forthe projection of a trihedral strip in Fig. 7a, know-ing the heights of points 1, 2, 3 and 4 producesa height propagation since, for i = 1, 3, 5, 7, . . . , the3D locations of points i, i + 1 and i + 2 determinethe plane where the points i + 4 and i + 5 lie and,thus, their height on the spatial reconstruction. How-ever, this plane becomes numerically ill-determinedas the angle α between the points i, i +1 and i +2 ap-proaches 180◦ and, when this happens, small errors

CEa This sentence is a little hard to follow. Do you mean that some projections must use TT-transformations?


12 11

9 10

8 7

5 6

4 3

1 2

α

αβ

3

3.5

0 10 20 30 40 50 60 70 80 90 100

2.5

2

1.5

1

0.5

0

Point number

r = 0.004

r = 0.003

r = 0.002

r = 0.001

ρ60

70

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0 10 20 30 40 50 60 70 80 90 100

50

40

30

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Point number

r = 0.004

r = 0.003

r = 0.002

r = 0.001

ρ

a

b

c

d

e f

Fig. 7a–f. Error propagations. Given the projection of a trihedral strip (a), the Z coordinates of points 1, 2, 3 and 4 triggera height propagation along the whole strip. Experiments have been done with the strips in b and c, also shown in d togetherwith their spatial reconstructions. e,f The maximum relative error ρ of the computed heights on the two strips respectivelywhen the XY coordinates of the points are perturbed within a circle of radius r . See the text for details

on previous heights may accumulate rapidly alongthe height propagation. We can see this effect on thetwo strips in Fig. 7b,c, whose angles are α = β =120◦ and α = 175◦, β = 10◦, respectively, with all in-terior edges of length 1. Their spatial reconstructions

are shown in Fig. 7d, for z1 = 0.01, z2 = z3 = 0.05and z4 = 0.1. For the two strips respectively, theplots in Fig. 7e,f show the maximum relative errorρ in the height of point i, when the XY coordi-nates of all points are perturbed within a circle of


radius r around their nominal position. The boundson ρ have been obtained by randomly perturbingall points and then computing the height propaga-tion, repeating this process 10 000 times. The resultsshow that, for a fixed r , the errors accumulate lin-early as the propagation proceeds. Moreover, by lin-early varying r , the errors increase linearly as well.Note that, while for the first strip the errors keepmoderately low, they rapidly increase for the seconddue to the high degeneracy of the height propaga-tion. Thus, an algorithm to find height propagationsthat avoid these degeneracies is needed. The rest ofthe section presents one, based on finding the least-cost solution of cyclic AND/OR graphs [8]. We nowrecall some preliminary concepts about this kind ofgraphs.

4.1 Cyclic AND/OR graphs

An AND/OR directed graph G, can be regardedas a hierarchic representation of possible solutionstrategies for a major problem, represented as a rootnode, r , in G. Any other node v represents a subprob-lem of lower complexity whose solution contributesto solve the problem at hand.There are three types of nodes: AND nodes, ORnodes and TERMINAL nodes. Every node v has a setS(v) of successor nodes, possibly empty, to which itis connected in either of two ways:

– An AND node v is linked to all nodes si ∈ S(v)through directed AND arcs (v, si), meaning thatthe subproblem for v can be trivially solved onceall subproblems for the nodes in S(v) have beensolved.

– An OR node v is linked to all nodes si ∈ S(v)through directed OR arcs (v, si), meaning that thesubproblem for v can be trivially solved once anyone of the subproblems for the nodes in S(v) hasbeen solved.

– A TERMINAL node represents a yet-solved ortrivial subproblem and has no successors.

With this setting, a feasible solution to the problembecomes represented as a directed subgraph T of Gverifying:

– r belongs to T .– If v is an OR node and belongs to T , then exactly

one of its successors in S(v) belongs to T .– If v is an AND node and belongs to T , then every

successor in S(v) belongs to T .

– Every leaf node in T is a TERMINAL node.– T contains no cycle, it is a tree.

One can also assign a cost c(u, v) > 0 to every arc(u, v) in G and ask for the solution T with mini-mum overall cost C(T ) = ∑

(u,v)∈E(T ) c(u, v), whereE(T ) is the set of arcs of T . Note that, as defined,G can contain cycles. This turns out to be the maindifficulty for this optimization problem, which, inthe past, was usually tackled by a rather inefficienttrick: “unfolding” the cycles and applying standardAND/OR search methods for acyclic graphs. How-ever, explicit treatment of cycles has recently beenconsidered, and an efficient algorithm is achievedin [8].The search for an optimal height propagation is nextreduced to this model. This amounts to (1) construct-ing an AND/OR graph Ghp whose feasible solutionsdefine a height propagation, and (2) define a costfunction that promotes non-degenerate propagationsover degenerate ones.

4.2 Feasible height propagations

A height propagation can be defined by the followingrules, with the given straightforward translation intoAND/OR subgraphs.

R1: Four selected vertices of the projection triggerthe propagation. For this, we put a TERMINALnode for each of the triggering vertices.

R2: Every face in the polygonization can be de-termined once the heights of any three of itsvertices are determined. If deg( f ) denotes thenumber of vertices of face f , then there arec f = (deg( f )

3

)possible combinations of three

vertices determining f . If we put a node in Ghpfor every vertex, except for the four trigger-ing ones, then this rule is translated by addingan OR node for every face, linked to c f new“dummy-face” AND nodes, each representingone of the above combinations. Each dummy-face node is in turn linked with arcs to the threeinvolved vertices in the combination. Figure 8gives a schematic representation. The newly in-troduced vertex nodes have not been assigneda type yet. This type is induced by the followingrule.

R3: Except for the initial four vertices, the heightof every other vertex is determined once one ofits incident faces has a determined plane. This


Fig. 8a–c. AND/OR subgraphs for the propagation rules. AND nodes are indicated by joining all their emanating arcs. a Con-structed subgraph translating rule R2 for a quadrilateral face. Dummy-face nodes are shadowed in grey. Note that, actually,there is only one vertex node for each vertex in the trihedral mesh, but for clarity they are here duplicated. b Propagationwaves reaching a vertex. c Subgraph for rule R3, with an arc for each of the possibilities in b

implements the fact that the propagation wavefixing the height of a vertex can come from anyof its three incident faces (Fig. 8b). This rule canbe represented by setting each vertex node asOR type, and linking it to the face nodes of itsincident faces (Fig. 8c).

R4: The height propagation must reach all vertices.For this, we add a root AND node r to Ghp andlink it to all vertex nodes.

Note that a feasible solution tree of Ghp provides in-structions to derive a height propagation that reachesall vertices, starting at the four pre-specified heights.

4.3 Cost function

In order to penalize propagations using sets ofalmost-aligned vertices, we proceed as follows. Con-sider a height propagation that fixes a face-plane ffrom the point coordinates of three previously fixedvertices vi , v j and vk. We can simply penalize thecorresponding arcs in Ghp emanating from f by giv-ing them a cost that is inversely proportional to thearea of the triangle defined by vi , v j and vk in theprojection. The rest of arc costs are actually irrele-vant, but need to be positively defined [8]. In sum,for every directed arc (u, v) we define its cost asfollows:

1. c(u, v) = 1/ det(vi, v j , vk), if u is a dummy-faceAND node and v is any one of its descendants.Here, vi , v j and vk are the homogeneous coor-dinates of the vertices associated with the three

descendants of u, with a one in the last coordi-nate.

2. c(u, v) = 1, otherwise.

Once the least-cost solution T is found, the projec-tion can be made robust to slight vertex perturbationsas follows. At a vertex v receiving more than onepropagation wave, we put a T/TT-transformation onall faces fixing v, except on the one in the propaga-tion wave represented in T .

4.4 Complexity analysis

The worst-case complexity of computing the optimalsolution of a cyclic AND/OR graph with n nodes isO(n3) [8]. We now prove that the number of nodes inGhp grows linearly with the number of vertices of thetrihedral polygonal mesh.Let e, v and f be the number of edges, vertices andfaces of the given mesh. Then, 2e = 3v because themesh is trihedral. Moreover, if the mesh has h holes,with “the outside” of the mesh counting as a holetoo, then Euler’s relation says that v− e+ f = 2−h.From these two equalities the number of faces ofthe mesh can be written in terms of the number ofvertices and holes, f = v+4

2 − h. Let us now countthe number of nodes added by each of the rulesR1, . . . , R4:

– Rule R1 adds four vertex nodes.– Rule R2 adds one OR node for each face, amount-

ing to f = v+42 − h = O(v) total nodes, assum-

ing a constant number of holes. Also, for every


face f this rule adds c f = (deg( f )

3

)dummy-face

AND nodes. Although this number is clearly inthe worst case O(deg( f )3), if we divide the sumof face degrees by the number of faces, the aver-age face degree is six, at an increasing number ofrandomly placed vertices in the mesh:∑

allfaces deg( fi)

f= 3v

v+42 −h

= 6v

v+4−2h

which will keep the number of dummy-face ANDnodes linearly growing:(

6

3

)f = 20

(v+4

2−h

)= O(v)

– Rule R3 adds a linear number of OR vertex nodes.– Rule R4 only adds one AND node, the root.

Up to now we have assumed that the four ver-tices triggering the propagation are a priori selected.But other height propagations starting at other fourvertices could yield better height propagations. Totest all possibilities, we do not need to repeat theAND/OR search for every different combination offour vertices. Indeed, note that these vertices justfix the planes of the faces they belong to. So, anyother set of four vertices on these faces will yieldthe same optimal propagations, provided that two ofthem lie on the common edge. We can equivalentlythink of pairs of faces triggering the propagationand use their face nodes as TERMINAL in Ghp. Thechoice of TERMINAL vertices (instead of TERMI-NAL faces) was done to be coherent with previousexplanations. In sum, if one wants to search over allpossible starting places of propagation, then for eachpair of adjacent faces the AND/OR search needs tobe repeated. This amounts to solving e = 3

2v opti-mization problems in the worst case, meaning thatthe overall complexity will be O(v4), under the as-sumption that the face degree is six.

5 Conclusion

We have shown how trihedral mesh projections cancapture the spatial shape of a given object’s surface,once the heights of four vertices are known, and wehave given an algorithm to derive trihedral meshesfrom arbitrary input surfaces. We have also presenteda local strategy that takes a trihedral projection asinput and places some triangular faces at strategicplaces until its reconstructibility is made robust to

perturbations in its vertex coordinates. Finally, wehave found how to put these triangles so that thespatial reconstruction is performed in the most ac-curate way possible, avoiding height propagationsalong degenerate paths.One issue that must be solved is that errors accumu-late from one vertex to another along a height prop-agation, as shown in Fig. 7. The AND/OR searchalgorithm reduces these errors but, for the same pur-pose, one could additionally extend the number ofvertices for which a Z value is given. Optimizingthe position of such vertices is almost certainly in-tractable, but simple heuristics could be used to addvertices to a “specified Z” set whenever errors accu-mulated along a propagation path become unaccept-able. This point deserves further attention.The main issue currently concentrating our efforts isto find an enhancement of (or an alternative to) thetrihedrization algorithm presented in Sect. 2. Clearly,as observed in the bottom row of Fig. 4, the obtainedpolygonization contains small polygons, almost or-thogonal to the input surface. This kind of “staircaseeffect” may produce numerical instabilities when at-tempting to reconstruct the spatial shape from itsprojection, and should be avoided. A possible solu-tion to this problem may be found by using projectivepolarities [4]. It is well known that the projectivepolar of a triangulated polyhedron (with respect toa quadric) is a trihedral polyhedron. Thus, in princi-ple, to get a trihedral mesh of a surface S, one couldderive an auxiliary surface S∗, the polar of S througha polarity P [7], and then apply P−1 to a triangula-tion of S∗, to obtain a trihedral mesh approximatingS. We believe this is a promising method to increasethe smoothness of the obtained polygonization.

Acknowledgements. We are very grateful to an anonymous reviewer forhis helpful comments, and to Francisco Perez and Pablo Jimenez forfruitful discussions on several parts of the paper. This work has beenpartially funded by the Spanish CICYT under contracts TIC-2000-0696and TAP99-1086.

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Photographs of the authors and their biographies are given onthe next page.

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LLUIS ROS was born in Vi-lafranca del Penedès (Barcelona,Catalonia), on June 30, 1968.He received the mechanical en-gineering degree in 1992, andthe Ph.D. degree (with hon-ors) in industrial engineering in2000, both from the PolytechnicUniversity of Catalonia. From1993 to 1996 he worked withthe Control of Resources groupof the Cybernetics Institute inBarcelona, involved in the ap-plication of constraint logic pro-gramming to the control of elec-

tric and water networks. Since August 2000, he has been a Re-search Scientist in the Institut de Robòtica i Informàtica Indus-trial of the Spanish High Council for Scientific Research. Hiscurrent research interests are in geometry and kinematics, withapplications to robotics, computer graphics and machine vision.His web page is http://www-iri.upc.es/people/ros.

KOKICHI SUGIHARA re-ceived B.Eng., M.Eng. and Dr.Eng. degrees in MathematicalEngineering from the Univer-sity of Tokyo in 1971, 1973and 1980 respectively. In 1986he moved to the Department ofMathematical Engineering andInformation Physics of the Uni-versity of Tokyo, and he is nowa professor of the Departmentof Mathematical Informatics ofthe Graduate School of Infor-mation Science and Technologyof the University of Tokyo. His

research interests include computational geometry, computergraphics and computer vision. He is a member of the Infor-mation Processing Society of Japan, the Operational ResearchSociety of Japan, Japan SIAM, IEEE and ACM.

FEDERICO THOMAS wasborn in Barcelona, on October5, 1961. He received the B.S.degree in telecommunicationsengineering in 1984, and thePh.D. degree (with honors) incomputer science in 1988, bothfrom the Polytechnic Universityof Catalonia. Since March 1990,he has been a Research Scien-tist in the Institut de Robòticai Informàtica Industrial of theSpanish High Council for Sci-entific Research. His researchinterests are in geometry and

kinematics, with applications to robotics, computer graphicsand machine vision. He has published more than forty researchpapers in international journals and conferences. His web pageis http://www-iri.upc.es/people/thomas.

llu´ısros ,kokichisugihara federicothomas · 2 l. ros et al.: trihedral polygonizations 1 2a 2b...

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