International audienceWe present an exact method to compute the boundaries between umbra, penumbra and full-light regions cast on a plane by a set of disjoint convex polyhedra, some of which are light sources. This method builds on a recent characterization of topological visual event surfaces presented in a companion paper

Demouth, Julien

Goaoc, Xavier

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HAL Id: inria-00431544https://hal.inria.fr/inria-00431544Submitted on 12 Nov 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Computing Direct Shadows Cast by Convex PolyhedraJulien Demouth, Xavier GoaocTo cite this version:Julien Demouth, Xavier Goaoc. Computing Direct Shadows Cast by Convex Polyhedra. 25th Euro-pean Workshop on Computational Geometry - EuroCG 2009, Mar 2009, Brussels, Belgium. ￿inria-00431544￿Computing Direct Shadows Cast by Convex PolyhedraJulien Demouth∗ Xavier Goaoc†AbstractWe present an exact method to compute the bound-aries between umbra, penumbra and full-light regionscast on a plane by a set of disjoint convex polyhedra,some of which are light sources. This method buildson a recent characterization of topological visual eventsurfaces presented in a companion paper.1 IntroductionSince shadows play an important role in human un-derstanding of three-dimensional scenes from two-dimensional pictures [18, 23], their realistic renderinghas been a central problem in Computer Graphics forthe last decades [12]. While illumination simulationmethods such as photon mapping [16] now produceexcellent results, it is sometimes desirable to deter-mine shadow boundaries a priori to the illuminationsimulation, that is, to deduce it from the geometry ofthe scene. This is the case, for instance, when usingshadow volumes [5, 1] for realtime rendering or dis-continuity mesh [15, 17] to improve radiosity meth-ods [4, 20]. Unlike hard shadows cast by punctuallight sources, whose geometry is well understood [25],soft shadows cast by extended light sources still proveelusive. The classical approach traces shadow bound-aries back to visual event surface [22], that is pointsfrom which the view of the scene changes. The classi-cal visual event surfaces [13], however, capture muchmore than direct shadows, for example shadows castby indirect lighting. As a result, methods based onthem such as the discontinuity mesh [15, 17, 21, 9] orthe visibility skeleton [11, 10] produce sets of curvesthat are too large to be used in practice. In this ab-stract, we present a method that computes exactlythe shadows cast by a set of disjoint convex objects,some of which are light sources, ignoring the effectsof light reflection which are arguably less important(see, for example, [4]), and evaluate its performanceson a few types of random scenes.∗Dept. of Computer Sci. & Eng., POSTECH, Pohang, Ko-rea, julien@postech.ac.kr†VEGAS Project, INRIA Grand Est, Nancy, France,Xavier.Goaoc@loria.fr2 Characterization of shadow boundariesIn a companion paper, we studied visual events re-lated to changes in the topology of the apparent con-tour of a collection of convex sets [8]. Here, we firstshow that these events are sufficient to capture di-rect shadows. We then associate to each point ofa visual event surface a combinatorial structure, itsgerm, which characterizes the type of change occuringat that point. We show that only points with certaingerms can participate in each type of shadow bound-ary, and that these germs can be associated with thearcs of a graph encoding the visual event surfaces, thevisibility skeleton.Topological visual events. The visual event surfacesare defined as the locus of points at which the view ofthe scene undergoes a qualitative change. The notionsof “view” and “qualitative change” need to be definedcarefully, though, as different definitions capture dif-ferent phenomena. For polyhedral scenes, the viewis usually defined as the (abstract) graph obtainedby projecting the visible portions of the polyhedra’sedges, each element of the graph being labelled by itsthree-dimensional generators, and two views are qual-itatively equivalent if they are isomorphic, as labelledgraphs [13]. The corresponding visual event surfacesare of two types [13]: ev surfaces, consisting of originsof rays through a vertex and an edge, and eee sur-faces, formed by origins of rays through three edges(see Figure 1). Thus, the visual event surfaces of avee1e2e3Figure 1: Classical visual event surfaces for polyhe-dra: ev surface (left) and eee surface (right).scene consisting of a single convex polyhedron are theplanes supporting its faces. Yet, since the polyhedronis totally visible from anywhere in space, it inducesno shadow boundary at all.1These choices of “view” and “qualitative change”are sensitive to the changes in the view not only ofthe polyhedra, but also of their meshing: events oc-cur whenever a facet starts or ceases to be visible. Ina companion paper [8], we studied the visual eventscorresponding to the topological changes in the ap-parent contour of a collection of disjoint convex sets.Specifically, we define the view from a point as theset of directions of rays from that point that meet thescene tangentially and declare two such views equiva-lent if there exists an homeomorphism from the spaceof direction into itself that maps one view into theother. The associated “topological” visual event sur-faces suffice to capture direct shadows.Theorem 1 The boundary of the direct shadow castby a collection of disjoint convex objects is containedin the trace of visual event surfaces corresponding totopological changes in the apparent contour.Call a ray tritangent if it sees tangentially threeobjects and limiting bitangent if it sees tangentiallytwo objects and lies in a plane tangent to both ofthem. In the case of disjoint convex polyhedra, weshow [8, Theorem 3] that there are two types of suchvisual event surfaces: eee surfaces where the threeedges come from distinct polyhedra, that is originsof tritangent rays, and ev surfaces where the edgeand the vertex come from distinct polyhedra and spana plane tangent to both, that is, origins of limitingbitangent rays.Germ of a ray. Let ρ be a tritangent or limiting bi-tangent ray. Up to symmetries, the projection of thethe objects seen tangentially by ρ are locally equiv-alent (that is, homeomorphic) to one of eight basicsituations (see Figure 2) which we call the germ ofρ; we group the germs into four types, from 1 to 4.A ray has one germ for each triple (resp. pair) of1324Figure 2: The germs of ev surfaces (top) and eeesurfaces (bottom). Sectors covered by the projectionof an object are shaded in grey, lighter colors corre-sponding to objects closer to the origin of the ray.The numbers 1, . . . , 4 indicate the type of the germ(s)of the corresponding box.object to which it is tritangent (resp. limiting bitan-gent). Points that belong to a umbra/penumbra or apenumbra/full light boundary are origins of rays withprescribed germ types:Theorem 2 A point belongs to the boundary be-tween the umbra and the penumbra (resp. penumbraand full light) regions cast by a collection of convexsets only if it sees a light source tangentially along aray with germ of type 1 (resp. 2) or if it sees a lightsource along a ray with germ of type 3.Visibility skeleton. Let C denote a set of convexpolyhedra and A(C) the arrangement, in ray space,of the sets of tritangent and limiting bitangent rays.A(C) encodes the visual event surfaces of C in thesense that these surfaces are the origins of the raysthat make up the faces of A(C). Two rays ρ ⊂ ρ′such that the segment joining their origin does notintersect C belong to the same cell in A(C). We canthus “collapse” A(C) by identifying all such rays. Oneway to do that is to consider the quotient space R/ ∼where R is the space of rays and:(p+R+~u) ∼ (q+R+~v) ⇔ ~u = ~v = ±−→pq and [pq]∩C = ∅.The origins of all rays in a given equivalence classmake up a maximal free segment : a segment thatdoes not intersect C and is maximal for this prop-erty with respect to the inclusion1. There is thus anatural bijection betweenR/ ∼ and the space of max-imal free segments2; we call the image of A(C) underthis bijection the visibility skeleton of C relative totopological changes in the apparent contour (by anal-ogy to the classical visibility skeleton [11] defined toencode classical visual event surfaces). For a collec-tion of disjoint convex polyhedra in generic position3,this visibility skeleton is a graph whose arcs are maxi-mal free segments supported by tritangent or limitingbitangent rays to a given triple/pair of objects, andwhose nodes are maximal free segments supported bya line through two vertices (vv node), one vertex andtwo edges (vee node) or four edges (eeee node).Lemma 3 Two rays in the same facet of A(C) havethe same germ.As a consequence, all rays corresponding to a givenarc of the visibility skeleton have the same germ,which we call the germ of the arc. The characteriza-tion of Theorem 2 then extend to arcs of the visibilityskeleton, and thus to visual event surfaces.1Note that this differs from the usual definition in that amaximal free segment is not allowed to be tangent to an object.2This bijection is not an homeomorphism as the naturaltopologies on these spaces differ, but this has no consequencefor our purposes.3The same holds for nongeneric position but additional carein the definition is required.23 Algorithm outlineOur algorithm takes as input a set C of disjoint convexpolyhedra and a plane Π, and consists of 3 steps:1. Compute the visibility skeleton of C and thegerms of its arcs.2. Compute the arrangements, in the plane Π, ofthe curves traced by the surfaces swept by therelevant arcs of the skeleton. We compute twoarrangements: (a) with arcs of type 1 tangent toa light source in last position and arcs of type 3with an endpoint on a light source, and (b) onewith arcs of type 2 tangent to a light source.3. At each curve crossing, remove the arcs not onthe shadow boundary locally (see Figure 3).XXFigure 3: Two arcs incident to a crossing of twocurves in arrangement (a) do not belong to the shadowboundary (the shading indicates on which side of thearc the source becomes locally invisible). A similarcriterion holds for crossings in arrangement (b).The output consists of two collections of closed non-crossing loops which, by Theorem 2, contain theboundaries between, respectively, umbra and penum-bra – for (a) – and penumbra and full light – for (b).Testing if a given loop is an actual shadow boundarycan be done by computing the view of a single point.The visibility skeleton has size Θ(n2k2) [3] and canbe computed in time O(n2k2 log n) by a sweep-planealgorithm [14, Chapter 8]. The complexity of the ar-rangements (a) and (b) is O(n3k3) [7], so altogetherthe algorithm runs in O(n3k3 log n) time in the worstcase.4 Implementation and experimental resultsOur implementation was done in C++ using exactnumber types and geometrical objects from CGAL li-brary. The implementation assumes the objects arein general position in the sense that no 3 verticesare collinear, no 4 vertices (from distinct polyhedra)coplanar and no 10 vertices on a common quadricsurface. For Step 1, we compute the vv nodes bybrute-force enumeration and use an implementationby Zhang et al. [26] to compute the vee and eeeenodes; we retrieve the connectivity using local compu-tations. For Step 2, we use the 2D arrangement pack-age from CGAL [24], the arcs of conic curves inducedby eee arcs being described by a rational equationand two algebraic endpoints4. See [6, Chapter 6] fora more detailed discussion on the implementation.We tested three different types of random scenes. Ineach case, we chose a random collection of k disjointunit spheres, and for each pick p points on its surfaceand compute their convex hull. The three types ofdistribution of centers of spheres that we consider are:(i) k centers randomly distributed inside a cube,(ii) k centers randomly distributed on a constantnumber of horizontal slices of a cube,(iii) k− 1 centers randomly distributed on a constantnumber of horizontal slices of a cone with apexthe kth sphere, which is the light source.We run tests for k = 10, 30 and 50 and p rangingbetween 4 and the greatest power of 2 manageable.Disjointedness is ensured by throwing away centersthat violate it and generating them again. For eachvalue of k and p we created 10 scenes for type (i) anda single one for types (ii) and (iii). We analyzed ourmethod using three indicators:• gain1 = b−ab , where a denotes the size (number ofnodes) of our visibility skeleton and b that of theclassical visibility skeleton [11]. It measures thegain obtained by considering “topological” visualevents rather than the classical visual events.• gain2 = d−cd , where d denotes the number of arcsof our visibility skeleton and c the number of arcsused in Step 2. It measures the effectiveness ofthe filter based on germ conditions.• gain3 = e−fe , where e denotes the complexityof the arrangements (a) and (b) and f that ofthe output of Step 3. It measures the reductionin size obtained by storing only the (candidate)contours of direct shadows (versus keeping directdiscontinuities in the interior of the penumbra).Our measures show that all three gains are high,most of the time. Specifically,• gain1 increases with p, reaching 50% for p = 16and 75% for p = 64, independantly of k. Sincethe visibility skeleton has a rather large size, thisgain is significant; for instance, for k = 30 andp = 128, a = 290000 and b = 50000.• gain2 is above 85% for all tested scenes, and in-creases with k.• gain3 is below 10% for the limit between penum-bra and full-light, but above 70% for the limitbetween umbra and penumbra.4Note that a polygonal approximation of these arcs is per-formed at visualization.3A comparison of some outputs of our program withimages produced using the PBRT rayshooting pro-gram [19], represented in Figures 4 and 5 in the Ap-pendix, also suggests that the results are visually con-vincing. Due to space limitation, we refer to [6, Chap-ter 6] for a detailed account of the performances.5 ConclusionOur experiments suggest that the method we proposeis simple enough to be put in practice and leads to sig-nificant gains over the classical approach. Of course,these experiments are on small-scale (≤ 104 triangles)nonrealistic scenes. The main challenges to overcomethese limitations include the efficient computation ofeeee nodes and the understanding of “topological”visual events in nonconvex settings. A first progresson the latter was recently made by Batog [2]. Ourresults also suggest that the boundary of shadow re-gions is simple enough to be of interest in ComputerGraphics.References[1] U. Assarsson and T. Akenine-Möller. A Geometry-based Soft Shadow Volume Algorithm using Graph-ics Hardware. ACM Transactions on Graphics,22(3):511–520, 2003.[2] G. Batog. Événements visuels de polyèdres, Master’sthesis, MPRI, ENS Cachan, 2008.[3] H. Brönnimann, O. Devillers, V. Dujmović, H. Ev-erett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na andS. Whitesides. Lines and Free Line Segments Tangentto Arbitrary Three-dimensional Convex Polyhedra.SIAM Journal of Computing, 37(2):522–551, 2007.[4] M. F. Cohen and J. R. Wallace. Radiosity and Real-istic Image Synthesis. Academic Press, 1993.[5] F. C. Crow. Shadow Algorithms for ComputerGraphics. Computer Graphics, 11(2):242–248, 1977.[6] , J. Demouth. Événements visuels de convexes et lim-ites d’ombres. PhD thesis, Université Nancy 2, 2008.[7] J. Demouth, O. Devillers, H. Everett, M. Glisse,S. Lazard and R. Seidel. On the Complexity of Um-bra and Penumbra. Computational Geometry: The-ory and Applications, to appear, 2009.[8] J. Demouth and X. Goaoc. TopologicalChanges in the Apparent Contour of Con-vex Sets. Submitted version available athttp://www.loria.fr/∼goaoc/papers/TopVE.pdf.2009.[9] G. Drettakis and E. Fiume. A Fast Shadow Algo-rithm for Area Light Sources Using Backprojection.Proceedings of ACM Siggraph 94, pages 223–230,1994.[10] F. Duguet and G. Drettakis. Robust Epsilon Visi-bility. Proceedings of ACM SIGGRAPH 2002, pages567–575, 2002.[11] F. Durand, G. Drettakis and C. Puech. The Visibil-ity Skeleton: a Powerful and Efficient Multi-PurposeGlobal Visibility Tool. Proceedings of ACM Sig-graph 97, pages 89–100, 1997.[12] P. Dutré and K. Bala and P. Bekaert. AdvancedGlobal Illumination. A.K. Peters, 2006.[13] Z. Gigus and J. Malik. Computing the Aspect Graphfor Line Drawings of Polyhedral Objects. IEEETransactions on Pattern Analysis and Machine In-telligence, 12(2):113–122, 1990.[14] X. Goaoc. Structures de visibilité globales : taille, cal-cul et dégénérescences. PhD thesis, Université Nancy2, 2004.[15] P. S. Heckbert. Discontinuity Meshing for Radiosity.Proceedings of the Eurographics Workshop on Ren-dering, pages 203–215, 1992.[16] H. W. Jensen. Global Illumination Using PhotonMaps. Proceedings of the Eurographics Workshop onRendering, pages 21–30, 1996.[17] D. Lischinski, F. Tampieri and D. P. Greenberg. ADiscontinuity Meshing Algorithm for Accurate Ra-diosity. IEEE Computer Graphics and Applications,12(6):25–39, 1992.[18] P. Mamassian, D. C. Knill and D. Kersten. The Per-ception of Cast Shadows. Trends in Cognitive Sci-ences, 2(8):288–295, 1998.[19] M. Pharr and G. Humphreys. Physically Based Ren-dering: From Theory to Implementation. MorganKaufmann, 2004.[20] F. Sillion and C. Puech. Radiosity and Global Illumi-nation. Morgan Kaufmann, 1994.[21] A. J. Stewart and S. Ghali. Fast Computation ofShadow Boundaries Using Spatial Coherence andBackprojections. Proceedings of ACM Siggraph 94,pages 231–238, 1994.[22] S. J. Teller. Computing the Antipenumbra of an AreaLight Source. Proceedings of ACM SIGGRAPH 92,pages 139–148, 1992.[23] L. R. Wanger. The Effect of Shadow Quality onthe Perception of Spatial Relationships in ComputerGenerated Images. Computer Graphics, 25(2):39–42,1992.[24] R. Wein, E. Fogel, B. Zukerman andD. Halperin. 2D Arrangements. CGAL Userand Reference Manual, Release 3.3, 2007.http://www.cgal.org/Manual/3.3/doc html/cgal manual/packages.html#Pkg:Arrangement2.[25] A. Woo, P. Poulin and A. Fournier. A Survey ofShadow Algorithms. IEEE Computer Graphics andApplications, 10(6):13–32, 1990.[26] L. Zhang, H. Everett, S. Lazard, C. Weibel andS. Whitesides. On the Size of the 3D Visibility Skele-ton: Experimental Results. Proceedings of the Six-teenth Annual European Symposium on Algorithms(ESA 2008), to appear, 2008.4A IllustrationsFigure 4: Four triangles, one of which is a light source. Top: output of our algorithm (left) and arrangementsobtained after Step 2 (right). Middle: details of the arrangements. Bottom: the three regions obtained by ourmethod (left) and the same scene rendered using a raytracer (right).Figure 5: Thirty convex polyhedra, two of which are light sources. Top: output of our algorithm (left) and zoomon a detail (right). Bottom: the three regions obtained by our method (left) and the same scene rendered usinga raytracer (right).5

Computing Direct Shadows Cast by Convex Polyhedra

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Computing Direct Shadows Cast by Convex Polyhedra

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