Hoops: 3D Curves as Conservative Occluders for Cell-Visibility

Transcription

1 EUROGRAPHICS 2001 / A. Chalmers and T.-M. Rhyne (Guest Editors) Volume 20 (2001), Number 3 Hoops: 3D Curves as Conservative Occluders for Cell-Visibility Pere Brunet, Isabel Navazo, Jarek Rossignac and Carlos Saona-Vázquez Abstract Most visibility culling algorithms require convexity of occluders. Occluder synthesis algorithms attempt to construct large convex occluders inside bulky non-convex sets. Occluder fusion algorithms generate convex occluders that are contained in the umbra cast by a group of objects given an area light. In this paper we prove that convexity requirements can be shifted from the occluders to their umbra with no loss of efficiency, and use this property to show how some special non-planar, non-convex closed polylines that we call "hoops" can be used to compute occlusion efficiently for objects that have no large interior convex sets and were thus rejected by previous approaches. 1. Introduction Despite the continuous improvement in the bandwidth of graphics hardware, navigation of very complex models remains an open problem of interest to graphics researchers. Software solutions to this shortcoming abound, and can be grouped into three families: level of detail, image-based rendering and visibility culling. This paper introduces a new tool that can be used to improve culling performance of most current visibility culling algorithms. Research on visibility culling first focused on ways to compute conservatively whether a set of objects are hidden by one single convex object called the occluder. Local visibility algorithms perform occlusion culling at each viewpoint while the user is navigating. Approximate cell visibility algorithms subdivide the navigational space into cells and estimate the set of visible objects for each cell by sampling viewpoints locations in it. Recently, several researchers proposed to use synthesized convex shapes which can be safely substituted for the original occluders and that can be combined to synthesize a larger set (which may be safely substituted for several occluders). But occluder convexity has remained a necessary condition for the computation of individual occluders. In many Polytechnic University of Catalonia G.V.U., Georgia Institute of Technology situations, convexity of occluders yields faster culling algorithms and has been a prerequisite for occluder fusion. This paper proposes new results for occlusion computation. From a theoretical point of view, it relaxes the conditions occluders must meet in order to allow to infere cell visibility from local visibility at its vertices. If an object S is hidden by an occluder O from the vertices of a convex cell C, then S is hidden from all points of C. On a more practical side, new algorithms are introduced that compute suitable non-convex occluders from arbitrary scenes. More importantly, though concave these occluders retain the main properties of convex occluders regarding visibility: occlusion speed and cell visibility computability. Finally, we also show how these occluders can be stored compactly as special polylines we call hoops and how hoops from disjoint sources can be fused together. Hoops can be integrated effortlessly into almost all visibility culling algorithms for enhanced occlusion culling. Our tests show that hoops can be synthesized from non-convex sets that do not contain any large convex sets, thus improving occlusion performance. Section 2 reviews current approaches to visibility culling, occluder synthesis and occluder fusion. Our theoretical contributions to visibility are introduced next in section 3. Section 4 exploits those results to elaborate a practical algorithm to synthesise occluders from heavily tessellated scenes and store them as hoops. Fusion of hoops is addressed next in section 5. Finally, sections 6 and 7 show empiric results and discuss conclusions and future work. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

2 2. Previous Work Back in 1993, Teller and Hanrahan 17 extended previous ideas of Teller 16 and Teller and Séquin 18 to present an algorithm that computed visibility in architectural environments. Linear-programming techniques were used to compute interroom visibility. Later, Coorg and Teller 5, 6 introduced the concepts of separating and supporting planes to conservatively decide visibility between two convex sets separated by a convex polygon. At each viewpoint, supporting and separating planes were used to cull the kd-tree of the scene with a set of occluders that had been selected previously in a preprocess. Instead of sets of planes, Hudson et al. 10 computed shadow frusta to cull the scene. Greene, Kass and Miller 9 and Zhang et al. 20 also computed visibility locally at each viewpoint, but computations were done in image space with hierarchical generalisations of the z-buffer. Instead of performing occlusion culling at each viewpoint, several authors proposed to partition the space into cells and computing visibility of the cell as a conservative estimation of the union of the visible sets of its inner points. Cohen-Or et al. 4 used a uniform grid to partition navigational space, whereas Saona-Vázquez et al. 14 proposed an octree. Klosowski and Silva 11 presented a non-conservative, realtime approach to visibility. They employ a heuristic metric that estimates the likelihood of a polygon being visible. Andújar et al. 2 also presented an algorithm that traded accuracy for speed. It was accomplished by means of a generalisation of potentially visible sets, called hardly visible sets, that allows to cull geometry that is partially occluded. Additionally, the authors showed how hardly visible sets could be used to improve level of detail. As discussed in the previous section, almost all visibility algorithms depend heavily on the tesselation and on the convexity of its models. Occluder synthesis algorithms provide alternative tessellations of potentially good occluders to increase the efficiency of occlusion culling. Zhang 19 and Law and Tan 13 used simplification algorithms to accelerate the culling process. As convexity was not an issue, their applicability is more limited than more recent methods. Instead of simplification, several authors have opted for converting the original bounded representation to a hierarchical volumetric representation (i.e., an octree). Andújar, Saona- Vázquez and Navazo 1 used a seed-like algorithm to generate sets of densely overlapping boxes contained into the black nodes of the octree. Schaufler et al. 15 presented a similar synthesis algorithm that also included occluder fusion (see below). Bernardini, Klosowski and El-Sana 3 generated view-dependent occluders by considering the faces of black nodes as potential occluders. Finally, until very recently visibility computation was limited to occlusion due to a single connected convex occluder (except for frame buffer based occlusion techniques). That is to say, an object that was occluded by the joined action of two blockers but not by either of them alone was always classified as visible. Fusion of occluders may be achieved by extending occluders inside previously computed umbras. If an occluder O 2 intersects the umbra U 1 of occluder O 1, then extending O 2 into U 1 yields a fusioned occluder that culls geometry occluded by O 1 and O 2 together. Even better, this procedure can be repeated iteratively with other occluders. Koltun, Chrysanthou and Cohen-Or 12 have proposed this approach in 2.5D space using supporting and separating lines to compute umbras, and so do Schaufler et al., 15 but in 3D using volume octrees. In an utterly different fashion, Durand et al. 8 project occluders into image space and accomplish occluder fusion by reprojecting on several planes. It is the only cell visibility algorithm that can cope with concave sets, although with some limitations. Namely, a plane has to be found whose intersection with the occluder is a singly connected surface. Even then, and as happens also with imagespace local visibility algorithms, concave occlusion is much slower than convex occlusion. A solution, sketched by Durand, 7 was to generate a set of convex occcluders whose union approximated the concave set conservatively. In order for this union to be of any use, the sets of the union must overlap. As we will see, hoops satisfy these requirements completely. 3. Cell Occlusion and Hoops This section discusses efficient ways of computing cell visibility. Given a partition of the space into cells, the objective is to compute a conservative estimation of the visible set of each cell. The visible set is the set of potentially visible scene polygons from any viewpoint in the cell (the visible set of a cell is also the complement of the set of polygons that are invisible from all possible viewpoints in the cell). In what follows we will first show that occluder convexity is a sufficient but not necessary condition for efficient cell occlusion. In fact, convexity of the umbra from the cell is enough to compute the visible set of the cell. Then, by extending the notion of umbra to non-planar polylines, we will introduce the concept of hoops and provide efficient tests to check umbra convexity and hoop conditions in linear time. The main idea derives from the fact that non-convex objects can generate convex shadows from certain locations of the viewpoint q (see figure 1). In these cases, a 3D polyline (hoop) generating the same convex shadow as O can be used in the occlusion tests instead of using the initial object O, figure 1-d and 1-e. This section presents the basic definitions and properties of hoops, whereas the hoop synthesis and fusion algorithms are introduced in sections 4 and Shadows We start with the definitions of shadows and umbrae. In what follows, the convex hull of a set of points p 1,..., p k will

3 Figure 1: A non-convex object (a). The object is a nonconvex occluder from a general viewpoint (b), but is almost convex from the view in (c). A q-hoop is a closed polyline that is seen as convex from the viewpoint q (d), although it is really non-planar and non-convex, (e). Definition 2 The shadow from a set C R m of a set O R m linearly separable from it (figure 2-b), is S(V,O) = {p R m \ O q C : qp O } = S(q, O) q C As we will see, it will prove useful to consider also an extended version of the shadow that includes the occluder and the region between the occluder and the viewing points, figure 2-c and 2-d. Definition 3 Let q R m,o R m be a point and a set that are linearly separable. The extended shadow of O from q is S * (q,o) = {p R m ray(q, p) O } = {p R m x O,λ > 0 : p = q + λ qx} Definition 4 Let O,C R m be two linearly separable sets. The extended shadow of O from C is S * (C,O) = S * (q,o) q C The next part of this section deals on how to infer cell visibility from the visibility properties of a few points, such as the cell vertices Cell Occluders Definition 5 Given a set O R m and a bounded polytope C R m, we will say that O is a cell occluder for C, or just a C-occluder, if the following proposition holds for any other set S O hides S from every vertex of C S is hidden by O from every point in C Figure 2: The shadow of a pointset O from a point q (a), the shadow of the same pointset from a set C (b), and the corresponding extended shadows from q and C, (c), (d). be denoted by CH(p 1,..., p k ) and the segment between two points p and q by pq. Definition 1 Assume we have a point q and a set O both in R m. Assume that there exists a plane linearly separating q and O. Then the shadow of O from q is, S(q,O) = {p R m p O qp O } If we extend this definition to sets of points, we define the umbra (Teller, 16 Saona 14 ), figure 2-a. The following result shows that, assuming convexity of the extended shadow of O from C, the cell visibility can be directly computed from the visibility properties of the vertices of C (observe that the convexity requirement for the extended shadow is weaker than the usual requirement of O being convex, figure 2-d) Theorem 1 Let O,C R m, where C = CH(q 1,...,q n) are linearly separable and such that q C : S * (q,o) is convex. Then O is a C-occluder: for every point x that is linearly separable from C O n x S(q i,o) q C : x S(q,O) i=1 Proof. We will demonstrate it by induction on the dimension of the polytope C,k = dim(c) < n. Let q be a point in C. Assume that x n S(q i,o) i=1 k = 1) Suppose q 1,q 2 are the extremal points of the segment C. Let H 1 be the plane separating C from O and H 2 the plane separating O from x (see figure 3). We know that

4 Figure 3: Example of lemma 1 for a segment. The drawing shows a 2D section by the plane q 1,q 2,x. Note that O need not be connected. Figure 4: The closed polyline defined by the sequence of edges (a) of the L-shaped object is not self-intersecting, but it is not q-simple from the view in (b). exist two points in O, p 1, p 2, that lie on rays xq 1 and xq 2 because x is hidden by O from all q i. As q, p 1, p 2 are on different edges of the triangle q 1 xq 2, the segments qx and p 1 p 2 intersect in a point y. Then, because p 1, p 2 O p 1, p 2 S * (q,o) by convexity of S * we have y S * (q,o) and then x S * (q,o). As H 2 separates x from O, if x is in the extended shadow of q from O then it must also be in the regular shadow S(q,O). k > 1) Let p 1, p 2 R m now be two different points on two (k 1)-facets f 1, f 2 of C such that point q CH(p 1, p 2 ) = p 1 p 2. Then, n x S(q i,o) induction x 2 S( f i,o) k>k 1 i=1 i=1 2 x S(p i,o) induction x S(p 1 p 2,O) k>2 i=1 x S(q,O) The consequence of theorem 1 is that, independently of its shape, we can conclude that a certain set O is a C-occluder simply from visibility tests from the vertices of C (provided that the extended shadow is convex). On the other hand, theorem 1 is also true in the case of a general-shaped polyhedral cell C, due to the fact that the vertices of CH(C) are always also vertices of C and that C CH(C). We will discuss now how polylines can be used as C- occluders for objects that are potentially good occluders despite not containing any large convex set (like the object in figure 1). First we should decide what we consider that a polyline occludes. This decission is not straightforward because it is unclear what the interior of a polyline is. And we must be careful about what we define a polyline to hide because it should be consistent with our previous definitions of shadows and umbrae. More precisely, we would like the shadow of a polyline P to be the same as the shadow of any object whose silhouette happens to be P. A first intuitive approach could be considering the shadow to be the cone bounded by the semiinfinite triangles from the viewing point and through each of the polyline edges. However in the general case this cone is not well defined. In particular, it may happen that the viewpoint sees the polyline as self-intersecting, as in the case depicted in figure 4 (observe that the projection of the polyline in figure 4-b is a right-turning polyline; but it is not convex because it is not a simple polygon). The following definition generalises the concept of simple polygon to non-planar polylines. Definition 6 Let P be a non-self-intersecting closed polyline and q a point. We say that P is simple from q, or just q- simple, if every line through q intersects P at one point at most and P and q are linearly separable. Property 1 P is q-simple if and only if the projection of P from q onto a plane separating P and q is a simple polygon. Proof. Let us assume that P is q-simple. Then, any projection line from q intersects P at only one point, which means that the projection of P is a simple polygon. On the other hand, if we assume that the projection of P is a simple polygon, then all straight lines from q passing through a point of the projected polygon will intersect P at one point. Now we can define safely the extended shadow of a polyline.

5 Definition 7 Let P be a polyline and q R 3 a point such that P is q-simple. We define S * (q,p), the extended shadow of P from q, as the cone bounded by the semiinfinite triangles from q and every two consecutive vertices of P. Property 2 Let P be a polyline and q R 3 a point such that P is q-simple. Then P belongs to the boundary of the extended shadow, P δs * (q,p). The extended shadow of a polyline from a region can be defined accordingly. It is clear that the previous definitions are consistent with that of a regular set. In this case, Property 3 Should P be the silhouette of a set S from the viewpoint q, then S * (q,p) = S * (q,s). Brunet, Navazo, Rossignac and Saona-Vázquez / Hoops However, although testing view-dependent simplicity of a polyline from a single point is an easy matter, simplicity from a cell is somewhat more complicated. Besides, if we want to actually use polylines as occluders, we need tools to check for the convexity of their extended shadow from a cell. We will introduce next the convex appearance polytope that generalises polygonal convexity to non-planar polylines. Definition 8 Let P be a polyline in R 3 with vertices [p 1,..., p n] such that there are not three consecutive colinear vertices. The convex appeareance polytope of P, denoted CAP(P), is defined as the (possibly empty) open intersection of the positive half-spaces bounded by the supporting planes of every three consecutive vertices of P. n CAP(P) = H + (p i, p i 1 p i p i p i+1 ) i=1 where p 0 = p n, p n+1 = p 1 The convex appearance polytope of a simple closed polyline is shown in figure 5. The non-planar closed polyline P in figure 5-a is seen as a simple, left-turning, closed and apparently convex polyline when it is seen from a viewpoint q, as shown in figure 5-b. CAP(P) (see figure 5-c ) defines the possible locations of the viewpoint q such that P appears as a left-turning polygon in its projection from q. Property 4 CAP(P) is a convex polytope (figure 5-c ), possibly empty. In the case of P being a planar, convex polygon, CAP(P) is the positive halfspace defined by the plane supporting P. In the case of P being a planar, non-convex polygon, CAP(P) is empty Hoops Unfortunately, inclusion in CAP(P) is not sufficient by itself to ensure convexity of the shadow, and simplicity of the projection of P must also be required. We will define hoops to be polylines that are seen as convex and simple from a set of viewpoints. Definition 9 Let q R 3 be a point and P a polyline. We say that P is a q-hoop iff P is q-simple and q CAP(P). Figure 5: A 3D polyline defined on a cube (a). The polyline P appears as convex when viewed from q (b). The convex appearance polytope of P, CAP(P), is an infinite three- sided pyramid (c). Definition 10 Let C be a subset of R 3 such that it is linearly separable from P. Then P is a C-hoop iff C CAP(P) and P is C-simple (P appears as a simple polygon from all viewpoints in C). Lemma 1 Let P be a polyline that does not self-intersect and q R 3 a point. Then, P is a q-hoop S * (q,p) is convex Proof. Let Π be a plane separating P from q. As P is q- simple, Π S * (q,p) = Q is a simple polygon that is the projection of P from q onto Π. Because of q being in CAP(P), we know that every three consecutive vertices of its projection Q share the same leftturn orientation (figure 5-b). Besides, Q is simple because P was q-simple. We can conclude then that Q is a convex polygon. On the other hand, as Q is the projection of P from q, then S * (q,q) = S * (q,p). As Q is convex, so is its extended shadow, and so is S * (q,p). Corollary 1 If P is a q-hoop, then CH(P) S * (q,p) (1) Proof. Straightforward as S * (q,p) is convex and P S * (q,p). Corollary 2 If P is a q-hoop, then S * (q,ch(p)) = S * (q,p) (2)

6 Proof. It is enough to show that x S * (q,ch(p)) x S * (q,p) and that follows from equation 1. Before hoops can be used as cell occluders, we must be sure that we can test whether a polyline is a hoop for a given cell. Inclusion of the cell in the convex appereance polytope of the polyline is easy to check by testing all the vertices of C against the halfspaces defined by P s pairs of consecutive edges. As for simplicity, we will now prove that checking whether P is a hoop for a cell C can also be ensured by checking only the vertices of the cell. The proof runs by showing that hoops are the silhouettes of convex sets. Definition 11 Let O R 3 be a convex set and q R 3 a point linearly separable from it. We define V(q,O) as the convex polytope defined by intersection of the halfspaces contaning q of the supporting planes of the 2-facets of O that are visible from q. We can define also I(q,O) using the supporting planes of the invisible faces of O. Property 5 The sets V(q,O) and I(q,O) are linearly separable. In most cases, the convex set CAP(P) coincides with V(q,O) for points q in CAP(P) and O = CH(P), figure 5- c. Similarly, the convex set CAP(P ) (P being the inversion of polyline P ) also coincides with I(q,O) for points q in CAP(P ) and O = CH(P). Lemma 2 If P is a q-hoop, then P is the silhouette of the convex hull of P when viewed from q. Proof. It follows from equation 2 and from P being on the boundary of S * (q,p) that all the edges of P are either edges of the silhouette of CH(P) from q, or are on faces of CH(P) that are colinear with q. Say Π 0 separates P from q, and consider the convex polygon Q = Π 0 S * (q,p). Suppose E is a set of edges of P that lie on a face of CH(P) colinear with q and whose supporting plane is Π. The edges of E must project onto an edge of Q on the line Π 0 Π so, as P is simple, they must form a connected sub-polyline [e 1,...,e k ] of P. Moreover, this sub-polyline can have just one edge e; if there are but two of them then Π is one of the planes that define CAP(P). And this cannot be, as it would mean that q CAP(P) Π and CAP(P) is an open set. Finally, as Π CH(P) = e, the convex hull of P cannot have a face on Π, so e must be a silhouette edge. Corollary 3 If P is a q-hoop, then q is either in V(q,CH(P)) or in I(q,CH(P)). Proof. Straightforward as P must be the silhouette of CH(P) by lemma 2. We are now ready to be able to ascertain P-simplicity from a region when we know that P appears as a simple polygon from the region vertices. The following Lemma gives the necessary results for this: Lemma 3 Let P be a closed polyline and Q a set of points q 1,...,q n linearly separable from it. Then, if P is a q i -hoop for all points in Q, then it is also a CH(Q)-hoop. Proof. We will prove it by induction on k the dimension of the convex hull of Q. Let q be a point in CH(Q) and P = [p 1,..., p n]. CASE k = 2) Let q 1,q 2 be the endpoints of the segment CH(Q). q 1,q 2 CAP(P) convexity q CAP(P) By lemma 2 P is the silhouette of CH(P) from q 1 and q 2. Consider the disjoint sets V = V(q 1,CH(P)) and I = I(q 1,CH(P)), and suppose that q 1 V. Point q 2 must be either in V or in I as it also sees P as the silhouette of CH(P). But it cannot be in I, because then the orientation of q 2 and any two consecutive edges of P would be the opposite of that of q 1 and the same two vertices, contradicting that q 1,q 2 CAP(P). By convexity, point q must also be in V, and P is thus the silhouette of CH(P) when viewed from q. Therefore, P must be q-simple. CASE k > 2) Let p 1, p 2 R m be two different points on two (k 1)-facets f 1, f 2 of C such that q CH(p 1, p 2 ) = p 1 p 2. Then, P : Q-hoop induction P : { f 1, f 2 }-hoop k>k 1 P {p 1, p 2 }-hoop induction P : q-hoop k>2 The main consequence from Lemma1 and Lemma 3 is that if we have a convex cell C contained in CAP(P), we can guarantee that the extended shadow S * will be convex from any viewpoint in C, by simple tests only on the vertices of C. Now we are ready to state that the hoop property as we defined it can be used in practice to employ polylines as valid view-dependent cell occluders. Theorem 2 Let P be a closed polyline, Q a set q 1,...,q n of points linearly separable from it such that P is a q i -hoop for all i, and C = CH(Q). Then P is a C-occluder. Proof. By lemmas 3 and 1, we know that P is CH(Q)- simple and thus that S * (CH(Q),P) is a convex set. Then, P is a CH(Q)-occluder by theorem 1. The main consequence is that non-convex closed polylines can define convex shadows and can be valid cell occluders. Given a convex cell C, we must only find a closed polyline P being a C-hoop (testing if C is in CAP(P) and cheking that P is a q i -hoop for the C vertices, according to definition 10). Then, we know that P is a valid C-occluder and that its extended shadow from any viewpoint in C is convex. A simple algorithm for computing C-hoops from scene objects will be presented in the next section. C-hoops generated from a certain object O can be stored compactly, can

7 be easely integrated in visibility culling algorithms and give a good conservative visibility estimation of O at a low computational cost. 4. Hoop Synthesis Given an object O with a well-defined interior and a cell C linearly separable from it, our goal is to compute a C-hoop P that is a C-occluder and that does not hide anything that M did not. In practice, O will be a polyhedron and C an isothetic box. A first possible solution, based on T heorem 2, would be to compute a C-hoop P such that CH(P) O. However, this is not a good solution in most practical cases: in concave objects such as the one presented in figure 1, the requirement that CH(P) O produces too small C-hoops. Observing that a C-hoop P is also a q-hoop for any point q in C, it is clear from Lemma 2 and figure 5 that the projection of P coincides with the silhouette of CH(P) from any point in the cell C and that this silhouette is invariant from any point in C. This leads us to a second and better solution: we can compute a C-hoop P such that a connected surface S exists fulfilling S O and silhouette(s) = silhouette(p) from any viewpoint in C. In other words, P will be a valid C-hoop if we can find a surface S within O that exactly fills out the empty region delimited by P as seen from C. In our scheme, we have a top-down seed algorithm that uses an octree representation of the original object O. As with any seed-like algorithm, we need a way to enlarge the seed that preserves view-dependent cell occlusion properties throughout the process. The following definitions will help us to discriminate between suitable and non-suitable candidates, Definition 12 Given a cubic cell C, a surface with border S, an object O and a closed polyline P, we will say that S C-fills P within O iff S O and the silhouette of S coincides with the silhouette of P from any possible viewpoint in the cell C. Definition 13 Popping vertex. Let p be a point, P a closed polyline, C a cubic cell, and O an object with a well-defined interior such that P is a C-hoop and there exists a surface S that C-fills P within O. Then p pops P from C iff the following three conditions hold 1. The point p is outside all the eight shadows S * (q i,p), where q i are the vertices of C. 2. Let [p 1,..., p n] be the vertices of P. Then there exists another polyline Q = [p 1,..., p i k, p, p i+l,..., p n], k + l > 1 such that Q is also a C-hoop. 3. There exists a surface S such that S C-fills P within O. The whole general procedure is shown in algorithm 1. We assume that a classical octree representation has been computed from O. The octree Oct is an input parameter of the function because it is only computed once for the whole scene and is then used for every Cell Occlusion computations. We will use the silhouettes of the black nodes of the octree as seeds, for they are both convex and fully contained into the original set. As we want occluders to be as big as possible, it makes sense to consider the biggest black nodes of the octree first. Once the n biggest black nodes are selected as seeds (where n is an input parameter entered by the user), the algorithm proceeds to generate a C-hoop for each of them. Observe that a seed black node is valid only if it has the same silhouette from all the vertices from C (in the case that none of the seed black nodes is valid, we subdivide the visibility cell C). Finally, the occlusion power of the different generated C-hoops is measured by the area of a section of their umbra from C, and the C-hoop with the biggest area is selected. function Cell_Occluder(O,Oct,C,numSeeds) returns Polyline var seeds: array[1..numseeds] of node CandidateVertices: list of 3DPoint P,Poccluder: Polyline seeds:= biggest_nodes(oct,numseeds) CandidateVertices:= black_vertices(oct) Poccluder := nil for i := 1 to numseeds do P:= silhouette(c,seeds[i]) for i:=1 to size(candidatevertices) do P:= pop(candidates[i], P, Oct,C) end for Poccluder:= MaximumShadowArea(P,Poccluder) end for return Poccluder Algorithm 1: Generation of a C-hoop for an object O from a view cell C. The algorithm and the funtion pop guarantee that P is always a C-hoop with vertices within O, and that there exists a surface S that C-fills P within the octree Oct and the object O. Initially, S is a surface filling the silhouette of the seed black nodes. The black_vertices function traverses the octree and returns a list of the vertices of its black nodes such that those vertices corresponding to bigger nodes appear first. We exploit the fact that all vertices of the octree black nodes are inside the object O, and that smaller nodes are located closer to the object boundary. This fact allows an automatic progressive enlargement and refinement of the C-hoop, figure 6. The MaximumshadowArea function selects the biggest polyline between two candidates by measuring the area of a section of the umbra with a plane separating the cell from the polylines. The silhouette function first determines which faces of the seed are visible from every vertex of the cell and then returns their silhouette. The key part of the procedure is located at the f or loop that traverses the list of candidate verc The Eurographics Association and Blackwell Publishers 2001.

8 tices. The current cell occluder is enlarged by popping it with the vertices in the list of candidates. Due to the way in which this list was created, this loop is equivalent to traversing the octree level by level using the occluder from level k as the seed to generate a larger one in level k + 1. The pop function in algorithm 2 runs a battery of tests before adding the candidate vertex p to the current occluder. First, p is tested for inclusion into the umbra of o from C (this requires only a simple test because the shadow is convex, Lemma 1). If it is located inside, it is rejected as it cannot contribute to enlarge it. Next, function insertion_points classifies the supporting planes of the faces of the umbra using the signed distance to p. This classification allows to determine where should p be inserted and which existing edges of o would have to be eliminated, if any. Besides, should cell occlusion be preserved, the three new edges so introduced have to be seen as convex from C. The three calls to the function convex_edge perform this simple test and ensure that this condition is fulfilled. At this point, we know that the new polyline is also a C-hoop. The only remaining issue is the existence of a surface inside the original object. We knew such a surface to exist for the previous hoop. For each of the edges that are going to be erased, the algorithm generates a triangle formed by the two vertices of the edge and the popping vertex. If all the triangles so generated are inside the octree, we know that an inner surface exists for the new hoop. Note that a negative answer to this test does not necessary mean that no inner surface exists. Nevertheless, this test is much cheaper that an exhaustive search and our empiric results have shown it to work well in practice. function pop(p,p,oct,c) returns Polyline s:= Shadow(C,P) if outside(p,s) then < i, j >:= insertion_points(p, s,c) b 1 := convex_vertex(p[i], p,p[ j],c) b 2 := convex_vertex(p[i 1],P[i], p,c) b 3 := convex_vertex(p,p[ j],p[ j + 1],C) if b 1 b 2 b 3 then if inside(triangles(p, P[i],..., P[ j]), Oct) then o:= polyline(o[1]...p[i], p,p[ j]...p[n]) endif end if end if return o Algorithm 2: Popping algorithm Observe that the algorithm guarantees that there exists a surface S that C-fills the hoop P, without never computing it. The surface S exists for the initial hoop built on the black seed nodes, and the triangle-in-octree inside tests guarantee that S exists in every subsequent popping iteration. Figure 6 shows several steps of the hoop popping algorithm in a particular example. The figure shows a section of the occluder and its octree as seen from a viewpoint in the cell. In 6-a, the concave occluder is shown together with the black nodes of its octree representation. Using one of the largest black nodes as a seed node 6-b, we have three possible candidate vertices in the coarsest octree level. The initial hoop - silhouette of the seed black node in 6-b - is enlarged by popping vertices in 6-c, 6-d and 6-e. To enlarge this intermediate hoop 6-e, new candidate vertices from the black nodes in the next octree level 6-f must be tested for popping. Out of the five candidate vertices in 6-f, only two of them can be used to enlarge the hoop in 6-g. The final hoop would now be enlarged using candidate vertices at deeper octree levels. Hoops can be efficiently used for visibility testing, due to the convexity of their extended shadow. Assuming that P is a C-hoop, the following algorithm tests if a given point Z is hidden by P from C: function InShadow(Z,P,C) returns boolean i:= 1 InExtendedShadow:= true while InExtendedShadow and i <= P.numVertices do pl:= EnclosingPlane(P,i,C) InExtendedShadow:= InPositiveHal f space(z, pl) i:= i + 1 end while U:= Umbragon(C, P) at this point, InExtendedShadow is true iff Z belongs to the extended shadow S * (C,P) return InExtendedShadow and InPositiveHal f space(z,u) Algorithm 3: Visibility testing algorithm The function EnclosingPlane returns the oriented plane containing the i-vertex of the hoop P and its next vertex in P, such that all vertices q i of the cell C are either in pl or in its positive halfspace. On the other hand, the Umbragon U is any oriented plane usable for conservative visibility computations of the occluder P from C (for any point q such that InPositiveHalfspace (q, U), we know that q is in the shadow S(C,P) ). Possible Umbragon options include triangulations of the hoop P, the shadow of CH(P) from C, or a plane cutting the hoop shadow and passing through the farthest point of P as seen from C. In our present implementation, we use a hoop triangulation. Observe that the information associated to the hoop P (the enclosing planes and its umbragon) can be computed only once and used for all the visibility computations from the cell C. Computational Cost. The number of edges of the polyline at each step can be considered bounded by a constant. The list of candidate vertices can be generated in O(n), where n stands for the number of nodes of the octree. The number of vertices of the octree is also O(n). The first tests in the pop function take constant time and the triangle-inoctree test has O(logn) cost. So the cost of pop is O(logn),

9 (a) Concave object in red and the black nodes of its octree (b) Seed node and the three candidate vertices of level 1. (c) First popping. The occluder is depicted in blue. (d) Second popping. (e) Third popping. (f) The candidate vertices of level 2. Valid ones are in black. (g) The occluder after all vertices up to level two have been popped. Figure 6: Occluder synthesis from a concave set viewed from a cell located on a plane parallel to the sheet. and that makes the cost of the whole occluder synthesis procedure to be O(nlogn). 5. Hoop Fusion In the last year several papers have been presented dealing with occluder fusion. The goal is to compute occlusion due to the combined action of several disjoint blockers. In this section we will discuss how hoops can be merged to this purpose. Our proposal uses a property first noted by Schaufler et al. 15 and Koltun et al.. 12 An occluder A can be extended safely into the umbra of another occluder B and the resulting occluder will hide anything that was invisible due to the presence of both A and B. This procedure can be iterated with as many occluders as wanted. Our hoop fusion algorithm computes joint occlusion incrementally. Given a set of candidate occluders {O i } an initial occluder O 1 is selected first using some metric like the solid angle. Once its hoop has been synthesised in the way explained in the previous section, its umbra U 1 is computed and the scene is culled with it. The solid angle metric is used again to select another occluder O 2 among the candidates that intersect U 1 (the intersection test is easy because of the convexity of U 1 ). The isothetic bounding box B 2 of O 2 is enlarged as much as possible until it surpasses U 1. An octree representation of B 2 U 1 is then computed. This octree can then be merged easily with that of O 2. The resulting octree is then used to generate another hoop using the algorithm of the previous section. The only alteration to it is that seeds are restricted to be in the O 2 part of the octree. Again, the umbra U 2 of O 2 can be computed easily and used to cull the scene further. Note that, as U 2 is obtained from the fusion of O 1 and O 2 (see figure 7), it occludes as much as O 1 and O 2 together, except for the zone between them. This does not affect occlusion performance as any object inside this zone was previously culled by U 1. This fusing procedure is iterated until none of the remaining candidates intersects the current umbra. When this happens, the candidate with the biggest solid angle is selected from the remaining candidates in {O i }, its hoop and its umbra are computed and the fusion process starts again with the rest of candidates. The whole process ends when there are no candidates left. 6. Results We have implemented the synthesis algorithm and carried on some tests with the models of St. Paul Cathedral and a

10 Figure 7: Hoop fusion. From left to right: two concave unconnected sets; the hoop of the closest set and its umbra; the intersection of the extended box of the second set and the union of the umbra of the previous hoop and the second set; the hoop generated from the octree of the intersection. folding screen to show how our algorithm behaves with nonconvex bulky objects, figure 9 and figure 10. We have tested the algorithm from two hundred cells placed randomly on spheres surrounding the model and using one hundred seeds at each cell, resulting in hoops in total. The results of these runs on the Cathedral model are depicted in figure 9 for two specific cells. As seen in the images, the synthesised hoops are quite tight. We have observed that using a small number of hoops for a concave object gives better results at a very small computational and storage cost. If the C-hoop generation algorithm in section 4 is modified in a way that it returns the N polygons with the highest shadow area instead of simply returning the polygon with the absolute maximum, the cell visibility computations can be based on these N hoops, giving a much tighter estimation of the extended shadow of the initial occluder, figure 9. We have observed that computing and storing the best N = 5 hoops gives good results for moderately concave occluders. The figure 11 shows the first 5 hoops obtained from the view-cell. It can be observed that hoops appear as convex when seen from a viewpoint close to the cell (left image) but they are not convex from other viewpoints (right image). In order to estimate the minimum number of seeds to obtain significant results, we performed four experiments from each cell, selecting the hoop with the largest shadow area among the total of the 100 generated hoops, and the hoops with the largest shadow area among the first five, ten and fifteen generated hoops. Figure 8 shows the cell visibility ratios for the Cathedral between the first and the others. The conclusion is that a low number of seeds (of the order of 10) gives already a hoop with good cell visibility, in the range of 80 to 90 percent of the visibility obtained with the hoop generated from 100 seeds. As a second example, the algorithm has been tested on the model of a folding screen (a twisted object whose occlusion Figure 8: Improvement on the size of occluders with different number of seeds. Results with five, ten and fifteen seeds are measured against results with one hundred seeds. cannot be computed by previous approaches). The results of the hoop generation on this model are shown in figure 10, that presents the maximum shadow area hoop obtained from 100 seeds. The tests also showed that the sizes of the computed occluders are small enough. The mean and standard deviation of the number of edges of the occluders were 13 and 5 for the folding screen and 21 and 4 for the cathedral. 7. Conclusions The main contributions of our work can be summarised as follows: We have enlarged the domain of occluders by shifting convexity requirements from the candidate occluder to its shadow. This allows to improve occlusion culling by using as occluders pointsets that would have been rejected by previous approaches. For instance, objects that are thin and twisted.

11 We have introduced a fast criterion to decide whether a candidate occluder generates convex shadows from every viewpoint in a cell. This criterion can be tested in linear time with respect to the number of vertices of the occluder and the cell. A new occluder synthesis algorithm that makes use of all these theoretical results has been presented. It generates view-dependent, non-convex occluders from a concave set independently of the tessellation. Furthermore, these occluders can be stored cheaply as hoops, special polylines that preserve most of the occlusion power of the original object. We have presented also an occluder fusion algorithm to merge hoops from separate occluders thus allowing computation of occlusion caused by joint action of many disconnected sets. Hoops can be easily integrated into existing visibility algorithms for improved occlusion performance. Even most recent algorithms that do not require convexity can take advantage of hoops to increase computation speed. We consider three different areas for future work: We would like to study whether extended shadow convexity is a necessary condition for cell visibility, or if further relaxations can be achieved. We are interested also in exploring mechanisms to speed up visibility culling. For instance, by making use of occluder coherence from cell to cell by using the current occluder as a seed for the next cell. Or by computing a set of cell occluders for each object such that every cell is guaranteed to have a valid occluder in the set. And finally, we would like to expand the synthesis algorithm to polygon meshes without an interior. Figure 9: Cell occluders of the cathedral from two different cells. On the left, the eight-level octree as viewed from the cell. On the right, the cubic cell and some hoops (in red) superimposed against its silhouette. 8. Acknowledgments The fourth author has been partially supported by an FPI grant of the Spanish Ministry of Education and Science. The second author thanks Spanish Ministry of Science and Technology for supporting her summer stay at Georgia Tech University. This work has also been partially supported by the project TIC C The authors would like to thank N. Aznar, E. Monclus, M. Soler and A. Vinacua for their help on the preparation of this paper. References 1. Carlos Andújar, Carlos Saona-Vázquez, and Isabel Navazo. LOD visibility culling and occluder synthesis. Computer-Aided Design, 32(13): , November Carlos Andújar, Carlos Saona-Vázquez, Isabel Navazo, and Pere Brunet. Integrating occlusion culling with levels of detail through hardly-visible sets. Computer Graphics Forum (Eurographics 2000), 19(3): , August Figure 10: Cell occluders of the folding screen from two different cells. On the left, the eight-level octree as viewed from the cell. On the right, the best hoop (in red) superimposed against its silhouette 3. Fausto Bernardini, James T. Klosowski, and Jihad El- Sana. Directional discretized occluders for accelerated occlusion culling. Computer Graphics Forum (Eurographics 2000), 19(3): , August Daniel Cohen-Or, Gadi Fibich, Dan Halperin, and Eyal Zadicario. Conservative visibility and strong occlusion for viewspace partitioning of densely occluded scenes. Computer Graphics Forum, 17(3): , Satyan Coorg and Seth Teller. Temporally coherent conservative visibility. In Proceedings of the Twelfth Annual Symposium On Computational Geomc The Eurographics Association and Blackwell Publishers 2001.

12 etry, pages 78 87, New York, May ACM Press. 6. Satyan Coorg and Seth Teller. Real-time occlusion culling for models with large occluders. In Proceedings of the Symposium on Interactive 3D Graphics, pages ACM Press, April Frédo Durand. 3D Visibility: Analytical study and applications. PhD thesis, Université Joseph Fourier, Frédo Durand, George Drettakis, Joëlle Thollor, and Claude Puech. Conservative visibility preprocessing using extended projections. In SIGGRAPH 2000 Proceedings, pages , July Ned Greene, Michael Kass, and Gavin Miller. Hierarchical Z-buffer visibility. Computer Graphics (SIG- GRAPH 93 Proceedings), 27: , T. Hudson, D. Manocha, J. Cohen, M. Lin, K. Hoff, and H. Zhang. Accelerated occlusion culling using shadow frusta. In Proceedings of the Thirteenth Annual Symposium on Computational Geometry, pages 1 10, June James T. Klosowski and Cláudio T. Silva. Rendering on a budget: A framework for time-critical rendering. In IEEE Visualization, pages , Vladlen Koltun, Yiorgos Chrysanthou, and Daniel Cohen-Or. Virtual occluders: An efficient intermediate PVS representation. In Eurographics Workshop on Rendering, pages 59 70, June Fei-Ah Law and Tiow-Seng Tan. Preprocessing occlusion for real-time selective refinement. In Proceedings of the 1999 Symposium on Interactive 3D Graphics, pages 47 53, Carlos Saona-Vázquez, Isabel Navazo, and Pere Brunet. The visibility octree. A data structure for 3D navigation. Computers & Graphics, 23(5): , Gernot Schaufler, Julie Dorsey, Xavier Decoret, and François X. Sillion. Conservative volumetric visibility with occluder fusion. In SIGGRAPH 2000 Proceedings, pages , July Seth Teller. Visibility Computation in Densely Occluded Polyhedral Environments. PhD thesis, CS Dept., UC Berkeley, Seth Teller and Pat Hanrahan. Global visibility algorithms for illumination computations. Computer Graphics (SIGGRAPH 93 Proceedings), 27: , Seth J. Teller and Carlo H. Séquin. Visibility preprocessing for interative walkthroughs. Computer Graphics (SIGGRAPH 91 Proceedings), 25(4):61 69, July Hansong Zhang. Effective occlusion culling for the interactive display of arbitrary models. PhD thesis, University of North Carolina at Chapel Hill, Hansong Zhang, Dinesh Manocha, Tom Hudson, and Kenneth E. Hoff III. Visibility culling using hierarchical occlusion maps. Computer Graphics (SIGGRAPH 97 Proceedings), 31:77 88, August 1997.

13 Figure 11: The original Cathedral model with five hoops computed for a specific viewcell. On the left, the image viewed from a point close to the viewcell. On the right, the view from an arbitrary viewpoint. In this last case the hoops are not convex.