Efficient Algorithms for Computing Conservative Portal Visibility Information

Transcription

1 EUROGRAPHICS 2000 / M. Gross and F.R.A. Hogood (Guest Editors) Volum9 (2000), Number 3 Efficient Algorithms for Comuting Conservative Portal Visibility Information W. F. H. Jiménez, C. Eserança and A. A. F. Oliveira L.C.G. - Laboratório de Comutação Gráfica, COPPE - Sistemas / UFRJ, Rio de Janeiro, Brazil Abstract The number of olygons in realistic architectural models is many more than can be rendered at interactive frame rates. Tyically, however, due to occlusion by oaque surfaces (e.g., walls), only small fractions of such models are visible from most viewoints. This fact is used in many oular methods for rerocessing visibility information which assume a scene model subdivided into convex cells connected through convex ortals. These methods need to establish which cells or arts thereof are visible to a generalized observer located within each cell. The geometry of this information is termed a visibility volume and its comutation is usually quite comlex. Conservative aroximations of viewing volumes, however, are simler and less exensive to comute. In this aer we resent techniques and algorithms which ermit the comutation of conservative viewing volumes incrementally. In articular, we describe an algorithm for comuting the viewing volumes for a given cell through a sequence of m ortals containing a total of n edges in O m nµ time.. Introduction Realistic scenes such as architectural models of furnished buildings may consist of several million olygons. They tyically contain large connected elements of oaque material (e.g., walls), so that from most vantage oints only a small fraction of the model can be seen. A visibility algorithm aimed at reducing the number of rimitives rendered can exloit this roerty. Following rior work 2 6 0, we make use of a satial subdivision scheme that divides such models along the occluding rimitives into cells and ortals. A cell is a convex olyhedral volume of sace; a ortal is a transarent convex 2D region on a cell boundary that connects adjacent cells. Cells can only see other cells through ortals. The ortals are stored in a list along with the identifiers for the two neighboring cells to which the ortal leads, and both cells have a reference to the ortal. In this way we are constructing an adjacency grah in which two cells (vertices) are adjacent (share an edge) if and only if there is a ortal connecting them. It is frequently convenient to comute Visibility information offline, so it can be associated with each cell for later use in an interactive rendering hase. Given a generalized observer, i.e., an observer constrained to lie within the cell, but free to move anywhere inside it and to look in any direction, the visibility information can be categorized into cell-to-cell, cell-to-region and cell-to-object visibility, corresonding to the set of cells, the viewing volume, or the set of objects which are visible to that observer. Comuting the viewing volume also known as the antienumbra for a sequence of ortals means to characterize the volume illuminated by the light source (i.e., the first ortal in the sequence) within the target cell (i.e., the cell reached through the last ortal in the sequence). Clearly, a oint belongs to the viewing volume if and only if it belongs to a line stabbing the sequence. Comuting this volume for a sequence of a single ortal is trivial: the antienumbra in this case is the entire halfsace beyond the ortal, intersected with the reached cell which is, of course, an immediate neighbor of the source cell. Since we are only considering convex cells, then this intersection yields exactly the volume of the reached cell. Comuting the antienumbra of ortal sequences of length two or more involves interactions among vertices and edges of different ortals and requires both linear and quadratic imlicit rimitives (line swaths) to correctly describe the illuminated volume. Teller 0 found an analytic solution for this ortal-ortal visibility roblem but his aroach is mathematically and comutationally comlex, requiring hours of rerocessing time for large modc The Eurograhics Association and Blackwell Publishers Published by Blackwell Publishers, 08 Cowley Road, Oxford OX4 JF, UK and 350 Main Street, Malden, MA 0248, USA.

2 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information els and it is not yet sufficiently robust for use on comlex models. Consequently, he develoed a simler algorithm that comutes a olygonal volume which conservatively estimates the exact visible region. In other words, this volume, which we will refer to henceforth as a conservative viewing volume is guaranteed contain the exact viewing volume. In this aer we describe a new aroach for simultaneously comuting the conservative aroximations of cell-tocell and cell-to-region visibilities for each cell of the subdivision. This aroach is simler than that described by Teller 0 and, at the same time, makes it ossible to comute the visibility information for the whole scene incrementally. It uses an adatation of the Visibility Skeleton 3 to encode all ossible visibility changes of a sequence of ortals into a grah structure. The Visibility Skeleton is a owerful tool which can be used to solve numerous different roblems which require global visibility information; it is easy to build and it is well-adated to on-demand or lazy construction. The central comonents of the Visibility Skeleton are line swaths sets of critical lines and extremal stabbing lines, which are the foci of all visibility changes in a scene. All modifications of visibility in a olygonal scene can be described by extremal lines, and a set of line swaths which are necessarily adjacent to these lines. We resent a descrition of our adatation of the Visibility Skeleton in Section 2. In Section 3 we describe in detail: an algorithm to comute the viewing volume for two ortals, an algorithm to comute a conservative aroximation of the viewing volume for three ortals and, finally, an algorithm that handles sequence of four or more ortals. In Section 4 we show the results obtained by alying these to a single synthetic scene. Our conclusions and suggestions for further work are resented in Section The Portal Visibility Skeleton The Visibility Skeleton (VS) devised by Durand, Drettakis and Puech 3 is a tool for solving many different roblems which require global visibility information. The Portal Visibility Skeleton (PVS) introduced here borrows the idea of encoding visual events into a grah structure, but whereas the VS tries to handle general olyhedral scenes, the PVS focuses on ortals and cells. Not all of the visual events encoded by VS and by PVS are the same. Thus, some visual events exist in VS but not in PVS and vice-versa. Also, some visual events are encoded in a different way. Many global visibility algorithms have been roosed in the recent literature, in articular those which comute asect grahs 2, antienumbra 9 or discontinuity 5 meshing 4. In many of those, visibility changes, also known as visual events, have been characterized by critical line sets or line swaths and by extremal stabbing lines. Following the rocedure used for the Visibility Skeleton 3, we encode these visual events in a grah structure that can describe comletely all ossible visibility relationshis along a sequence of ortals. The nodes of the grah structure reresent extremal stabbing lines while arcs corresond to line swaths. In this aer, we do not describe the PVS in full, but only those arts of the PVS which are needed for the analysis of interactions between airs of ortals. As is exlained in Section 3, the conservative antienumbra estimations for arbitrary sequences of ortals can be derived incrementally from such airwise analysis. 2.. Extremal stabbing lines Following rior work 3 8 9, we define an extremal stabbing line to be incident on four ortal edges. Although the Visibility Skeleton defines a host of extremal stabbing line tyes, we must consider only two: vertex-vertex (or VV) lines, and ortal-vertex-vertex (PVV) lines. Notice that this simlification is ossible because our algorithms will examine only interactions between airs of ortals. Thus, for instance, extremal stabbing lines that touch four edges (tye EEEE) are not exlicitly comuted. The VV lines (reresented by VV nodes in the grah structure) corresond to the interaction of two vertices (see Figur). The PVV lines (reresented by PVV Nodes) corresond to the interaction of two vertices and, each lying on a different ortal, but both on the same lane as one of the two ortals, which we refer to as. Without loss of generality, we assume that is one of the vertices of. There are five ossible configurations for PVV lines, deending on the relative osition of the line with resect to the vertices of (see Figur). 2e v Figur: A VV node is adjacent to four EV arcs defined by a vertex and edges of the other vertex: and and and Line swaths A swath is the surface swet by extremal stabbing lines when they are moved after relaxing exactly one of the four edge constraints defining the line. In general, a swath can either be lanar (if the line remains tight on a vertex) or a regulus, whose three generator lines embed three edges. Once again, we can simlify our discussion of the Portal Visibility c The Eurograhics Association and Blackwell Publishers 2000.

3 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information e v v v (a) (b) ' e2 ' (c) (d) (e) Figur: A PVV node. In (a) and (b) vertex is on the same lane as ortal. Unlike the case deicted in (b), in (a) all vertices of are on the same closed sub-lane defined by line. In (c), (d) and (e) edge is on the same lane as ortal. In (c) and (d) but not in (e) all vertices of are on the same closed sub-lane defined by. Finally, in (c) edge and are on the same closed sub-lane defined by, and in (d) they are on different sub-lanes. Skeleton by considering only lanar swaths, since no stabbing lines embedding three edges will ever occur between two ortals. In Figur, we see the three ossible arc tyes: an EV line swath (a), a PV line swath relating a ortal to one vertex of another ortal (b), and a PE line swath relating a ortal to an edge of another ortal (c). It should be clear that PV lines occur when the ortal is on the same lane as a vertex of the other ortal, and PE lines occur when the ortal is on the same lane as some edge of the other ortal. In the uer art of each figure we show three views (with changes in visibility), as seen from a viewoint located above the scene and, in front of, on, or behind the line swath. 3. Comuting the Conservative Viewing Information We make the following assumtions: the inut is a list of m oriented convex olygons (ortals), P... P m. The first olygon P is the light source, and all others are holes. The olygons are disjoint, and ordered in the sense that the negative halfsace determined by the lane of P i contains all olygons P j, i j m. Thus, an observer looking along a stabbing line from the light source would see the vertices of each olygon arranged in counterclockwise order. We assume that each ortal P i is laced between cells C i and C i. Thus, it is clear that we may initialize the cellto-cell visibility information of cell C i with links to C i and C i, for all i m. The cell-to-cell information of cells C and C m is initialized with links to C 2 and C m, resectively. Given two consecutive ortals P i and P i, as we are only considering convex cells, then cell C i 2 is directly visible from source cell C i. Thus, we can add cells C i 2 and C i 2 to the set of cells that form the cell-to-cell visibility information of cell C i, for all 2 i m. Also, we can add cells C 3 and C m 2 to the cell-to-cell information of cells C and C m, resectively. c The Eurograhics Association and Blackwell Publishers 2000.

4 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information v e v e ve v e (a) (b) (c) Figur: (a) While the eye traverses line swath ve, vertex v crosses the edge e. (b) In front of swath v we see the front side of, on the swath we see a line, and behind we see the other side of. (c) The swath e is similar to the v case. In general, for cells C i and C i k to be visible to each other, for k 2, there must be at least one stabbing line through ortals P i, P i... P i k. In other words, mutual visibility between cells C i and C i k requires the viewing volume for the ortal sequence P i, P i... P i k to be non-null. Cell-to-object visibility information can be easily comuted by first determining in which cell each object lies and, if that cell is visible from the source cell, then the object is visible only if it falls within the viewing volume for the sequence of ortals connecting the two cells. All it remains now is to be able to comute the viewing volume (antienumbra) for ortal sequences of arbitrary length. There is a key observation shown by Teller 0 which is imortant for our algorithm: if we consider any air of ortals P i, P j, i j m and any edge e on one of them (Figur), then edge e uniquely defines a searating lane such that P i and P j lie in different halfsaces. This searating lane is sanned by edge e, and some vertex v on the other ortal. Clearly, one halfsace of this lane (the halfsace containing the ortal P j ) must contain the antienumbra of any ortal sequence containing this ortal air. As shown in Section 2.2, edge e and vertex v define a swath, which will be called a searating swath. Similarly, its corresonding arc in the grah structure will be called a searating arc. First we resent an efficient algorithm to comute the viewing volume for two ortals, next the algorithm to comute the conservative aroximation of the viewing volume for three ortals, and finally, the algorithm to comute a conservative aroximation of the viewing volume for a sequence of four or more ortals. 3.. The viewing volume for two ortals In order to comlete the visibility information for this case, we need only to comute the cell-to-region viewing volume for the ortal sequence P,. It should be noted that when only two ortals are involved, the viewing volume will be bounded by searating lanes only, and thus the rocedure described below will actually comute the exact viewing volume. We begin with the simlest case, that is, when no vertex of one ortal is on the same lane as the other ortal. Here, all extremal stabbing lines are Vertex-Vertex lines and, in the grah structure, each node has four adjacent arcs reresenting VE swaths, as shown in Figura. Consider the edges of every ortal as directed segments, following the order of its vertices, that is, a counterclockwise order as shown in Figurb. Then we can imose this direction on every swath defined by the edges and on their corresonding arcs in the grah structure. In Figurb the directed searating swaths and their corresonding directed searating arcs in the grah structure are drawn in solid lines. These searating swaths form the border of the viewing volume, and the nodes in the grah structure on which they are incident reresent the Vertex-Vertex extremal stabbing lines. We call these nodes the bounding nodes of the viewing volume. c The Eurograhics Association and Blackwell Publishers 2000.

5 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information e Figur: The lane through edge e, searating ortals P and. v P i P j the aroriate vertex of the first ortal would take O n µ time. Since each edge of both ortals uniquely defines a searating swath with some vertex on the other ortal and is art of the viewing volume, then the total number of searating swaths that form the viewing volume is O n n 2 µ. In the second art of the algorithm, each time a bounding node is reached, we need to comute two swaths and check which one is the searating swath. Thus, the total time to identify the searating swaths is O 2 n n 2 µµ, which is also the total worst-case time comlexity of the algorithm. Note that this is considerably better than the O n n 2 µ 2 µ algorithm roosed by Teller 0. Finally, we have to consider the cases where a vertex (Figure 6a) or an edge (Figure 6b) of one ortal lies on the same lane as the second ortal. In such cases, the traversal of the grah is similar, but when a PVV node is reached through an VE arc, we must choose the PV or PE arc leaving from it in order to reach the next PVV node. Comuting the viewing volume for a sequence of two ortals means to find, in the grah structure, all the bounding nodes and all the searating arcs. We can do that by finding one bounding node and then following the directed searating arcs to reach the other bounding nodes. Note that each node has two incoming and two outgoing arcs. This means that when a given node is reached, we need to decide which of the two outgoing arcs is a searating arc. If both are searating arcs, then we can choose either. Our algorithm has two arts. The first art consists of choosing the first searating arc. For this, we ick any edge of one ortal and find a searating swath defined by it and some vertex on the other ortal. This searating swath corresonds to the first searating arc to be traversed by the algorithm and is stored in a list. The second art of the algorithm consists of reeatedly finding another searating arc which leaves the bounding node reached by the last arc in the list. When the first bounding node is reached again, the algorithm stos, and the list contains the comuted antienumbra. Let n and n 2 be the number of edges of the two ortals. Since any edge defines a searating arc, then the first art of the algorithm entails only searching for an aroriate vertex. Given an edge e of the first ortal and a vertex v i of the second ortal, then e and v i define a searating lane if and only if v i and v i both lie on the same side of the lane but all remaining vertices of the first ortal (which are not endoints of e) lie on the oosite side. This can be determined merely by testing v i and v i, since the relative ositions of the vertices of the first ortal with resect to e are known in advance. Thus, even if all vertices of the second ortal need to be tested, the first art of the algorithm can be erformed in O n 2 µ time. Of course, we could have chosen an edge of the second ortal, in which case locating 3.2. The conservative viewing volume for three ortals The rationale used in the algorithm to comute the conservative viewing volume for three ortals can be better understood by referring to the two-dimensional examle shown in Figure 7. In our discussion, we denote by V P i P j (or V P j P i ) the exact viewing volume formed by the lines stabbing ortals P i and P j. Thus, in Figure 7a, P and defines a exact viewing volume called V P (or V P ). The exact viewing volume V P is formed by all lines stabbing ortals P and and, if some of these lines stab ortal as well, then they form the conservative viewing volume for ortals P, and, which we will term V P. All lines that form V P stab ortals and and are a subset of V. Similarly, all lines that form V P stab ortals P and and are a subset of V P. Thus we can say that V P is formed by the intersection of V P, V and V P, (Figure 7b). If art of falls outside V P then the lines of V P will stab only the art of that falls within V P (Figure 7b). We denote this art of as ortal P3. Thus, V P and V P P3 are the same volume (Figure 7c). Similarly, if art of P falls outside V P3 then the lines of V P P3 will stab only the art of P that falls within V P3 (Figure 7c). This art of P is a ortal denoted by P. Thus, V P P3 and V P P3 are the same volume (Figure 7d). We also notice that V P and V P P3 are the same volume. Finally, we see that V P P3 and V P P 3 are the same volume (Figure 7d), that is, V P P 3 V P V P P 3 V P 3 V P P 3 In order to understand this, notice that since P3 is totally c The Eurograhics Association and Blackwell Publishers 2000.

6 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information e 2 v 2 e 3 v5 v6 v3 e 6 v v v 6 6 6v4e e 6 v2 e 6 (a) Figur: (a) Extremal stabbing lines, swaths and grah structure for two ortals. (b) The directed swaths and directed grah structure. (b) included in V P, then V P P 3 is a subset of V P. Similarly, since P is totally included in V P 3, then V P P 3 is a subset of V P 3. This way, we may comute the conservative aroximation of the viewing volume for the sequence of three ortals P, and as though we had only two ortals: P and P 3. The algorithm is described schematically below. We use the following notation: Volume ViewVolume3 P i P j P k µ is the function for comuting the conservative viewing volume for ortals P i, P j and P k. Volume ViewVolume2 P i P j µ is the function for comuting the exact viewing volume for ortals P i and P j, as described in Section 3.. Portal IntersectVolumePortal V Pµ denotes a function to comute ortal P clied by viewing volume V, that is, P V. The algorithm can be best understood by referring to a simle examle in two dimensions as shown in Figure 7. It consists of the following stes: Volume ViewVolume3 P µ begin V P ViewVolume2 P µ P 3 IntersectVolumePortal V P µ V P 3 ViewVolume2 P 3 µ P IntersectVolumePortal V P 3 P µ V P P 3 ViewVolume2 P P 3 µ return V P P 3 end It should be noted that, in ractice, the first ste of the algorithm is seldom necessary since the value of V P will be available from revious comutations. The time comlexity of the algorithm is O nµ, where n is the total number of edges in P, and. This can be inferred by noticing that: i. The time comlexity of function ViewVolume2 is linear on the number of edges of its arguments. ii. Function IntersectVolumePortal can be imlemented in a manner that is analogous to any algorithm for comuting the intersection of convex olygons. The comlexity of this roblem was shown to be 7 O n n 2 µ, where n and n 2 are the number of edges of the oerand olygons. iii. The comlexity of conservative viewing volumes, i.e., the number of searating swaths that form them, was shown to be not greater than twice the total number of edges in the ortal sequenc0. This is due to the fact that each edge can contribute with no more than 2 searating swaths. iv. Since the comlexity of the final viewing volume V P P 3 is no greater than 2n, then ortals P and P 3 will sum no more than 2n edges. Since all functions used in the algorithm have time comlexity linearly deendent on the number of edges/swaths of their arguments, and given that all arguments have comlexity no greater than 2n, then we may safely assume that the algorithm is linear on n The conservative viewing volume for sequences of four or more ortals Now we comute the conservative aroximation of the viewing volume for a sequence of m oriented ortals P... c The Eurograhics Association and Blackwell Publishers 2000.

7 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information v e2 e 6 e 6 e 6 v6 e 6 (a) Figure 6: Directed swaths and directed grah structure when (a) vertex is on the same lane as ortal, and (b) edge is on the same lane as ortal. (b) P m, m 3. We do this in a rogressive way, that is, once we know the conservative viewing volume for the sequence P... P i, the conservative viewing volume for the sequence P... P i may be comuted with one more call to function ViewVolume3. Given the first two ortals P and, we can comute the viewing volume V P as described in Section 3.. Next, when ortal is added, we use the rocess described in Section 3.2 to comute the conservative aroximation of the viewing volume for this sequence of three ortals, which is the volume V P P 3. When the fourth ortal P 4 is added, we comute the conservative aroximation of the viewing volume for this sequence of four ortals as though we had only three ortals: P, P 3 and P 4, as follows: V P 4 ViewVolume3 P P 3 P 4µ The remaining ortals can be rocessed in the same way. Thus, in general, we may write: 2µ V P m Pm 3µ ViewVolume3 P m P m Pmµ µ The algorithm is resented schematically below: Volume ViewVolume P P mµ begin for i 2 to m do if i 2 then V P ViewVolume2 P µ else if i 3 then V P P 3 ViewVolume3 P µ else 2µ V P i Pi end if end for 2µ return VP m Pm end 3µ ViewVolume3 P i P i P iµ µ 2µ Notice that the resulting viewing volume V P m Pm will contain no more than 2n searating swaths 0, where n is the total number of edges in ortals P... P m. It is safe to say 3µ that ortals P m and P m µ may have not more than 2n edges each. Thus, the last call to function ViewVolume3 will send O nµ time. We may safely assume that all other calls to functions ViewVolume2 and ViewVolume3 are also bounded by O nµ. Overall, m such calls are erformed and thus we infer that the worst case time comlexity of the algorithm is bounded by O m nµ. Observe that this is, in most cases, better than the algorithm roosed by Teller which runs in O n 2 µ time. 4. Imlementation and construction statistics The algorithms described above has been imlemented in C++ and alied to a synthetic test scene comrising 72 square cells arranged in a 8 by 9 rectangular array (see Figc The Eurograhics Association and Blackwell Publishers 2000.

8 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information V VP VP P VP VP VP P (a) (b) V VP V VP VP VP VP VP P P P (c) Figure 7: (a) P and define a viewing volume called V P (or V P ). (b) V P is formed by the intersection of V P, V and V P. (c) V P P3 and V P are the same volume, (d) V P P 3 and V P P3 are the same volume. (d) ure 8). All adjacent cells in the scene are connected by convex olygonal ortals ranging from 3 to 0 edges. The alication comuted cell-to-cell and cell-to-region visibility information for all cells and all ortal sequences of the test scene in roughly 28 seconds on a Pentium-II-based comuter running at 300MHz. A total of 5772 viewing volumes were found, i.e., the conservative cell-to-cell visibility grah for the scene contains that number of edges. Detailed statistics are shown in Tabl. Although this simle test scene cannot be regarded as a tyical architectural model, the obtained results seem to confirm our estimation that the roosed algorithm comutes the conservative viewing volume for sequences of m ortals in time which varies linearly with m. In fact, since the average comlexity of the viewing volumes does not vary overmuch in the scene, the average time required to rocess another ortal in a sequence (i.e., calls to ViewVolume3) also remains fairly constant. We believe that this situation is not uncommon in most tyical scenes, that is, as a sequence of ortals grows longer, the comlexity of the resulting conservative viewing volume tends to stabilize. As a consequence, for such scenes, the total time required to comute the conservative antienumbra for a sequence of ortals will vary linearly with the length of the sequence. This is evidenced grahically in the chart resented in Figure 9. Figure 8: A simle test scene. The viewing volumes of all ortal sequences starting at the red cell are shown. c The Eurograhics Association and Blackwell Publishers 2000.

9 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information Sequence Number Avg. total Avg. comlexity of Avg. time to Avg. total time length of number of antienumbra add one ortal for comuting (ortals) sequences ortal edges (number of swaths) to sequence (s) sequences in sequence antienumbrae (s) Tabl: Results obtained by our samle imlementation when alied to the test scene shown in Figure Conclusions We have resented several techniques and algorithms that make ossible the comutation of conservative visibility information for scenes comosed of convex cells and ortals. We introduced the Portal Visibility Skeleton, an adatation of the Visibility Skeleton roosed by Durand et al. 3 and used it to solve the roblem of comuting the antienumbra for a sequence of two ortals. We resented an algorithm for comuting conservative antienumbrae for ortal sequences of arbitrary length which has time comlexity that is no worse than reviously known aroaches and, in some cases, it fares considerably better. We are currently investigating the ossibility of comuting the conservative visibility volume for sequences of three ortals using the PVS. Although the asymtotic comlexity of the algorithm should remain the same, we hoe to obtain a faster algorithm than the one resented herein. We also lan on using the PVS to comute exact visibility volumes. References. A.J.Steward and S.Ghali. Fast comutation of shadow boundaries using satial coherence and backrojections. Comuter Grahics : Orlando. SIG- GRAPH 94 Proc. July Time (s) 0,040 0,035 0,030 0,025 0,020 0,05 0,00 0,005 0, Sequence length (Portals) Avg.time to add ortal to sequences (s) Avg. total time for comuting sequence antienumbra (s) Figure 9: Average times taken by antienumbra comutations as a function of the sequence length. 2. C.B.Jones. A New Aroach to the Hidden Line Problem. The Comuter Journal 4(3): 232. August F.Durand, G.Drettakis, and C.Puech. The Visibility Skeleton: A Powerful and Efficient Multi-Purose Global Visibility Tool. Turner Whitted. Comuter Grahics : Los Angeles, Addison Wesley. Anc The Eurograhics Association and Blackwell Publishers 2000.

10 Jiménez, Eserança and Oliveira / Conservative Portal Visibility Information nual Conference Series. SIGGRAPH 97 Proc. August , 9 4. G.Drettakis and E.Fiume. A Fast Shadow Algorithm for Area Light Sources Using Backrojection. Comuter Grahics : Orlando. SIGGRAPH 94 Proc. July H.Plantinga and C.R.Dyer. Visibility, Occlusion, and the Asect Grah. International Journal of Comuter Vision 5(2): J.Airey. Increasing Udate Rates in the Building Walkthrough System with Automatic Model-Sace Subdivision and Potentially Visible Set Calculations. TR # UNC-CH CS Deartament. Ph.D Thesis. July J.O Rourke. Comutational Geometry in C. Cambridge University Press M.Pellegrini. Stabbing and Ray Shooting in 3- Dimensional Sace. Proceedings of the Sixth Annual Symosium on Comutational Geometry : Jun S.J.Teller. Comuting the antienumbra of an area light source. Comuter Grahics 26(4): Chicago. SIGGRAPH 92 Proc. July 992a S.J.Teller. Visibility Comutation in Densely Occluded Polyhedral Environments. TR #92/708. UC Berkeley CS Deartment. Ph.D Thesis. 992b., 2, 4, 5, 6, 7. Z.Gigus, J.Canny, and R.Seidel. Efficiently comuting and reresenting asect grahs of olyhedral objects. IEEE Trans. on Pat. Matching & Mach. Intelligence 3(6). Jun Z.Gigus and J.Malik. Comuting the asect grah for the line drawings of olyhedral objects. IEEE Transactions on Pattern Analysis and Machine Intelligence 2(2): February c The Eurograhics Association and Blackwell Publishers 2000.