Grouping of Patches in Progressive Radiosity

Transcription

1 Grouing of Patches in Progressive Radiosity Arjan J.F. Kok * Abstract The radiosity method can be imroved by (adatively) grouing small neighboring atches into grous. Comutations normally done for searate atches are now alied to these grous. Grous receive energy from the environment, and can shoot energy into the environment. The radiosities of the atches in the grou are derived from the energy the grou receives. Grouing small atches reduces the number of form factor comutations, reduces aliasing effects, and imroves the convergence in rogressive radiosity. This aer resents methods for grouing small atches in rogressive radiosity algorithms with ray tracing based form factor comutations. 1 Introduction The rogressive radiosity algorithm (Cohen et al 88) comutes the global illumination of a scene by calculating the interreflection between atches. In each iteration ste of the algorithm one of the atches (the one with the most unshot energy) is selected to shoot its energy into the scene. The contribution of the shooting atch (= source atch) to an energy receiving atch is determined by the form factor, the fraction of energy leaving the source atch that reaches the receiving atch. This form factor deends on geometry and mutual visibility of these atches. The accuracy of the results of the radiosity rocess deends on the discretization of the environment into atches. Small atches give the most accurate results, but also require many form factor comutations. It is therefore imortant to choose the correct discretization. Adative subdivision methods that find the correct atch subdivision for each iteration ste, based on the relative osition and orientation of the source atch (Hanrahan et al 91; Languénou et al 92), may find an otimum between accuracy and number of form factor comutations. Surfaces close to the source are subdivided into more atches than atches far away. These methods, however, ignore that a single surface can be too small to be treated as a atch. A surface can be much smaller than the atch size threshold normally used for deciding to discretize a surface into atches. In these cases it is useful to take surfaces, that are too small and that are close to each other, together as one 'macro'- * Faculty of Technical Mathematics and Informatics, Delft University of Technology, Julianalaan 132, 2628BL Delft, The Netherlands. arjan@duticg.twi.tudelft.nl

2 atch. We will call such a macro -atch, that reresents several small neighboring surfaces, a grou, and erform the radiosity calculations for this grou, instead of for the searate surfaces. Usually, a grou consists of many small atches that form one very accurately modelled object. Examles of such detailed objects are lants, keyboards, etc., so the objects that make a scene look realistic. Use of a grou as resented in this aer shows some resemblance with the radiosity method for bummaed surfaces (Chen and Wu 88). They describe a method that takes into account the variation of the normal at a single bum-maed atch using a erturbed form factor in a hemi-cube based rogressive radiosity method. In our method we distribute the energy that a grou receives over all its atches, where we also take into account the orientations of the atches. However, there are no restrictions to the orientations of the atches in a grou, and there are no restrictions to the shae of the grou. This aer resents methods to use grouing of surfaces in ray tracing based rogressive radiosity algorithms. Section 2 shows the need for grouing. Section 3 describes how to grou atches and section 4 describes how the grouing is used during the shooting rocess. Results and conclusions are resented in section 5. 2 Motivation for grouing Two methods exist for form factor calculation based on ray tracing. The first method (Wallace et al 89) traces rays directed from the source atch to oints on all other atches, to determine whether this oint receives energy from the source. The other method (Malley 88; Sillion and Puech 89) traces rays in a cosine distribution from the source atch into the scene. So the rays are not directed to secific atches or oints in the scene. If a ray hits a atch, then the radiosity of this atch is udated. In both methods, the tracing of the ray is the most time consuming art. Fig 1. a) Object with many small atches in directed shooting, b) Aliasing on the keyboard due to insufficient samling density in undirected shooting

3 For both ray tracing based form factor calculation methods, grouing small atches has advantages. Consider the scene of figure 1. A light source illuminates a comuter from a reasonable distance. The keyboard consists of 496 quadrilateral atches. When the directed method is used, and radiosities are calculated for the vertices of the atches, then 1984 rays must be traced to calculate the form factors for all samle oints that are facing the light source, although the mutual visibility for all these samle oints will be the same. This is an enormous waste of effort. When the undirected method is used, an enormous amount of rays must be traced from the light source to avoid aliasing effects caused by inaroriate ray density (see figure 1b). Another roblem with very small atches is that they are almost never selected to be the source atch, because their unshot energy is always less than the unshot energy of the larger atches in the scene. However, the total unshot energy of a number of neighboring small atches can be quite considerable. It is therefore useful to consider atches, that are close to each other, as one shooting grou, so that they are selected together as source during the rogressive radiosity. 3 Grouing atches We define grouing as the rocess of regarding a number of surfaces that are close to each other as one atch during the radiosity rocess. Before the radiosity rocess starts, the scene for which the illumination must be calculated can be scanned for clusters of small atches that are close to each other, where small and close to each other are defined relatively to the size of the total scene. Small atches in such a cluster are groued. Grouing is referably done adatively, i.e. before each iteration ste a decision is made which atches must be groued. Patches for which the area is small comared to the distance to the source atch (so when the solid angle is small) are selected to be groued. For undirected shooting this means that atches that will not receive enough rays (when no occlusion is taken into account) are groued. Patches that have a large area or are close to the source (so have a solid angle that is larger than a re-defined threshold solid angle) are not groued and are treated as individual atches. The easiest way of grouing atches is to relace a grou of atches by its bounding box. The lanes of the bounding box act as atches like in the local environment or virtual wall techniques (Xu et al 89). The reflectance of the bounding box lanes is comuted by averaging the reflectance of the atches in the grou rojected onto the lanes. During the shooting rocess, the atches within the grou are ignored. At the end of the rogressive radiosity rocess, the bounding box lane radiosities are converted to atch radiosities for the atches within the grou. This method can be imlemented easily in a radiosity algorithm.

4 A number of disadvantages, however, makes this method less attractive. The accuracy of the method is affected by the fact that the radiosities are stored only for 6 directions (corresonding with the bounding box lanes) although the number of orientations in the grou may be much larger. So when reconstructing the atch radiosities from the grou radiosities large errors may be made. Also, when other atches, that do not belong to the grou, are artially within the bounding box of the grou, then the radiosities for these atches may be incorrect, because samle oints may be hidden by the grou lanes, although not hidden by the grou atches themselves. Another disadvantage is that the bounding box exists in the scene. When visibility between two atches is determined, and a grou is situated between these atches, then the shadow will have the shae of the bounding box instead of the grou, if no secial actions are erformed when a grou bounding box atch is hit by a ray. The bounding box aroach has too many disadvantages. It is therefore worthwhile to investigate grouing methods where radiosities are stored directly at the atches in the grou, and where no extra (bounding box lane) atches are introduced. 4 Grouing methods When a grou receives energy from a source atch, then this energy must be distributed over the individual atches in the grou, taking into account the orientation of each atch. When a grou is selected to shoot its unshot energy, then the unshot energy of the different atches in the grou must be converted to the unshot energy of the grou. To simlify these transformations we make a few assumtions: We only cluster atches in a grou when the distance of the grou to the source atch is large comared to the size of the grou; otherwise it is better to use the individual atches. This means that we can assume the distance and the direction of the atches to the source atch to be constant. A grou has a constant visibility, so all atches in the grou have the same occlusion. This assumtion has the consequence that shadow boundaries are not visible on a grou. There is no self occlusion, so atches within a grou will not generate shadows on other atches within the grou. The choice of shooting method (directed or undirected) influences the way grouing is used. We therefore give different methods for grouing in undirected and directed shooting. 4.1 Grouing in undirected shooting In undirected shooting (Malley 88; Sillion and Puech 89) a number of rays is traced from a source atch into the environment. A distribution function (e.g. cosine) determines the directions of the rays. Each ray leaving the source reresents some

5 energy E r. Normally, when a ray hits a atch, then this atch receives all the energy, and the radiosity of this atch is udated. The larger the solid angle of the atch with resect to the source oint, the more rays will hit the atch, and so the more energy the atch will receive. In our algorithm, when a ray hits a atch of a grou, we distribute the energy E r over all atches in the grou in such a way that each atch receives an amount of energy that is roortional to the solid angle between the source oint and the atch. The energy a atch receives is equal to the energy of the ray times the solid angle of the atch divided by the total solid angle of the grou. Patches not facing the source do not receive any energy contribution. The total solid angle for the grou can be calculated in two ways. We can calculate the exact geometrical solid angle for the entire grou, or we can use the sum of the solid angles of all atches in the grou. When the geometrical solid angle is used, then the illumination of the illuminated atches is calculated correctly. However, the atches that should not be illuminated because of occlusion, caused by atches within the grou, also receive an energy contribution because we do not take self-occlusion within a grou into account. Too much energy will be calculated because of overlaing solid angles (see figure 2). Therefore, the sum of the energy received by all atches in the grou will be larger than the energy of the ray E r. When later on the grou is selected to shoot its energy, too much energy is shot into the environment. Fig 2. Overlaing solid angles When the sum of the solid angles of all atches in the grou is used, then the total energy that is assigned to the grou is correct. Overlaing solid angles are taken into account. However, the total energy is sread over all atches, so the energy of the really illuminated atches will be lower then exected, resulting in darker atches. Both calculation methods for the solid angle of the grou are not otimal. We suggest the following solution. The geometrical solid angle is used for calculation of the radiosities (that are used for dislay), so the shading for the really illuminated atches is correct. The sum of solid angles is used for calculation of the unshot radiosities of

6 the atches in the grou, so that the energy that is shot from the grou in later iteration stes is correct. Therefore the energy a atch in a grou receives is where E w = w E max(0,ω ) = ω r g w = weight of atch within grou = solid angle formed by source oint and atch g = solid angle formed by source oint and grou The solid angle from a source oint and a atch is given by where ω = A cosθ r 2 = angle between atch normal (n ) and direction to source A = area of atch r = distance between source oint and atch Given the assumtions that the distance of the source to all the atches in the grou is constant and that the direction from all atches in the grou to the source is constant (= d g ), the solid angle can be aroximated with cosθ A ( dg n ) ω = = 2 2 r r So the weight of a atch is equal to w = max( 0, A w ( d n g g )) w g = total weight of the grou During the shooting rocess a grou can be selected to shoot its energy, because its unshot energy is larger than the unshot energy of the non-groued atches and of the other grous. From the selected grou we shoot as many rays as we would do normally when a atch is selected to shoot its unshot energy. A decision now has to be made from which atches in the grou the rays will be shot into the environment. We use a method in which the number of rays that originate from a atch is roortional to the unshot energy of this atch. The larger the unshot energy of the

7 atch, the more rays originate from it. The number of rays shot from a atch in the grou R g is therefore where R g = E E g R E = unshot energy of atch in grou E g = total unshot energy of grou = E R = number of rays normally shot from a non-groued atch With this method, most rays are shot from the most radiant arts of the grou. This method also allows diffuse interreflection within a grou. 4.2 Grouing in directed shooting Directed shooting (Wallace et al 89) from a source starts with subdividing the source atch into delta areas. The form factor of each delta area with resect to each of the receiving vertices of the atches in the environment is comuted. The visibility is tested by tracing a ray from the center of the delta area to a vertex. Fig 3. Visibility determination between source oints and grou When the contribution of the source to a grou must be calculated, we use the following method. For each oint (center of delta area) on the source atch we determine one or a few random oints on the grou. Points are only chosen on atches that face the source. Rays are traced to determine the visibility between the source oint and these oints on the grou (intersections of the ray with atches of the grou can be ignored). The visibility of the grou from a source oint is given by the fraction of rays that reach the atches of the grou (figure 3). The radiosities of each vertex in the grou can now be calculated using the form factor from the source atch

8 to the vertex given the estimated visibility of the grou. Note that the radiosities of the individual atches are now calculated correctly because they are not based on E r, contributions as in undirected shooting, but on an exact form factor calculation. When a grou is selected to shoot its energy, a number of source oints must be determined on the grou, just as is done for a source atch. We choose oints on the boundary of the grou. Each of these oints has a direction ointing out of the grou. The directions must have a good distribution, so that all directions can be reresented with the chosen directions. First, the unshot energy of the source oints is calculated from the unshot energy of the atches of the grou. The unshot energy E i, of source oint i, due to a atch is given by (see figure 4) where E i = max(0,cosθ i ) E max(0,cosθ ) n j= 1 j = max(0,( ni n )) E max(0,( n n )) i = angle between shooting direction (n i ) and atch normal (n ) E = unshot energy of atch n j= 1 n = the number of chosen source oints j Fig 4. Calculation of atch contribution to source oint energy Just like the weight calculation in undirected shooting, we must comensate for the fact that some atches in the grou may be occluded from the source oint by other atches, and so will not contribute to the source oint. E i is therefore reduced by a self-occlusion factor. Instead of many source atches reresenting the grou, now only n shooting oints exist. Each virtual atch of a source oint sends its energy into the environment as a normal source atch. A drawback of this method is that no interreflection between the atches within a grou is calculated, because energy is shot from the boundaries of the grou, so no energy ends u in this grou. This can be solved by erforming a normal directed

9 shooting for atches within the grou, where each atch in the grou is successively selected to shoot its energy, but only to the other atches in the grou. 5 Results and conclusion The described grouing methods are imlemented in our rogressive radiosity rogram. For the moment the grouing of surfaces is done manual. However, adative grouing can be done. When a atch in a grou is large enough with resect to the osition of the source, then this atch will temorarily not be art of the grou and will be treated as a searate atch. Reconsider the scene of figure 1. Grouing methods were tested both for the directed and undirected shooting. The results of directed shooting are shown in figure 5. The left icture is made without grouing. The number of rays to be traced in each iteration ste to determine the form factors for the keyboard is 1984 (= the number of vertices on the keyboard). For the right icture, the atches of the monitor (excet the screen atch) and the atches of the keyboard were groued into two grous. For these grous we can choose the number of visibility tests ourselves. We used only one visibility test. This did not affect the quality of the image, excet for small shadow details caused by occlusion within a grou, e.g. near the lower right corner of the screen. Figure 6 and 7 show the results for undirected shooting. Although the number of rays originating from the light source that hit the keyboard is only 691 (for figure 6) and 7464 (for figure 7) no aliasing effects are visible when grouing is used (right ictures). Even for a smaller number of rays the shading of the keyboard will be smooth, because the energy of a ray that hits the grou is distributed over all atches in the grou, instead of only over the atch that is hit. When the atches are groued, the robability that the grou is hit, and so that all atches receive an equal contribution, increases considerably. To converge to a solution, the tests that used grouing required only 4 iterations, before a re-defined sto criterion was reached. Without grouing resectively 30, 44 and 43 iteration stes were needed for figures 5 to 7. Grouing small atches is a useful extension for the rogressive radiosity method. It reduces the number of rays for ray tracing based form factor comutation. The use of grous in an undirected shooting method reduces the aliasing caused by an inaroriate resolution of the shooting rays. The number of shooting rays can be reduced. Adative grouing gives an otimum between accuracy and efficiency. Adative grouing fits well in an adative discretization method (Languénou et al 92). When surfaces are too small to discretize, and even too small to act as individual atches, for instance because they are too far away from the source atch and will not be hit by enough rays, then they can be combined into a grou.

10 Using a grou as a source for an iteration in the rogressive radiosity algorithm makes the radiosity rocess converge faster. Some roblems remain to be solved. We need a method for automating the grouing, as now atches are assigned to grous manually. Another roblem is that shadows generated by obstruction of other atches as well as from atches within the grou are not taken into account because a constant visibility for the whole grou is assumed. Areas that should be in shadow are illuminated. The total energy that the grou receives and reflects, however, is correct. Thus, for dislay the shading of the grou is not comletely correct, but the error for the indirect illumination of other atches is negligible. Acknowledgements Many thanks to Erik Jansen and Theo Verelst for the discussions about grouing methods. Li-Shen Sheng made the Macintosh model used for testing. References Chen, H., Wu, E.H. (1990), An Efficient Radiosity Solution for Bum Texture Generation, Comuter Grahics 24(4): , Siggrah 90. Cohen, M.F., Chen, E.S., Wallace, J.R., Greenberg, D.P. (1988), A Progressive Refinement Aroach to Fast Radiosity Image Generation, Comuter Grahics 22(4): 75-84, Siggrah 88. Hanrahan, P., Saizman, D., Auerle, L. (1991), A Raid Hierarchical Radiosity Algorithm, Comuter Grahics 25(4): , Siggrah 91. Languénou, E., Bouatouch, K., Tellier, P. (1992), An Adative Discretization Method for Radiosity, Comuter Grahics Forum 11(3): , Eurograhics 92. Malley, T.V. (1988), A Shading Method for Comuter Generated Images, Master's Thesis, Univ. of Utah, June Sillion, F., Puech, C. (1989), A General Two-Pass Method Integrating Secular and Diffuse Reflection, Comuter Grahics 23(3): , Siggrah 89. Wallace, J.R., Elmquist, K.A., Haines, E.A. (1989), A Ray Tracing Algorithm for Progressive Radiosity, Comuter Grahics 23(3): , Siggrah 89. Xu, H., Peng, Q.S., Liang, Y.D. (1989), Accelerated Radiosity Method for Comlex Environments, Eurograhics 89,

11 Fig 5. Directed shooting; left without grouing, right with grouing Fig 6. Undirected shooting, 10 5 rays er shooting oint; left without grouing, right with grouing Fig 7. Undirected shooting, 10 6 rays er shooting oint; left without grouing, right with grouing