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Improving Form Factor Accuracy by Projecting Onto the Hemisphere Base

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Improving Form Factor Accuracy by Projecting Onto the Hemisphere Base View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace at University of West Bohemia Improving Form Factor Accuracy by Projecting onto the Hemisphere Base Vicente Rosell Roberto Vivó Computer Graphics Section Computer Graphics Section Computer Science Dep. Computer Science Dep. Universidad Politécnica de Valencia (Spain) Universidad Politécnica de Valencia (Spain) [email protected] [email protected] ABSTRACT In this work we introduce a new method for computing Form Factors in Radiosity. We demostrate how our method improves on existing projective techniques such as the hemicube. We use the Nusselt analog to directly compute form factors by projecting the scene onto the unit circle. We compare our method with other form factor computation methods. The results show an improvement in the quality/speed ratio using our technique. Key words Global Illumination, radiosity, form factor, Z-buffer, simple plane, polar plane, projection techniques. 1. INTRODUCTION single plane projection. It is also introduced a new Radiosity is one of the most important techniques for algorithm, the polygonal projection onto the base of the synthesis of realistic images with global the hemisphere method (PPBH), which is compared illumination. One of the keys of the application of the with their precedents. After the previous work radiosity method is the computation of the form review, the basis of the method is presented in section factors. The form factor of a patch i to another patch 3. In section 4, we give a description of the j specifies the fraction of total energy emitted from i experiments between the studied methods, showing that arrives at patch j. the results in section 5. We finish with relevant conclusions and the new opened fields for the future. The formula of the calculation of the form factor of a patch i to another j is : 2. PREVIOUS WORK 1 θ coscos θ One of the most studied methods has been the F = ∫∫ ji dAdAV (1) , ji AA 2 , jiji Ai ij π r hemicube algorithm introduced by Cohen and Greenberg [Cohen85]. They proposed to locate a Vi,j is the visibility function and the rest of terms hemicube over the shooting patch so that the other are geometric magnitudes. Since it is not possible to patches can be projected on its five faces. Each one find an analytical solution, the integral is solved by of the faces take advantage of the Z-buffer to solve approximation. The importance of the form factor the visibility problem. So, each face is a pixel buffer speed up calculation lies on the high cost of this where each cell has its individual form factor value, process within the radiosity method (90%) and its called delta form factor, so that if a polygon is visible 2 apparent complexity O(N ). from a pixel it increases its form factor in the In this work we compile, classify and evaluate corresponding delta-value. different methods for the computation of the form Nelson Max [Max95] made a deep study about this factors, in particular, those based on the use of a method. Following the same scheme of hemicube, the method of the cubical tetrahedron appeared Permission to make digital or hard copies of all or part of afterwards [Jeffrey91]. This technique solely uses this work for personal or classroom use is granted without three half faces of hemicube to project the scene. fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this Following the tendency of the reduction of the notice and the full citation on the first page. To copy number of projection faces, Sillion and Puech otherwise, or republish, to post on servers or to redistribute [Sillion89] presented the single plane method. In that to lists, requires prior specific permission and/or a fee. proposal the scene only project on one plane whose WSCG POSTERS proceedings extension varies according to the energy which we WSCG’2004, February 2-6, 2004, Plzen, Czech Republic. want to gather. The distribution of the sampling Copyright UNION Agency – Science Press points must be chosen in such a way the delta form Now we have to project each edge onto de factors of the cells was approximately equal. hemisphere and then onto its base. In order to do the In a later work Recker et al. [Recker90] developed double projection in one step, we only evaluate the the method of the single plane derived directly from X,Y coordinate of the intersection between the plane the hemicube. The plane where the scene projects passing through the origin and the edge and the corresponds with the top face of the hemicube and its hemisphere (Figure 1). The intersection with the extension follows the same rule that the method of hemisphere will be a circle arc and its orthographic Sillion above. In this approach each cell has its own projection an ellipse arc. delta form factor. In the same work, they introduced the method of the extended single plane, developing the idea that in the central part of the plane greater precision is required. However, these methods share a common problem of rectangular partition of the plane. Recently, Vivó et al. [Vivo01] have introduced a new cell distribution for the form factors calculation using a polar plane. They take advantage of the geometry coherence of the problem. Their method improves the quality of the image and it has a similar time cost than the same Figure 1. Edge projection onto the hemiesphere family of methods. base Some new methods have been based in the use of the Being the imaginary plane that passes through the projection onto the hemisphere or unit circle edge, π : Ax+By+Cz=0 , and the formula of the proposed by Gatenby and Hewit [Gatenby91] with a hemisphere, x2+y2+z2=1 (z>=0), the final equation hemisphere discretization method when it is split in for Y coordinate of an edge projected point would be: triangular regions with nearly equal areas, or the ray tracing strategy utilized by Doi and Itoh [Doi98] −+−ABx C x2222() A ++++ B C B 22 C y = (2) when a surface element is subdivided into small B22+ C triangular patches. A solid angle criterion is used to guarantee accuracy; when the angle is larger than the Equation 2 is a curved line easy to calculate and it user tolerance, recursive triangulation is applied. gives the exact value of the edge projection in the base. The projected vertex of the segment are the In this paper we present a method to compute form final values of the curve. Obviously, if we do the factors projecting onto the base of the hemisphere. It same for each polygon edge we will obtain a list of a is based in the Nusselt analog. We will show how the linked curve segments. The region closed by the computing time and the quality of the results can be segment list is the polygon projection. According to better than other methods. the Nusselt analog the area of this projection relative 3. DESCRIPTION OF THE METHOD. to the hemisphere base area divided by π is exactly The Polygonal Projection onto the Base of the its form factor. Hemisphere (PPBH) method computes the form Discrete method factor integral measuring the area covered by each Given that equal areas on the hemisphere base visible polygon projected onto the unit circle. This correspond to equal form factors, we can superpose a area is computed by a double projection: first onto square grid where each cell would account the same the hemisphere and then orthogonally down onto its form factor. So, only knowing the length of the base base. The relative area occupied accounts exactly for cell, the delta form factor would also be known and the form factor so we can expect an accuracy no further storage. In fact, whatever decomposition is improvement instead of using rectangular techniques. valid if all cells have the same area. We can control Next we present a geometrical description of the the cell size to adjust the precision of the PPBH method. After that, we describe two different computation. Each cell will be a sample point. approaches to compute the projected area: a discrete We have chosen a square grid covering the unit and the continuous algorithm. circle. Each element in the grid accounts the same Base of the method form factor which could be easily calculated. We use the central point of the shooting patch as That square grid is used as a Z-Buffer where we can origin of a local coordinate system and we transform store the current depth and polygon id. In such a way each vertex to this reference system. the visibility problem is solved processing every polygon at a time. The ids of visible polygons are Traditionally, the problem has been solved in the eventually available in the buffer when the whole object space by means of a kind of algorithms known scene had been processed. Thus, the form factor as area-sorting algorithms. Those algorithms maintain value of each polygon is approximated by the number a list of visible parts of polygons clipping new of own cells. It must be notice that the irradiation is polygons against the visible list. totally computed in one projection step because of all In addition, we have built a BSP tree structure to polygons are projected onto the hemisphere base. The ensure next polygon resides behind the visible graph. visibility problem is solved by storing the distance In such a way the whole process speeds up. between the nearest projected point and origin.
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