Parallel Radiosity: Evaluation of Parallel Form Factor Calculations and a Static Load Balancing Algorithm

Transcription

1 Parallel Radiosity Evaluation of Parallel Form Factor Calculations and a Static Load Balancing Algorithm Akira Uejima 1 and Katsuhiro Yamazaki 2 1 Kobe University Kobe JAPAN uej imaemaestro cs. kobe-u ac. jp Ritsumeikan University Shiga JAPAN yamazaki~hpc cs. rit sumes ac. jp Abstract. Although the radiosity algorithm can generate photo-realistic images due to global illumination effects a large amount of form factor calculations are required. This paper describes how to parallelize the radiosity algorithm by subdividing hemispheres into multiple elements and allocating them statically to multiple processors. An enhanced communication procedure is proposed where partial hemisphere data at each processor is communicated and the complete hemisphere data prepared on all processors. In this procedure the size of the communication data is independent of the number of elements. In addition the load balancing efficiency of our static load balancing algorithm is evaluated. On the distributed memory parallel computer AP1000+ which has 64 processors the speedup is 20.4 ~ 21.8 for benchmark scenes and 35.0 ~ 40.2 for classroom scenes. The load balancing efficiency is 0.93 ~ Introduction The radiosity algorithm[l] can generate photo-realistic images taking account of mutual reflections between multiple polygons. However an enormous amount of computations are required to calculate the form factors. This amount increases as the number of primitives that compose a scene increases. Parallel versions of the radiosity algorithm have been studied by a number of researchers[2-8]. In our previous work we parallelized the algorithm on a multi-transputer system where hemicubes are subdivided into small parts that are allocated to each processor by dynamic/static load balancing algorithms[9-11]. The performance by both methods is almost the same when the resolution of the hemicubes is a multiple of the number of processors. In the other situations the parallel efficiency by the static load balancing algorithm is 12% less than that by the dynamic one. Therefore we adopted the static load balancing algorithm for the parallelizing method on the distributed memory parallel computer AP It is important to use an algorithm that can balance the workload; otherwise an imbalance among multiple processors could create a situation where many processors are idle during different phases of the algorithm.

2 182 By our method subdivided hemispheres are allocated to each processor using a two-dimensional interleaving method. Therefore even a static load balancing algorithm can almost equalize the loads on all the processors. The parallel efficiency is defined as T(1)/nT(n) where T(1) is the processing time on one processor and T(n) is the processing time on n processors. In our previous study a problem occurred where the parallelization efficiency decreased when the scene size became large[9-12]. This was because communication of the partial form factor data which is proportional to the number of elements is required for parallel form factor calculations. We improved the communication procedure in parallel form factor calculations in order to reduce that problem. The partial hemisphere data distributed to each processor is communicated in order to gather all the hemisphere data and then the form factors are calculated simultaneously on all the processors. Large-scaie benchmark scenes (maximum polygons) and classroom scenes are processed using two procedures (communication of partial form factor data and communication of partial hemisphere data). The two communication procedures and the load balancing algorithms are evaluated. In this paper three benchmark scenes and three realistic classroom scenes are processed on the distributed memory parallel computer AP The performance improvements by the enhanced communication method are discussed and static load balancing by subdividing the hemispheres is evaluated. 2 A Parallel Radiosity Algorithm 2.1 A Procedure for Computing Radiosity The surfaces of the objects in a scene are defined with rectangle and triangle polygons. Each polygon is subdivided into patches and each patch is subdivided into elements of regular interval[12]. Progressive refinement[13] is an efficient approach to calculate radiosity. In this approach the patch that has the greatest unshot radiosity is chosen and the radiation from this patch to all other elements is calculated. This step is repeated until the unshot radiosity energies converge. This approach is advantageous because intermediate results are obtained at any point and less memory is required. When radiation is calculated the form factor for each element is required. This is the proportion of optical energy from a patch that reaches the element. The form factors are calculated by the hemisphere algorithm[14]. As shown in Figure 1 an imaginary hemisphere is put around the radiation patch in question and the surface is subdivided into small meshes or grid cells. The delta form factor for each grid cell is pre-calculated easily using geometry and stored in a lookup table on each processor. In each step the form factors are calculated by the following procedure. All the elements are projected onto the hemisphere removing the hidden surfaces. Then the form factor for an element is approximated by summing the delta form factors of the grid cells occupied by the element. In Figure 1 Fij is calculated by summing the delta form factors of the hatched grid cells.

3 183 The important point to note concerning our hemisphere algorithm[12] is that polygons rather than elements are projected onto the hemisphere. Then the element corresponding to each grid cell is determined and the element ID is stored in the item buffer of the hemisphere. The form factor for each element is calculated based on these results. tj Polygons Include_Geome tries lii~iii!i~iiilii~iili~iilii~{lii~ ~ 132bytes -Attributes l 9 l Patches i~l ii 2i31415] 6171i~i ~ Include -ARadiosities Elements 16bytes lis]4j i [ ] ] i i{i ~ t- 8bytes Fig. 1. The hemisphere. Fig. 2. Allocation of scene descriptions. 2.2 Allocation of Scene Descriptions As shown in Figure 2 all processors have a copy of all the polygon and patch data which is distributed by an interleaving method. The element data is allocated to the processor that has the patch containing the element. In this method all processors have the polygon data needed by the hemisphere algorithm and thus transfers of scene descriptions are unnecessary for the parallel form factor calculations. Furthermore patch and element data is distributed to each processor so each processor can search the patch that has the greatest unshot radiosity and update the radiosity of all the elements in parallel[12]. Each cell processor requires (132NpI + 16[Npt/Npr] + 8[NF_l/Npr])bytes of memory where Npt Npt and NEt are the numbers of polygons patches and elements respectively and Npr is the number of processors. 2.3 Parallel Computation of Form Factors As shown in Figure 3 we use two procedures for parallel form factor calculations. In this figure parallel processing by eight processors is shown with the hatched parts denoting calculation by one processor. One procedure is based on communication of partial form factor data and the other is based on communication of partial hemisphere data. In both procedures each processor draws all NEI elements existing in the scene on its own part of a hemisphere. After that in the former each processor partially calculates the form factors and all processors get the complete form factor data by communicating the partial calculations among themselves. In the latter all processors get the complete hemisphere data by communicating partial hemisphere data among the processors. Each processor then calculates the form factors simultaneously.

4 184 Draw Nm elements on hatched /parts of each hemisphere. -- c g ri d ce ii s -- ri cells Broadcast... ial hemisphere Partial hemisphere O[] [] [] (I) ~ l~~u~ate partial form factors. NBelement s (i} Step A Communications. DDDD Broadcast 2 D~D [] DDQD Partial form factors (obtained from hatched parts of hemisphere) (2) (8) mdmb Broadcast Step B Communications. ~ N E... Complete form factors ~ DDD Broadcast ~]-[] D D N[]D[] (Fin&].)[] [] [] [] Q[]9 Complete hemisphere.~... mbmb I Step B ~ Calculate complete form factors. /elements Complete fo~ factors (a)communicating partial form factor data. (b)co~unicating partial hemisphere data Fig. 3. Parallel form factor calculations using two communication procedures. Communications of Partial Form Factor Data As shown in Figure 3(a) each processor partially calculates the form fartors obtained from their own part of the hemisphere (Step A). Then all processors get the complete form factor data by communicating between the cells and by broadcasting (Step B). First the partial form factor data on two processors is added step-by-step and the complete form factor data is obtained at the upper left processor. The data is then broadcast to the other cells. Therefore [log 2 Npr] stages for gathering the data and one stage for broadcasting are required where Np~ is the number of processors. For example for parallel processing with eight processors four communication stages are required (Figure 3(a)). The size of each communication data packet becomes 4NEibytes where NEI is the number of elements because the number of elements in a partial or complete form factor data set is NEt and the size of a floating-point value is 4bytes. In this method Step A can be parallelized but the size of each communication data packet is proportional to the number of elements in Step B. This reduces performance when the number of elements is large[12 15].

5 185 Communications of Partial Hemisphere Data As shown in Figure 3(b) all processors get the complete hemisphere data by communicating partial hemisphere data among the processors (Step A). Then each processor calculates the complete form factors simultaneously by adding the delta form factors (Step B). During the communication of step A each processor broadcasts the partial hemisphere data in sequence. Therefore Npr stages for broadcasting data are required where Npr is the number of processors. For example for parallel processing with eight processors eight communication stages are required (Figure 3(b)). The size of the communication data packet for each processor becomes 4Nc/Np~bytes where Arc is the number of grid cells on a hemisphere and Npr is the number of processors since the number of grid cells on one processor is Nc/Np~ and the data size for one grid cell is 4bytes. This method has the advantage that the size of each communication data packet depends on the number of grid cells on a hemisphere instead of the number of elements. 3 Load Allocation by Subdividing Hemispheres 3.1 Dynamic Load Balancing versus Static Load Balancing During the form factor calculations the drawing time of each processor may be unbalanced because the number of polygons drawn by one processor is not constant. Therefore a load balancing algorithm that can efficiently balance each processor's drawing time is required. For dynamic load balancing allocation to each processor is decided at run-time by considering the loads on each processor. Load balancing is achieved even in the situation where the difference between individual processing times is large. However overheads arise since the tasks are allocated dynamically. In static load balancing the load balancing efficiency may be worse than the former because a constant number of tasks are allocated to the processors but no overhead is required to allocate the tasks during run-time. In our previous work a dynamic and a static load balancing algorithm were compared on a multi-transputer system[10 11]. Using the static algorithm the parallel efficiency is 12% less than the dynamic one in the worst case if the number of hemicube subdivisions is not a multiple of the number of processors. In the other case however differences between the two algorithms are very small. If the same number of tasks is allocated to the processors by an interleaved method a balanced load can be achieved even using the static load balancing algorithm. Hence we use the static one in this paper. 3.2 Static Load Balancing by Subdividing Hemispheres Subdivided hemispheres are allocated to each processor with a two-dimensional interleaving method. For parallel processing with eight processors of which the IDs are (0 0) ~ (3 1) a grid cell (r 0) = (p t) on a hemisphere is allocated to each processor (t mod 4p rood 2). Figure 4 shows a hemisphere with resolution (Re Ro) = (8 32). For example processor (0 1) draws the hatched parts of the hemisphere.

6 186 il ~" ~iiii ~ 2 Fig. 4. Subdivision of hemispheres. 8 4 Experiments 4.1 A Distributed Memory Parallel Computer AP1000+ We use a distributed memory parallel computer the AP1000+ developed by Fujitsu Laboratories Ltd. It has 64 cells or processing elements. We apply a SPMD (Single Program Multiple Data) model to the parallel implementation of the radiosity algorithm so each cell's processor has the same code. The elements are connected by the following three types of network (see Figure 5) - T-net (torus network); wormhole routing is used for point to point communications between cells. - B-net (broadcast network) for broadcast communications and data collections. - S-net (synchronization network) for barrier synchronizations and status communications. ~ ~ ~(50Mbyte/s T-net (25Mbyte/s ) Fig. 5. AP1000+ system configuration.

7 187 Each cell consists of a SuperSPARC processor (50MHz) 64M bytes local memory a message controller(msc+) a B-net interface(bif) and a routing controller(rtc). The time needed for message transmission between cells depends on the size of the messages but it is independent of the distance between source and destination. Normal message communications are asynchronous. This means that succeeding operations on the sending side can be executed without waiting for the completion of the sending process. The file inputs/outputs are processed on the host computer joined to the cells. Parallel programs on the AP1000+ can call the following functions - point to point communication for host to cell or cell to cell - broadcast communication for host to all cells or one cell to the other cells - distribution of data from host to all cells and collection of data from all cells to host - search for a maximum/minimum value or calculation of a total value on all cells and - synchronization. Table 1. Benchmark scenes Scene ~Cubesl#Polygons ~Patches ~Elements ~Iterations Data(MB) B B B Table 2 Classroom scenes. Scene ~Polygons C C C #Patches#Elements #Iterations Data(kB) Experimental Method Processing of Benchmark Scenes Three benchmark scenes as shown in Table 1 were processed then the two procedures for parallel form factor calculation at three hemisphere resolutions compared. Progressive radiosity refinement is repeated 100 times. Each scene has a light source and a number of cubes in a room with a ceiling a floor and four walls. Each cube has six polygons and each patch is subdivided into two patches which are subdivided into 2 2 elements. The

8 188 ceiling the floor and the four walls are subdivided into 5 x 5 patches and each patch is subdivided into 2 x 2 elements. Six polygons for each light source are subdivided into 2 x 2 patches. Figure 6 shows the rendering results for the benchmark scenes using the radiosity algorithm. While benchmark scene B3 required a total of 23M bytes of memory only 13.4M bytes of memory was required by each processor with parallel processing using 64 processors. Processing of Classroom Scenes Three realistic classroom scenes as shown in Table 2 were processed then the two procedures for parallel form factor calculations were compared and the load balancing efficiency evaluated. Radiosity refinement was continued until the energies converged for the three color-bands (R G and B). These three classrooms have different room sizes and different number of fluorescent lights desks and chairs. Scene C1 C2 and C3 are classrooms with seats for and 90 people respectively. Figure 7 shows the rendering results for the classroom scenes using the radiosity algorithm. (a)scene B1. (b)scene B2. (c)scene B3. Fig. 6. Benchmark scenes. (a)scene C1. (b)scene C2. (c)scene C3. Fig. 7. Classroom scenes.

9 Experimental Results and Discussion Evaluation of Parallel Form Factor Calculations (a)processing of Benchmark Scenes Execution time versus number of processors is shown in Figure 8. Benchmark scenes B1 B2 and B3 were processed using the two communication procedures. The resolution of the hemispheres is ( ). As the number of processors increased the rate of decrease of processing time decreased. This is because the size of the communication data packets increases in proportion to the number of elements and the ratio of overheads for communication increases especially when the number of processors is large. The graphs show that communication of partial hemisphere data is better than partial form factor communication. The difference between the two procedures increases in proportion to the number of processors. The speedup using 64 processors is 9.5 -~ 11.1 for communication of partial form factor data and 20.4 ~ 21.8 (almost twice the former) for communication of partial hemisphere data. While the cubes of these three benchmark scenes are placed regularly we have confirmed that the difference in processing times is less than 2% even when the cubes are not placed regularly. (b)processing of Classroom Scenes Execution time versus number of processors is shown in Figure 9. The resolution of the hemispheres is ( ). The time difference between the two procedures increases in proportion to the complexity of the scenes. Therefore communication of partial hemisphere data becomes effective. Because the classroom scenes C1 C2 and C3 are smaller than the benchmark scenes the difference between the two communication procedures is smaller. Communication of partial hemisphere data is better except when the number of processors is below 16 for scene C1. In other words communication of partial hemisphere data is effective except in the case when both the scene size and the number of processors are small. The speedup using 64 processors is for communication of partial form factor data and ~ 40.2 for communication of partial hemisphere data. Because the classroom scenes are smaller than the benchmark scenes the difference of the speedup using the two communication procedures is smaller. Evaluation of the Load Balancing The load balancing efficiency e is Tml/T.= where Tmi~ is the minimum processing time and Tma= is the maximum over all the processors e becomes 1 when the load balancing is perfect and gets smaller as load imbalance increases. We evaluated the average the maximum and the minimum values of e for every distribution using progressive refinement. Evaluation of the load balancing for scene C2 based on the number of processors is shown in Figure 10. The resolution of the hemisphere is ( ) for all instances. ~ becomes smaller (load balancing efficiency decreases) as the number of the processors increases. However the average value of c is 0.99 for 4 processors and 0.94 for 64 processors indicating that the load balancing succeeded.

10 190 1OO00 50O0 4OOO 3OOO 2OOO \. Partial B1 +o ""-. Partial Bt ~ Pallial C1 -o--- Partial C1 i..= 5o0 4oo 3oo 2oo loo i i i i i Number of processoes ~30O0 i"= 2O0O O0 4OO 3OO i i i i i Number of processors Fig. 8. Comparison of the two communication procedures(benchmark scenes). Fig. 9. Comparison of the two communication procedures(classroom scenes). Scene i C1 i C2 C3 r 1 i i 1 1 B-... B [I Q Maximum ---~--- Average Minimum ~-- i i i i i Number of processors O.4 02 t ~ Maximum Average Minimum i i L i ~ i t T (... )Tl *'4 (12ss121 I Resolution 116~176 (leo64o)/ ) / (192768) ( ) (192768) Fig. 10. Evaluation of load balancing based on the number of processors(scene C2). Fig. 11. Evaluation of load balancing based on scenes and resolutions(64 processors). 2OOO0 1OOO k } ) 50OO O0 4OO = i = i = Number of processors Fig. 12. Comparison based on the resolution of the hemispheres.

11 191 Evaluation of the load balancing based on scenes and resolutions is shown in Figure processors are used in this evaluation. In each scene e becomes larger as the resolutions of the hemispheres increase. This is because the difference of the drawing time among the grid cells on the hemispheres becomes smaller and the number of processing elements becomes larger as the resolutions of the hemispheres increase. The average and the maximum value of r are almost constant 0.93 ~ ~ 0.99 respectively when resolutions are the same regardless of the scenes. However c is small ~ 0.71 for scene C3. This is a consequence of the following differences between the drawing times among grid cells on the hemispheres increases because the number of polygons is large and the probability that load imbalance appears increases because the amount of necessary refinement becomes large. Comparison Based on the Resolution of Hemispheres Comparison of execution time based on resolution of the hemispheres is shown in Figure 12. Benchmark scene B3 is processed using communication of partial hemisphere data and three resolutions ( ) ( ) and ( ). Generally during form factor calculations using the hemisphere method finer resolution are required as the size of each element becomes smaller. The processing time for one processor using resolution ( ) is 1.65 times longer than the time using resolution ( ). However it is 1.40 times for 64 processors. The speedup using 64 processors is for resolution ( ) and for resolution ( ). In other words parallel efficiency gets higher as the resolution improves. This is because the increase in drawing time is larger than the corresponding increase in communication time. 5 Related Work Baum et al. described a parallelization algorithm by subdividing hemicubes[2]. They used a producer-consumer model (one processor as producer and the others as consumers) on the shared memory multiprocessor workstation SGI 4D/280 GTX. The producer draws polygons on hemicubes using a hardware renderer and then allocates subdivided hemicubes to each consumer. Each consumer accumulates delta form factors distributes the radiosity energy and selects the next shooting patch in parallel. Scene descriptions are accessed by the producer only centralized on one place. To improve the performance processing by the producer and the consumers overlap each other. Sturzlinger et al. described a processing method for scenes including polygons and elements on the distributed memory parallel computer ncube2s which has 256 processors[7]. They used ray casting for form factor calculations. Patch data are distributed to each processor and visibility tests are parallelized. They tried to achieve load balancing using a dynamic algorithm in the processor-groups. They estimated the speedup to be about 20 for 64 processors and about 50 for 256 processors.

12 192 Renambot et al. described a processing method for scenes including 1 million polygons and 1.3 million elements on the distributed-shared memory parallel computer SGI Origin 2000 which has 32 processors[8]. They subdivided environments into small sub-environments and allocated them to each processor. Each processor sends visibility masks which are the results of the intersection checking between rays from the source and objects in the environments to neighboring processors in sequence. The speedup is 21. In our study drawing elements to hemispheres was parallelized and an enhanced communication procedure using partial hemisphere data communication was developed to improve the performance. Then large-scale scenes including patches and elements were processed. 6 Conclusions This paper described how to parallelize the radiosity algorithm by subdividing hemispheres. Two procedures were used for parallel form factor calculation. Communication of partial hemisphere data is better than communication of partial form factor data. Load balancing efficiencies were evaluated based on the average the maximum and the minimum values. For evaluation of the parallel form factor calculations it was seen that communication of partial hemisphere data is better for all the benchmark scenes and the speedup is ~ This is almost twice as much as our previous method which used communication of partial form factor data. For realistic classroom scenes communication of partial hemisphere data is better except when both the scene size and the number of processors are small. The speedup is ~ For evaluation of the load balancing it was seen that load balancing efficiency decreases as the number of processors increase. The average value is ~ 0.96 for 64 processors which indicates that load balancing succeeded. Load balancing efficiency increases as the resolutions of the hemispheres increase. Representing the processing times by formulas and parallelizing the multipath rendering method should be done in the future. Acknowledgments The authors would like to thank Professor Hidekatsu Tokumaru and Professor Tohru Watanabe for valuable comments and discussion. Special thanks to Fujitsu Laboratories Ltd. and Tomita Laboratory of Kyoto University for allowing us to use their parallel computers. Finally thanks are due to Mr. Gaute Lambertsen for reading the manuscript and making a number of helpful suggestions. References Goral C. Torrance K. Greenberg D. and Battaile B. Modeling the Interaction of Light Between Diffuse Surfaces Computer Graphics~ Vol. 18 No. 3 (1984)

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