Radiosity. Early Radiosity. Page 1

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1 Page 1 Radiosity Classic radiosity = finite element method Assumptions Diffuse reflectance Usually polygonal surfaces Advantages Soft shadows and indirect lighting View independent solution Precompute for a set of light sources Useful for walkthroughs Early Radiosity From Goral, Torrance, Greenberg, Battaile 1984

2 Page 2 Early Radiosity From Cohen, Chen, Wallace and Greenberg 1988 First Radiosity Pictures... Parry Moon and Domina Spencer (MIT), Lighting Design, 1948

3 Page 3 Finite Element Method The Radiosity Equation Assume diffuse reflection only Solve for radiosity (2D function) x da θ x x x θ da Bx ( ) = B( x) + ρ( xex ) ( ) e Bx ( ) = Be( x) + ρ ( x) Fxx (, ) Bx ( ) da 2 M cosθ cosθ Fxx (, ) = Vxx (, ) 2 π x x

4 Page 4 Classic Radiosity Algorithm Mesh Surfaces into Elements Compute Form Factors Between Elements Solve Linear System for Radiosities Reconstruct and Display Solution Simple Room Scene Table in room sequence from Cohen and Wallace

5 Page 5 Basic Functions Piecewise constant basis functions Ni( x) Express radiosity as sum of basis functions Bx ( ) = BN( x) i B ( x) = EN ( x) e i i i ρ( x) = ρ N ( x) i i i i i Constant radiosity assumption Derivation Convert integral equation to matrix equation B( x) = Be( x) + ρ ( x) F( x, x ) BjNj( x ) da M2 B A E A ρ B F( x, x ) N ( x) N ( x ) dada = + i i i i i j i j j Ai Aj BN i i( x) = BN i i( x) + ρini( x) Bj F( x, x ) Nj( x ) da j i i i j A BiNi( x) = BN i i( x) + ρini( x) Bj F( x, x ) Nj( x ) da da j i i i j A

6 Page 6 Form Factor Form Factor cosθ ocosθi AiFij = AjFji = V( x, x ) dada 2 π x x Ai Aj Summation F ij = 1 j Form factor is the percentage of light leaving i that makes it to j Classic Radiosity Power balance BA = EA + ρ BAF i i i i i j i ij j B = E + ρ FB i i i ij j j Linear system of equations 1 ρ1f11 ρ1f12 ρ1f1 n B1 E1 ρ2f21 1 ρ2f22 ρ2f 21 B 2 E 2 = ρ F ρ F 1 ρ F B E n n1 n n2 n nn n n

7 Page 7 Form Factors Hemicube Algorithm First radiosity algorithm to deal with occlusion 1. Render scene from the point of view of each vertex/element 2. Compute delta form factors contribution from each pixel Render source elements from POV of receiving element F dai, Aj = F p Aj p Typical resolution: 32x32

8 Page 8 Hemicube Delta Form Factors r x y 2 2 = r = 1+ y + z 2 2 cosφ = x y + 1 cosφ = y + z 2 2 A F = π ( x + y + 1) A F = π (1 + y + z ) Hemicube Algorithms Advantages + First practical method -> Patent! + Use existing rendering systems; Hardware + Computes row of form factors in O(n) Disadvantages - Computes differential-finite form factor - Aliasing errors due to sampling Randomly rotate/shear hemicube - Proximity errors - Visibility errors - Expensive to compute a single form factor

9 Page 9 Solving Solve [F][B] = [E] Direct methods: O(n 3 ) Gaussian elimination Goral, Torrance, Greenberg, Battaile, 1984 Iterative methods: O(n 2 ) Energy conservation diagonally dominant iteration converges Gauss-Seidel, Jacobi: Gathering Nishita, Nakamae, 1985 Cohen, Greenberg, 1985 Southwell: Shooting Cohen, Chen, Wallace, Greenberg, 1988

10 Page 10 Gathering for(i=0; i<n; i++) B[i] = Be[i]; while(!converged ) { for(i=0; i<n; i++) { E[i] = 0; for(j=0; j<n; j++) E[i] += F[i][j]*B[j]; B[i] = Be[i]+rho[i]*E[i]; } } Row of F times B Calculate one row of F and discard Successive Approximation L e K L e K K Le K K K Le L e L + K L e e 2 Le + K Le 3 Le + K Le

11 Page 11 Shooting Brightness order Column of F times B for(i=0; i<n; i++) { B[i] = db[i] = Be[i]; while(!converged ) { set i st db[i] is the largest; for(j=0;j<n;j++) if(i!=j) { db =rho[j]*f[j][i]*db[i]; db[j] += db; B[j] += db; } db[i]=0; } } Progressive Radiosity (a) (b) (a) Traditional Gauss-Seidel iteration of 1, 2, 24 and 100. (b) Progressive Refinement (PR) iteration of 1, 2, 24 and 100. From Cohen, Chen, Wallace, Greenberg 1988

12 Page 12 Meshing Accuracy Reference Solution Uniform Mesh Table in room sequence from Cohen and Wallace

13 Page 13 Artifacts A. Blocky shadows B. Missing features C. Mach bands D. Inappropriate shading discontinuities E. Unresolved discontinuities Error Image Increasing Resolution

14 Page 14 Adaptive Meshing Discontinuity Mesh From Baum et al.

15 Page 15 Discontinuity Mesh From Campbell et al. Discontinuity Meshing From Lischinski, Tampieri, Greenberg 1992

16 Page 16 Hierarchical Radiosity Summary Remember assumptions Diffuse reflectance Polygons Difficult to relax assumptions Computation challenges Meshing Complex input geometry Complexity due to shadows Dense coupling O(n 2 ) matrix elements HR leads to O(n) algorithm (ignoring discontinuities)