The Fast Multipole Method and the Radiosity Kernel

Transcription

1 and the Radiosity Kernel The Fast Multipole Method Sharat Chandran sharat January 8, 2006 Page 1 of 43 (Joint work with Alap Karapurkar and Nitin Goel) 1 Copyright c 2005 Sharat Chandran Jan 9 11

2 Overview What s FMM about? Page 2 of 43

3 Overview What s FMM about? Point-based rendering Page 2 of 43

4 Overview What s FMM about? Point-based rendering Insight into the method Page 2 of 43

5 Overview What s FMM about? Point-based rendering Insight into the method The algorithm and our contributions Page 2 of 43

6 Page 2 of 43 Overview What s FMM about? Point-based rendering Insight into the method The algorithm and our contributions Limitations, conclusion future work and 2 Copyright c 2005 Sharat Chandran Jan 9 11

7 1. The Fast Multipole Method The Fast Multipole Method Speeds up matrix-vector product sums of certain types Naive method is quadratic: O(n 2 ) FMM solves the problem in linear time: O(n) Page 3 of 43

8 1. The Fast Multipole Method The Fast Multipole Method Speeds up matrix-vector product sums of certain types Naive method is quadratic: O(n 2 ) FMM solves the problem in linear time: O(n) Page 3 of 43 Tricks of the trade: Math, Precision, Data Structures

9 1. The Fast Multipole Method The Fast Multipole Method Speeds up matrix-vector product sums of certain types Naive method is quadratic: O(n 2 ) FMM solves the problem in linear time: O(n) Page 3 of 43 Tricks of the trade: Math, Precision, Data Structures Sometimes can get even better results Move from O(n 3 ) to O(knlogn) or even O(kn) Solve impossible problems :-) 3 Copyright c 2005 Sharat Chandran Jan 9 11

10 Point-Based Rendering Performance of graphics card has increased tremendously Points are available as input data Want to go beyound flat polygons Page 4 of 43

11 Point-Based Rendering The Fast Multipole Method Performance of graphics card has increased tremendously Points are available as input data Want to go beyound flat polygons Rendering options Page 4 of 43 Local illumination models... (surfels?) Ray-tracing and photon mapping Diffuse global illumination (this talk)

12 Point-Based Rendering The Fast Multipole Method Performance of graphics card has increased tremendously Points are available as input data Want to go beyound flat polygons Rendering options Page 4 of 43 Local illumination models... (surfels?) Ray-tracing and photon mapping Diffuse global illumination (this talk) Method works for traditional patch models too 4 Copyright c 2005 Sharat Chandran Jan 9 11

13 So what s the Problem? Lots of points make the problem intractable. 1GB RAM = bytes n 2 = 2 30 n = 32K In reality, we may desire a larger number Space is a constraint Page 5 of 43

14 So what s the Problem? Lots of points make the problem intractable. 1GB RAM = bytes n 2 = 2 30 n = 32K In reality, we may desire a larger number Space is a constraint Page 5 of 43 If solution involves inverting a matrix O(n 3 ) time, that s also a constraint What if we could solve the problem in O(n) space and O(n) time? 5 Copyright c 2005 Sharat Chandran Jan 9 11

15 Illumination Models Are Key to Rendering Ray tracing produces stunning pictures but Pictures appear too good. Takes too long to produce reasonable images Page 6 of 43

16 Illumination Models Are Key to Rendering Ray tracing produces stunning pictures but Pictures appear too good. Takes too long to produce reasonable images Flat shading is quick but unreal Page 6 of 43

17 Illumination Models Are Key to Rendering Ray tracing produces stunning pictures but Pictures appear too good. Takes too long to produce reasonable images Flat shading is quick but unreal Page 6 of 43 Flat shading can be remarkably improved by discretizing environment into patches and computing intensity for each patch. 6 Copyright c 2005 Sharat Chandran Jan 9 11

18 Global Illumination In a Hurry The Fast Multipole Method Radiosity produces photorealistic pictures that can handle non specular scenes Enables color bleeding effects View independent representation Starting point is the general energy balance equation for the radiance L(x,θ 0,φ 0 ) = L e (x,θ 0,φ 0 ) + Ω ρ bd(x,θ 0,φ 0,θ,φ)L i (x,θ,φ)cosθ dω Page 7 of 43

19 Global Illumination In a Hurry The Fast Multipole Method Radiosity produces photorealistic pictures that can handle non specular scenes Enables color bleeding effects View independent representation Starting point is the general energy balance equation for the radiance L(x,θ 0,φ 0 ) = L e (x,θ 0,φ 0 ) + Ω ρ bd(x,θ 0,φ 0,θ,φ)L i (x,θ,φ)cosθ dω Page 7 of 43 Point-based rendering is in some sense easier For diffuse surfaces, radiosity is a popular quantity to compute In the sequel we assume occlusion is handled separately 7 Copyright c 2005 Sharat Chandran Jan 9 11

20 Gauss-Jordan: B (k+1) i Iterative Solution = E i + N j=1(ρ i F i j )B (k) j (some i) X B i. X = X E i. X + X X X X... X X. X Page 8 of 43 Step 1 1 Step Copyright c 2005 Sharat Chandran Jan 9 11

21 Use Iterative Methods The Fast Multipole Method Step Step 2 Page 9 of 43

22 Use Iterative Methods The Fast Multipole Method Step Step 2 Page 9 of 43 Southwell: For all j β (k+1) i Results in B (k+1) j β 1 β i. β n = β (k) i = + N j=1(ρ j F i j )r (k) i E i. + X B j + ρ j F i j A i A j B i X X X X X... 9 Copyright c 2005 Sharat Chandran Jan 9 11

23 Page 10 of 43 Introduction to FMM Suppose we have a collection of N points in 2D x i {i = 1,2,...N} and we want to evaluate f (x j ) = N i=1 α i (x j x i ) 2 j = 1,...,N Each evaluation requires O(N). Since there are N evaluations, straightforward method takes time O(N 2 ) for j = 1 to N for i = 1 to N sum [ j ] + = alpha [ i ] ( x [ j ] x [ i ] ) ( x [ j ] x [ i ] ) Can we do better? 10 Copyright c 2005 Sharat Chandran Jan 9 11

24 Expand f as A Faster Solution: This is Not FMM f (x j ) = N i=1 = x 2 j (α i x 2 j + α i x 2 i 2α i x j x i ) ( N i=1α i ) + ( N α i xi 2 i=1 ) ( N ) 2x j α i x i i=1 Page 11 of 43

25 Expand f as A Faster Solution: This is Not FMM f (x j ) = N i=1 = x 2 j (α i x 2 j + α i x 2 i 2α i x j x i ) ( N i=1α i ) + ( N α i xi 2 i=1 ) ( N ) 2x j α i x i i=1 Page 11 of 43 Precompute )(in O(N) time) ( ) ) β = ( Ni=1 α i and γ = N i=1 α i xi 2 and δ = ( Ni=1 α i x i Then each evaluation is f (x j ) = x 2 j β + γ 2x jδ and we have the O(N) code for i = 1 to N β += alpha[i]; for i = 1 to N γ += alpha[i] * x[i] * x[i]; for i = 1 to N δ += alpha[i] * x[i]; for j = 1 to N sum[j] = x[j]*x[j]*β + γ - 2*x[j]*δ 11 Copyright c 2005 Sharat Chandran Jan 9 11

26 Page 12 of A Faster Solution: This is Not FMM Key point : f (x j ) is written as a sum of products using analytical manipulations f (x j ) = = N 3 i=1 k=1 3 A k (x j ) k=1 α i A k (x j )B k (x i ) ( N i=1 ) α i B k (x i ) }{{} precomputed A 1 (x) = x 2 A 2 (x) = 1 A 3 (x) = 2x B 1 (x) = 1 B 2 (x) = x 2 B 3 (x) = x Effect of the data point is independent of where it is going to be used (in management, work done is people independent ). 12 Copyright c 2005 Sharat Chandran Jan 9 11

27 Points To Note The Fast Multipole Method Starting point is a sum of the form f (x) = N i=1 w(y i )K(x,y i ) Faster solution because of analytical rearrangement But not all problems admit such a solution Consider f (ω) = N i=1 w(y i )e 2πiω N 1 Celebrated Discrete Fourier Transform Page 13 of 43

28 Points To Note The Fast Multipole Method Starting point is a sum of the form f (x) = N i=1 w(y i )K(x,y i ) Faster solution because of analytical rearrangement But not all problems admit such a solution Consider f (ω) = N i=1 w(y i )e 2πiω N 1 Celebrated Discrete Fourier Transform Fast solutions are obtained due to the nature of the kernel FMM provides a fast solution by trading accuracy for speed but requires Page 13 of 43

29 Points To Note The Fast Multipole Method Page 13 of 43 Starting point is a sum of the form f (x) = N i=1 w(y i )K(x,y i ) Faster solution because of analytical rearrangement But not all problems admit such a solution Consider f (ω) = N i=1 w(y i )e 2πiω N 1 Celebrated Discrete Fourier Transform Fast solutions are obtained due to the nature of the kernel FMM provides a fast solution by trading accuracy for speed but requires Four types of factorization formulae Analysis of convergence of analytical expression Data structure issues

30 Points To Note The Fast Multipole Method Page 13 of 43 Starting point is a sum of the form f (x) = N i=1 w(y i )K(x,y i ) Faster solution because of analytical rearrangement But not all problems admit such a solution Consider f (ω) = N i=1 w(y i )e 2πiω N 1 Celebrated Discrete Fourier Transform Fast solutions are obtained due to the nature of the kernel FMM provides a fast solution by trading accuracy for speed but requires Four types of factorization formulae Analysis of convergence of analytical expression Data structure issues x is said to be the point of evaluation of the sources y i 13 Copyright c 2005 Sharat Chandran Jan 9 11

31 3. Step 1: Multipole Expansion The Fast Multipole Method Direct evaluation requires N interactions at runtime Page 14 of 43

32 3. Step 1: Multipole Expansion The Fast Multipole Method Direct evaluation requires N interactions at runtime Page 14 of 43 Can we reduce this to one interaction? 14 Copyright c 2005 Sharat Chandran Jan 9 11

33 Multipole Expansion: Math The Fast Multipole Method Page 15 of 43 To evaluate f at x due to y i {i = 1...N} f ( x) = N i=1 w( y i )K( x, y i ) If we can factorize the kernel as K( x, y) = p j=1 A j( Ox)B j ( Oy) Substituting this expansion as the kernel f ( x) = = = N i=1 w( y i ) p p A j ( Ox) j=1 j=1 ( N i=1 p A j ( Ox)M j (O) j=1 A j ( Ox)B j ( Oy i ) w(y i )B j ( Oy i ) } {{ } M j (O) ) Y X O Preprocessing cost is proportional to the number of points, but run time cost is O(1) 15 Copyright c 2005 Sharat Chandran Jan 9 11

34 The Radiosity Kernel Is Not Easy To Factorize n x n y y r y x r x O Writing the difference in energy B(x) E(x) Page 16 of 43 = = y k B(y) y=y 1 A x y k B(y) y=y 1 A x A x A x ρ(x) cosθ 1 cosθ 2 A y r 2 da x da y ρ(x) [ n y.( r x r y )][ n x.( r y r x )] A y π r y r x 4 da x da yi 16 Copyright c 2005 Sharat Chandran Jan 9 11

35 Key Contribution: Factorization for the Radiosity Kernel The Fast Multipole Method Representing vectors as 3x1 matrices, r = (x,y,z) x y = r z r 1. r 2 = r t 1r 2 = r t 2r 1 Expand the expression in the numerator [ n y.( r x r y )] [ n x.( r y r x )] = r t xn y r t yn x r t xn x r t xn y r t yn y r t yn x + r t yn y r t xn x Page 17 of 43 Define receiver matrices RM and the source matrices SM n y r t r t x y SM( r y ) = n y n x r t yn y r t RM( r x ) = r t xn x r t x y r t r yn t xn x y n x r t x [ n y.( r x r y )][ n x.( r y r x )] = RM( r x ) SM( r y ) 17 Copyright c 2005 Sharat Chandran Jan 9 11

36 Key Contribution: Factorization for the Radiosity Kernel The Fast Multipole Method For r y < r x (Hausner, 1997) Page 18 of 43

37 Key Contribution: Factorization for the Radiosity Kernel The Fast Multipole Method Page 18 of 43 For r y < r x (Hausner, 1997) 1 r y r x 4 = n=0 [n/2] n 2 j en j j=0 m= n+2 j { 1 We have the factored radiosity kernel B(x) E(x) = R m n j( r x ) = M m n j(o) = [n/2] n 2 j n=0 j=0 m= n+2 j 1 A x y k y=y 1 ρ(x) A x rx n+4 rx n+4 Yn 2 m j(θ x,φ x ) Y m } { r n e j nr m n j( r x ) M m n j(o) n 2 j(θ x,φ x )RM( r x )da x } yyn 2 m j (θ y,φ y ) A y B(y)r n yy m n 2 j (θ y,φ y )SM( r y )da y Two observations r y < r x p = 18 Copyright c 2005 Sharat Chandran Jan 9 11

38 The N logn Algorithm (2 Dimensions) The Fast Multipole Method Assumption: Particles are uniformly distributed in a square Consider a uniform hierarchical subdivision of space X X Level 0 X Page 19 of 43 Level 1 Level 2 19 Copyright c 2005 Sharat Chandran Jan 9 11

39 The N logn Algorithm (2 Dimensions) Nearest neighbors share a vertex Well separated Boxes are on the same level and are not nearest neighbors X X Page 20 of 43 Calculate multipole moments at the center of each box of Level 2 Multipole Expansion cannot be evaluated at nearest neighbors Recursion! 20 Copyright c 2005 Sharat Chandran Jan 9 11

40 The N logn Algorithm (2 Dimensions) The Fast Multipole Method Nearest neighbors share a vertex Well separated Boxes are on the same level and are not nearest neighbors X X Page 21 of 43 Calculate multipole moments at the center of each box of Level 2 Multipole Expansion cannot be evaluated at nearest neighbors Particles we have not yet accounted for in their interaction with box X are in children of the near neighbors of X s parent (at level 3). Recursion! 21 Copyright c 2005 Sharat Chandran Jan 9 11

41 4. The Fast Multipole Method of a box i consists of children of near neighbors of i s parent which are well separated from i. Maximum size of the interaction list is 27 X Page 22 of Copyright c 2005 Sharat Chandran Jan 9 11

42 The N logn Algorithm (2 Dimensions) The Fast Multipole Method Algorithm Subdivide till each box contains not more than M particles Number of leaves N/M Depth d of quad-tree log 4 (N/M) Total number of boxes d i=1 4 i 4N/3M At each level, calculate multipole moments at each box N p Page 23 of 43 At each level, for each particle, evaluate the multipole expansion of all boxes in its owners interaction list 27N p For the last level, for each particle calculate directly its interaction with all particles in its owners nearest neighbors O(N) Total Cost O(N logn)!

43 The N logn Algorithm (2 Dimensions) The Fast Multipole Method Algorithm Subdivide till each box contains not more than M particles Number of leaves N/M Depth d of quad-tree log 4 (N/M) Total number of boxes d i=1 4 i 4N/3M At each level, calculate multipole moments at each box N p Page 23 of 43 At each level, for each particle, evaluate the multipole expansion of all boxes in its owners interaction list 27N p For the last level, for each particle calculate directly its interaction with all particles in its owners nearest neighbors O(N) Total Cost O(N logn)! We are accessing each particle at every level Can we do better? 23 Copyright c 2005 Sharat Chandran Jan 9 11

44 5. Step 2: Translation of Multipole Expansion Why recompute multipole moments for each particle at different centers? Page 24 of 43

45 5. Step 2: Translation of Multipole Expansion Why recompute multipole moments for each particle at different centers? Page 24 of 43 Reuse computation, shift multipole centers 24 Copyright c 2005 Sharat Chandran Jan 9 11

46 Translation of Multipole Expansion : Math The Fast Multipole Method Page 25 of 43 We have the multipole moments due to N particles at O as M j (O) = N i=1 w(y i )B j ( Oy i ) To find the moments at O given the moments at O, if we can expand B j as B j ( O y i ) = p k=1 B k( Oy i )α j k ( OO ) Then the multipole moment at O will be M j (O ) = = = N i=1 N w(y i )B j ( O y i ) p i=1 k=1 w(y i )B k ( Oy i )α j k ( OO ) p M k (O)α j k ( OO ) k=1 Coefficients can be computed in O(p 2 ) time 25 Copyright c 2005 Sharat Chandran Jan 9 11 O O y j

47 Page 26 of 43 Contribution: Translation Theorem for the Radiosity Kernel B(x) E(x) = M m n j(o ) = n=0 [n/2] j=0 n k/2 k=0 s=0 e j n k e j n n 2 j m= n+2 j k 2s m 1 = k+2s e j nr m n j( r x ) M m n j(o ) j max j = j min J(...) OO k Y m 1 k 2s (OO )TM(OO ) M m 1+m n k, j (O)

48 Page 26 of 43 Contribution: Translation Theorem for the Radiosity Kernel B(x) E(x) = M m n j(o ) = n=0 [n/2] j=0 n k/2 k=0 s=0 e j n k e j n n 2 j m= n+2 j k 2s m 1 = k+2s e j nr m n j( r x ) M m n j(o ) j max j = j min J(...) OO k Y m 1 k 2s (OO )TM(OO ) M m 1+m n k, j (O) This one was rough... Used a theorem from Sack Copyright c 2005 Sharat Chandran Jan 9 11

49 N logn to N : Translation of Multipole Expansion The Fast Multipole Method Computing multipole moments at all levels was O(N logn) Revised Procedure Compute multipole moments at the lowest level O(N) Shift and aggregate multipole moments upwards till Level 2 O( d i=3 4 i p 2 ) O(4p 2 N/3M) Page 27 of 43 But we are still evaluating multipole expansions for each particle at each level 27 Copyright c 2005 Sharat Chandran Jan 9 11

50 6. Multipole moments represent field outside a cluster in a constant number of coefficients M(O ) 1 M(O ) 2 M(O ) 3 Page 28 of 43

51 6. Multipole moments represent field outside a cluster in a constant number of coefficients M(O ) 1 M(O ) 2 M(O ) 3 Page 28 of 43 Can external multipole moments be combined into a constant number of coefficients to represent the field inside a cluster? M(O ) 1 M(O ) 2 M(O ) 3 28 Copyright c 2005 Sharat Chandran Jan 9 11

52 Local Expansion : Math The Fast Multipole Method Page 29 of 43 To calculate f at x we now have f ( x) = p i=1 A i( Ox)M i (O) Now suppose we make A i degenerate in x and O about x 0 as A i ( Ox) = p l=1 C l( x 0 x)βl i( Ox 0 ) Substituting this expansion as A i, we have p f ( x) = C l ( p x 0 x)( βl i ( ) Ox 0 )M i (O) l=1 i=1 } {{ } L l ( x 0 ) = p l=1 C l ( x 0 x)l l ( x 0 ) x x 0 L l ( x 0 ) is the lth local expansion coefficient about x 0 and is computed in O(p 2 ) time O Multipole Center 29 Copyright c 2005 Sharat Chandran Jan 9 11

53 Page 30 of 43 Contribution: Local Expansion for the Radiosity Kernel If Ox 0 > x 0 x B(x) E(x) = k/2 k 2s k=0 s=0 m 1 = k+2s E ksm1 (x 0 x) L ksm1 (Ox 0 ) E ksm1 (x 0 x) = 1 A x A x ρ(x) x 0 x k Y m 1 k 2s (x 0x)RM(x 0 x)da x L ksm1 (Ox 0 ) = [n/2] n 2 j n=0 j=0 m= n+2 j l max enj(...)f j n jm ksm 1 l 2 (Ox 0 ) Mn m j(o) l 2 =l min F n jm ksm 1 l 2 (Ox 0 ) = Ox 0 n 4 k Y m m 1 l 2 (Ox 0 )TM(Ox 0 ) 30 Copyright c 2005 Sharat Chandran Jan 9 11

54 Page 31 of 43 N logn to N : Local Expansion At each level, multipole expansions of each box in its interaction list was evaluated at a particle 27N plogn Revised Procedure For each box at each level, combine multipole moments of the boxes in its interaction list into local coefficients at its center and evaluate for each particle d i=3 27(4 i p 2 ) + N p) 36p 2 N/M + N plogn Why evaluate at each level? 31 Copyright c 2005 Sharat Chandran Jan 9 11

55 7. Step 4: Translation of Local Expansion Collect local expansion coefficients but do not evaluate M(O ) 1 M(O ) 2 M(O ) 3 Page 32 of 43

56 7. Step 4: Translation of Local Expansion Collect local expansion coefficients but do not evaluate M(O ) 1 M(O ) 2 M(O ) 3 Shift the center of the local expansion of a box to each of its children Page 32 of Copyright c 2005 Sharat Chandran Jan 9 11

57 Translation of Local Expansion : Math To calculate f at x, we now have f ( x) = p i=1 C i ( x 0 x)l i ( x 0 ) x 0 To calculate f at x in terms of a shifted center x 1, we expand r i as C i ( x 0 x) = p j=1 C j( x 1 x)γ i j ( x 0 x 1 ) x x 1 Page 33 of 43 Substituting this expansion of C i f ( x) = p j=1 C j ( p x 1 x)( i=1 γ i j( x 0 x 1 )L i ( x 0 ) ) = p j=1 C j ( x 1 x)l j (x 1 ) Thus we can evaluate f in terms of the translated local expansion coefficients L j (x 1 ) = p i=1 γi j ( x 0 x 1 )L i ( x 0 ) in O(p 2 ) time 33 Copyright c 2005 Sharat Chandran Jan 9 11

58 Contribution: Local Expansion Translation for the Radiosity Kernel Page 34 of 43 B(x) E(x) = L k s m 1 (x 0x 1 ) = k/2 k 2s k=0 s=0 m 1 = k+2s k/2 k 2s k=k s=0 m 1 = k+2s E ksm1 (x 1 x) L ksm1 (x 0 x 1 ) j max j = j min ( 1) k+m 1 e j k k e s k J(...) x 1 x 0 k k Y m 1 +m 1 k k 2 j (x 1 x 0 )TM(x 0 x 1 ) L ksm1 (Ox 0 ) 34 Copyright c 2005 Sharat Chandran Jan 9 11

59 Page 35 of 43 N logn to N : Local Expansion Translation For each box at each level, combine multipole moments of the boxes in its interaction list into local coefficients at its center and evaluate for each particle 36p 2 N/M + N plogn Revised Procedure For each box at each level, combine multipole moments of the boxes in its interaction list into local coefficients and shift them to its children d i=3 27(4 i p 2 ) + 4 i p 2 36p 2 N/M + 4p 2 N/3M At the last level, evaluate the local expansion in each of the boxes at its constituent particles and compute direct interaction with near neighbors 9M 2 (N/M) + (pm)(n/m) = 9NM + pn We are in O(N)! 35 Copyright c 2005 Sharat Chandran Jan 9 11

60 8. Step 1 Construct an Octree Subdivide the space containing the whole system as an octree until the leaves contain not more than a constant number of bodies k Level 0 Page 36 of 43 Level 1 Level 0 36 Copyright c 2005 Sharat Chandran Jan 9 11

61 The Fast Multipole Algorithm Step 2 Compute s Two cells are nearest neighbors if they are at the same refinement level and are separated by not more than one cell. Two cells are well separated if they are at the same refinement level and are not nearest neighbors With each cell i, associate an interaction list consisting of children of nearest neighbors of i s parents which are well separated from cell i X X Page 37 of Copyright c 2005 Sharat Chandran Jan 9 11

62 The Fast Multipole Algorithm The Fast Multipole Method Step 3 Compute Multipole Moments (Upward) Compute the multipole moments M m n for each leaf cell, at the center of the cell, due to the particles contained within the cell. For a non leaf cell, translate and aggregate the multipole moments of its children to its center. This is repeated at each level in an upward pass at the end of which, we have the multipole moments in each cell due to the particles contained in that cell. Page 38 of Copyright c 2005 Sharat Chandran Jan 9 11

63 The Fast Multipole Algorithm The Fast Multipole Method Step 4 Compute Local Expansion Coefficients (Downward) Starting from the root cell, for each cell at level l, the multipole moments of all the cells in its interaction list are translated to local expansion coefficients about the center of the cell. The local expansion coefficients of the parent are then translated to the center of this cell and aggregated. Page 39 of Copyright c 2005 Sharat Chandran Jan 9 11

64 The Fast Multipole Algorithm Page 40 of 43 Step 5 Final Evaluation For each particle in the system, evaluate the local expansion using the local expansion coefficients of the cell to which it belongs. Interaction with all particles in its nearest neighbors and its parent cell are computed directly. The contribution from these sources is added to get the radiosity at that particle. 40 Copyright c 2005 Sharat Chandran Jan 9 11

65 Handling Occlusions Visibility is a point to point phenomenon, and is not analytical Page 41 of Copyright c 2005 Sharat Chandran Jan 9 11

66 Page 42 of 43 Handling Occlusions in FMM Procedure Modify(Box A) { visible interactionlist(a)=null for each box B old interactionlist(a){ state=visibility(a,b) if equals(state,valid) then visible interactionlist(a).include(b) else if equals(state,partial) then { if(notleaf(a)) for each a child(a) for each b child(b) interactionlist(a).include(b)}}} 42 Copyright c 2005 Sharat Chandran Jan 9 11

67 Handling Occlusions in FMM The Fast Multipole Method Procedure visibility(box A, Box B) { visible=0; for each cell a leafcell(a) for each cell b leafcell(b){ if FacingEachOther(a,b) then { result=shootanddetect(a,b) if equals(result,0) then Increment(visible,1) }} if equals(visible,0) return(invalid) else if equals(visible,leafcell(a).size*leafcell(b).size) return(valid); else return(partial) } Page 43 of 43 Procedure Generate(Box A){ Modify(A) for each a child(a) Generate(a)} 43 Copyright c 2005 Sharat Chandran Jan 9 11

68 Page 44 of 43 Conclusion A quick introduction to FMM and Global Illumination (GI) Reduction of the GI problem to problems similar to FMM Four new theorems for the radiosity kernel Also supporting implementation 44 Copyright c 2005 Sharat Chandran Jan 9 11