Quasi-Monte Carlo Methods in Photorealistic Image Synthesis

1. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis Alexander Keller Leonhard Grünschloß

2. QMC Methods in Photorealistic Image Synthesis Image Synthesis Light Transport Ray Tracing Quasi-Monte Carlo Points Quasi-Monte Carlo Rendering Techniques Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 2

3. Image Synthesis Photorealistic image synthesis by light transport simulation find all light transport paths that connect camera and light sources then sum up their contribution Image courtesy Delta Tracing with NVIDIA iray. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 3

4. Image Synthesis Light transport radiance = source radiance + transported radiance L = Le +Tf L Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 4

5. Image Synthesis Light transport radiance = source radiance + transported radiance L = Le +Tf L Fredholm integral equation of the 2nd kind L(x,ω) = e− R d(x,−ω) 0 σt (x−sω)ds · Le(xb,ω)+ Z S 2 −(xb) fr (ωi ,xb,ω)L(xb,ωi )cosθi dωi + Z d(x,−ω) 0 e− R s 0 σt (x−tω)dt Le,v (x −sω,ω) + σs(x −sω) Z S 2 p(x −sω,ωi ·ω)L(x −sω,ωi )dωi ds Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 4

6. Image Synthesis Light transport radiance = source radiance + transported radiance L = Le +Tf L Fredholm integral equation of the 2nd kind ∂V x∂V x−sω ω x Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 5

7. Image Synthesis Assemble paths by tracing rays from origin x into direction ω find first intersection h(x,ω) with boundary determine mutual visibility V(x,y) of two points x and y Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 6

8. Image Synthesis Assemble paths by tracing rays from origin x into direction ω find first intersection h(x,ω) with boundary determine mutual visibility V(x,y) of two points x and y ray-primitive intersection – surfaces often approximated by tessellation, for example triangles – ill-posed problem, unless original surface can be reconstructed – consistent numerics to ameliorate numerical issues Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 6

9. Image Synthesis References P. Shirley, Morley: Realistic Ray Tracing, 2nd Ed., AK Peters 2003. A. Glassner: An Introduction to Ray Tracing, Morgan Kaufmann, 1989. C. Wächter: Quasi-Monte Carlo Light Transport Simulation by Efficient Ray Tracing, PhD thesis, 2007. J. Hanika: Spectral Light Transport Simulation using a Precision-Based Ray Tracing Architecture, PhD thesis, 2010. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 7

10. QMC Methods in Photorealistic Image Synthesis Image Synthesis Quasi-Monte Carlo Points Halton sequence and Hammersley points (t,s)-sequences and (t,m,s)-nets in base b Rank-1 lattice sequences and rank-1 lattices Implementation Quasi-Monte Carlo Rendering Techniques Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 8

11. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

12. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. consistent integro-approximation by uniform sampling I(x) = Z [0,1)s f(x,y)dy Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

13. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. consistent integro-approximation by uniform sampling I(x) = Z [0,1)s f(x,y)dy = lim n→∞ 1 n n−1 ∑ i=0 f(x,yi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

14. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. consistent integro-approximation by uniform sampling I(x) = Z [0,1)s f(x,y)dy = lim n→∞ 1 n n−1 ∑ i=0 f(x,yi ) ≈ 1 n n−1 ∑ i=0 f(x,yi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

15. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. consistent integro-approximation by uniform sampling I(x) = Z [0,1)s f(x,y)dy = lim n→∞ 1 n n−1 ∑ i=0 f(x,yi ) ≈ 1 n n−1 ∑ i=0 f(x,yi ) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

16. Quasi-Monte Carlo Points Quasi-Monte Carlo methods ’For every randomized algorithm, there is a clever deterministic one.’ Harald Niederreiter, Claremont, 1998. consistent integro-approximation by uniform sampling I(x) = Z [0,1)s f(x,y)dy = lim n→∞ 1 n n−1 ∑ i=0 f(x,yi ) ≈ 1 n n−1 ∑ i=0 f(x,yi ) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q pseudo-random low discrepancy Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 9

17. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

18. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

19. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

20. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

21. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t t Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

22. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t t properties – subsequent points that “fall into biggest holes” – not completely uniform distributed (CUD) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

23. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t t d d d d properties – subsequent points that “fall into biggest holes” – not completely uniform distributed (CUD) – contiguous blocks of stratified points xi for kbm ≤ i (k +1)bm −1 for each block the Φb(i) are equidistant for each block the integers bbm Φb(i)c are a permutation of {0,...,bm −1} Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

24. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t t t t t t properties – subsequent points that “fall into biggest holes” – not completely uniform distributed (CUD) – contiguous blocks of stratified points xi for kbm ≤ i (k +1)bm −1 for each block the Φb(i) are equidistant for each block the integers bbm Φb(i)c are a permutation of {0,...,bm −1} Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

25. Quasi-Monte Carlo Points Radical inversion van der Corput sequence in base b Φb : N0 → Q∩[0,1) i = ∞ ∑ l=0 al (i)bl 7→ Φb(i) := ∞ ∑ l=0 al (i)b−l−1 – example Φ2(i): t t t t t t t t t t t t t t t t properties – subsequent points that “fall into biggest holes” – not completely uniform distributed (CUD) – contiguous blocks of stratified points xi for kbm ≤ i (k +1)bm −1 for each block the Φb(i) are equidistant for each block the integers bbm Φb(i)c are a permutation of {0,...,bm −1} Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 10

26. Quasi-Monte Carlo Points Radical inversion based points for dimensions s 1 let the bj be co-prime, for example the j-th prime number Halton sequence xi := Φb1 (i),...,Φbs (i) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (Φ2(i),Φ3(i))63 i=0 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 11

27. Quasi-Monte Carlo Points Radical inversion based points for dimensions s 1 let the bj be co-prime, for example the j-th prime number Halton sequence Hammersley point sets xi := Φb1 (i),...,Φbs (i) xi := i n ,Φb1 (i),...,Φbs−1 (i) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (Φ2(i),Φ3(i))63 i=0 ( i 64 ,Φ2(i))63 i=0 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 11

28. Quasi-Monte Carlo Points Radical inversion based points for dimensions s 1 let the bj be co-prime, for example the j-th prime number Halton sequence Hammersley point sets xi := Φb1 (i),...,Φbs (i) xi := i n ,Φb1 (i),...,Φbs−1 (i) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (Φ17(i),Φ19(i))63 i=0 ( i 64 ,Φ2(i))63 i=0 – correlations in low dimensional projections Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 11

29. Quasi-Monte Carlo Points Radical inversion based points for dimensions s 1 let the bj be co-prime, for example the j-th prime number Halton sequence Hammersley point sets xi := Φb1 (i),...,Φbs (i) xi := i n ,Φb1 (i),...,Φbs−1 (i) a a a a a a a a a a a a a a a a a q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (Φ17(i),Φ19(i))63 i=0 ( i 64 ,Φ2(i))63 i=0 – correlations in low dimensional projections Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 11

30. Quasi-Monte Carlo Points Scrambling algorithm: start with H = Is and for each axis j 1. slice H into bj equally sized volumes H1,H2,...,Hbj along the axis 2. permute these volumes 3. for each Hh recursively repeat the procedure with H = Hh Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 12

31. Quasi-Monte Carlo Points Scrambling algorithm: start with H = Is and for each axis j 1. slice H into bj equally sized volumes H1,H2,...,Hbj along the axis 2. permute these volumes 3. for each Hh recursively repeat the procedure with H = Hh many variants, simplifications, and generalizations – example: XOR-scrambling unit square [0,1)2 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 12

32. Quasi-Monte Carlo Points Scrambling algorithm: start with H = Is and for each axis j 1. slice H into bj equally sized volumes H1,H2,...,Hbj along the axis 2. permute these volumes 3. for each Hh recursively repeat the procedure with H = Hh many variants, simplifications, and generalizations – example: XOR-scrambling bit 1 of x Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 12

35. Quasi-Monte Carlo Points Scrambling algorithm: start with H = Is and for each axis j 1. slice H into bj equally sized volumes H1,H2,...,Hbj along the axis 2. permute these volumes 3. for each Hh recursively repeat the procedure with H = Hh many variants, simplifications, and generalizations – example: XOR-scrambling all bits of x Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 12

36. Quasi-Monte Carlo Points Scrambling algorithm: start with H = Is and for each axis j 1. slice H into bj equally sized volumes H1,H2,...,Hbj along the axis 2. permute these volumes 3. for each Hh recursively repeat the procedure with H = Hh many variants, simplifications, and generalizations – example: XOR-scrambling all bits of x and y Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 12

37. Quasi-Monte Carlo Points Scrambled radical inversion example: deterministic permutations σb by Faure i = ∞ ∑ j=0 aj (i)bj 7→ ∞ ∑ j=0 σb(aj (i))b−j−1 – b is even: Take 2σb 2 and append 2σb 2 +1 – b is odd: Take σb−1, increment each value ≥ b−1 2 and insert b−1 2 in the middle σ2 = (0,1) σ3 = (0,1,2) σ4 = (0,2,1,3) σ5 = (0,3,2,1,4) σ6 = (0,2,4,1,3,5) . . . Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 13

38. Quasi-Monte Carlo Points Scrambled radical inversion example: deterministic permutations σb by Faure i = ∞ ∑ j=0 aj (i)bj 7→ ∞ ∑ j=0 σb(aj (i))b−j−1 – b is even: Take 2σb 2 and append 2σb 2 +1 – b is odd: Take σb−1, increment each value ≥ b−1 2 and insert b−1 2 in the middle σ2 = (0,1) σ3 = (0,1,2) σ4 = (0,2,1,3) σ5 = (0,3,2,1,4) σ6 = (0,2,4,1,3,5) . . . q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q → q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 13

39. Quasi-Monte Carlo Points Properties of radical inversion subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

40. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

41. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

42. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

43. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

44. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

45. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

46. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: van der Corput sequence xi = Φ2(i) q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

47. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

48. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

49. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

50. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

51. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

52. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

53. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

54. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

55. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

56. Quasi-Monte Carlo Points Properties of radical inversion in dimensions s 1 subsequent points that “fall into biggest holes” stratification invariant under scrambling contiguous blocks of stratified points xi for k s ∏ j=1 b mj j ≤ i (k +1) s ∏ j=1 b mj j −1 example: Halton sequence xi = (Φ2(i),Φ3(i)) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 14

57. Quasi-Monte Carlo Points (t,s)-sequences and (t,m,s)-nets in base b elementary interval E := s ∏ j=1 aj blj , aj +1 blj ⊆ Is for integers lj ≥ 0 and 0 ≤ aj blj with volume λs(E) = ∏s j=1 1 b lj = 1 b ∑s j=1 lj For two integers 0 ≤ t ≤ m, a finite point set of bm points in s dimensions is called a (t,m,s)-net in base b, if every elementary interval of volume λs(E) = bt−m contains exactly bt points. For t ≥ 0, an infinite point sequence is called a (t,s)-sequence in base b, if for all k ≥ 0 and m ≥ t, the vectors xkbm ,...,x(k+1)bm−1 ∈ Is form a (t,m,s)-net. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 15

58. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

59. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

60. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

61. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

62. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

63. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

64. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

65. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

66. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

67. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

72. Quasi-Monte Carlo Points (t,s)-sequences are sequences of (t,m,s)-nets in base b example: stratification properties of the Sobol’ (0,2)-sequence in base 2 – the sequence of (0,3,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – the sequence of (0,4,2)-nets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – all components of the Sobol’ sequence are (0,1)-sequences in base 2 ⇒ deterministic LHS Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 16

73. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

74. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

75. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q a Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

76. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

77. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

78. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

79. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

80. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q generator vectors – Korobov form (1,a,a2,a3,...) – rare constructions example: Fibonacci lattice with n = Fk and (g0,g1) = (1,Fk−1) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

81. Quasi-Monte Carlo Points Rank-1 lattices given generator vector (g0,...,gs−1) ∈ Ns xi := i n (g0,...,gs−1) mod [0,1)s q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q generator vectors – Korobov form (1,a,a2,a3,...) – rare constructions example: Fibonacci lattice with n = Fk and (g0,g1) = (1,Fk−1) – usually tabulated coefficients a or gj search by certain criteria, e.g. maximized minimum distance, projections, . . . component by component construction (CBC) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 17

82. Quasi-Monte Carlo Points Rank-1 lattice sequences replace i n by radical inverse xi = φb(i)·(g0,...,gs−1) mod [0,1)s Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 18

83. Quasi-Monte Carlo Points Rank-1 lattice sequences replace i n by radical inverse xi = φb(i)·(g0,...,gs−1) mod [0,1)s xkbm ,...,x(k+1)bm−1 form a shifted lattice – shift ∆ in the k +1st block for n = bm φb(i +kbm )·g = φb(i)+φb(kbm ) ·g = φb(i)·g+φb(k)b−m−1 g | {z } =:∆ q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 18

84. Quasi-Monte Carlo Points Rank-1 lattice sequences replace i n by radical inverse xi = φb(i)·(g0,...,gs−1) mod [0,1)s xkbm ,...,x(k+1)bm−1 form a shifted lattice – shift ∆ in the k +1st block for n = bm φb(i +kbm )·g = φb(i)+φb(kbm ) ·g = φb(i)·g+φb(k)b−m−1 g | {z } =:∆ q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 18

85. Quasi-Monte Carlo Points Rank-1 lattice sequences replace i n by radical inverse xi = φb(i)·(g0,...,gs−1) mod [0,1)s xkbm ,...,x(k+1)bm−1 form a shifted lattice – shift ∆ in the k +1st block for n = bm φb(i +kbm )·g = φb(i)+φb(kbm ) ·g = φb(i)·g+φb(k)b−m−1 g | {z } =:∆ q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 18

86. Quasi-Monte Carlo Points Rank-1 lattice sequences replace i n by radical inverse xi = φb(i)·(g0,...,gs−1) mod [0,1)s xkbm ,...,x(k+1)bm−1 form a shifted lattice – shift ∆ in the k +1st block for n = bm φb(i +kbm )·g = φb(i)+φb(kbm ) ·g = φb(i)·g+φb(k)b−m−1 g | {z } =:∆ q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q similar to (t,s)-sequences – for b and gj relatively prime, φb(i)gj mod [0,1) are (0,1)-sequences Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 18

87. Quasi-Monte Carlo Points Uniformity of a point set Pn := {x0,...,xn−1} ∈ [0,1)s maximum minimum distance on torus dmin(Pn) := min 0≤in min ijn kxj −xi kT q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 19

90. Quasi-Monte Carlo Points Uniformity of a point set Pn := {x0,...,xn−1} ∈ [0,1)s maximum minimum distance on torus dmin(Pn) := min 0≤in min ijn kxj −xi kT q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 19

108. Quasi-Monte Carlo Points Uniformity of a point set Pn := {x0,...,xn−1} ∈ [0,1)s maximum minimum distance on torus dmin(Pn) := min 0≤in min ijn kxj −xi kT low discrepancy D∗ (Pn) := sup A=∏s j=1[0,aj )⊆[0,1)s Z [0,1)s χA(x)dx − 1 n n−1 ∑ i=0 χA(xi ) ∈ O logs n n Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 19

109. Quasi-Monte Carlo Points Uniformity of a point set Pn := {x0,...,xn−1} ∈ [0,1)s maximum minimum distance on torus dmin(Pn) := min 0≤in min ijn kxj −xi kT low discrepancy D∗ (Pn) := sup A=∏s j=1[0,aj )⊆[0,1)s Z [0,1)s χA(x)dx − 1 n n−1 ∑ i=0 χA(xi ) ∈ O logs n n Let (X,B,µ) be an arbitrary probability space and let M be a nonempty subset of B. A point set Pn of n elements of X is called (M ,µ)-uniform if n−1 ∑ i=0 χM (xi ) = µ(M)·n for all M ∈ M , where χM (xi ) = 1 if xi ∈ M, zero otherwise. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 19

110. Quasi-Monte Carlo Points Low discrepancy, minimum distance, and (M ,µ)-uniformity examples Hammersley Sobol‘ sequence Larcher-P’hammer Rank-1 lattice q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q dmin = 0.0883883 dmin = 0.0883883 dmin = 0.176777 dmin = 0.25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q dmin = 0.0220971 dmin = 0.0441942 dmin = 0.0883883 dmin = 0.125973 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 20

111. Quasi-Monte Carlo Points Lesson learned consistency realized by uniform sequences, which may be deterministic – Halton, (t,s)-, and rank-1 lattice sequences more uniform than random samples can be – deterministic samples can be constructed as progressive Latin hypercube samples common principle: radical inversion – isomorphy to permutations Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 21

112. Quasi-Monte Carlo Points Efficient generation of the Faure-scrambled Halton sequence efficient implementations of radical inversion double IntegerRadicalInverse(int Base, int i) { int numPoints, inverse; numPoints = 1; for(inverse = 0; i 0; i /= Base) { inverse = inverse * Base + (i % Base); numPoints = numPoints * Base; } return (double) inverse / (double) numPoints; } double RadicalInverse(const int Base, int i) { double Digit, Radical, Inverse; Digit = Radical = 1.0 / (double) Base; Inverse = 0.0; while(i) { Inverse += Digit * (double) (i % Base); Digit *= Radical; i /= Base; } return Inverse; } Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 22

113. Quasi-Monte Carlo Points Efficient generation of the Faure-scrambled Halton sequence creating one look-up table for multiple digits results in compact and branchless code – example: σ5 = (0,3,2,1,4) σ5 ×σ5 =          (0,0) (0,3) (0,2) (0,1) (0,4) (3,0) (3,3) (3,2) (3,1) (3,4) (2,0) (2,3) (2,2) (2,1) (2,4) (1,0) (1,3) (1,2) (1,1) (1,4) (4,0) (4,3) (4,2) (4,1) (4,4)          Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 23

114. Quasi-Monte Carlo Points Efficient generation of the Faure-scrambled Halton sequence creating one look-up table for multiple digits results in compact and branchless code – example: σ5 = (0,3,2,1,4) σ5 ×σ5 =          (0,0) (0,3) (0,2) (0,1) (0,4) (3,0) (3,3) (3,2) (3,1) (3,4) (2,0) (2,3) (2,2) (2,1) (2,4) (1,0) (1,3) (1,2) (1,1) (1,4) (4,0) (4,3) (4,2) (4,1) (4,4)          ∼ =          0 3 2 1 4 25 28 27 26 29 10 13 12 11 14 5 8 7 6 9 20 23 22 21 24          Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 23

115. Quasi-Monte Carlo Points Efficient generation of the Faure-scrambled Halton sequence creating one look-up table for multiple digits results in compact and branchless code – example: σ5 = (0,3,2,1,4), for b = 5 and 3 digits, i.e. σ5 ×σ5 ×σ5 static const unsigned short perm5[] = { 0, 75, 50, 25, 100, 15, 90, 65, 40, 115, 10, 85, 60, 35, 110, 5, 80, 55, 30, 105, 20, 95, 70, 45, 120, 3, 78, 53, 28, 103, 18, 93, 68, 43, 118, 13, 88, 63, 38, 113, 8, 83, 58, 33, 108, 23, 98, 73, 48, 123, 2, 77, 52, 27, 102, 17, 92, 67, 42, 117, 12, 87, 62, 37, 112, 7, 82, 57, 32, 107, 22, 97, 72, 47, 122, 1, 76, 51, 26, 101, 16, 91, 66, 41, 116, 11, 86, 61, 36, 111, 6, 81, 56, 31, 106, 21, 96, 71, 46, 121, 4, 79, 54, 29, 104, 19, 94, 69, 44, 119, 14, 89, 64, 39, 114, 9, 84, 59, 34, 109, 24, 99, 74, 49, 124 }; inline float halton5(const unsigned index) { return (perm5[index % 125u] * 1953125u + perm5[(index / 125u) % 125u] * 15625u + perm5[(index / 15625u) % 125u] * 125u + perm5[(index / 1953125u) % 125u]) * (0x1.fffffep-1 / 244140625u); // For results 1. } Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 24

116. Quasi-Monte Carlo Points Digital (t,s)-sequences in base b use a generator matrix Cj for each component finite precision m x (j) i =      b−1 . . . b−m      T Cj      a0(i) . . . am−1(i)      | {z } multiplication in Fb ∈ [0,1) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 25

117. Quasi-Monte Carlo Points Digital (t,s)-sequences in b = 2 x (j) i =      2−1 . . . 2−m      T Cj      a0(i) . . . am−1(i)      | {z } multiplication in F2 ∈ [0,1) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 26

118. Quasi-Monte Carlo Points Digital (t,s)-sequences in b = 2 x (j) i =      2−1 . . . 2−m      T Cj      a0(i) . . . am−1(i)      | {z } multiplication in F2 ∈ [0,1) example: vectorized radical inverse double RI(uint i, uint r = 0) { for (uint k = 0; i; i = 1, ++k) if (i 1) r ^= C[k]; // SIMD addition of column return (double) r / (double) (1 M); } Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 26

119. Quasi-Monte Carlo Points Digital (t,s)-sequences in b = 2 x (j) i =      2−1 . . . 2−m      T Cj      a0(i) . . . am−1(i)      | {z } multiplication in F2 ∈ [0,1) example: vectorized radical inverse by Larcher and Pillichshammer double RI LP(uint i, uint r = 0) { for (uint v = 1u 31; i; i = 1, v |= v 1) if (i 1) r ^= v; return (double) r / (double) 0x100000000ull; } Cj =         1 1 1 ... 0 1 1 ... 0 0 1 ... . . . . . . . . . ...         Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 26

120. Quasi-Monte Carlo Points Practical considerations radical inverse components easily “run out of bits”            1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ...                       1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 ...            φ2 second Sobol’ component changing the most significant index bits changes the result – slightly for Halton points – completely for Sobol’ points Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 27

121. Quasi-Monte Carlo Points Practical considerations radical inverse components easily “run out of bits”            1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ...                       1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 ...            φ2 second Sobol’ component changing the most significant index bits changes the result – slightly for Halton points – completely for Sobol’ points can use tiling on image plane for Halton Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 27

122. Quasi-Monte Carlo Points Hierarchical sample warping traverse kd-tree and rescale samples (probabilistic traversal) Images from P. Clarberg, W. Jarosz, T. Akenine-Möller, and H. Wann Jensen: “Wavelet Importance Sampling: Efficiently Evaluating Products of Complex Functions” Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 28

123. Quasi-Monte Carlo Points How to select split planes want to minimize expected number of traversal steps E[S] – Huffman trees are optimal Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 29

124. Quasi-Monte Carlo Points How to select split planes want to minimize expected number of traversal steps E[S] – Huffman trees are optimal heuristic to preserve spatial connectivity: split by minimizing estimated entropy H E[S] ≈ 1+pleft ·H(Tleft)+pright ·H(Tright) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 29

125. Quasi-Monte Carlo Points How to select split planes want to minimize expected number of traversal steps E[S] – Huffman trees are optimal heuristic to preserve spatial connectivity: split by minimizing estimated entropy H E[S] ≈ 1+pleft ·H(Tleft)+pright ·H(Tright) – use SAT for finding split planes efficiently traversal depths (small / large) input mid-split entropy heuristic optimal kd-tree Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 29

126. Quasi-Monte Carlo Points Implementation F. Kuo: Sobol sequence generator matrices. http://web.maths.unsw.edu.au/~fkuo/sobol/index.html F. Kuo: Lattice rule generating vectors. http://web.maths.unsw.edu.au/~fkuo/lattice/index.html Fast Sobol’ sequence generator (including pixel enumeration), inverse matrices, and Faure scrambled Halton sampler http://gruenschloss.org/ Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 30

127. Quasi-Monte Carlo Points References H. Niederreiter: Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Pennsylvania, 1992. C. Lemieux: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics, New York, 2009. J. Dick, F. Pillichshammer: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, 2010. proceedings of the MCQMC conference series. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 31

128. QMC Methods in Photorealistic Image Synthesis Image Synthesis Quasi-Monte Carlo Points Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization Motion blur Scattering Connecting path segments by shadow rays Connecting path segments by proximity Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 32

129. Quasi-Monte Carlo Rendering Techniques Path tracing generation of path segments P L Eye Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 33

130. Quasi-Monte Carlo Rendering Techniques Path tracing connecting path segments by a shadow rays P L Eye – works for L 2 – weak singularity needs to be handled Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 33

131. Quasi-Monte Carlo Rendering Techniques Path tracing connecting path segments by proximity P L Eye – required for singular surfaces 6∈ L 2, so-called “SDS” paths – requires multiple photons to be efficient Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 33

132. Quasi-Monte Carlo Rendering Techniques Path tracing efficient algorithms require both connection techniques P L Eye – simple algorithm: path space partitioning shadow rays to connect diffuse-diffuse photon mapping to connect diffuse-specular Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 33

133. Quasi-Monte Carlo Rendering Techniques Path tracing mapping the components of a quasi-Monte Carlo point sequence s i x0 x1 x2 x3 x4 x5 x6 x7 x8 – consistent integro-approximation LP = lim n→∞ 1 n n−1 ∑ i=0 fP (xi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 34

134. Quasi-Monte Carlo Rendering Techniques Path tracing mapping the components of a quasi-Monte Carlo point sequence s i x0 x1 x2 x3 x4 x5 x6 x7 x8 – consistent integro-approximation LP = lim n→∞ |P| n n−1 ∑ i=0 χP (xi,1,xi,2)f(xi,1,xi,2,xi,3,xi,4,xi,5,...) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 34

135. Quasi-Monte Carlo Rendering Techniques Path tracing mapping the components of a quasi-Monte Carlo point sequence s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 – consistent integro-approximation LP = lim n→∞ |P| n n−1 ∑ i=0 χP (xi,1,xi,2)f(xi,1,xi,2,xi,3,xi,4,xi,5,...,yi,1,yi,2,yi,3,yi,4,...) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 34

136. Quasi-Monte Carlo Rendering Techniques Path tracing mapping the components of a quasi-Monte Carlo point sequence s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 – consistent integro-approximation LP ≈ |P| n n−1 ∑ i=0 χP (xi,1,xi,2)f(xi,1,xi,2,xi,3,xi,4,xi,5,...,yi,1,yi,2,yi,3,yi,4,...) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 34

137. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

138. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

139. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

140. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

141. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

142. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

143. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

144. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

145. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

146. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

147. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

148. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

149. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

150. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

151. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

152. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

153. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

154. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

155. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

156. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

157. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

158. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

159. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

160. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

161. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

162. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – one sequence to cover whole image stratified, minimum distance, Latin hypercube, and low discrepancy Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

163. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization quasi-Monte Carlo integro-approximation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q – one sequence to cover whole image stratified, minimum distance, Latin hypercube, and low discrepancy – adaptively enumerate samples in elementary intervals in parallel example: Halton sequence and (0,2)-sequences in base b Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 35

164. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization allows for consistent stratified adaptive sampling Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 36

165. Quasi-Monte Carlo Rendering Techniques Anti-aliasing and deterministic parallelization partition one low discrepancy sequence along one dimension – projection yields multiple low discrepancy sequences – for Sobol’ and φ2 this results in simple leap-frogging for 2m partitions Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 37

175. Quasi-Monte Carlo Rendering Techniques Motion blur simulate aperture interval – too expensive: different time per image sample – desirable: one instant in time for whole (0,m,2)-net Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 38

176. Quasi-Monte Carlo Rendering Techniques Motion blur splitting by replicating the function bm times along temporal dimension f → f f f f Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 39

177. Quasi-Monte Carlo Rendering Techniques Motion blur splitting by replicating the function bm times along temporal dimension f → f f f f Theorem: Given an s-dimensional (M ,µ)-uniform point sequence (xi ,ti ), where the component ti is a (0,1)-sequence generated by an upper triangular matrix, Z [0,1)s f(x,t)dtdx = Z [0,1)s bm −1 ∑ j=0 χ[ j bm , j+1 bm ) (t)f(x,bm t −j)dtdx Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 39

178. Quasi-Monte Carlo Rendering Techniques Motion blur splitting by replicating the function bm times along temporal dimension f → f f f f Theorem: Given an s-dimensional (M ,µ)-uniform point sequence (xi ,ti ), where the component ti is a (0,1)-sequence generated by an upper triangular matrix, Z [0,1)s f(x,t)dtdx = Z [0,1)s bm −1 ∑ j=0 χ[ j bm , j+1 bm ) (t)f(x,bm t −j)dtdx = lim n→∞ 1 n n−1 ∑ i=0 f(xi ,tbi/bmc) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 39

179. Quasi-Monte Carlo Rendering Techniques Splitting can increase efficiency by enumerating components at different speeds s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 40

180. Quasi-Monte Carlo Rendering Techniques Scattering evaluation and simulation of scattering Phong model silver metallic paint violet rubber Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 41

270. Quasi-Monte Carlo Rendering Techniques Scattering evaluation and simulation of scattering Phong model silver metallic paint violet rubber Measured data from “A Data-Driven Reflectance Model”, W. Matusik, H. Pfister, M. Brand and L. McMillan, ACM Transactions on Graphics 22, 3(2003), 759-769. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 41

271. Quasi-Monte Carlo Rendering Techniques Scattering evaluation and simulation of scattering Phong model silver metallic paint violet rubber Measured data from “A Data-Driven Reflectance Model”, W. Matusik, H. Pfister, M. Brand and L. McMillan, ACM Transactions on Graphics 22, 3(2003), 759-769. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 41

272. Quasi-Monte Carlo Rendering Techniques Scattering construction of alias map in O(n) by 40 lines of sequential code 0 1 2 3 4 5 ⇒ 0 1 2 3 4 5 a0 = 4 (a1 = 1) a2 = 1 a3 = 2 a4 = 1 a5 = 2 density alias map Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 42

273. Quasi-Monte Carlo Rendering Techniques Scattering construction of alias map in O(n) by 40 lines of sequential code 0 1 2 3 4 5 ⇒ 0 1 2 3 4 5 a0 = 4 (a1 = 1) a2 = 1 a3 = 2 a4 = 1 a5 = 2 density alias map SIMD sample generation in O(1) – superior to inversion method with O(log2 n) divergent memory accesses – perfectly fits permutations generated by low discrepancy sequences Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 42

274. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases 0 1 2 3 4 5 a0 = 4 (a1 = 1) a2 = 1 a3 = 2 a4 = 1 a5 = 2 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

275. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases Phong model diffuse lobe glossy lobe = + Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

276. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases Phong model diffuse lobe glossy lobe = + evaluation scattering Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

277. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases violet rubber diffuse lobe glossy lobe = + evaluation scattering Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

278. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases Phong model = + evaluation scattering Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

279. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases violet rubber = + evaluation scattering Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

280. Quasi-Monte Carlo Rendering Techniques Scattering partition BRDF data into integration and scattering by alias map more diffuse: evaluate/average clamped density more glossy: simulate aliases silver metallic = + evaluation scattering Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 43

281. Quasi-Monte Carlo Rendering Techniques Simulation of light sources already explored for light sources on the OMPF forum investigations with respect to deterministic low discrepancy sampling Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 44

282. Quasi-Monte Carlo Rendering Techniques Simulation of light sources already explored for light sources on the OMPF forum investigations with respect to deterministic low discrepancy sampling Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 44

283. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays trace light paths and store vertices (xj ,Lj (xj → ·))M j=1 as point lights Light Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 45

284. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays trace light paths and store vertices (xj ,Lj (xj → ·))M j=1 as point lights Light Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 45

285. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays trace camera paths Light Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 45

286. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays trace shadow rays to illuminate camera paths by points lights Light Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 45

287. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays trace shadow rays to illuminate camera paths by points lights Light L(y,z) ≈ Le(y,z)+ M ∑ j=1 Lj (xj → y)fr (xj ,y,z)V(xj ,y) cosθxj cosθy |xj −y|2 – numerical issues for small distances |xj −y|2 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 45

288. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays biased result of only bounding the weak singularity Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 46

289. Quasi-Monte Carlo Rendering Techniques Connecting path segments by shadow rays consistent simulation algorithm Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 47

290. Quasi-Monte Carlo Rendering Techniques Connecting path segments by proximity consistent photon mapping s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 LP = lim n→∞ 1 n n−1 ∑ i=0 fP(xi ,yi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 48

291. Quasi-Monte Carlo Rendering Techniques Connecting path segments by proximity consistent photon mapping s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 LP = lim n→∞ 1 n n−1 ∑ i=0 χB(r(n))(h(xi )−h(yi )) πr2(n) ·f0 P(xi ,yi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 48

292. Quasi-Monte Carlo Rendering Techniques Connecting path segments by proximity consistent photon mapping s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 LP = lim n→∞ 1 n n−1 ∑ i=0 χB(r(n))(h(xi )−h(yi )) πr2(n) ·f0 P(xi ,yi ) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 48

293. Quasi-Monte Carlo Rendering Techniques Connecting path segments by proximity consistent photon mapping s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 LP = lim n→∞ 1 n n−1 ∑ i=0 1 bm bm −1 ∑ k=0 χB(r(n))(h(xi )−h(ybmbi/bmc+k )) πr0(xi )2 ·n−α ·f0 P(xi ,ybmbi/bmc+k ) – variation of trajectory splitting, relies on (0,1)-sequences Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 48

294. Quasi-Monte Carlo Rendering Techniques Comparison at identical number of paths stochastic progressive photon mapping Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 49

295. Quasi-Monte Carlo Rendering Techniques Comparison at identical number of paths consistent block-wise QMC photon mapping (Sobol’ sequence) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 49

296. Quasi-Monte Carlo Rendering Techniques Comparison at identical number of paths stochastic progressive photon mapping Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 49

297. Quasi-Monte Carlo Rendering Techniques Comparison at identical number of paths consistent block-wise QMC photon mapping (Sobol’ sequence) Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 49

298. Quasi-Monte Carlo Rendering Techniques References E. Veach: Robust Monte Carlo Methods for Light Transport Simulation, PhD Thesis, 1997. A. Glassner: Principles of Digital Image Synthesis, Morgan Kaufmann, on the internet for free. Pharr, Humphreys: Physically Based Rendering, Morgan Kaufmann 2010, 2nd ed. T. Kollig, A. Keller: Illumination in the Presence of Weak Singularities. A. Keller: Myths of Computer Graphics. A. Keller: Quasi-Monte Carlo Image Synthesis in a Nutshell. proceedings of the MCQMC conference series. Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 50

299. Quasi-Monte Carlo Rendering Techniques Lessons learned deterministic consistent image synthesis by quasi-Monte Carlo integro-approximation – push button – simple to parallelize and strictly reproducible – more efficient Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 51

300. Quasi-Monte Carlo Rendering Techniques Lessons learned deterministic consistent image synthesis by quasi-Monte Carlo integro-approximation – push button – simple to parallelize and strictly reproducible – more efficient efficient algorithms for low discrepancy sequences p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 51

301. Quasi-Monte Carlo Rendering Techniques Lessons learned deterministic consistent image synthesis by quasi-Monte Carlo integro-approximation – push button – simple to parallelize and strictly reproducible – more efficient efficient algorithms for low discrepancy sequences p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p efficient algorithms to enumerate and use low discrepancy sequences s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 s1 s2 i x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 x7 y7 x8 y8 Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 51

302. Results Push-button rendering in NVIDIA iray R brilliant results without tweaking parameters Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 52

303. Results Push-button rendering in NVIDIA iray R even for pre-visualization Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 52

304. Results Push-button rendering in NVIDIA iray R even for pre-visualization Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 52

305. Results Push-button rendering in NVIDIA iray R physically based modeling with real lights and surfaces Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 52

308. Thank you for your attention ! Slides and course notes http://gruenschloss.org Credits Nikolaus Binder, Ken Dahm, Carsten Wächter, and Thomas Kollig Image credits Delta Tracing, Jeff Patton, 2020, Daniel Simon Acknowledgements David Luebke, Sebastian Sylwain, Luca Fascione Quasi-Monte Carlo Methods in Photorealistic Image Synthesis 53

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