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PATH CONSTRUCTION Iliyan Georgiev Solid Angle MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION PATH INTEGRAL FRAMEWORK Pixel value Z I j = f j ( x ) d x P Z Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N i =1

  1. PATH CONSTRUCTION Iliyan Georgiev Solid Angle MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

  2. PATH INTEGRAL FRAMEWORK Pixel value Z I j = f j ( x ) d x P Z Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N i =1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 2

  3. PATH INTEGRAL FRAMEWORK Pixel value Z I j = f j ( x ) d x P Z Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N i =1 path contribution path pdf MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 3

  4. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 4

  5. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera response MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 5

  6. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ response phase MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 6

  7. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ geometry response phase MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 7

  8. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ geometry transmittance response phase MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 8

  9. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ emitted geometry transmittance response phase radiance MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 9

  10. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 i =1 Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ emitted geometry transmittance response phase radiance MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 10

  11. PATH INTEGRAL FRAMEWORK Pixel value 1 x k − 1 Z I j = f j ( x ) d x 1 x k P Z x 0 Pixel estimator P N h I j i = 1 f j ( x i ) X p ( x i ) N 0 x 1 ideally i =1 proportional ) ∝ Path contribution "Y # f j ( x ) = W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) L e ( x k , x k − 1 ) i camera BSDF/ emitted geometry transmittance response phase radiance MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 11

  12. UNIDIRECTIONAL PATH SAMPLING ω 1 x 0 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 12

  13. UNIDIRECTIONAL PATH SAMPLING distance sampling ω 1 p ( t 1 | x 0 0 x 1 p ( t 1 | x 0 , ω 1 ) ∝ T ( x 0 , x 1 ) MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 13

  14. UNIDIRECTIONAL PATH SAMPLING ω 2 direction sampling p ( ω 2 | x 1 ) ∝ f s ( x 1 ) x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 14

  15. UNIDIRECTIONAL PATH SAMPLING ω 2 distance x 2 sampling p ( t 2 | p ( t 2 | x 1 , ω 2 ) ∝ T ( x 1 , x 2 ) x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 15

  16. UNIDIRECTIONAL PATH SAMPLING 4 x 5 3 x 4 A series of distance and direction sampling decisions x 2 x 0 0 x 1 2 x 3 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 16

  17. UNIDIRECTIONAL PATH SAMPLING cannot render A series of distance and 4 x 5 illumination from 3 x 4 direction sampling decisions point light sources x 2 high variance when light x 0 0 x 1 sources are small 2 x 3 not importance sampled "Y # # p ( x ) ∝ W j ( x 0 , x 1 ) f s ( x i ) G ( x i , x i +1 ) T ( x i , x i +1 ) ) L e ( x k , x k − 1 ) i MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 17

  18. EXPLICIT LIGHT SAMPLING x 2 x 0 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 18

  19. EXPLICIT LIGHT SAMPLING x 2 x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 19

  20. EXPLICIT: TRANSMITTANCE x 2 x 0 0 x 1 p ( t 1 | x 0 ) ∝ T ( x 0 , x 1 ) MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 20

  21. EXPLICIT: TRANSMITTANCE x 2 1 / G ( x 1 , x 2 ) = k x 1 , x 2 k 2 2 [0 , 1 ] T ( x 0 , x 1 ) ∈ [0 , 1] x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 21

  22. EXPLICIT: EQUIANGULAR x 2 x 0 1 p ( t 1 | x 0 ) / G ( x 1 , x 2 ) = k x 1 , x 2 k 2 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 22

  23. EXPLICIT: EQUIANGULAR x 2 uniform angular distribution x 0 1 p ( t 1 | x 0 ) / G ( x 1 , x 2 ) = k x 1 , x 2 k 2 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 23

  24. Transmittance sampling, 16 spp Equiangular sampling, 16 spp

  25. Equiangular sampling Transmittance sampling MIS combination

  26. UNIDIRECTIONAL + NEXT EVENT 2 x 3 x 2 x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 26

  27. Transmittance connections Equiangular connections

  28. UNIDIRECTIONAL + NEXT EVENT 2 x 3 x 2 angular singularity x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 28

  29. JOINT PATH SAMPLING 2 x 3 x 0 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 29

  30. JOINT PATH SAMPLING 2 x 3 x 2 x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 30

  31. JOINT PATH SAMPLING 2 x 3 g : n i p l m a s h a t p t n o i J 
 f d p n t i j o e i b c r s r e P ) 1 
 a v i s d f p l n a x 2 o i i t n d o c e i v e r D ) 2 
 n i o a t z l i n a i g a r m d f p t n o i j v e s i s c e c u s 
 n i d n e i a b t o r e a s a l n i o i t d n C o 3 ) r d e r o s e e r v r e x 0 0 x 1 TRADITIONAL : prescribes conditional pdfs, no explicit control over joint pdf JOINT SAMPLING : prescribe joint pdf, conditional pdfs derived from it MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 31

  32. JOINT PATH SAMPLING 2 x 3 x 2 x 0 0 x 1 joint pdf p ( x 1 , x 2 ) ∝ G ( x 0 , x 1 ) G ( x 1 , x 2 ) G ( x 2 , x 3 ) MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 32

  33. JOINT PATH SAMPLING 2 x 3 1 p ( t 1 ) / k x 3 � x 1 k x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 33

  34. JOINT PATH SAMPLING 2 x 3 ω 2 Cancels singularity at θ 2 θ 2 = 0 x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 34

  35. JOINT PATH SAMPLING 2 x 3 ω 2 1 p ( t 2 ) / k x 3 � x 2 k 2 x 2 equiangular pdf x 0 0 x 1 MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 35

  36. JOINT PATH SAMPLING 2 x 3 x 2 x 0 0 x 1 joint pdf p ( x 1 , x 2 ) ∝ G ( x 0 , x 1 ) G ( x 1 , x 2 ) G ( x 2 , x 3 ) MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 36

  37. JOINT PATH SAMPLING 2 x 3 x 2 x 0 0 x 1 via tabulation joint pdf p ( x 1 , x 2 ) ∝ G ( x 0 , x 1 ) G ( x 1 , x 2 ) G ( x 2 , x 3 ) f s ( x 1 ) f s ( x 2 ) MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION — PATH CONSTRUCTION 37

  38. path lengths 1-3 isotropic phase function Transmittance Equiangular Joint sampling

  39. path lengths 1-8 isotropic phase function Transmittance Equiangular Joint sampling

  40. path lengths 1-3 anisotropic phase function Transmittance connections Joint tabulated path sampling

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