Biased Monte Carlo Ray Tracing:

Transcription

1 Biased Monte Carlo Ray Tracing: Filtering, Irradiance Caching and Photon Mapping Dr. Henrik Wann Jensen Stanford University May 24, 2001

2 Unbiased and consistent Monte Carlo methods Unbiased estimator: E{X} =... Consistent estimator: lim E{X} N... Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 2

3 Unbiased and consistent: A very simple example Unbiased estimator: 1 N N i=1 f(ξ i ) Consistent estimator: 1 N + 1 N i=1 f(ξ i ) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 3

4 Path tracing (unbiased) 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 4

5 Path tracing (unbiased) 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 5

6 Path tracing (unbiased) 100 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 6

7 How can we remove this noise?)

8 How can we remove this noise?) More samples (slow convergence, σ 1/ N)

9 How can we remove this noise?) More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.)

10 How can we remove this noise?) More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling

11 How can we remove this noise?) More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling Filtering

12 How can we remove this noise?) More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling Filtering Caching and interpolation Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 7

13 Stratified sampling 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 8

14 Quasi Monte-Carlo (Halton sequence) 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 9

15 Fixed (Random) Sequence 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 10

16 Filtering: idea Noise is high frequency

17 Filtering: idea Noise is high frequency Remove high frequency content Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 11

18 Unfiltered image 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 12

19 3x3 lowpass filter 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 13

20 3x3 median filter 10 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 14

21 Energy preserving filters

22 Energy preserving filters Distribute noisy energy over several pixels

23 Energy preserving filters Distribute noisy energy over several pixels Adaptive filter width Diffusion style filters Splatting style filters Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 15

24 Problems with filtering Everything is filtered (blurred) Textures Highlights Caustics...

25 Problems with filtering Everything is filtered (blurred) Textures Highlights Caustics... Solution: Try to filter the noisy part of the illumination Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 16

26 Caching Techniques

27 Caching Techniques Irradiance caching : Compute irradiance at selected points and interpolate. Photon mapping : Density estimation and importance sampling using a precomputed flux representation. Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 17

28 Box: direct illlumination Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 18

29 Box: global illlumination Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 19

30 Box: indirect irradiance Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 20

31 Irradiance caching: idea Greg Ward, Francis Rubinstein and Robert Clear: ``A Ray Tracing Solution for Diffuse Interreflection''. Proceedings of SIGGRAPH Idea: Irradiance changes slowly interpolate. Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 21

32 Irradiance sampling E(x) = 2π L (x, ω ) cos θ dω

33 Irradiance sampling E(x) = = L (x, ω ) cos θ dω 2π 2π π/2 0 0 L (x, θ, φ) cos θ sin θ dθ dφ

34 Irradiance sampling E(x) = = 0 L (x, ω ) cos θ dω 2π 2π π/2 π T P θ t = sin 1 ( 0 T t=1 t ξ T ) L (x, θ, φ) cos θ sin θ dθ dφ P p=1 L (θ t, φ p ) and φ p = 2π p ψ P Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 22

35 Irradiance change ɛ(x) E x (x x 0) + E θ (θ θ 0) } {{ } } {{ } position rotation

36 Irradiance change ɛ(x) E x (x x 0) + E θ (θ θ 0) } {{ } } {{ } position rotation ( ) 4 x x 0 E N(x) π x N(x 0 ) avg } {{ } } {{ } position rotation Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 23

37 Irradiance interpolation w(x) = 1 ɛ(x) 1 x x 0 1 N(x) N(x 0 ) x avg + E i (x) = i w i (x)e(x i ) w i (x) i Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 24

38 Irradiance caching algorithm Find all irradiance samples with w(x) > q if (samples found) interpolate else compute new irradiance sample Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 25

39 Box: irradiance gradients 1000 sample rays, w>10 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 26

40 Box: irradiance cache positions 1000 sample rays, w>10 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 27

41 Box: irradiance gradients 1000 sample rays, w>20 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 28

42 Box: irradiance cache positions 1000 sample rays, w>20 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 29

43 Box: irradiance gradients 5000 sample rays, w>10 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 30

44 Box: irradiance cache positions 5000 sample rays, w>10 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 31

45 Photon Mapping Two-pass method: Pass 1 : Build a photon map using photon tracing Pass 2 : Render the image using the photon map Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 32

46 A simple test scene Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 33

47 Building the Photon Map: Photon Tracing Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 34

48 Photons Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 35

49 The photon map datastructure The photons are stored in a left balanced kd-tree struct photon = { float position[3]; rgbe power; char phi, theta; short flags; } // power packed as 4 bytes // incoming direction Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 36

50 Radiance estimate L(x, ω) = Ω f r (x, ω, ω)l (x, ω ) cos θ dω

51 Radiance estimate L(x, ω) = = Ω Ω f r (x, ω, ω)l (x, ω ) cos θ dω f r (x, ω, ω) dφ2 (x, ω ) dω cos θ da cos θ dω

52 Radiance estimate L(x, ω) = = = Ω Ω Ω f r (x, ω, ω)l (x, ω ) cos θ dω f r (x, ω, ω) dφ2 (x, ω ) dω cos θ da cos θ dω f r (x, ω, ω) dφ2 (x, ω ) da

53 Radiance estimate L(x, ω) = = = f r (x, ω, ω)l (x, ω ) cos θ dω Ω f r (x, ω, ω) dφ2 (x, ω ) Ω dω cos θ da cos θ dω f r (x, ω, ω) dφ2 (x, ω ) Ω da n f r (x, ω p, ω) Φ p(x, ω p) πr 2 p=1 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 37

54 Radiance estimate L Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 38

55 Reflection inside a metal ring photons / 50 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 39

56 Caustics on glossy surfaces photons / 100 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 40

57 Cognac glass Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 41

58 Cube caustic Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 42

59 Caustic from a glass sphere photons / 50 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 43

60 Caustic from a glass sphere Path tracing 1000 paths/pixel Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 44

61 Caustic from a glass sphere in Grace Cathedral Using lightprobe from Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 45

62 Direct visualization of the radiance estimate photons / 50 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 46

63 Direct visualization of the radiance estimate photons / 500 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 47

64 Fast estimate 200 photons / 50 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 48

65 Only use photons for indirect irradiance photons / 500 photons in radiance estimate Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 49

66 Two photon maps global photon map caustics photon map Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 50

67 Rendering Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 51

68 Rendering: direct illumination Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 52

69 Rendering: specular reflection Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 53

70 Rendering: caustics Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 54

71 Rendering: indirect illumination Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 55

72 Two-pass method Radiance = direct illumination + specular reflection/transmission + caustics + soft indirect irradiance Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 56

73 Rendering Equation Solution L r (x, ω) = = f r (x, ω, ω)l i (x, ω ) cos θ i dω i Ω x Ω x f r (x, ω, ω)l i,l (x, ω ) cos θ i dω i + Ω x f r,s (x, ω, ω)(l i,c (x, ω ) + L i,d (x, ω )) cos θ i dω i + Ω x f r,d (x, ω, ω)l i,c (x, ω ) cos θ i dω i + Ω x f r,d (x, ω, ω)l i,d (x, ω ) cos θ i dω i. Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 57

74 Box global photons, caustic photons Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 58

75 Fractal box global photons, caustic photons Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 59

76 Sphereflake caustic Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 60

77 Little Matterhorn Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 61

78 Mies house (swimmingpool) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 62

79 Mies house (3pm) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 63

80 Mies house (6pm) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 64

81 Mies house (7pm) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 65

82 Participating media Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 66

83 Participating media: photon tracing Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 67

84 The volume photon map Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 68

85 The volume radiance estimate L ( ω )L(x, ω) = n p=1 p(x, ω p, ω) Φ p(x, ω p) 4 3 πr3 Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 69

86 Rendering participating media Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 70

87 Volume caustic Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 71

88 Rising smoke Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 72

89 Rising smoke Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 73

90 Subsurface scattering Skin Marble Actually most materials Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 74

91 Subsurface scattering Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 75

92 David (subsurface scattering) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 76

93 David (subsurface scattering) Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 77

94 Diana the Huntress

95 Diana the Huntress

96 Diana the Huntress Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 78

97 Diana the Huntress Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 79

98 Diana the Huntress: no subsurface scattering Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 80

99 Diana the Huntress: subsurface scattering Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 81

100 More information Biased Monte Carlo ray tracing: filtering, irradiance caching and photon mapping 82