Biased Monte Carlo Ray Tracing

Transcription

1 Biased Monte Carlo Ray Tracing Filtering, Irradiance Caching, and Photon Mapping Henrik Wann Jensen Stanford University May 23, 2002

2 Unbiased and Consistent Unbiased estimator: E{X} =... Consistent estimator: lim E{X} N...

3 Unbiased and Consistent Unbiased estimator: 1 N N i=1 f(ξ i ) Consistent estimator: 1 N + 1 N f(ξ i ) i=1

4 Unbiased Methods Variance (noise) is the only error This error can be analyzed using the variance (i.e. 95% of samples are within 2% of the correct result)

5 Path Tracing (Unbiased) 10 paths/pixel

6 Path Tracing (Unbiased) 10 paths/pixel

7 Path Tracing (Unbiased) 100 paths/pixel

8 How Can We Remove This Noise

9 The World is Diffuse! Arnold Rendering

10 The World is Diffuse! Arnold Rendering

11 The World is Diffuse! Arnold Rendering

12 Noise Reduction/Removal More samples (slow convergence, σ 1/ N)

13 Noise Reduction/Removal More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.)

14 Noise Reduction/Removal More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling

15 Noise Reduction/Removal More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling Filtering

16 Noise Reduction/Removal More samples (slow convergence, σ 1/ N) Better sampling (stratified, importance, qmc etc.) Adaptive sampling Filtering Caching and interpolation

17 Stratified Sampling Latin Hypercube: 10 paths/pixel

18 Quasi Monte-Carlo Halton-Sequence: 10 paths/pixel

19 Fixed (Random) Sequence 10 paths/pixel

20 Filtering: Idea Noise is high frequency

21 Filtering: Idea Noise is high frequency Remove high frequency content

22 Unfiltered Image 10 paths/pixel

23 3x3 Lowpass Filter 10 paths/pixel

24 Unfiltered Image 10 paths/pixel

25 3x3 Median Filter 10 paths/pixel

26 Energy Preserving Filters

27 Energy Preserving Filters Distribute noisy energy over several pixels

28 Energy Preserving Filters Distribute noisy energy over several pixels Adaptive filter width [Rushmeier and Ward 94] Diffusion style filters [McCool99] Splatting style filters [Suykens and Willems 00]

29 Problems With Filtering Everything is filtered (blurred) Textures Highlights Caustics...

30 Caching Techniques

31 Caching Techniques Irradiance caching : Compute irradiance at selected points and interpolate. Photon mapping : Trace ``photons'' from the lights and store them in a photon map, that can be used during rendering.

32 Box: Direct Illlumination

33 Box: Global Illlumination

34 Box: Indirect Irradiance

35 Irradiance Caching: Idea ``A Ray Tracing Solution for Diffuse Interreflection''. Greg Ward, Francis Rubinstein and Robert Clear: Proc. SIGGRAPH'88. Idea: Irradiance changes slowly interpolate.

36 Irradiance Sampling E(x) = 2π L (x, ω ) cos θ dω

37 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

38 Irradiance Sampling E(x) = = 0 L (x, ω ) cos θ dω 2π 2π π/2 π T P 0 T t=1 L (x, θ, φ) cos θ sin θ dθ dφ P p=1 L (θ t, φ p ) θ t = sin 1 ( ) t ξ T and φ p = 2π p ψ P

39 Irradiance Change ɛ(x) E x (x x 0) + E θ (θ θ 0) } {{ } } {{ } position rotation

40 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

41 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

42 Irradiance Caching Algorithm Find all irradiance samples with w(x) > q if (samples found) interpolate else compute new irradiance sample

43 Box: Irradiance Caching 1000 sample rays, w>10

44 Box: Irradiance Cache Positions 1000 sample rays, w>10

45 Box: Irradiance Caching 1000 sample rays, w>20

46 Box: Irradiance Cache Positions 1000 sample rays, w>20

47 Box: Irradiance Caching 5000 sample rays, w>10

48 Box: Irradiance Cache Positions 5000 sample rays, w>10

49 Caustics Pathtracing 1000 paths/pixel

50 A simple test scene

51 Rendering

52 Photon Tracing

53 Photons

54 Radiance Estimate L(x, ω) = Ω f r (x, ω, ω)l (x, ω ) cos θ dω

55 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

56 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

57 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

58 Radiance Estimate L

59 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

60 Rendering: Caustics

61 Caustic from a Glass Sphere Photon Mapping: photons / 50 photons in radiance estimate

62 Caustic from a Glass Sphere Path Tracing: 1000 paths/pixel

63 Sphereflake Caustic

64 Reflection Inside A Metal Ring photons / 50 photons in radiance estimate

65 Caustics On Glossy Surfaces photons / 100 photons in radiance estimate

66 HDR environment illumination Using lightprobe from

67 Cognac Glass

68 Cube Caustic

69 Global Illumination photons / 50 photons in radiance estimate

70 Global Illumination photons / 500 photons in radiance estimate

71 Fast estimate 200 photons / 50 photons in radiance estimate

72 Indirect illumination photons / 500 photons in radiance estimate

73 Global Illumination

74 Global Illumination global photon map caustics photon map

75 Photon tracing Photon emission Photon scattering Photon storing

76 Photon emission Given Φ Watt lightbulb. Emit N photons. Each photon has the power Φ N Watt. Photon power depends on the number of emitted photons. Not on the number of photons in the photon map.

77 What is a photon? Flux (power) - not radiance! Collection of physical photons A fraction of the light source power Several wavelengths combined into one entity

78 Diffuse point light Generate random direction Emit photon in that direction // Find random direction do { x = 2.0*random()-1.0; y = 2.0*random()-1.0; z = 2.0*random()-1.0; } while ( (x*x + y*y + z*z) > 1.0 );

79 Example: Diffuse square light - Generate random position p on square - Generate diffuse direction d - Emit photon from p in direction d // Generate diffuse direction u = random(); v = 2*π*random(); d = vector( cos(v) u, sin(v) u, 1 u );

80 Surface interactions The photon is Stored (at diffuse surfaces) and Absorbed (A) or Reflected (R) or Transmitted (T ) A + R + T = 1.0

81 Photon scattering The simple way: Given incoming photon with power Φ p Reflect photon with the power R Φ p Transmit photon with the power T Φ p

82 Photon scattering The simple way: Given incoming photon with power Φ p Reflect photon with the power R Φ p Transmit photon with the power T Φ p Risk: Too many low-powered photons - wasteful! When do we stop (systematic bias)? Photons with similar power is a good thing.

83 Russian Roulette Statistical technique Known from Monte Carlo particle physics Introduced to graphics by Arvo and Kirk in 1990

84 Russian Roulette Probability of termination: p

85 Russian Roulette Probability of termination: p E{X}

86 Russian Roulette Probability of termination: p E{X} = p 0

87 Russian Roulette Probability of termination: p E{X} = p 0 + (1 p)

88 Russian Roulette Probability of termination: p E{X} = p 0 + (1 p) E{X} 1 p

89 Russian Roulette Probability of termination: p E{X} = p 0 + (1 p) E{X} 1 p = E{X}

90 Russian Roulette Probability of termination: p E{X} = p 0 + (1 p) E{X} 1 p = E{X} Terminate un-important photons and still get the correct result.

91 Russian Roulette Example Surface reflectance: R = 0.5 Incoming photon: Φ p = 2 W r = random(); if ( r < 0.5 ) reflect photon with power 2 W else photon is absorbed

92 Russian Roulette Intuition Surface reflectance: R = incoming photons with power: Φ p = 2 Watt Reflect 100 photons with power 2 Watt instead of 200 photons with power 1 Watt.

93 Russian Roulette Very important! Use to eliminate un-important photons Gives photons with similar power :)

94 Sampling a BRDF f r (x, ω i, ω o ) = w 1 f r,1 (x, ω i, ω o ) + w 2 f r,2 (x, ω i, ω o )

95 Sampling a BRDF f r (x, ω i, ω o ) = w 1 f r,d + w 2 f r,s r = random() (w 1 + w 2 ); if ( r < w 1 ) reflect diffuse photon else reflect specular

96 Rendering

97 Direct Illumination

98 Specular Reflection

99 Caustics

100 Indirect Illumination

101 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.

102 Features Photon tracing is unbiased Radiance estimate is biased but consistent The reconstruction error is local Illumination representation is decoupled from the geometry

103 Box global photons, caustic photons

104 Box: Global Photons global photons

105 Fractal Box global photons, caustic photons

106 Cornell Box

107 Indirect Illumination

108 Little Matterhorn

109 Mies house (swimmingpool)

110 Mies house (3pm)

111 Mies house (6pm)

112 More Information Foreword by Pat Hanrahan The creation of realistic three-dimensional images is central to computer graphics. Photon mapping, an extension of ray tracing, makes it possible to efficiently simulate global illumination in complex scenes. Photo mapping can simulate caustics (focused light, such as shimmering waves at the bottom of a swimming pool), diffuse inter-reflections (e.g., the `bleeding' of colored light from a red wall onto a white floor, giving the floor a reddish tint), and participating media (e.g., clouds or smoke). This book is a practical guide to photon mapping; it provides both the theory and the practical insight necessary to implement photon mapping and simulate all types of direct and indirect illumination efficiently. A K PETERS LTD. Realistic Image Synthesis Using Photon Mapping Realistic Image Synthesis Using Photon Mapping Jensen Henrik Wann Jensen AK PETERS Henrik Wann Jensen Realistic Image Synthesis Using Photon Mapping Foreword by Pat Hanrahan henrik@graphics.stanford.edu