The Rendering Equation. Computer Graphics CMU /15-662, Fall 2016

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1 The Rendering Equation Computer Graphics CMU /15-662, Fall 2016

2 Review: What is radiance? Radiance at point p in direction N is radiant energy ( #hits ) per unit time, per solid angle, per unit area perpendicular to N. radiant energy solid angle projected area

3 Intuition: Radiance and Irradiance irradiance radiance in direction ω angle between ω and normal

4 Incident vs. Exitant Radiance INCIDENT EXITANT In both cases: intensity of illumination is highly dependent on direction (not just location in space or moment in time).

5 The Rendering Equation Core functionality of photorealistic renderer is to estimate radiance at a given point p, in a given direction ω o Summed up by the rendering equation (Kajiya): outgoing/observed radiance emitted radiance (e.g., light source) incoming radiance angle between incoming direction and normal point of interest direction of interest all directions in hemisphere scattering function incoming direction Key challenge: to evaluate incoming radiance, we have to compute yet another integral. I.e., rendering equation is recursive.

6 Recursive Raytracing Basic strategy: recursively evaluate rendering equation!

7 Renderer measures radiance along a ray y o, d Pinhole x At each bounce, want to measure radiance traveling in the direction opposite the ray direction.

8 Renderer measures radiance along a ray y o, d Pinhole x Radiance entering camera in direction d = light from scene light sources that is reflected off surface in direction d.

9 How does reflection of light affect the outgoing radiance?

10 Reflection models Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident (same) side without change in frequency Choice of reflection function determines surface appearance

11 Categories of reflection functions Ideal specular Perfect mirror Ideal diffuse Uniform reflection in all directions Glossy specular Majority of light distributed in reflection direction Retro-reflective Reflects light back toward source Diagrams illustrate how incoming light energy from given direction is reflected in various directions.

12 Materials: diffuse

13 Materials: plastic

14 Materials: red semi-gloss paint

15 Materials: Ford mystic lacquer paint

16 Materials: mirror

17 Materials: gold

18 Materials

19 Models of Scattering How can we model scattering of light? Many different things that could happen to a photon: - bounces off surface - transmitted through surface - bounces around inside surface - absorbed & re-emitted - What goes in must come out! (Total energy must be conserved) In general, can talk about probability* a particle arriving from a given direction is scattered in another direction *Somewhat more complicated than this, because some light is absorbed!

20 Hemispherical incident radiance At any point on any surface in the scene, there s an incident radiance field that gives the directional distribution of illumination at the point Consider view of hemisphere from this point

21 Diffuse reflection Exitant radiance is the same in all directions Incident radiance Exitant radiance

22 Ideal specular reflection Incident radiance is flipped around normal to get exitant radiance Incident radiance Exitant radiance

23 Plastic Incident radiance gets flipped and blurred Incident radiance Exitant radiance

24 Copper More blurring, plus coloration (nonuniform absorption across frequencies) Incident radiance Exitant radiance

25 Scattering off a surface: the BRDF Bidirectional reflectance distribution function Encodes behavior of light that bounces off surface Given incoming direction ω i, how much light gets scattered in any given outgoing direction ω o? Describe as distribution f r (ω i ω o ) why less than or equal? where did the rest of the energy go?! Helmholtz reciprocity Q: Why should Helmholtz reciprocity hold? Think about little mirrors bv (Szymon Rusinkiewicz)

26 Radiometric description of BRDF For a given change in the incident irradiance, how much does the exitant radiance change?

27 Example: Lambertian reflection Assume light is equally likely to be reflected in each output direction Z L o (! o )= H 2 f r L i (! i ) cos i d! i =f r ZH 2 L i (! i ) cos i d! i f r = c =f r E f r =

28 Example: perfect specular reflection [Zátonyi Sándor]

29 Geometry of specular reflection Top-down view ~n (looking down on surface)! i! o i o i o = o = i! o +! i = 2 cos ~n = 2(! i ~n)~n! o =! i + 2(! i ~n)~n

30 Specular reflection BRDF L i ( i, i) L o ( o, o) i o L o ( o, o) =L i ( o, o ± ) Dirac delta f r ( i, i; o, o) = (cos i cos o ) cos i ( i o ± ) Strictly speaking, f r is a distribution, not a function In practice, no hope of finding reflected direction via random sampling; simply pick the reflected direction!

31 Transmission In addition to reflecting off surface, light may be transmitted through surface. Light refracts when it enters a new medium.

32 Snell s Law Transmitted angle depends on index of refraction of medium incident ray is in and index of refraction of medium light is entering. Medium *! i ~n Vacuum Air (sea level) Water (20 C) Glass Diamond * index of refraction is wavelength dependent (these are averages)! t i sin i = t sin t

33 Law of refraction! i ~n i sin i = t sin t q cos t = 1 sin 2 t = = s 1 s 1 i t i t 2 sin 2 i 2 (1 cos 2 i )! t Total internal reflection: When light is moving from a more optically dense medium to a less optically dense medium: > 1 t Light incident on boundary from large enough angle will not exit medium. i 1 i t 2 (1 cos 2 i ) < 0

34 Optical manhole Only small cone visible, due to total internal reflection (TIR)

35 Fresnel reflection Many real materials: reflectance increases w/ viewing angle [Lafortune et al. 1997]

36 Snell + Fresnel: Example Reflection (Fresnel) Refraction (Snell)

37 Without Fresnel (fixed reflectance/transmission)

38 Glass with Fresnel reflection/transmission

39 Anisotropic reflection Reflection depends on azimuthal angle Results from oriented microstructure of surface e.g., brushed metal

40 Subsurface scattering Visual characteristics of many surfaces caused by light entering at different points than it exits - Violates a fundamental [Jensen et al 2001] assumption of the BRDF [Donner et al 2008]

41 Scattering functions Generalization of BRDF; describes exitant radiance at one point due to incident differential irradiance at another point: S(x i,! i,x o,! o ) Generalization of reflection equation integrates over all points on the surface and all directions(!) Z Z L(x o,! o )= A H 2 S(x i,! i,x o,! o ) L i (x i,! i ) cos i d! i da

42 Translucent materials: Jade

43 Translucent materials: skin

44 Translucent materials: leaves

45 BRDF

46 BSSRDF

47 Ok, so scattering is complicated! What s a (relatively simple) algorithm that can capture all this behavior?

48 The reflection equation dl r (! r )=f r (! i!! r )dl i (! i ) cos i Z L r (p,! r )= H 2 f r (p,! i!! r ) L i (p,! i ) cos i d! i

49 The reflection equation Key piece of overall rendering equation: Z L r (p,! r )= H 2 f r (p,! i!! r ) L i (p,! i ) cos i d! i Approximate integral via Monte Carlo integration Generate directions sampled from some distribution! j p(!) Compute the estimator 1 N NX j=1 f r (p,! j!! r ) L i (p,! j ) cos j p(! j ) To reduce variance p(!) should match BRDF or incident radiance function

50 Estimating reflected light // Assume: // Ray ray hits surface at point hit_p // Normal of surface at hit point is hit_n Vector3D wr = -ray.d; // outgoing direction Spectrum Lr = 0.; for (int i = 0; i < N; ++i) { Vector3D wi; // sample incident light from this direction float pdf; // p(wi) generate_sample(brdf, &wi, &pdf); // generate sample according to brdf Spectrum f = brdf->f(wr, wi); Spectrum Li = trace_ray(ray(hit_p, wi)); // compute incoming Li Lr += f * Li * fabs(dot(wi, hit_n)) / pdf; } return Lr / N;

51 The rendering equation y! i L o (p,! o )! o p Pinhole x Z L o (p,! o )=L e (p,! o )+ H 2 f r (p,! i!! o ) L i (p,! i ) cos i d! i Now that we know how to handle reflection, how do we solve the full rendering equation? Have to determine incident radiance

52 Key idea in (efficient) rendering: take advantage of special knowledge to break up integration into easier components.

53 Path tracing: overview Partition the rendering equation into direct and indirect illumination Use Monte Carlo to estimate each partition separately - One sample for each - Assumption: 100s of samples per pixel Terminate paths with Russian roulette

54 Direct illumination + reflection + transparency Image credit: Henrik Wann Jensen

55 Global illumination solution Image credit: Henrik Wann Jensen

56 Review: Monte Carlo integration Z b Definite integral What we seek to estimate a f(x)dx Random variables X i is the value of a random sample drawn from the distribution Y i is also a random variable. p(x) X i p(x) Y i = f(x i ) Z b Expectation of f E[Y i ]=E[f(X i )] = f(x) p(x)dx a Estimator Monte Carlo estimate of Assuming samples X i Z b a f(x)dx drawn from uniform F N = b N a NX i=1 Y i pdf. I will provide estimator for arbitrary PDFs later in lecture.

57 Basic Monte Carlo estimator Assume uniform probability density (for now) X i U(a, b) p(x) = 1 b a

58 Näive direct lighting estimate Uniformly-sample hemisphere of directions with respect to solid angle Z p(!) = 1 E(p) = L(p,!) cos d! 2 Z L(p,!) d! Estimator: Z X i p(!) Y i = f(x i ) Y i = L(p,! i )cos i da F N = 2 N NX i=1 Y i

59 Näive direct lighting estimate Uniformly-sample hemisphere of directions with respect to solid angle Z E(p) = L(p,!) cos d! Z Given surface point p Z For each of N samples: Generate random direction:! i! i Compute incoming radiance arriving L i at p from direction: Compute incident irradiance due to ray: de i = L i cos i Accumulate 2 N de i into estimator

60 Uniform hemisphere sampling Generate random direction on hemisphere (all directions equally likely) p(!) = 1 2 Direction computed from uniformly distributed point on 2D plane: q q ( 1, 2 )=( cos(2 2), sin(2 2), 1 ) Try at home: derive from the inversion method [Arvo]

61 Hemispherical solid angle Light source sampling, 100 sample rays (random directions drawn uniformly from hemisphere) Occluder (blocks light)

62 Why is the image in the previous slide noisy?

63 Incident lighting estimator uses different random directions in each pixel. Some of those directions point towards the light, others do not. (Estimator is a random variable)

64 Idea: don t need to integrate over entire hemisphere of directions (incoming radiance is 0 from most directions) Only integrate over the area of the light (directions where incoming radiance is non-zero)

65 Direct lighting: area integral Z E(p) = L(p,!) cos d! Integral over directions Z E(p) = ZA0 L o (p 0,! 0 ) V (p, p 0 ) Z cos cos 0 p p 0 2 da0 Change of variables to integral over area of light A 0 p 0 dw = da p 0 p 2 = da0 cos p 0 p 2 0! 0 =p p 0 Binary visibility function: 1 if p is visible from p, 0 otherwise (accounts for light occlusion)! =p 0 p Outgoing radiance from light point p, in direction w towards p p

66 Light source area sampling, 100 sample rays If no occlusion is present, all directions chosen in computing estimate hit the light source. (Choice of direction only matters if portion of light is occluded from surface point p.)

67 Hemispherical solid angle sampling, 100 sample rays (random directions drawn uniformly from hemisphere)

68 Light source area sampling, 100 sample rays

69 Estimating indirect lighting

70 The reflection equation y! i L o (p,! o )! o p Pinhole x Z L o (p,! o )=L e (p,! o )+ H 2 f r (p,! i!! o ) L i (p,! i ) cos i d! i Need to know incident radiance. So far, have only computed incoming radiance from scene light sources.

71 Accounting for indirect illumination y! i L o (p,! o )! o p Pinhole x Z L o (p,! o )=L e (p,! o )+ H 2 f r (p,! i!! o ) L i (p,! i ) cos i d! i Incoming light energy from direction! i may be due to light reflected off another surface in the scene (not an emitter)

72 Path tracing: indirect illumination Z H 2 f r (! i!! o ) L o,i (tr(p,! i ),! i ) cos i d! i Sample incoming direction from some distribution (e.g. proportional to BRDF): Recursively call path tracing function to compute incident indirect radiance Monte Carlo estimator:! i p(!) f r (! i!! o ) L o,i (tr(p,! i ),! i ) cos i p(! i )

73 p Direct illumination

74 p One-bounce global illumination

75 p Two-bounce global illumination

76 p Four-bounce global illumination

77 p Eight-bounce global illumination

78 p Sixteen-bounce global illumination

79 Wait a minute When do we stop?!

80 Russian roulette Idea: want to avoid spending time evaluating function for samples that make a small contribution to the final result Consider a low-contribution sample of the form: L = f r(! i!! o ) L i (! i ) V (p, p 0 ) cos i p(! i ) V (p, p 0 ) Stanford CS348b, Spring 2014

81 Russian roulette L = f r(! i!! o ) L i (! i ) V (p, p 0 ) cos i p(! i ) apple fr (! i!! o ) L i (! i ) cos i L = p(! i ) V (p, p 0 ) If tentative contribution (in brackets) is small, total contribution to the image will be small regardless of V (p, p 0 ) Ignoring low-contribution samples introduces systemic error - No longer an unbiased estimator Instead, randomly discard low-contribution samples in a way that leaves estimator unbiased Stanford CS348b, Spring 2014

82 Russian roulette New estimator: evaluate original estimator with probability p rr, reweight. Otherwise ignore. Same expected value as original estimator: p rr E apple X p rr + E[(1 p rr )0] = E[X] Stanford CS348b, Spring 2014

83 No Russian roulette: 6.4 seconds Stanford CS348b, Spring 2014

84 Russian roulette: terminate 50% of all contributions with luminance less than 0.25: 5.1 seconds Stanford CS348b, Spring 2014

85 Russian roulette: terminate 50% of all contributions with luminance less than 0.5: 4.9 seconds Stanford CS348b, Spring 2014

86 Russian roulette: terminate 90% of all contributions with luminance less than 0.125: 4.8 seconds Stanford CS348b, Spring 2014

87 Russian roulette: terminate 90% of all contributions with luminance less than 1: 3.6 seconds Stanford CS348b, Spring 2014

88 Next time: Variance reduction how do we get the most out of our samples?

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