Monte Carlo Integration of The Rendering Equation. Computer Graphics CMU /15-662, Spring 2017
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1 Monte Carlo Integration of The Rendering Equation Computer Graphics CMU /15-662, Spring 2017
2 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.
3 Basic Monte Carlo estimator Assume uniform probability density (for now) X i U(a, b) p(x) = 1 b a
4 Estimating indirect lighting
5 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.
6 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)
7 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 )
8 p Direct illumination
9 p One-bounce global illumination
10 p Two-bounce global illumination
11 p Four-bounce global illumination
12 p Eight-bounce global illumination
13 p Sixteen-bounce global illumination
14 Wait a minute When do we stop?!
15 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 )
16 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
17 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]
18 No Russian roulette: 6.4 seconds
19 Russian roulette: terminate 50% of all contributions with luminance less than 0.25: 5.1 seconds
20 Russian roulette: terminate 50% of all contributions with luminance less than 0.5: 4.9 seconds
21 Russian roulette: terminate 90% of all contributions with luminance less than 0.125: 4.8 seconds
22 Russian roulette: terminate 90% of all contributions with luminance less than 1: 3.6 seconds
23 Pseudocode using Russian Roulette L o (p,! o )=L e (p,! o )+ Z Z H2 f r (! i!! o ) L o,d (tr(p,! i ),! i ) cos i d! i + H 2 f r (! i!! o ) L o,i (tr(p,! i ),! i ) cos i d! i Spectrum pathtrace(ray ray) { Intersection isect = scene.intersect(ray); Vector2D wo = -ray.d; Spectrum Lo = isect.le(wo); // surface emission in direction wo } Lo += estimate_direct_lighting(isect, wo); // (see code on earlier slide, but do not // include Le in estimate) Vector2D wi; float pdf; generate_direction_sample(isect.brdf, wo, &wi, &pdf); // random direction to sample indirect Spectrum f = isect.brdf.f(wo, wi); float terminateprobability = 1.f - f.rho(); // termination probability based on // reflectance (averaged over spectrum). Lower // reflectance = high chance of terminating if (RandomFloat() < terminateprobability) return Lo; return Lo + ((f * pathtrace(ray(isect.p, wi)) * Dot(wi, isect.n) / (pdf * (1-terminateProbability))); 23CMU /662, Spring 2016
24 Pseudocode using Russian Roulette L o (p,! o ) L e (p,! o )+ f r(! i!! o ) L o,d (tr(p,! i ), p(! i )! i ) cos i + f r (! 0 i!! o) L o,i (tr(p,! 0 i ), p(! 0 i )!0 i ) cos 0 i P(choosing w i not_terminating) P(not terminating) Spectrum pathtrace(ray ray) { Intersection isect = scene.intersect(ray); Vector2D wo = -ray.d; Spectrum Lo = isect.le(wo); // surface emission in direction wo } Lo += estimate_direct_lighting(isect, wo); // (see code on earlier slide, but do not // include Le in estimate) Vector2D wi; float pdf; generate_direction_sample(isect.brdf, wo, &wi, &pdf); // random direction to sample indirect Spectrum f = isect.brdf.f(wo, wi); float terminateprobability = 1.f - f.rho(); // termination probability based on // reflectance (averaged over spectrum). Lower // reflectance = high chance of terminating if (RandomFloat() < terminateprobability) return Lo; return Lo + ((f * pathtrace(ray(isect.p, wi)) * Dot(wi, isect.n) / (pdf * (1-terminateProbability))); 24CMU /662, Spring 2016
25 One sample per pixel 25 CMU /662, Spring 2016
26 32 samples per pixel 26 CMU /662, Spring 2016
27 1024 samples per pixel 27 CMU /662, Spring 2016
28 Next time: Variance reduction how do we get the most out of our samples?
29 What you should know: What is Russian Roulette? How does Russian Roulette help us to obtain unbiased samples in path tracing? If we choose to continue tracing a ray in a Russian Roulette scenario, we will reweight the value provided by this ray by some factor. What is the weighting factor and why is it used? Another point of view on the Rendering Equation is to sum up light energy contributions from emitted light, paths of length 1, paths of length 2, etc. Be prepared to explain what effects you would see when tracing out paths of various lengths (color bleeding, reflections, caustics, etc) 29 CMU /662, Spring 2016
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