Rendering Equation & Monte Carlo Path Tracing I

Rendering Equation & Monte Carlo Path Tracing I CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1

Announcements Homework 1 due tonight! Programming Assignment 1 is out and will be due in 1.5 weeks on May 2 CS295, Spring 2017 Shuang Zhao 2

Previous Lectures The behavior of light Radiometry Monte Caro integration General framework Sampling from PDFs CS295, Spring 2017 Shuang Zhao 3

Today s Lecture Rendering equation Describe the distribution of light at equilibrium Monte Carlo Path Tracing I An unbiased numerical solution to the rendering equation CS295, Spring 2017 Shuang Zhao 4

The Rendering Equation CS295: Realistic Image Synthesis Rendering Equation & Monte Carlo Path Tracing I CS295, Spring 2017 Shuang Zhao 5

Recap: Radiometry Measurement of light energy Radiometric quantities Energy Power (radiant flux) Irradiance & radiosity Radiance CS295, Spring 2017 Shuang Zhao 6

Light Transport Goal Describe steady-state radiance distribution in virtual scenes Assumptions Geometric optics Achieves steady state instantaneously CS295, Spring 2017 Shuang Zhao 7

Radiance at Equilibrium Radiance values at all points in the scene and in all directions expresses the equilibrium 5D Light-field We only consider radiance on surfaces (4D) Assuming no volumetric scattering or absorption CS295, Spring 2017 Shuang Zhao 8

Rendering Equation (RE) RE describes the distribution of radiance at equilibrium RE involves: Scene geometry Light source info. Surface reflectance info. Radiance values at all surface points in all directions (Known) (Unknown) CS295, Spring 2017 Shuang Zhao 9

Rendering Equation (RE) = + = + CS295, Spring 2017 Shuang Zhao 10

Rendering Equation Incoming radiance CS295, Spring 2017 Shuang Zhao 11

Rendering Equation (Invariant of radiance along lines) CS295, Spring 2017 Shuang Zhao 12

Monte Carlo Path Tracing I CS295: Realistic Image Synthesis Rendering Equation & Monte Carlo Path Tracing I CS295, Spring 2017 Shuang Zhao 13

Path Tracing (Version 0) Estimating L r using MC integration: Draw ω i uniformly at random p(ω i ) = 1/(2π) CS295, Spring 2017 Shuang Zhao 14

Path Tracing (Version 0) radiance(x, ω): return emittedradiance(x, ω) + reflectedradiance(x, ω) reflectedradiance(x, ω): ω i = uniformrandom(n x ) y = RayTrace(x, ω i ) return 2.0 * π * radiance(y, -ω i ) * brdf(x, ω i, ω) * dot(n x, ω i ) CS295, Spring 2017 Shuang Zhao 15

uniformrandom uniformrandom(n): z = rand() r = sqrt(1.0 - z * z) φ = 2.0 * π * rand() x = r * cos(φ) y = r * sin(φ) [u, v, w] = createlocalcoord(n) return x * u + y * v + z * w [x, y, z] distributes uniformly on the hemisphere around [0, 0, 1] createlocalcoord(n) returns a local (orthogonal) coordinate system around n CS295, Spring 2017 Shuang Zhao 16

Path Tracing (Version 0.1) Estimating L r using MC integration: Randomly sample ω i using a probability density proportional to i.e., CS295, Spring 2017 Shuang Zhao 17

Path Tracing (Version 0.1) Estimating L r using MC integration: CS295, Spring 2017 Shuang Zhao 18

Path Tracing (Version 0.1) radiance(x, ω): return emittedradiance(x, ω) + reflectedradiance(x, ω) reflectedradiance(x, ω): ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) return π * radiance(y, -ω i ) * brdf(x, ω i, ω) CS295, Spring 2017 Shuang Zhao 19

uniformrandompsa uniformrandompsa(n): z = sqrt(rand()) r = sqrt(1.0 - z * z) φ = 2.0 * π * rand() x = r * cos(φ) y = r * sin(φ) [u, v, w] = createlocalcoord(n) return x * u + y * v + z * w [x, y] distributes uniformly on a unit disc centered at [0, 0] createlocalcoord(n) returns a local (orthogonal) coordinate system around n CS295, Spring 2017 Shuang Zhao 20

Path Tracing (Version 0.1) radiance(x, ω): return emittedradiance(x, ω) + reflectedradiance(x, ω) reflectedradiance(x, ω): ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) return π * radiance(y, -ω i ) * brdf(x, ω i, ω) CS295, Spring 2017 Shuang Zhao 21

Path Tracing (Version 0.1) radiance(x, ω): rad = emittedradiance(x, ω) ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) rad += π * radiance(y, -ω i ) * brdf(x, ω i, ω) return rad CS295, Spring 2017 Shuang Zhao 22

Path Tracing (Version 0.1) Light source CS295, Spring 2017 Shuang Zhao 23

Path Tracing (Version 0.1) Light source CS295, Spring 2017 Shuang Zhao 24

Path Tracing (Version 0.5) Monte Carlo Path Tracing I CS295, Spring 2017 Shuang Zhao 25

Path Tracing (Version 0.1) radiance(x, ω): rad = emittedradiance(x, ω) ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) rad += π * radiance(y, -ω i ) * brdf(x, ω i, ω) return rad Problem: infinite recursion! CS295, Spring 2017 Shuang Zhao 26

Avoid Infinite Recursion Idea 1: bounding the depth radiance(x, ω, depth): if depth > maxdepth: return 0.0 rad = emittedradiance(x, ω) ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) rad += π * radiance(y, -ω i, depth + 1) * brdf(x, ω i, ω) return rad Problem: biased CS295, Spring 2017 Shuang Zhao 27

Avoid Infinite Recursion Idea 2: Russian roulette Pick survival probability, a constant within (0, 1) radiance(x, ω): rad = emittedradiance(x, ω) if rand() < survivalprobability: ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) rad += π * radiance(y, -ω i ) * brdf(x, ω i, ω) / survivalprobability return rad Why does this work? CS295, Spring 2017 Shuang Zhao 28

Russian Roulette For any random variable X, let for some 0 < p < 1, then Pros: estimators of Y will always terminate Cons: increased variance (noise) CS295, Spring 2017 Shuang Zhao 29

Path Tracing (Version 0.5) radiance(x, ω, depth): rad = emittedradiance(x, ω) if depth <= rrdepth: p = 1.0 else: p = survivalprobability if rand() < p: ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) rad += π * radiance(y, -ω i, depth + 1) * brdf(x, ω i, ω) / p return rad CS295, Spring 2017 Shuang Zhao 30

Path Tracing (Version 1.0) Monte Carlo Path Tracing I CS295, Spring 2017 Shuang Zhao 31

Challenge Small light source leads to high noise Because the probability for a light path to hit the light source is small 64 samples per pixel CS295, Spring 2017 Shuang Zhao 32

Idea: Separating Direct & Indirect Indirect Direct CS295, Spring 2017 Shuang Zhao 33

Idea: Separating Direct & Indirect radiance(x, ω, depth): return emittedradiance(x, ω) + reflectedradiance(x, ω, depth) reflectedradiance(x, ω, depth): return directradiance(x, ω) + indirectradiance(x, ω, depth) CS295, Spring 2017 Shuang Zhao 34

Direct Radiance Direct CS295, Spring 2017 Shuang Zhao 35

Indirect Radiance Indirect CS295, Spring 2017 Shuang Zhao 36

Summary: Direct + Indirect L = L e + L r This idea is usually called next-event estimation (more on this later) CS295, Spring 2017 Shuang Zhao 37

Estimating Indirect Radiance Almost identical to our reflected radiance estimator version 0.5 indirectradiance(x, ω, depth): if depth <= rrdepth: p = 1.0 else: p = survivalprobability if rand() < p: ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) return π * reflectedradiance(y, -ω i, depth + 1) * brdf(x, ω i, ω) / p else: return 0.0 CS295, Spring 2017 Shuang Zhao 38

Estimating Direct Radiance Idea: sampling the light source Solid angle integral area integral Change of measure CS295, Spring 2017 Shuang Zhao 39

Area Integral A e : surfaces of all light sources V(x, y): visibility term describing if x and y are mutually visible (visible) (occluded) CS295, Spring 2017 Shuang Zhao 40

Area Estimator Sample y uniformly from A e p(y) = 1/S e S e denotes the surface area of A e E.g., for one spherical light source with radius r, S e = 4πr 2 CS295, Spring 2017 Shuang Zhao 41

Estimating Direct Radiance directradiance(x, ω): [y, pdf] = luminairesample() ω i = normalize(y - x) r 2 = dot(y - x, y - x) return emittedradiance(y, -ω i ) * brdf(x, ω i, ω) * visibility(x, y) * dot(n x, ω i ) * dot(n y, -ω i ) / (r 2 * pdf) CS295, Spring 2017 Shuang Zhao 42

Path Tracing (Version 1.0) reflectedradiance(x, ω, depth): return directradiance(x, ω) + indirectradiance(x, ω, depth) directradiance(x, ω): [y, pdf] = luminairesample() ω i = normalize(y - x) r 2 = dot(y - x, y - x) return emittedradiance(y, -ω i ) * brdf(x, ω i, ω) * visibility(x, y) * dot(n x, ω i ) * dot(n y, -ω i ) / (r 2 * pdf) indirectradiance(x, ω, depth): if depth <= rrdepth: p = 1.0 else: p = survivalprobability if rand() < p: ω i = uniformrandompsa(n x ) y = RayTrace(x, ω i ) return π * reflectedradiance(y, -ω i, depth + 1) * brdf(x, ω i, ω) / p else: return 0.0 CS295, Spring 2017 Shuang Zhao 43

Path Tracing (Version 1.0) reflectedradiance(x, ω, depth): [y 1, pdf] = luminairesample() ω 1 = normalize(y 1 - x) r 2 = dot(y 1 - x, y 1 - x) reflrad = emittedradiance(y 1, -ω 1 ) * brdf(x, ω 1, ω) * visibility(x, y 1 ) * dot(n x, ω 1 ) * dot(n y, -ω 1 ) / (r 2 * pdf) if depth <= rrdepth: p = 1.0 else: p = survivalprobability if rand() < p: ω 2 = uniformrandompsa(n x ) y 2 = RayTrace(x, ω 2 ) reflrad += π * reflectedradiance(y 2, -ω 2, depth + 1) * brdf(x, ω 2, ω) / p return reflrad CS295, Spring 2017 Shuang Zhao 44

Equal-Sample Comparison (Both at 64 sample paths per pixel) Path tracing version 0.5 Path tracing version 1.0 CS295, Spring 2017 Shuang Zhao 45

Next Lecture Monte Carlo Path Tracing II CS295, Spring 2017 Shuang Zhao 46