GAMES Webinar: Rendering Tutorial 2. Monte Carlo Methods. Shuang Zhao

Transcription

1 GAMES Webinar: Rendering Tutorial 2 Monte Carlo Methods Shuang Zhao Assistant Professor Computer Science Department University of California, Irvine GAMES Webinar Shuang Zhao 1

2 Outline 1. Monte Carlo integration A powerful numerical tool for estimating complex integrals 2. Rendering equation The physical framework governing light transport 3. Path tracing Basically, Path integral formulation 5. Advanced methods GAMES Webinar Shuang Zhao 2

3 Monte Carlo Integration GAMES Webinar: Rendering Tutorial 2 GAMES Webinar Shuang Zhao 3

4 Why Monte Carlo? The gold standard approach to solve the RTE Advantages: Provides estimates with quantifiable uncertainty Adaptable to systems with complex geometries GAMES Webinar Shuang Zhao 4

5 Monte Carlo Integration A powerful framework for computing integrals Numerical Nondeterministic (i.e., using randomness) Scalable to high-dimensional problems GAMES Webinar Shuang Zhao 5

6 Random Variables (Discrete) random variable X Possible outcomes: x 1, x 2,, x n with probability masses p 1, p 2,, p n such that E.g., fair coin Outcomes: x 1 = head, x 2 = tail Probabilities: p 1 = p 2 = ½ GAMES Webinar Shuang Zhao 6

7 Random Variables (Continuous) random variable X Possible outcomes: with probability density function (PDF) p(x) satisfying Cumulative density function (CDF) P(x) given by GAMES Webinar Shuang Zhao 7

8 Strong Law of Large Numbers Let x 1, x 2,, x n be n independent observations (aka. samples) of X Sample mean Actual mean GAMES Webinar Shuang Zhao 8

9 Example: Evaluating π Unit circle S Let X be a point uniformly distributed in the square Let, then Circle area = π Square area = 4 GAMES Webinar Shuang Zhao 9

10 Example: Evaluating π Unit circle S Simple solution for computing π: Generate n samples x 1,, x n independently Circle area = π Square area = 4 Compute Live demo GAMES Webinar Shuang Zhao 10

11 Example: Evaluating π GAMES Webinar Shuang Zhao 11

12 Integral f(x): one-dimensional scalar function GAMES Webinar Shuang Zhao 12

13 Deterministic Integration Quadrature rule: Scales poorly with high dimensionality: Needs n m bins for a m-dimensional problem We have a high-dimensional problem! GAMES Webinar Shuang Zhao 13

14 Monte Carlo Integration: Overview Goal: Estimating Idea: Constructing random variable Such that is called an unbiased estimator of But how? GAMES Webinar Shuang Zhao 14

15 Monte Carlo Integration Let p() be any probability density function over [a, b] and X be a random variable with density p Let, then: To estimate : strong law of large numbers GAMES Webinar Shuang Zhao 15

16 Monte Carlo Integration Goal: to estimate Pick a probability density function Generate n independent samples: Evaluate for j = 1, 2,, n Return sample mean: GAMES Webinar Shuang Zhao 16

17 How to pick density function p()? In theory (Almost) anything In practice: Uniform distributions (almost) always work As long as the domain is bounded Choice of p() greatly affects the effectiveness (i.e., convergence rate) of the resulting estimator GAMES Webinar Shuang Zhao 17

18 Monte Carlo Integration Hello, World! Estimating Algorithm: Draw i.i.d x 1, x 2,, x n uniformly from [0, 1) Return Live demo GAMES Webinar Shuang Zhao 18

19 Monte Carlo Integration Hello, World! Estimating GAMES Webinar Shuang Zhao 19

20 Multiple Importance Sampling To estimate Assume there are n probability densities p 1, p 2,, p n to sample x. Then, is an unbiased estimator of for all x with whenever as long as: GAMES Webinar Shuang Zhao 20

21 Weighting Functions The balance heuristic Then, How good is the new estimator? As long as there exists a good estimator for, will also be good GAMES Webinar Shuang Zhao 21

22 Example: MIS Consider the problem of evaluating: denotes the indicator function with two probability densities f: normal distribution with mean 0 and variance σ 2 g: uniform distribution between a and b GAMES Webinar Shuang Zhao 22

23 Example: MIS σ = 1; a = 1.9, b = 2.0 GAMES Webinar Shuang Zhao 23

24 Example: MIS σ = 0.05; a = -3.0, b = 3.0 GAMES Webinar Shuang Zhao 24

25 The Rendering Equation GAMES Webinar: Rendering Tutorial 2 GAMES Webinar Shuang Zhao 25

26 Radiance Radiant energy at x in direction ω: A 5D function per projected surface area per solid angle : Power GAMES Webinar Shuang Zhao 26

27 Light Transport Goal Describe steady-state radiance distribution in virtual scenes Assumptions Geometric optics Achieves steady state instantaneously GAMES Webinar Shuang Zhao 27

28 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 GAMES Webinar Shuang Zhao 28

29 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) GAMES Webinar Shuang Zhao 29

30 Rendering Equation (RE) = + = + GAMES Webinar Shuang Zhao 30

31 Rendering Equation Incoming radiance GAMES Webinar Shuang Zhao 31

32 Rendering Equation (Invariant of radiance along lines) GAMES Webinar Shuang Zhao 32

33 Monte Carlo Path Tracing GAMES Webinar: Rendering Tutorial 2 GAMES Webinar Shuang Zhao 33

34 Path Tracing (Version 0) Estimating L r using MC integration: Draw ω i uniformly at random p(ω i ) = 1/(2π) GAMES Webinar Shuang Zhao 34

35 Path Tracing (Version 0) Light source Light path GAMES Webinar Shuang Zhao 35

36 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 GAMES Webinar Shuang Zhao 36

37 Idea: Separating Direct & Indirect Indirect Direct GAMES Webinar Shuang Zhao 37

38 Direct Illumination Direct GAMES Webinar Shuang Zhao 38

39 Indirect Illumination Indirect GAMES Webinar Shuang Zhao 39

40 Summary: Direct + Indirect L = L e + L r Non-recursive This idea is usually called next-event estimation Recursion GAMES Webinar Shuang Zhao 40

41 Estimating Direct Illumination Idea: sampling the light source Solid angle integral area integral Change of measure GAMES Webinar Shuang Zhao 41

42 Path Tracing (with NEE) Light source Shadow ray Shadow ray Light path GAMES Webinar Shuang Zhao 42

43 Path Integral Formulation GAMES Webinar: Rendering Tutorial 2 GAMES Webinar Shuang Zhao 43

44 Path Integral Formulation (Preview) Goal: rewriting the measurement equation as an integral of the form where form denotes individual light paths for the GAMES Webinar Shuang Zhao 44

45 Benefits Express the measurement I as an integral instead of an integral equation Provide a unified framework for deriving advanced estimators of I GAMES Webinar Shuang Zhao 45

46 Recap: Change of Measure From solid angle to area: We now apply this idea to the RE as well as the measurement equation GAMES Webinar Shuang Zhao 46

47 Path Integral Formulation of the RE By repeatedly expanding L, we get: where for any : GAMES Webinar Shuang Zhao 47

48 Applying the Path Integral Formulation The path integral can be estimated via Monte Carlo integration framework: What density p to use? How to draw a sample path from p? GAMES Webinar Shuang Zhao 48

49 Recap: Local Path Sampling Path tracing Without next-event estimation One transport path at a time With next-event estimation Multiple transport paths at a time GAMES Webinar Shuang Zhao 49

50 Advanced Methods GAMES Webinar: Rendering Tutorial 2 GAMES Webinar Shuang Zhao 50

51 Advanced Methods Bidirectional path tracing (BDPT) Metropolis light transport (MLT) GAMES Webinar Shuang Zhao 51

52 Bidirectional Path Tracing Build light transport paths by connecting two sub-paths starting from the light source and the sensor, respectively [Veach 1997] Use multiple importance sampling (MIS) to properly weight each path GAMES Webinar Shuang Zhao 52

53 Building Transport Paths For any, create light sub-path and sensor sub-path (using local path sampling) The full path two pieces: is obtained by concatenating these GAMES Webinar Shuang Zhao 53

54 Building Transport Paths Remark: there is more than one sampling technique for each path length (k + 1) techniques for paths with k vertices [Veach 1997] GAMES Webinar Shuang Zhao 54

55 Building Transport Paths For each s and t, the construction of a probability density Similar to the unidirectional case, equals the product of densities of sampling both sub-paths gives GAMES Webinar Shuang Zhao 55

56 Building Transport Paths Using multiple importance sampling, we can combine all these path sampling schemes, resulting in: where w s,t is the weighting function Balance heuristic GAMES Webinar Shuang Zhao 56

57 Example: Modified Cornell Box Area light facing up The scene is lit largely indirectly Cornell Box Mirrored ball Camera facing a mirrored ball The scene is observed indirectly Difficult to render with unidirectional path tracing GAMES Webinar Shuang Zhao 57

58 Example: Modified Cornell Box (Rendered in equal-time) Path tracing Bidirectional path tracing GAMES Webinar Shuang Zhao 58

59 Metropolis light transport (MLT) A Markov Chain Monte Carlo (MCMC) framework implementing the Metropolis- Hasting method first proposed by Veach and Guibas in 1997 Capable of efficiently constructing difficult transport paths Lots of ongoing research along this direction GAMES Webinar Shuang Zhao 59

60 Metropolis-Hasting Method A Markov-Chain Monte Carlo technique Given a non-negative function f, generate a chain of (correlated) samples X 1, X 2, X 3, that follow a probability density proportional to f Main advantage: f does not have to be a PDF (i.e., unnormalized) GAMES Webinar Shuang Zhao 60

61 Metropolis-Hasting Method Input Non-negative function f Probability density g(y x) suggesting a candidate for the next sample value x, given the previous sample value y The algorithm: given current sample X i 1. Sample X from g(x i X ) 2. Let 3. Set X i+1 to X with prob. a; otherwise, set X i+1 to X i Start with arbitrary initial state X 0 GAMES Webinar Shuang Zhao 61

62 Path Mutations The key step of the MLT Given a transport path, we need to define a transition probability to allow sampling mutated paths based on Given this transition density, the acceptance probability is then given by GAMES Webinar Shuang Zhao 62

63 Desirable Mutation Properties High acceptance probability should be large with high probability Large changes to the path Ergodicity (never stuck in some-region of the path space) should be non-zero for all with Low cost GAMES Webinar Shuang Zhao 63

64 Path Mutation Strategies [Veach & Guibas 1997] Bidirectional mutation Path perturbations Lens sub-path mutation [Jakob & Marschner 2012] Manifold exploration [Li et al. 2015] Hamiltonian Monte Carlo GAMES Webinar Shuang Zhao 64

65 Bidirectional Path Mutations Basic idea Given a path the vertices l and m satisfies, pick l, m and replace with Image plane Image plane GAMES Webinar Shuang Zhao 65

66 Path Perturbation Slightly modify the direction x m x m-1 (at random) Trace a ray from x m with this new direction to form the new sub-path Center of projection Image plane Specular interface Diffuse surface GAMES Webinar Shuang Zhao 66

67 Results BDPT [Veach 1997] MLT (equal-time) GAMES Webinar Shuang Zhao 67

68 Results BDPT [Veach 1997] MLT (equal-time) GAMES Webinar Shuang Zhao 68

69 Summary Physics-based rendering is a rich field Things we have covered Light transport model Rendering equation (reflection and refraction) Monte Carlo solutions Path tracing Bidirectional path tracing Metropolis light transport GAMES Webinar Shuang Zhao 69

70 Topics we have not covered Radiative transfer (sub-surface scattering) Unbiased methods Primary sample space & multiplexed MLT Gradient-domain PT/MLT Biased methods Photon mapping Lightcuts Monte Carlo denoising Diffusion approximations GAMES Webinar Shuang Zhao 70