Metropolis Light Transport

Transcription

1 Metropolis Light Transport CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1

2 Announcements Final presentation June 13 (Tuesday) at 4:00 pm in ICS 180 Each project is supposed to have a minute presentation CS295, Spring 2017 Shuang Zhao 2

3 Previous Lectures Rendering equation Radiative transfer equation Unbiased Monte Carlo solutions Path tracing Adjoint particle tracing Bidirectional path tracing CS295, Spring 2017 Shuang Zhao 3

4 Today s Lecture 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 long this direction MLT is capable of solving both the rendering equation (RE) and the radiative transfer equation (RTE). We will focus on the former CS295, Spring 2017 Shuang Zhao 4

5 Recap: 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) CS295, Spring 2017 Shuang Zhao 5

6 Recap: 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 and draw 3. If, set X i+1 to X ; otherwise, set X i+1 to X i Start with arbitrary initial state X 0 CS295, Spring 2017 Shuang Zhao 6

7 The Problem We focus on estimating the pixel values of a virtual image where intensity I (j) of pixel j is Image plane CS295, Spring 2017 Shuang Zhao 7

8 The Problem We focus on estimating the pixel values of a virtual image where intensity I (j) of pixel j is h (j) varies per pixel and is called the filter function f stays identical for all pixels CS295, Spring 2017 Shuang Zhao 8

9 Example Filter Functions Box Filter h (j) Image plane Gaussian Filter h (j) Image plane CS295, Spring 2017 Shuang Zhao 9

10 Estimating Pixel Values We have seen that if we can draw N path samples according to some density function p, then Particularly, if we take, namely with b being the normalization factor, then CS295, Spring 2017 Shuang Zhao 10

11 Estimating Pixel Values Challenges How to obtain? Monte Carlo integration How to draw samples from? Metropolis-Hasting method CS295, Spring 2017 Shuang Zhao 11

12 Metropolis Light Transport (MLT) Overview Phase 1: initialization (estimating b) Draw N seed paths from some known density p 0 (e.g., using bidirectional path tracing) Set Pick a small number (e.g., one) of representatives from and apply Phase 2 to each of them Phase 2: Metropolis Starting with a seed path, apply the Metropolis-Hasting method to generate samples according to f CS295, Spring 2017 Shuang Zhao 12

13 Metropolis Phase Overview (pseudocode) Metropolis_Phase(image, x seed ): x = x seed for i = 1 to N: y = mutate(x) a = acceptanceprobability(x y) if rand() < a: x = y recordsample(image, x) CS295, Spring 2017 Shuang Zhao 13

14 Path Mutations The key step of the Metropolis phase 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 CS295, Spring 2017 Shuang Zhao 14

15 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 CS295, Spring 2017 Shuang Zhao 15

16 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 CS295, Spring 2017 Shuang Zhao 16

17 Bidirectional Path Mutations Basic idea Given a path the vertices l and m satisfies, pick l, m and replace with Image plane Image plane CS295, Spring 2017 Shuang Zhao 17

18 Deletion Probability Image plane l and m are sampled as follows: Draw integer k d from some probability mass function p d,1 [k d ]. This number captures the length of deleted sub-path (i.e., m - l) Draw l from another probability mass function p d,2 [l k d ] to avoid low acceptance probability and set m to l + k d (more on this at the end of today s lecture) The joint probability p d for drawing (l, m) is CS295, Spring 2017 Shuang Zhao 18

19 Addition Probability Image plane The deleted sub-path is then replaced by adding l and m vertices on each side. To determine l and m : Draw integer k a from p a [k a ]. This integer determines the new sub-path length (i.e., k a = l + m + 1) Draw l uniformly from {0, 1,, k a - 1} and set m to k a l Let p a [l, m ] denote the joint probability for drawing (l, m ) After obtaining l and m, the two corresponding subpaths are generated via local path sampling, yielding the new path CS295, Spring 2017 Shuang Zhao 19

20 Parameter Values Veach [1997] proposed the following parameters: Deletion parameters p d,1 [1] = 0.25, p d,1 [2] = 0.5, p d,1 [k] = 2 -k for k > 2 (before normalization) p d,2 [l k d ] to be discussed later Addition parameters (given k d ) p a,1 [k d ] = 0.5, p a,1 [k d ± 1] = 0.15, p a,1 [k d ± j] = 0.2(2 -j ) for j > 2 (before normalization) CS295, Spring 2017 Shuang Zhao 20

21 Evaluating Transition Probability Image plane Image plane The probability for transitioning from to is CS295, Spring 2017 Shuang Zhao 21

22 Bidirectional Mutation: Example Image plane Original path: Mutation parameters: l = 1, m = 2 (deletion); l = 1, m = 0 (addition) Mutated path: The probability to accept equals where CS295, Spring 2017 Shuang Zhao 22

23 Bidirectional Mutation: Example Image plane, where Recall that does not involve as it is captured by the filter function h (j) CS295, Spring 2017 Shuang Zhao 23

24 Bidirectional Mutation: Example can be generated from in two ways Image plane Image plane Thus, CS295, Spring 2017 Shuang Zhao 24

25 Bidirectional Mutation: Example Image plane, where To obtain from using bidirectional path mutation, we need l = 1, m = 3 and l = m = 0. Thus, CS295, Spring 2017 Shuang Zhao 25

26 Path Perturbations Smaller mutations Useful for finding nearby paths with high contribution CS295, Spring 2017 Shuang Zhao 26

27 Path Perturbations Basic idea: choosing a sub-path and moving the vertices slightly Three types of perturbations Lens Caustic Multi-chain CS295, Spring 2017 Shuang Zhao 27

28 Path Perturbation: Lens Replace sub-paths (x 0,, x m ) of the form ES*D(D L) Randomly move the endpoint x 0 on the image plane to z 0 Trace a ray through z 0 to form the new sub-path Center of projection Diffuse surface Image plane Specular surface Diffuse surface CS295, Spring 2017 Shuang Zhao 28

29 Path Perturbation: Lens To draw z 0 : First, sample a distance R using Then, uniformly sample z 0 from the circle which is center at x 0 and has radius R The mutation is immediately rejected if ray tracing through z 0 fails to generate a new sub-path with exactly the same form (i.e., ES*D(D L)) Otherwise, the acceptance probability is evaluated in a way similar to the bidirectional mutation case CS295, Spring 2017 Shuang Zhao 29

30 Path Perturbation: Caustic Replace sub-paths (x 0,, x m ) of the form EDS*(D L) 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 CS295, Spring 2017 Shuang Zhao 30

31 Path Perturbation: Multi-Chain Replace sub-paths of the form ES + DS + D(D L) Lens perturbation is applied for ES + D The direction of the DS + edge in the original sub-path is perturbed The new direction is then used to complete the DS + D(D L) segment of the new sub-path (using ray tracing) Image plane Diffuse surface Specular interface Diffuse surface CS295, Spring 2017 Shuang Zhao 31

32 Lens Sub-Path Mutation Used to stratify samples over the image plane Each pixel should get enough sample paths Replace lens sub-paths of the form ES*(D L) Similar to lens perturbation, but draw z 0 from a different density Center of projection Diffuse surface Image plane Specular surface CS295, Spring 2017 Shuang Zhao 32

33 Selecting Between Mutation Types Path mutations strategies introduced so far: Bidirectional mutation Lens, caustic, multi-chain perturbations Lens sub-path mutation Choose one randomly in each iteration CS295, Spring 2017 Shuang Zhao 33

34 Refinements Direct lighting It is more efficient to estimate direct illumination with standard methods (e.g., area & BSDF sampling combined using MIS) and apply MLT only for indirect illumination Importance sampling for mutation probabilities For increasing the average acceptance probability CS295, Spring 2017 Shuang Zhao 34

35 Improving Acceptance Rates Recall: Observation: given a path and, can be partially evaluated without constructing can be fully evaluated can be partially evaluated CS295, Spring 2017 Shuang Zhao 35

36 Improving Acceptance Rates Image plane Let k a = m - l - 1, then Unknown Known Set the unknown term to one and get a weight w l,m for each mutation CS295, Spring 2017 Shuang Zhao 36

37 Improving Acceptance Rates Given a path, we can evaluate the weights for several possible mutation strategies and use these weights to sample one Can be used to obtain p d,2 for bidirectional mutations Given k d, simply make CS295, Spring 2017 Shuang Zhao 37

38 [Veach 1997] Results BDPT MLT (equal-time) CS295, Spring 2017 Shuang Zhao 38

39 [Veach 1997] Results BDPT MLT (equal-time) CS295, Spring 2017 Shuang Zhao 39

40 Closing Notes Realistic image synthesis is a rich field Things we have covered Light transport models Rendering equation (reflection and refraction) Radiative transfer (sub-surface scattering) Monte Carlo solutions Path tracing, adjoint particle tracing Bidirectional path tracing Metropolis light transport Volume path tracing CS295, Spring 2017 Shuang Zhao 40

41 Closing Notes Related topics we have not covered Anisotropic radiative transfer Unbiased methods Primary sample space & multiplexed MLT Gradient-domain PT/MLT Biased methods Photon mapping Lightcuts Monte Carlo denoising Diffusion approximations CS295, Spring 2017 Shuang Zhao 41

42 Closing Notes Related topics we have not covered Appearance acquisition Measuring optical properties (e.g., BRDFs) from realworld samples Appearance fabrication Creating physical materials with desired optical properties CS295, Spring 2017 Shuang Zhao 42