Metropolis Light Transport. Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire

Transcription

1 Metropolis Light Transport Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire

2 Overview of Presentation 1. Description of necessary tools (Path Space, Monte Carlo Integration, Rendering Equation) 2. A description of the Metropolis Light Transport algorithm. 3. Details into the workings of some parts of the algorithm, especially mutations

3 MLT Samples Paths --A Path is a sequence of points in the scene along which light travels --MLT samples paths as a whole --The set of all scene paths is called Path Space Veach and Guibas. Metropolis Light Transport. (1997) Figure 1. So what is a path? As should be apparent, a path is a feasible path of light that can travel from a light source to the camera. How does this differ from the recursive ray tracer we built in the earlier graphics lab? A path simulates light particles as they bounce around the scene (these bounces are calculated as random directional values), picking up all the values they will need to solve the rendering equation; they hit a color of surface with a high reflectivity (energy of ray after hitting surface), with some surface graininess (reflection/refraction), etc etc. In a ray tracer light rays don t actually physically bounce; light rays are fired from the camera into the scene, but where those light rays intersect, they fire multiple rays to every light in the scene, and then computes the pixel value based on the surfel properties of the object with the amount of light that pixel is receiving from all the lights in the scene. This means that ray tracing can actually only compute direct lighting in reality. All other effects, such as caustics and global illumination, are based on separate, non-physically based equations. In other words, what our recursive ray tracer in Lab 3 did was that it didn t actually compute global or indirect illumination. It only considered the possibility that some other non-light source object might be potentially emitting light and contributing energy to other surfaces. With a paths, all of those things are inherently capturered.

4 Monte Carlo Integration: Approximating an integral through random sampling f(x) Wikipedia, CC License x1 x3 x2 Monte Carlo Integration involves approximating an integral through a random sampling. How does this make sense? Well, by Riemann Summation, we have the below limit is true, and we can approximate it by a finite number of samples. Now, most times, this is interpreted as uniform samples, but why not take the samples at random instead. This is the premise of Monte Carlo Sampling. In the diagram, you see an example function f(x), which we want to know the integral of. If we don t know the formula for f, how can we integrate it? Well, we know about Riemann Summation, and how that can be approximated by a finite sum. However, the actual Riemann sum can be approximated by taken a finite number of samples of the domain chosen uniformly at random, as you see in the diagram

5 Non-Uniform Sampling uniform sampling: f(x) Sampling with probability density p(x): Sometimes it makes sense to sampling the domain non-uniformly, In that case, we have to modify Monte Carlo by considering expected value sums of f weighted by the inverse probability density, as shown below, the expected value of f with this weight, if we sample according to the density, is actually precisely the integral, so this works out very well.

6 Our Integral is the Rendering Equation L: transported radiance fs: BSDF G: geometry term M: Surfaces in scene da: area measure So, we mentioned Monte Carlo Integration, how should we apply it? We need to approximate an integral, and this integral is the integral of the rendering equation. Now, here, the domain is points in the scene, and the integral is recursive, which can lead to some issues. Now, there are some solutions that have been found to this using monte carlo integration

7 We Convert the Integral to Path Space We is the lens filter function (Product Measure) We take the rendering equation, which is defined infinitely recursively as an integral over points in the scene, and then we expand the integral by writing out the steps of the recursion. Now, we observe that in all cases, the integrand is simply the contribution of the path from x0 to xn, so we can express this entire integral as simply an integral over path space, with f the path contribution function, and mu an appropriate measure, which expands to the proper product. A path simulates light particles as they bounce around the scene (these bounces are calculated as random directional values), picking up all the values they will need to solve the rendering equation; they hit a color of surface with a high reflectivity (energy of ray after hitting surface), with some surface graininess (reflection/refraction), etc etc. As light continues to bounce around, it bends or splits into multiple light rays depending on the material properties (glass, diamond, etc) that it interacts with. Those rays also bounce around, performing the same function. Eventually, all these rays hit a light source, giving the final piece of the rendering puzzle; the initial amount of energy. The equation is complete and the computer renders the pixel s final color value based on the sum total of that equation.

8 Overview of the Metropolis Algorithm --Method of sampling a hard to sample distribution --Provably unbiased convergence --Random Walk using Markov Chain of states. The Metropolis Algorithm is a way of sampling hard-to-sample probability distributions, that dates back to a paper in Chemical Physics in 1953 by Metropolis et. al. If set up properly, sampling in this way will lead to an unbiased result (essentially because we always sample proportional to the desired distribution) The Metropolis Algorithm works by sampling the space through a Markov Chain of states: establishing probabilities for transitioning to the next state.

9 Random Walks and Markov Chains --Moving from point to point in a space with some random probability of travelling between adjacent points. --Markov Chains encode these probabilities of going from state to state. Osgood, Brad. EE261 The Fourier Transform and its Applications Fall 2007, Course Materials. (CC License) A brief summary of the idea of a random walk, with an illustration in R^3. A random walk is just going through some space, from adjacent point to adjacent point, with some probability of walking in any direction this idea is made much more abstract in the case of Phase Space, but is fairly easy to visualize in R^3, and a visualization is given here. To give a very brief summary, Markov Chains are a way of encoding these probabilities of travelling between adjacent states, and computing the probability of being in any given state at a given point in time

10 Metropolis Framework for MLT --Random walk through Path space through mutations. --Seeded by path tracing --Combine samples for Monte Carlo approximation Veach and Guibas. Metropolis Light Transport. (1997) pg. 2. So, we described Monte Carlo Processes, now, what is the framework of the Metropolis Light Transport Algorithm, and how does it apply to Monte Carlo? Well, we are trying to approximate that integral over path space, so we want to take a finite number of samples, chosen randomly. However, it makes intuitive sense that we do not want to sample uniformly, as most paths will have 0 contribution. So, we sample by picking an initial path out of some distribution: we get this distribution by doing path tracing, as that gives us a lot of paths which we know have non-zero transport. Then, we proceed to do a random walk around path space, mutating paths through insertions and deletions, and accepting only based on a calculated acceptance probability that is proportional to the contribution of that path.

11 MLT is Unbiased --With infinite samples, our computation converges to the desired integral. --Unbiased because of: 1. Ergodicity 2. Acceptance Probabilities Stationary Distribution The MLT algorithm is unbiased, which means that it actually will converge in the limit. Two properties that contribute to this are ergodicity, meaning it can never be stuck, and that our acceptance propabilities are chosen such that we get a stationary distribution, so once we reach a correct sampling distribution, we never deviate from it.

12 Bidirectional Mutations 3 Delete a subpath (2, 6) As we described, MLT involves sampling path space through random walks where we get from one path to another by mutation. But, what are these mutations, how do we do them, etc? Well, the most crucial type of mutation is a bidirectional mutation. For a bidirectional mutation, we delete some subpath (chosen at random with some probability) of the initial path. Then, we need to fill in the gap, so we choose at random some number of edges to add from each side. Then we scatter, so, sampling BSDFs, in order to pick new internal points in the path, growing towards the center Finally, we reconnect the two paths if the two points are mutually visible: otherwise, the path will have 0 contribution, and will be immediately thrown out.

13 Bidirectional Mutations 3 Delete a subpath (2, 6) Choose # to add on each side L: 2, R:

14 Bidirectional Mutations 3 Delete a subpath (2, 6) Choose # to add on each side L: 2, R: 2 Sample BSDF (scatter) to get new points

15 Bidirectional Mutations 3 Delete a subpath (2, 6) Choose # to add on each side L: 2, R: 2 Sample BSDF (scatter) to get new points Reconnect (or throw out)

16 Reasons for MLT over BDPT Veach and Guibas. Metropolis Light Transport. (1997) MLT really shines in certain scenes. The key comes from the fact that we perform a random walk, which probabilistically spends most of its time in regions of Path Space with large contributions. So, in this scene (taken from Veach s paper), all the light comes from a light on the ceiling of the other room, through a tiny crack through the ajar door. In BDPT most bounces will contribute no light, but here, once we have a path that goes through the door, we will mutate that path and get contributions from that path and nearby paths, so that we are mostly looking at paths that actually contribute

17 Citations Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. Equations of state calculations by fast computing machines. Journal of Chemical Physics 21 (1953), Osgood, Brad. EE261 The Fourier Transform and its Applications Fall 2007, Course Materials. (CC License) Van Dam, Andries, and Steven K. Feiner. Computer graphics: principles and practice. Pearson Education, Veach, E., & Guibas, L. J. (1997). Metropolis light transport. SIGGRAPH, 31, 65 76

18 Markov Chains: a Summary This slide describes the basic notion of Markov Chains, using finite dimensional examples. Markov chains encode the probabilities that a system, that is in some state, transitions to some other state. The chain is the chain of states that occurs by applying the transition matrix a number of times, so from an initial vector of probabilities that the system begins in a state, we can obtain vectors of probabilities after 1,2,...,n states. This therefore allows the building of a random walk, in the sense that you know the probability a random walk would end up at any given point after a certain number of steps.

19 Other Mutation Schemes: Perturbations --Other than bidirectional mutations, other mutations that are useful in sampling path space effectively are lens and caustic perturbations, which involve taking some number of points in the path from the lens size and slightly adjusting the locations of the points There are other types of mutations common in use with Metropolis Light Transport, the main ones involve perturbations of points.

20 Eliminating Start-up bias --Starting at a single random path will likely result in a black image: the initial path will have 0 contribution. --To fix this, we start with a distribution of contributing paths (from path tracing) --Also, we run multiple trials of the algorithm, and average together the results (i.e, multiple initial paths) --Direct Illumination computed via simple ray tracing

21 Results!?