It’s still not perfect, far from it in fact, but it’s progress none the less. I’ve been reading a lot lately about Metropolis Light Transport, Manifold Exploration, Multiple Importance Sampling (they do love their M names) and it’s high time I started implementing some of them myself.
So it’s with great sadness that I am retiring my PRT project which began over a year ago, all the way back at the start of my dissertation. PRT is written in Java, for simplicity, and was designed in such a way that as I read new papers about more and more complex rendering techniques I could easily drop in a new class, add a call to the render loop, or even replace the main renderer all together with an alternative algorithm which still called upon the original framework.
I added many features over time from Ray Tracing, Photon Mapping, Phong and Blinn-Phong shading, DOF, Refraction, Glossy Surfaces, Texture Mapping, Spacial Trees, Meshes, Ambient-Occlusion, Area Lighting, Anti-Aliasing, Jitter Sampling, Adaptive Super-Sampling, Parallelization via both multi-threading and using the gpu with OpenCL, Path Tracing, all the way up top Bi-Directional Path Tracing.
But the time has taken it’s toll and too much has been added on top of what began as a very simple ray tracer. It’s time to start anew.
My plans for the new renderer is to build it entirely in C++ with the ability to easily add plugins over time like the original. Working in C++ gives a nice benefit that as time goes by I can choose to dedicate some parts of the code to the GPU via CUDA or OpenCL without too much overhead or hassle. For now though the plan is to rebuild the optimized maths library and get a generic framework for a render in place. Functioning renderers will then be built on top of the framework each implementing different feature sets and algorithms.
Continuing on from experiments last week with Metropolis Hastings sampling for probability distributions I decided to implement Accept-Reject sampling. Accept-Reject seems to deliver well distributed results faster than Metropolis Hastings does but in the long run can easily introduce biased samples if the proposal distribution q(x) is not well tuned to the specific probability distribution function. MH on the other hand, is far more resilient to an improperly chosen proposal distribution as it will still converge to an unbiased result in most cases, albeit by taking a longer time to converge than it normally would. Accept-Reject also seems to have issues towards the truncations of it’s function causing under-sampling of values close to the lower and upper bounds.
A little update on what I’ve been learning about lately.
Metropolis-Hastings is an algorithm for sampling random values out of a probability distribution. Effectively, for a function f(x) you wish to simulate where the curve is higher it is more probable that a random number will be chosen from there. Uniform sampling of the range would mean that all x values on the graph have an even probability of being chosen, this could be simulated by saying f(x) = 1.
However, if the function f(x) represents a curve then the number of samples in high energy (probability) regions needs to be more dense. This can be seen in the simulations of f(x) = sqrt(x-1) and f(x) = sin(1/x) below.
Metropolis-Hastings is best suited to problems where Direct Sampling and other more efficient solutions are not available. Because it only relies on the availability of a computable function f(x) and a proposal density Q(x|x*) M-H can correctly sample unusual probability distributions that would else-wise be difficult or impossible to compute. This can be seen below in the simulations of the sine function f(x) = sin(x) + 1.5.
Matlab code for Metropolis-Hastings Monte Carlo Integration (for generating samples)
Below is the Matlab code used to generate the above graph. By modifying the function f(x), the sample count N, the proposal distribution and it’s variance qV, and the upper and lower truncation bounds LB & HB any PDF can be simulated.
Work continues and it’s going well! Over the last week I have been teaching myself Bayesian Statistics which allows you to think about probability in terms of degrees of belief. It is also predicated on the concept of parameters, or prior known data, allowing you to update the probability of something happening in the future. This allows for convergent solutions on unknown probabilities which useful in graphics in terms of being able to fully sample arbitrary probability distributions.