Gan, Z. (2018):
Variance of photon mapping density estimation
Computer graphics is a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. The major subfields in computer graphics might be geometry, animation, imaging, topology and rendering. The rendering generates images from a model. It may simulate light transport to create realistic images or it may create non-photorealistic images that have a particular artistic style. The rendering techniques can be classified into local illumination techniques and global illumination techniques. The local illumination algorithms are very fast, but images rendered using global illumination algorithms often appear more photorealistic than those using only local illumination algorithms. There are many algorithms used in global illumination. Photon mapping is a very popular two-pass global illumination algorithm: the first pass is casting photons from the light source and saving the information of reflection in the so called photon map; in the second pass, the brightness of the pixels are estimated from the photon map, this pass is also called radiance estimate. Many studies have successfully tried to improve the radiance estimation, for example, the radiance estimation can be improved by adding a filter of the distance. Because the error analysis of rendering algorithms is beneficial for understanding their behavior, this thesis extended the framework presented by Dr. García and presents an analysis of the variance in common used filtering kernels in the context of photon mapping density estimation. We use the joint distribution of order statistics to calculate the variance value in both 2D and 3D case for the constant kernel, the epanechnikov kernel, the silverman kernel (also called quartic kernel), the cone filter and the triangle kernel, which is a special case of the cone filter. Corresponding to our theoretical study, we have implemented a scene consisting of a planar unit disc, a very simple model, which illuminated by a directional light source for each kernel and calculated the signal to noise ratio of each kernel. After it we show the signal-to-noise ratio of our theoretical results and its computational cost compares to the empirical results and computational time. The theoretical comparision among different filters allow us to choose the best kernel for our needs. Besides, we could offer a threshold, the estimation of computational cost of algorithms which stop after error is smaller than the threshold.