Noise Dependence on the Number of Rays in Bidirectional Stochastic Ray Tracing with Photon Maps

The classical Monte Carlo ray tracing is a powerful technique for modeling almost all effects in geometric optics; however, it can be prohibitively slow in many cases, such as generation of images seen by an objective or camera with a point aperture. For this reason, numerous modifications of this technique are used, among which is the bidirectional stochastic ray tracing with photon maps. A drawback of all stochastic methods is the undesirable noise. The noise level, i.e., the variance of the pixel luminance calculated for one iteration step, depends on various parameters, such as the number of rays traced from the light source and from the camera, the method of merging their trajectories, the integration sphere radius, etc. The choice of the optimal parameters makes it possible to minimize the noise level for the given computation time. This is the topic of the current paper. It is shown that the variance of the pixel luminance is the sum of three functions scaled by the reciprocal of the number of rays tracedfrom the light source and from the camera, where the functions themselves are independent on the number of rays. Therefore, given these functions, one can predict the noise for any number of rays and thus find the optimal set of parameters. The calculation of these functions based on the data obtained by ray tracing is a nontrivial problem. The paper proposes a practical method for their calculation, and demonstrates that a single such calculation is able to predict the variance for an arbitrary number of rays. Therefore, the noise can be minimized due to the optimal choice of the number of rays.