Calculation of MIS Weights for Bidirectional Path Tracing with Photon Maps

Abstract

A widely used method for noise reduction in Monte-Carlo ray tracing is combing different means of sampling, known as multiple importance sampling (MIS). For bi-directional Monte-Carlo ray tracing with photon maps (BDPM), the join paths are obtained by merging camera and light sub-paths, and since several light paths are checked again the same camera path, and vice versa, the join paths obtained are not statistically independent. Thus the noise in this method obeys laws different from those in simple classic Monte-Carlo with independent samples so the weights that minimize that noise must also be calculated differently. This paper drives that weights for the simplest case when we mix contribution from only two vertices of camera ray. It shows that the weights obey an integral equation which is qualitatively different from the well-known MIS formulae for uncorrelated samples. Besides that, even if forget the integral operator, the weights depend on the integration sphere radius and the number of light rays used. The integral equation is solved analytically in a closed form and it is demonstrated how to perform the necessary calculations in BDPM.

Calculation of Noise Components for Bidirectional Path Tracing with Photon Maps

“…"Diffuse depth" i.e. the maximal allowed number of diffuse events for camera ray [9] was BDD=1. Scene image is shown in Figure 1.…”

“…where C(X (F ) , X (B) ) is the contribution to pixel luminance from "merging" the light path X (F ) and camera path X (B) , its average C is obviously the limiting pixel luminance L, and • F , • B denote the averages over the ensemble of light and camera paths, respectively. Notice this law holds regardless of whether we use Multiple Importance Sampling [9], [11], [4] or not which only affects the values of C 2 , C 2 F B and C 2 B F . Calculation of C 2 is easy: whenever we add C to the pixel luminance, we also add C 2 to the "error image".…”

“…The level of noise in this method mainly depends on the method of random scattering of the forward and backward rays, on choosing the vertex to merge (or, in other words, in which vertex of the camera path to use photon maps for estimation of the illumination) and, at last, on the number of forward and backward rays traced during one iteration. Most publications are devoted to the first two means, for example, [9], [7], [10] or [8], while less attention had been paid to the number of rays. Meanwhile it is an important factor and frequently it happens that e.g.…”