We investigate new method for calculation of radiance of scattering medium by bi-directional Monte-Carlo ray tracing with photon maps. Usually photons are collected by an integration spheres at the ends of camera ray segments, or a cylinder along that segments. Meanwhile in our method several integration spheres are distributed at random along the first camera ray segment. The rest segments do not collecting photons. The method optimal for a particular scene is the one which produces the least noise, so one need to be able to estimate it. In this paper an analytic calculation of noise in the general bi-directional Monte-Carlo ray tracing is derived and then applied to the proposed method. Then the analytic estimates of noise can be used to find optimal parameters and/or to choose between single integration sphere, multiple integration spheres and integration cylinders.2010 Mathematics Subject Classification: 78-04, 65C05, 65C20.

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Multiple importance sampling (MIS) is a well-known method for noise reduction in Monte-Carlo ray tracing. It weights contributions from merging camera and light paths in different vertices. Since noise strongly depends on these weights, the problem of the optimal choice of weight to reach the minimal noise is very important. For bi-directional Monte-Carlo ray tracing with photon maps (BDPM), different join paths are not statistically independent because several light paths are checked against the same camera path and vice versa. As a result, the optimal weights which minimize the noise functional in the classic Monte Carlo ray tracing and in BDPM are different. In this paper we calculate weights for the simple reduced case of just two strategies, i.e. merging at just two vertices of camera ray. We show that these weights obey an integral equation which is qualitatively different from the well-known MIS formulae for uncorrelated samples. The integral equation is solved analytically in a closed form and one can see that the MIS weights for BDPM algorithm depend on the number of rays and scene geometry. In this paper we also correctly take into account the direct illumination to pixel luminance.