Abstract: In this paper we introduce the potential equation that along with the rendering equation forms an adjoint system of equations and provides a mathematical frame work for all known approaches to illumination computation based on geometric optics. The potential equation is more natural for illumination computations that simulate light propagation starting from the light sources, such as progressive radiosity and particle tracing. Using the mathematical handles provided by this framework and the random-walk solution model, we present a number of importance sampling schemes for improving the computation of flux estimation. Of particular significance is the use of approximately computed potential for directing a majority of the random walks through regions of importance in the environment, thus reducing the variance in the estimates of luminous flux in these regions. Finally, results from a simple implementation are presented to demonstrate the high-efficiency improvements made possible by the use of these techniques.