A Random Walk Approach for Light Scattering in Material

Seeing means perceiving light from an external source which was scattered in a material medium and reemitted toward the observer’s eye. Thereby colors are the result of wavelength dependent absorption processes. The physical basis of color as molecular or atomic phenomenon is state-of-the-art in science, both in experimental and in theoretical respect. But to the authors knowledge, the deriving of macroscopic applications therefrom, for instance printing, is not really successful because of the huge complexity of corresponding models. Accordingly, state-of-the-art technologies like color management systems represent entirely empirical approaches which can be considered as scientific capitulation to the practical problems of color prediction. Therefore the ongoing challenge of modeling scattering affects is to optimize the compromise between simplicity of the use and the accuracy of prediction. For printing the widely used theory of KUBELKA and MUNK dates back to 1931 [1]. In the course of time this original theory was extended by several authors, see PHILIPS–INVERNIZZI [2] for a recent survey. In SIMON [3] the authors started the trial to give a stochastic interpretation of KUBELKA and MUNK’s theory. Now, this paper presents the next step on this trail containing several extensions and, in particular, a combinatorial analysis of the underlying distributions and solutions for the reflectance and the transmittance in case of finite layer thickness. The original work of KUBELKA and MUNK proposed a system of differential equations induced by a simplified model of light propagation, similar to the concepts of SCHUSTER [4] and GUREVIC̆ [5]. A horizontal colorant layer with thickness D is considered. We suppose that the paint material is homogeneous. Then any light inside the layer traveling in any direction can be divided into its vertical and horizontal components. For simplification, the horizontal components are ignored. Therefore only two