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We consider the scattering of light in participating media composed of sparsely and randomly distributed discrete particles. The particle size is expected to range from the scale of the wavelength to several orders of magnitude greater, resulting in an appearance with distinct graininess as opposed to the smooth appearance of continuous media. One fundamental issue in the physically-based synthesis of such appearance is to determine the necessary optical properties in every local region. Since these properties vary spatially, we resort to geometrical optics approximation (GOA), a highly efficient alternative to rigorous Lorenz—Mie theory, to quantitatively represent the scattering of a single particle. This enables us to quickly compute bulk optical properties for any particle size distribution. We then use a practical Monte Carlo rendering solution to solve energy transfer in the discrete participating media. Our proposed framework is the first to simulate a wide range of discrete participating media with different levels of graininess, converging to the continuous media case as the particle concentration increases.

We study the structure of Np $\mathcal N_{p}$-spaces in the ball. In particular, we show that any such space is Moebius-invariant and for 0 < p≤ n, all Np $\mathcal N_{p}$-spaces are different. Our results will be used in the study of operator theory on Np $\mathcal N_{p}$-spaces.