Legendre fluids: a unified framework for analytic reduced space modeling and rendering of participating media

In this paper, we present a unified framework for reduced space modeling and rendering of dynamic and non-homogenous participating media, like snow, smoke, dust and fog. The key idea is to represent the 3D spatial variation of the density, velocity and intensity fields of the media using the same analytic basis. In many situations, natural effects such as mist, outdoor smoke and dust are smooth (low frequency) phenomena, and can be compactly represented by a small number of coefficients of a Legendre polynomial basis. We derive analytic expressions for the derivative and integral operators in the Legendre coefficient space, as well as the triple product integrals of Legendre polynomials. These mathematical results allow us to solve both the Navier-Stokes equations for fluid flow and light transport equations for single scattering efficiently in the reduced Legendre space. Since our technique does not depend on volume grid resolution, we can achieve computational speedups as compared to spatial domain methods while having low memory and pre-computation requirements as compared to data-driven approaches. Also, analytic definition of derivatives and integral operators in the Legendre domain avoids the approximation errors inherent in spatial domain finite difference methods. We demonstrate many interesting visual effects resulting from particles immersed in fluids as well as volumetric scattering in non-homogenous and dynamic participating media, such as fog and mist.