Generalized wavelet product integral for rendering dynamic glossy objects

We consider real-time rendering of dynamic glossy objects with realistic shadows under distant all-frequency environment lighting. Previous PRT approaches pre-compute light transport for a fixed scene and cannot account for cast shadows on high-glossy objects occluded by dynamic neighbors. In this paper, we extend double/triple product integral to generalized multi-function product integral. We represent shading integral at each vertex as the product integral of multiple functions, involving the lighting, BRDF, local visibility and dynamic occlusions. Our main contribution is a new mathematical representation and analysis of multi-function product integral in the wavelet domain. We show that multi-function product integral in the primal corresponds to the summation of the product of basis coefficients and integral coefficients. We propose a novel generalized Haar integral coefficient theorem to evaluate arbitrary Haar integral coefficients. We present an efficient sub-linear algorithm to render dynamic glossy objects under time-variant all-frequency lighting and arbitrary view conditions in a few seconds on a commodity CPU, orders of magnitude faster than previous techniques. To further accelerate shadow computation, we propose a Just-in-time Radiance Transfer (JRT) technique. JRT is a new generalization to PRT for dynamic scenes. It is compact and flexible, and supports glossy materials. By pre-computing radiance transfer vectors at runtime, we demonstrate rendering dynamic view-dependent all-frequency shadows in real-time.