Specular Polynomials

Abstract

Finding valid light paths that involve specular vertices in Monte Carlo rendering requires solving many non-linear, transcendental equations in high-dimensional space. Existing approaches heavily rely on Newton iterations in path space, which are limited to obtaining at most a single solution each time and easily diverge when initialized with improper seeds. We propose specular polynomials, a Newton iteration-free methodology for finding a complete set of admissible specular paths connecting two arbitrary endpoints in a scene. The core is a reformulation of specular constraints into polynomial systems, which makes it possible to reduce the task to a univariate root-finding problem. We first derive bivariate systems utilizing rational coordinate mapping between the coordinates of consecutive vertices. Subsequently, we adopt the hidden variable resultant method for variable elimination, converting the problem into finding zeros of the determinant of univariate matrix polynomials. This can be effectively solved through Laplacian expansion for one bounce and a bisection solver for more bounces. Our solution is generic, completely deterministic, accurate for the case of one bounce, and GPU-friendly. We develop efficient CPU and GPU implementations and apply them to challenging glints and caustic rendering. Experiments on various scenarios demonstrate the superiority of specular polynomial-based solutions compared to Newton iteration-based counterparts.

Figures (12)

• Figure 1: Illustration of our problem setup and important symbols.
• Figure 2: (a) Product form of vertex constraints requires decomposing the ray in the local tangent frame of $\boldsymbol{x}_i$. (b) Recursive rational mappings represent $\boldsymbol{u}_{i+1}$ using a rational expression of $\boldsymbol{u}_i$.
• Figure 3: Overview of the pipeline. Taking vertex positions and normals as inputs, our pipeline systematically constructs specular polynomials, converts the multivariate systems into a univariate problem via constructing resultant matrices, and subsequently solves the univariate problem. Finally, it validates the solutions and generates admissible paths.
• Figure 4: Glints rendering featuring different types of specular chains. Top: R chains on a metal brush. Middle: RR chains in a scene with two specular dragons and a mirror. Bottom: TT chains passing through a relief, for which we also show glint-only images in the supplemental material.
• Figure 5: Equal-time comparisons (10 sec) on caustics rendering with Path Tracing (PT) Kajiya86, Practical Path Guiding (PPG) Muller17, the unbiased variant of Specular Manifold Sampling (SMS) Zeltner20 and Manifold Path Guiding (MPG) Fan23MPG on the Plane and Pool scenes. We report the samples per pixel (SPP) and quantitative errors in terms of mean square error (MSE).