Specular Polynomials

TL;DR

This paper introduces specular polynomials, a novel, deterministic, and GPU-friendly method for finding all valid specular light paths in rendering, overcoming limitations of traditional Newton iteration approaches.

Contribution

It reformulates specular path finding into polynomial systems, enabling complete and accurate solutions without iterative divergence issues.

Findings

Effective in rendering challenging caustics and glints.

Outperforms Newton iteration-based methods in completeness and stability.

Applicable to complex scenes with multiple specular bounces.

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.