Higher Order Algebraic Signed Distance Fields

Introduction: Signed distance functions (SDFs) are powerful implicit representations of curves and surfaces. Beyond encoding volume boundaries as their zero level-set, they also convey global geometric information about the scene as they map signed distances to all points in space. In real-time applications, their most common numerical representation is a regular grid of two or three dimensions. In conjunction with an interpolation method to infer a continuous approximation de ned on all points of space, these can be e ciently evaluated on the GPU such that even the most demanding applications can utilize them [4, 5]. Several authors proposed to use rst order approximations to the SDF at the samples, e.g. gradients [3] or plane equations [1]. In this paper, we present a straightforward generalization of this approach to higher orders and discuss various alternatives to the appropriate ltering of samples such that the inferred SDF reconstructs the given higher order derivatives at the samples. Our focus is on applications in high performance visualizations, as such, we prioritize run-time performance over optimal storage and restrict sampling topologies in our measurements to regular grids. We also discuss the shortcomings of this approach as means to decrease storage for complex shapes.