Globally adaptive and locally regular point discretization of curved surfaces

TL;DR

This paper introduces an algorithm for generating well-conditioned, adaptively distributed point discretizations on curved surfaces, optimizing for local regularity and global curvature adaptivity.

Contribution

The authors present a gradient descent-based algorithm that efficiently computes near-optimal surface point distributions with prescribed spacing and curvature adaptivity, using level-set methods.

Findings

Achieves low deviation from target spacing on various shapes

Rapidly converges to desired point distributions

Handles both parametric and non-parametric surfaces effectively

Abstract

Point discretization of curved surfaces is required in many applications ranging from object rendering to the solution of surface partial differential equations (PDEs). These applications often impose that surfaces are sampled with local regularity and global curvature adaptivity to maintain robustness and efficiency. Computing numerically well-conditioned point discretization is non-trivial, even for simple analytic curved surfaces. We present an algorithm for finding near-optimal surface point distributions governed by a prescribed length field on curved surfaces. The algorithm works by approximately minimizing a global potential over local point-point interactions. The optimization problem is solved using gradient descent, accelerated by line search to find optimal step sizes. We use a level-set method to describe the surface and perform all required projections without requiring additional surface-attractive forces. To further accelerate convergence, the algorithm dynamically fuses and inserts points where a local excess or lack of points is detected using an integral support measure. We test the proposed algorithm on a variety of shapes, ranging from parametric to non-parametric surfaces. We compute point distributions with different curvature adaptivity and show that the algorithm achieves low average deviation from the prescribed target spacing locally. Overall, the presented algorithm rapidly and robustly converges to the final number and distribution of surface points.