Analytic Methods for Geometric Modeling via Spherical Decomposition

TL;DR

This paper introduces a grid-free, efficient analytic method for geometric modeling that uses spherical decomposition and non-uniform FFTs to improve shape and motion analysis tasks.

Contribution

It proposes a novel, grid-free discretization approach for analytic correlations in shape modeling, enhancing computational efficiency over traditional grid-based FFT methods.

Findings

Outperforms uniform grid-based FFT methods in efficiency

Enables fusion of collision detection and shape analysis techniques

Provides a unified framework for shape and motion analysis

Abstract

Analytic methods are emerging in solid and configuration modeling, while providing new insights into a variety of shape and motion related problems by exploiting tools from group morphology, convolution algebras, and harmonic analysis. However, most convolution-based methods have used uniform grid-based sampling to take advantage of the fast Fourier transform (FFT) algorithm. We propose a new paradigm for more efficient computation of analytic correlations that relies on a grid-free discretization of arbitrary shapes as countable unions of balls, in turn described as sublevel sets of summations of smooth radial kernels at adaptively sampled 'knots'. Using a simple geometric lifting trick, we interpret this combination as a convolution of an impulsive skeletal density and primitive kernels with conical support, which faithfully embeds into the convolution formulation of interactions across different objects. Our approach enables fusion of search-efficient combinatorial data structures prevalent in time-critical collision and proximity queries with analytic methods popular in path planning and protein docking, and outperforms uniform grid-based FFT methods by leveraging nonequispaced FFTs. We provide example applications in formulating holonomic collision constraints, shape complementarity metrics, and morphological operations, unified within a single analytic framework.