LSMAT Least Squares Medial Axis Transform

TL;DR

This paper introduces a robust least squares-based medial axis transform that effectively handles noise and outliers, outperforming traditional methods in 2D and 3D object analysis.

Contribution

It formulates the medial axis transform as a least squares optimization problem, enhancing robustness and parallelizability compared to previous geometric approaches.

Findings

Outperforms state-of-the-art methods on synthetic data

Demonstrates robustness to noise and outliers

Applicable to both 2D and 3D objects

Abstract

The medial axis transform has applications in numerous fields including visualization, computer graphics, and computer vision. Unfortunately, traditional medial axis transformations are usually brittle in the presence of outliers, perturbations and/or noise along the boundary of objects. To overcome this limitation, we introduce a new formulation of the medial axis transform which is naturally robust in the presence of these artifacts. Unlike previous work which has approached the medial axis from a computational geometry angle, we consider it from a numerical optimization perspective. In this work, we follow the definition of the medial axis transform as "the set of maximally inscribed spheres". We show how this definition can be formulated as a least squares relaxation where the transform is obtained by minimizing a continuous optimization problem. The proposed approach is inherently parallelizable by performing independant optimization of each sphere using Gauss-Newton, and its least-squares form allows it to be significantly more robust compared to traditional computational geometry approaches. Extensive experiments on 2D and 3D objects demonstrate that our method provides superior results to the state of the art on both synthetic and real-data.