Efficient computation of topological integral transforms

TL;DR

Eucalc is a new software library that efficiently computes topological integral transforms like Euler characteristic and Radon transforms for weighted complexes, enabling exact, fast analysis of images in shape analysis applications.

Contribution

We introduce Eucalc, a novel C++ library that provides exact, efficient computation of topological transforms for weighted complexes, supporting both binary and grayscale images.

Findings

Eucalc reduces computational complexity by focusing on critical points.

The software handles both binary and grayscale images effectively.

Experimental results demonstrate Eucalc's efficiency and accuracy.

Abstract

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce Eucalc, a novel implementation of three topological integral transforms -- the Euler characteristic transform, the Radon transform, and hybrid transforms -- for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating Eucalc's efficiency.