Digital Euler Characteristic Transform

TL;DR

This paper introduces a fully digital algorithm for exact computation of the Euler Characteristic Transform (ECT), enabling precise shape analysis and inversion, with practical applications demonstrated in shape alignment tasks.

Contribution

The paper presents the first exact digital algorithm for ECT computation, along with an implementation that facilitates shape analysis, inversion, and alignment without information loss.

Findings

The algorithm computes ECT exactly up to computer precision.

The Ectoplasm package enables fast and practical shape analysis.

Application to shape alignment demonstrates the method's effectiveness.

Abstract

The Euler Characteristic Transform (ECT) of Turner et al. provides a way to statistically analyze non-diffeomorphic shapes without relying on landmarks. In applications, this transform is typically approximated by a discrete set of directions and heights, which results in potential loss of information as well as problems in inverting the transform. In this work we present a fully digital algorithm for computing the ECT exactly, up to computer precision; we introduce the Ectoplasm package that implements this algorithm, and we demonstrate this is fast and convenient enough to compute distances in real life data sets. We also discuss the implications of this algorithm to related problems in shape analysis, such as shape inversion and sub-shape selection. We also show a proof-of-concept application for solving the shape alignment problem with gradient descent and adaptive grid search, which are two powerful methods neither of which is possible using the discretized transform.