Digital tools for analyzing nondiffeomorphic shapes

The Euler Characteristic Transform (ECT) of Turner et al. provides a way to statistically analyze nondiffeomorphic 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 that this is fast and convenient enough to compute distances in real-life datasets. We also discuss the implications of this algorithm to related problems in shape analysis, such as shape inversion and subshape 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.