Non-congruent non-degenerate curves with identical signatures

TL;DR

This paper investigates the limitations of differential signatures in determining curve congruence, revealing that non-simple signatures can correspond to multiple non-congruent curves, and introduces a graph-based approach for congruence criteria.

Contribution

It demonstrates that equality of signatures does not guarantee congruence for non-simple curves and introduces a graph-based method to analyze congruence and symmetries.

Findings

Equality of signatures is not sufficient for congruence in non-simple curves.

A directed graph structure can generate non-congruent curves with identical signatures.

Congruence criteria can be formulated using the graph paths reflecting symmetries.

Abstract

While the equality of differential signatures (Calabi et al, Int. J. Comput. Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009). Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for non-degenerate planar curves, equality of Euclidean signatures implies congruence. We prove that while Hickman's claim holds for simple, closed curves with simple signatures, it fails for curves with non-simple signatures. In the later case, we associate a directed graph with the signature and show how various paths along the graph give rise to a family of non-congruent, non-degenerate curves with identical signatures. Using this additional structure, we formulate congruence criteria for non-degenerate, closed, simple curves and show how the paths reflect the global and local symmetries of the corresponding curve.