Non-congruent non-degenerate curves with identical signatures
This addresses a theoretical gap in computer vision and geometry for researchers, providing a counterexample and new criteria for curve congruence, though it is incremental in refining existing mathematical frameworks.
The paper tackles the problem of whether equality of Euclidean signatures implies congruence for non-degenerate planar curves, disproving a prior claim by showing it fails for curves with non-simple signatures, and introduces a directed graph method to generate families of non-congruent curves with identical signatures.
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.