  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.
