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 latter 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.
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07 April 2021
A Correction to this paper has been published: https://doi.org/10.1007/s10851-021-01028-0
Notes
It is very common for a simple curve to have non-simple signature.
In [13], the cogwheels are constructed using error functions (likely for computational reasons), and in fact, the permutation of cogs produces curves with slightly different signatures. In [12], the explicit formulas are not given, but the Mathematica code provided by the author shows that the construction was also done in terms of error functions. In both cases, replacing the error functions with smooth bump functions results in families of non-degenerate curves with identical signatures.
A map \(\gamma :I \rightarrow {\mathbb {R}}^2\) where I is an open subset of \({\mathbb {R}}\) is locally injective if for any \(t\in I\), there exists an open neighborhood J, such that \(\gamma |_J\) is injective.
The term piece has a different meaning in [12]. Also, if I is open, then a curve piece satisfies our definition of a curve, but we still use the term curve piece when we want to emphasize that the piece is a proper subset of a larger curve.
In [12], in a much more general setting, the symmetry index at p is defined as the cardinality of the set \(\text {sym}(\varGamma _p)/\text {sym}^*(\varGamma _p)\), where \(\text {sym}^*(\varGamma _p)\) denotes the subset of \(\text {sym}(\varGamma _p)\) consisting of elements that fix p. However, if p is not a point of self-intersection and \(\varGamma \) is oriented, then under the action of SE(2) the group \(\text {sym}^*(\varGamma _p)\) is trivial.
The change in the orientation sends \(\kappa (p)\) to \(-\kappa (p)\).
In [13], \(\kappa \) is assumed to be \(C^1\), but the proof is valid for a continuous function \(\kappa \).
Strictly speaking, we should call this signature the “special Euclidean signature,” because the full Euclidean group includes reflections and both \(\kappa \) and \(\dot{\kappa }\) change sign under reflections and are, therefore, not invariant. We also note that if we reverse a curve’s orientation then \(\kappa \) changes its sign, but \(\dot{\kappa }\) preserves the sign. Thus, change in the orientation induces the reflection of the signature about the vertical axis.
For a sufficient condition, see Corollary 5 in [13].
In [14], special Euclidean transformations are called proper Euclidean transformations. Euclidean transformation, \(\gamma _2 = g\cdot \gamma _1, g\in SE(2)\), if and only if their signature curves are identical \(S_1 = S_2\).
Except for the midpoint of “long” red segments, where it is 2, because the long red segment is an end-to-end attachment of two “short” red pieces.
A map between topological spaces is called proper if the preimage of every compact subset is compact. Any continuous map with a compact domain to \({\mathbb {R}}^2\) (and, more generally, to any Hausdorff space) is proper.
A terminology warning: please do not confuse the vertices of the simple planar curve \(\varGamma \) as in Definition 5 and vertices of the quiver \(\varDelta _S\).
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Acknowledgements
We are grateful to Peter Olver for pointing out the relationship between the cardinalities of local symmetry sets of a curve and multiplicities of the edges of its signature quiver and to Ekaterina Shemyakova for suggesting we use the term quiver for the graph associated with the signature. We would like to thank the anonymous reviewers for their exceptionally careful reading of the paper: Their remarks and questions were very helpful. We acknowledge NSF conference grant DMS-1952694 for providing travel funding to present an earlier version of this paper at DART X conference.
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Geiger, E., Kogan, I.A. Non-congruent non-degenerate curves with identical signatures. J Math Imaging Vis 63, 601–625 (2021). https://doi.org/10.1007/s10851-020-01015-x
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DOI: https://doi.org/10.1007/s10851-020-01015-x