General Theory of Image Normalization

We give a systematic, abstract formulation of the image normalization method as applied to a general group of image transformations, and then illustrate the abstract analysis by applying it to the hierarchy of viewing transformations of a planar object.Comment: 33 pages, plain tex, no figure

Metrics and Uniqueness Criteria on the Signatures of Closed Curves

This paper explores the paradigm of the differential signature introduced in 1996 by Calabi et al. This methodology has vast implications in fields such as computer vision, where these techniques can potentially be used to verify a person's handwriting is consistent with prior documents, or in medical imaging, to name a few examples. Motivated by examples provided by Hickman in 2011 and Musso and Nicolodi in 2009 regarding key failures in this invariant, we provide new criteria for the correspondence between a curve and its signature to be unique in a general setting. To show this result, we introduce new methods regarding the signature, particularly through the lens of differential equations, and the extension of the signature to include information on higher order derivatives of the curvature function corresponding to the curve and desired group action. We additionally show results regarding the robustness of the signature, showing that under a suitable metric on the space of subsets of $\mathbb{R}^n$, if two signatures are sufficiently close then so too will the corresponding equivalence classes of curves they correspond to, given certain conditions on these signatures

Non-congruent non-degenerate 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.Comment: 33 pages, 22 figures. Page 20: In the proof of Corollary 31 the notation for the length, $L_W$, of a reconstructed curve $\Gamma_W$ is introduced and defined. Page 23: The upper bound on the integral in equation (35) is updated to use $L_W$ instead of $L$ and the definition of $L_W$ is referred to. Page 23: The assumption "$m$ and $\xi$ are relatively prime" is added to Proposition 3