Closed surfaces with different shapes that are indistinguishable by the SRNF

Abstract

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The Square Root Normal Field Distance and Unbalanced Optimal Transport

“…The phenomenon of distinct closed surfaces that were indistinguishable by the SRNF shape pseudo-distance was first studied in [23]. As a result of Theorem 3.1, we obtain a full characterization of this phenomenon: Note that the Wasserstein-Fisher-Rao distance is a true distance and so the SRNF distance between two surfaces is zero if and only if they are mapped to the same measure by ψ.…”

“…The function d Imm is only a pseudo-metric due to the non-injectivity of Φ. Examples of this degeneracy have been discussed extensively in the recent article [23].…”

“…[27,22,36,30]. In a recent paper [23] certain degeneracy results for the resulting distance have been characterized, but a more detailed theoretical study of its properties is still missing; the second part of this article will contribute towards this aim.…”

“…Theorem 3.4 highlights the degeneracy of the SRNF distance: in the recent paper [23] it has been shown that there exist closed surfaces that are indistinguishable by the SRNF distance. Our result shows that for any closed surface, there exists a convex surface such that the SRNF distance cannot distinguish them.…”

Manifolds of mappings for continuum mechanics

“…This relative Moser theorem for Whitney manifold germs is modelled on the standard proof of Moser's theorem in [73,Theorem 31.13]. The proof of [52,Theorem 4] is for manifolds with corners, but it works without change for Whitney manifold germs.…”