Signature-inverse Theorem in Mesh Group-planes

TL;DR

This paper investigates the limitations of existing signature-inverse theorems for meshes, identifies conditions for their validity, and introduces a simplified version for closed meshes, enhancing invariant recognition methods.

Contribution

It demonstrates the invalidity of certain inverse theorems, classifies meshes to find conditions for their correctness, and proposes a simplified theorem for closed meshes.

Findings

Curvature-inverse and Signature-inverse theorems can be invalid for meshes.

Conditions are identified under which these theorems hold for specific mesh classifications.

A new Host Theorem simplifies the Signature-inverse Theorem for closed meshes.

Abstract

This is the second paper devoted to the numerical version of Signature-inverse Theorem in terms of the underlying joint invariants. Signature Theorem and its Inverse guarantee any application of differential invariant signature curves to the invariant recognition of visual objects. We first show the invalidity of Curvature-inverse and Signature inverse theorems, meaning non-congruent meshes may have the same joint invariant numerical curvature or signature. Then by classifying three and five point ordinary meshes respectively in the Euclidean and affine cases, we look for conditions in terms of the associated joint invariant signatures which make these theorems correct. Additionally, we bring forward The Host Theorem to provide a simpler version of Signature-inverse Theorem for closed ordinary meshes.