Complete $SE(3)$ invariants for a comeagre set of $C^3$ compact orientable surfaces in $\mathbb{R}^3$

TL;DR

This paper introduces degree four polynomial invariants for compact orientable surfaces in three-dimensional space, providing a stable method to reconstruct surfaces from these invariants for a broad class of smooth surfaces.

Contribution

The authors develop complete $SE(3)$ invariants for a large class of surfaces and provide an effective algorithm for surface reconstruction from these invariants.

Findings

Invariants are degree four polynomials in surface moments.

Reconstruction algorithm is numerically stable and effective.

Applicable to a comeagre set of $C^3$-surfaces.

Abstract

We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $\mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.