M\"obius Invariants of Shapes and Images

TL;DR

This paper develops M"obius invariants for shapes and images to improve recognition despite transformations, with applications in computer vision and biological imaging, introducing a new stable, noise-robust algorithm.

Contribution

It introduces a novel M"obius-invariant shape recognition algorithm and extends invariants to grey-scale images, enhancing robustness and applicability in vision tasks.

Findings

The algorithm is numerically stable and robust to noise.

It effectively recognizes shapes under M"obius transformations.

A new M"obius-invariant signature for grey-scale images is proposed.

Abstract

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images.