Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 080, 29 pages      arXiv:1603.09335

Möbius Invariants of Shapes and Images

Stephen Marsland a and Robert I. McLachlan b
a) School of Engineering and Advanced Technology, Massey University, Palmerston North, New Zealand
b) Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

Received April 01, 2016, in final form August 08, 2016; Published online August 11, 2016

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 Möbius 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 Möbius invariants, and then develop an algorithm by which shapes can be recognised that is Möbius- 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 Möbius-invariant signature of grey-scale images.

Key words: invariant; invariant signature; Möbius group; shape; image.

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