Differential and integral invariants under Mobius transformation

TL;DR

This paper develops differential and integral invariants under Mobius transformations for 2-D and 3-D shapes, aiding in handling non-rigid deformations in image and shape analysis.

Contribution

It introduces new differential expressions invariant under Mobius transformations and proposes integral invariants based on these, along with a conjecture on conformal invariants.

Findings

Two differential invariants under Mobius transformations are proposed.

Integral invariants under Mobius transformations are established.

A conjecture on the structure of conformal invariants is formulated.

Abstract

One of the most challenging problems in the domain of 2-D image or 3-D shape is to handle the non-rigid deformation. From the perspective of transformation groups, the conformal transformation is a key part of the diffeomorphism. According to the Liouville Theorem, an important part of the conformal transformation is the Mobius transformation, so we focus on Mobius transformation and propose two differential expressions that are invariable under 2-D and 3-D Mobius transformation respectively. Next, we analyze the absoluteness and relativity of invariance on them and their components. After that, we propose integral invariants under Mobius transformation based on the two differential expressions. Finally, we propose a conjecture about the structure of differential invariants under conformal transformation according to our observation on the composition of the above two differential invariants.