Differential and integral invariants under Mobius transformation
This work addresses non-rigid deformation in computer vision and geometry, but appears incremental as it builds on known transformation groups without demonstrating broad applications or SOTA results.
The paper tackles the problem of handling non-rigid deformation in 2-D images or 3-D shapes by focusing on Möbius transformations, proposing differential and integral invariants under these transformations, and conjecturing about the structure of differential invariants under conformal transformations.
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.