Anoptimisation over the M¨ obius group for an optimal solution in image registration

Abstract

 Image registration, the process of aligning two images to ensure  their appearances closely match, is a field profoundly influenced by the  pioneering contributions of D’Arcy Wentworth Thompson. In his influen tial work On Growth and Form, Thompson introduced the concept of im age transformation across species, emphasising the idea that deformations  between closely related species should exhibit simplicity. Specifically,  Thompson explored isogonal transformations—now known as conformal  transformations—suggesting that such deformations could be described  by finite dimensional groups. While the full set of diffeomorphisms or the  conformal group are too complex to be considered simple (as they are in finite), the eight-dimensional M¨obius group, a subgroup of the conformal  group, provides a more manageable framework. The M¨obius group com prises four fundamental transformations: scaling, rotation, translation, and  inversion, offering a balance between simplicity and mathematical expres siveness. In this study, we propose a novel approach to image registration using M¨obius diffeomorphisms. Additionally, an innovative method for se lecting the initial guess, a critical component in the optimisation process,  particularly within the gradient descent framework, has been introduced.  Through a series of numerical examples, we demonstrate the effectiveness  of this approach in achieving precise and efficient M¨obius-based image  registration. Our results highlight the potential of M¨obius transformations  as a robust tool for image alignment, bridging historical theoretical in sights with contemporary computational methodologies. This work not  only advances the field of image registration but also underscores the en during relevance of Thompson’s foundational ideas in modern scientific  research.