The process for the alignment of two images such that their appearances resemble is called the image registration. Image registration is widely studied in several fields such as remote sensing, computer vision, morphophonemics, medical imaging, etc. The field of image registration is inspired by the pioneer work of D’Arcy Thompson. In his seminal book, On Growth and form, he presented the idea of image transformation, between species. Over the past several years his book has been used as justification for the study of diffeomorphic deformations of images of different objects, a field that is now extremely well studied, particularly for the registration of medical images. Thompson’s examples are hand-drawn, and consist of a two dimensional outline of an animal (or part of an animal) and a grid superimposed on it. The deformation to another animal form is demonstrated by the picture of the new form and what is intended to be the same grid pushed forward so that the smoothness of the deformations can be clearly seen. This smoothness is one of the essential requirement for the diffeo-morphic image registration. Thompson, in his book, mentioned isogonal transformations for his several examples. Nowadays an isogonal transformation is termed as conformal transformation. Thompson’s principal point was that the deformations between closely related species should be simple. It has been suggested that this idea of simplicity can be interpreted in modern parlance as being examples of low dimensional groups. Clearly, the infinite dimensional diffeomorphism group or the conformal group would not be simple, but the six dimensional Möbius group (a subgroup of conformal group) would be. His claims of conformally-related change between species were investigated further by Rus-sian scientist Petukhov, who used Thompson’s grid method as well as computing the cross-ratio (which is an invariant of the Möbius group, a finite-dimensional subgroup of the group of conformal diffeomorphisms) to check whether sets of points in the images could be related by a Möbius transformation. Thmpson’s examples and Petukhov’s investigation motivated us to explore Möbius group. This Möbius group is the composition of four simple transformations that are scaling, rotation, translation and inversion. In this paper we present the image registration using Möbius diffeomorphisms. We present the novel idea for the selection of initial guess that plays a key role for the optimisation. The optimisation is based on the gradient descent algorithm with and without the labelled points (landmarks). Numerical examples illustrate the ability of the method to perform Möbius registration.