Iterative non-rigid image registration based on Möbius transformations

Abstract

The objective of this work is to present a new approach for non-rigid image registration, which is based on Möbius transformations. The Möbius transformations are conformal maps of the extended complex plane and can be expressed as composition of affine transformations and inversions. These transformations are general enough and can be obtained also by rigid motions of a sphere through 3D space and a stereographic projection to the complex plane. The equivalence between the motion of the sphere in the 3D space and the Möbius transformations is largely used in the registration model developed in this work. It allows the transformations to be parameterized with 6 parameters, each of which has clear physical meaning. In order to register image A to image B, image A is first stereographically projected on a sphere above it. A Möbius transformation is then applied by moving and rotating the sphere in the 3D space and performing a stereographic projection back to the plane. The similarity between the resulting image A' and image B is evaluated by a dedicated cost function (the sum of squared differences of pixel intensities in A' and B). The goal of the algorithm is to find iteratively a transformation that minimizes the cost function. The optimization procedure is over the 6 parameters, defining the position of the sphere (3 coordinates of the sphere's center and 3 Euler angles defining the orientation) and is performed using simulated annealing. The developed registration algorithm is applied to register heart images obtained in a typical SPECT heart imaging diagnostic study. We register stress SPECT heart image to the corresponding rest SPECT heart image of the same patient. The performance of the algorithm is evaluated qualitatively (with the aid of difference images) and promising results are obtained.