Introduction

Image registration requires finding an optimal transformation between an image pair, the source $S(\vec{x})$ and target $T(\vec{x})$. Single-level registration algorithms are divided between those which apply linear transformations and those which allow higher order deformations. Generally higher order deformations are performed after an initial registration by a linear method, so linear and non-linear are combined sequentially. In [1] we examined the application of hierarchies of data, warp and model, where complexity increases temporally with the progress of registration. It is rare that an algorithm allows simultaneous or parallel application of both linear and non-linear models within one image, so that only selected areas of the image deform. Many medical images contain regions representing both soft and hard tissue, and whereas the former often require high order deformations to achieve a good registration, in an intra-subject study the hard tissue regions should remain rigid. Registration of such image pairs requires algorithms where the model varies spatially within the image domain, using prior information on the variation of tissue types within the deforming image. These are instances of inhomogeneous non-linear registration algorithms. This paper classifies types of inhomogeneity and reviews those available in the literature. We then present three modifications to the fluid algorithm which introduce inhomogeneities into its application. Section 4 describes inhomogeneities in applying the force field and in computing the velocity field, and presents the varying-viscosity fluid registration algorithm. Finally, Sect. 5 shows results of application of these algorithms to 2- and 3-dimensional data.