Optimal registration of deformed images

Abstract

Motivated by the need to locate and identify objects in three dimensional CT images, an optimal registration method for matching two and three dimensional deformed images has been developed. This method was used to find optimal mappings between CT images and an atlas image of the same anatomy. Using these mappings, object boundaries from the atlas was superimposed on the CT images. A cost function of the form DEFORMATION-SIMILARITY is associated with each mapping between the two images. The mapping obtained by our registration process is optimal with respect to this cost function. The registration process simulates a model in which one of the images made from an elastic material is deformed until it matches the other image. The cross correlation function which measures the similarity between the two images serves as a potential function from which the forces required to deform the image are derived. The deformation part of the cost function is measured by the strain energy of the deformed image. Therefore, the cost function of a mapping is given in this model by the total energy of the elastic image. The optimal mapping is obtained by finding the equilibrium state of the elastic image, which by definition corresponds to a local minimum of the total energy. The equilibrium state is obtained by solving a set of partial differential equations taken from the linear theory of elasticity. These equations are solved iteratively using the finite differences approximation on a grid which describes the mapping. The image function in a spherical region around each grid point is described by its projections on a set of orthogonal functions. The cross correlation function between the image functions in two regions is computed from these projections which, serve as the components of a feature vector associated with the grid points. In each iteration step of the process, the values of the projections are modified according to the currently approximated deformation. The method was tested by registering several two and three dimensional image pairs. It can also be used to obtain the optimal mapping between two regions from a set of corresponding points (with and without error estimates) in these regions.

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