Algorithm for automatic nonlinear inter-subject registration M.I. Kesäniemi, A. Korvenoja, J. Lötjönen, K. Virtanen, M. Ollikainen, and R.J. Ilmoniemi BioMag Laboratory, Helsinki University Central Hospital, FIN-00029 HUS, Finland; Laboratory of Biomedical Engineering, Helsinki University of Technology, FIN-02015 HUT, Finland 1 Introduction In order to average or compare signals from functional brain images across subjects, e.g., in the case of fMRI, PET, or MEG, it is necessary to register the images with respect to each other. Registration techniques have been used also from TMS target area determination [1] to automatic segmentation of MRI images [2]. One of the advantages of spatially normalized images is that the locations of activation can be reported as Euclidean coordinates within a standard space. With segmentation, the transformation can be used to move the segments from one image to another: this way the geometrical knowledge of the anatomical information is automatically applied in the segmentation. Ultimately, inter-subject registration creates a correspondence map consisting of three parameters for each voxel in the image. These parameters represent the x, y, and z coordinates of the corresponding point in another image, typically a template image defined in the Talairach space. The number of free parameters would then be over 50 million for an MRI image of size 256×256×256. Automated inter-subject registration procedures typically overcome the parameter estimation problem by parameterizing the deformation field in a lower dimension. Techniques that align the images using affine 1 transformations are widely used with brain atlases such as the one of Talaraich & Tournoux. The 12-parameter affine transformation can be determined either according to a set of control points given by the user, or automatically by using the AIR method as described in [3]. One method is to represent the transformation as a linear combination of smooth basis functions [4]. One important property of a useful spatial transformation is homeomorphicity, i.e., the transformation has to be one-to-one. The affine transformations are always homeomorphic, so that every point in the registered image has a unique 1 An affine transformation is one that preserves straight and parallel lines. correspondence in the template image, and vice versa. The homeomorphism can not always be ensured with nonlinear transformations, except when the images are very close to each other [4]. We describe a method that can be used to determine a homeomorphic nonlinear transformation without floating point arithmetics. By using 8 and 16 bit integers throughout the program, the memory requirements can be decreased so that it is possible to determine a deformation with over 6 million parameters with a regular PC. The method is capable of handling large deformations between images, so it is not necessary to use an affine registration before calculating the nonlinear transformation matrices. 2 Mathematical model The similarity measure S used is the difference of the gray-levels of the corresponding points in the images g 1 and g 2 , (1) where t is the displacement vector defining the transformation. To simplify the equations and to ensure the symmetric processing of the images, the cost function E to be minimized is written as (2) In addition to the similarity-dependent term, a smoothness constraint minimizing the derivatives of the displacement field is included in the cost function [5]. The smoothness required is controlled by parameter α. 2.1 Minimization The necessary conditions for t to be an optimal solution can be found by the Euler–Lagrange equations [6], ( ) ( ) ( ) , 2 1 x t x x + − = g g S ( ) ( ) ( ) ( ) [ ] ∫∫∫ − − + = V g g E 2 2 1 x t x x t x . 3 1 3 1 2 x d dx dt i j j i ∑∑ = =         + α