A coordinate transformation is used to warp a source image to a transformed image to match a target brain (Fig. 2). The regional nature of spatial normalization determines the complexity of the coordinate transformation. For global SN the most common approach is to use a homogeneous 4x4 matrix, with three parameters, one each for rotation, translation, and scale, for coordinate transformations [1,32]. See the chapter "Spatial Transformation Models" for more detailed information about the 4x 4 transform matrix. This global SN transform is also called a nine-parameter or linear-affine transform. Additional regional anatomical matching can be achieved using higher order polynomials for coordinate transformations [57]. For practical purposes this approach is limited to polynomials of order 12 or less. Other methods, including the use of thin-plate splines [4] and deformable models [7], have the potential for additional regional feature matching. Finally, to achieve a maximum regional effect, 3D deformation fields are used with high degree-of-freedom regional transform methods [3,5,6,31,33,44]. A deformation field is a large matrix of 3D translations, one for each voxel in the source and target brain images. Because of the discrete nature of the images, coordinate transformations must be coupled with a 3D interpolation scheme.

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