This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.
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