A Nonlinear Variational Problem for Image Matching

Minimizing a nonlinear functional is presented as a way of obtaining a planar mapping that matches two similar images. A smoothing term is added to the nonlinear functional to penalize discontinuous and irregular solutions. One option for the smoothing term is a quadratic form generated by a linear differential operator. The functional is then minimized using the Fourier representation of the planar mapping. With this representation the quadratic form is diagonalized. Another option is a quadratic form generated via a basis of compactly supported wavelets. In both cases, a natural approximation scheme is described. Both quadratic forms are shown to impose the same smoothing. However, in terms of the finite dimensional approximations, it is easier to accommodate local deformations using the wavelet basis. Key words, image matching, movement compensation, nonlinear variational problem, spectral methods, wavelets AMS subject classifications. 68V10, 65M70, 49N60