Multidimensional Elastic Pattern Matching

In this paper, we present a theory of elastic images deformation and matching. Our theory is based on the physical theory of elastic deformation of a continuum, and is very general: it applies to matching n-dimensional images with n-dimensional templates. We derive the variational formulation of the optimal deformation problem, and the Euler-Lagrange equations to solve it. Also, we address the issue of invariance of matching with respect to a given transformation group. We use the Noether's theorem from mathematical physics to show that invariance with respect to an arbitrary transformation group|more properly: to its Lie algebra|can be \built in" the elastic properties of the template. Finally, we discuss the numerical solution of the Euler-Lagrange equation. We rst show that, under suitable hypotheses, the optimal solution can be obtained as the linear combination of a nite number of basis functions.