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This work deals with the problem of shape registration. Broadly speaking, the problem is that of establishing point-wise correspondences in two different shapes of arbitrary dimension and topology. This problem is a fundamental component in numerous image and vision applications. We propose a variational framework for a dense global-to-local 2D shape registration. Affine transformations are accounted for using vector distance functions. Based on this representation, a dissimilarity measure between the two shapes is minimized to recover the global matching parameters. The local coordinate transformation between the two shapes is explicitly estimated by solving a regularized non-linear PDE-based motion model. Various experimental results are presented and discussed to show the potential of the proposed framework with a finite element (FE)-based validation of its performance.