When we seek to compare shapes parameterized as sets of unlabeled points, we face the twin problems of estimating i) shape correspondence and ii) shape deformation. This chicken and egg problem can be formulated as joint pose and correspondence estimation with the deformation parameters playing the role of pose. Efficient optimization algorithms are available for simultaneously solving for both deformation and correspondence. Furthermore, effective energy functions can be obtained by eliminating one set of variables in terms of the other. When the correspondence variables are eliminated in this manner, we obtain a set of implicit correspondence objective functions which can be directly minimized w.r.t. the deformation parameters. Over the past few years, we have shown the efficacy of i) simultaneously solving for the correspondences and the deformation, ii) simultaneously clustering and matching the two shapes, iii) using information divergence measures to solve for the deformation without parameterizing the correspondences, and iv) finding a deformation which minimizes closed-form distances between two shape density functions.
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