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A common problem in many biomedical imaging studies is that of finding a correspondence between two plane curves which aligns their shapes. A mathematical formulation and solutions to this problem is proposed in this paper. The formulation exhibits desirable properties. It allows for one-to-one as well as non-one-to-one correspondences, it consistently compares shape, even in nonrigid situations, and it is completely symmetric with respect to the two curves. A numerical implementation of the algorithm for finding the optimal correspondence is also reported. The algorithm is used to estimate nonrigid motion of the endocardium in MRI image sequences of normal and post-infarct dog hearts. The return error (the difference between the starting and ending positions of a point) is used as a performance measure to evaluate the technique. Since heart motion is periodic, the return error is a measure of consistency of the algorithm. Preliminary applications to other data sets are reported as well.