Soft parametric curve matching in scale-space

We develop a softassign method for application to curve matching. Softassign uses deterministic annealing to iteratively optimize the parameters of an energy function. It also incorporates outlier rejection by converting the energy into a stochastic matrix with entries for rejection probability. Previous applications of the method focused on finding transformations between unordered point sets. Thus, no topological constraints were required. In our application, we must consider the topology of the matching between the reference and the target curve. Our energy function also depends upon the rotation and scaling between the curves. Thus, we develop a topologically correct algorithm to update the arc length correspondence, which is then used to update the similarity transformation. We further enhance robustness by using a scale-space description of the curves. This results in a curve-matching tool that, given an approximate initialization, is invariant to similarity transformations. We demonstrate the reliability of the technique by applying it to open and closed curves extracted from real patient images (cortical sulci in three dimensions and corpora callosa in two dimensions). The set of transformations is then used to compute anatomical atlases.