A novel algorithm for optimal matching of elastic shapes with landmark constraints

An important problem in statistical shape analysis is the matching of geometric features across shapes, known as registration. In short, given two objects, one wants to know the correspondence of points on one shape to points on another. Such a matching problem, with various levels of complexity, is present regardless of the shape's mathematical representation. A recent framework for shape analysis of n-dimensional curves combines an infinite-dimensional functional curve representation with landmark information encoding important curve features. In this setting, shape matching is performed by minimizing an objective function with constraints, which respect landmark correspondences. Currently, the minimizer in this approach is found using piecewise dynamic programming; this does not respect the smoothness requirement of the matching function. Thus, the solution is not really a member of the group of registration functions. In this work, we present a landmark-constrained gradient descent algorithm, which searches for a smooth matching function and respects landmark locations. We compare the proposed method to the previously used approach using examples from the MPEG-7 dataset.