An efficient recursive estimator of the Fréchet mean on a hypersphere with applications to Medical Image Analysis
چکیده
Finding the Riemannian center of mass or the Fréchet mean (FM) of manifold-valued data sets is a commonly encountered problem in a variety of fields of Science and Engineering including but not limited to, Medical Image Computing, Machine Learning, and Computer Vision. For instance, it is encountered in tasks such as, atlas construction, clustering, principal geodesic analysis etc. Traditionally, algorithms for computing the FM of the manifold-valued data require that the entire data pool be available apriori and not incrementally. When encountered with new data, the FM needs to be recomputed over the entire pool, which can be computationally as well as storage inefficient. A computational and storage efficient alternative is to consider a recursive algorithm for computing the FM which simply updates the previously computed FM when presented with a new data set. In this paper, we present such an alternative called the incremental Fréchet mean estimator (iFME) for data on the hypersphere. We prove the asymptotic convergence of iFME to the true FM of the underlying distribution from which the data samples were drawn. Further, we present several experiments demonstrating the performance on synthetic and real data sets.