Maseeh Mathematics + Statistics Colloquium Series Presents: Shape Alignment via Allen-Cahn Nonlinear-Convection

Speaker: Dr. Laurent Younes
Johns Hopkins University

Abstract: We present a phase field method on shape registration to align shapes of possibly different topology. A soft end-point optimal control problem is introduced whose minimum measures the minimal control norm required to align an initial shape to a final shape, up to a small error term. Inspired by level-set methods and large deformation diffeomorphic metric mapping, the controls spaces are integrable scalar functions to serve as a normal velocity and smooth reproducing kernel Hilbert spaces to serve as velocity vector fields. The paths in control spaces then follow an evolution equation which is a generalized convective Allen-Cahn. The existence of mild solutions to the evolution equation is proved, the minimums of the time discretized optimal control problem are characterized, and numerical simulations of minimums to the fully discretized optimal control problem are provided.

Biography: Laurent Younes is a professor in the Department of Applied Mathematics and Statistics and Director of the Centre for Imaging Science at Johns Hopkins University, that he joined in 2003, after ten years as a researcher for the CNRS in France. He is a former student of the Ecole Normale Supérieure (Paris) and of the University of Paris Orsay from which he received his Ph.D. in 1988. His work includes contributions to applied probability, statistics, graphical models, shape analysis and computational medicine. He is a fellow of the IMS, of SIAM and of the AMS.