Efficient numerical method for solution of L2 optimal mass transport problem.

摘要

Optimal Mass Transport (OMT) is an important problem with numerous applications in a wide range of fields such as econometrics, fluid dynamics, automatic control, transportation, statistical physics, shape optimization, expert systems and meteorology. More recently, it has been shown to also have application in image processing such as for non-rigid registration and morphing alongside other differential and variational methods based on fluid dynamics. In this thesis, we present a novel and efficient numerical method for the computation of the L2 optimal mass transport mapping in two and three dimensions. Our method uses a direct variational approach. We have formulated a new projection to the constraint technique that can yield a good starting point for the method as well as a second order accurate discretization to the problem. The numerical experiments demonstrate that our algorithm yields accurate results in a relatively small number of iterations that are mesh independent.;In the first part of the thesis, we develop the theory and implementation details of our proposed method. These include the reformulation of the Monge-Kantorovich problem using a variational approach and then using a consistent discretization in conjunction with the "discretize-then-optimize" approach to solve the resulting discrete system of differential equations. We also develop advanced numerical methods such as multigrid and adaptive mesh refinement to solve the linear systems in practical time for even 3D applications. In the second part we show the methods efficacy via application to various image processing tasks. These include image registration and morphing. We present application of (OMT) to registration in the context of medical imaging and in particular image guided therapy where registration is used to align multiple data sets with each other and with the patient. We believe that an elastic warping methodology based on the notion of mass transport is quite natural for several medical imaging applications where density can be a key measure of similarity between different data sets e.g. proton density based imagery provided by MR. We also present an application of two dimensional optimal mass transport algorithm to compute diffeomorphic correspondence maps between curves for geometric interpolation in an active contour based visual tracking application.