Georgia Institute of Technology College of Sciences

In this lecture, I will explain the greedy approximation algorithm on submodular function maximization due to Nemhauser, Wolsey, and Fisher. Then I will apply this algorithm to the problem of approximating an monotone submodular functions by another submodular function with succinct representation. This approximation method is based on the maximum volume ellipsoid inscribed in a centrally symmetric convex body. This is joint work with Michel Goemans, Nick Harvey, and Vahab Mirrokni.

In image guided procedures such as radiation therapies and computer-assisted surgeries, physicians often need to align images that are taken at different times and by different modalities. Typically, a rigid registration is performed first, followed by a nonrigid registration. We are interested in efficient registrations methods which are robust (numerical solution procedure will not get stuck at local minima) and fast (ideally real time). We will present a robust continuous mutual information model for multimodality regisration and explore the new emerging parallel hardware for fast computation. Nonrigid registration is then applied afterwards to further enhance the results. Elastic and fluid models were usually used but edges and small details often appear smeared in the transformed templates. We will propose a new inviscid model formulated in a particle framework, and derive the corresponding nonlinear partial differential equations for computing the spatial transformation. The idea is to simulate the template image as a set of free particles moving toward the target positions under applied forces. Our model can accommodate both small and large deformations, with sharper edges and clear texture achieved at less computational cost. We demonstrate the performance of our model on a variety of images including 2D and 3D, mono-modal and multi-modal, synthetic and clinical data.

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite entropy, we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L^1.