Georgia Institute of Technology College of Sciences

We propose a new fast algorithm for finding the global shortest path connecting two points while avoiding obstacles in a region by solving an initial value problem of ordinary differential equations (ODE's). The idea is based on the factthat the global shortest path possesses a simple geometric structure. This enables us to restrict the search in a set of feasible paths that share the same structure. The resulting search space is reduced to a finite dimensional set. We use a gradient descent strategy based on the intermittent diffusion (ID) in conjunction with the level set framework to obtain the global shortest path by solving a randomly perturbed ODE's with initial conditions.Compared to the existing methods, such as the combinatorial methods or partial differential equation(PDE) methods, our algorithm is faster and easier to implement. We can also handle cases in which obstacles shape are arbitrary and/or the dimension of the base space is three or higher.

Finite element approximations for the eigenvalue problem of the Laplace  operator are discussed. A gradient recovery scheme is proposed to enhance  the finite element solutions of the eigenvalues. By reconstructing the  numerical solution and its gradient, it is possible to produce more accurate  numerical eigenvalues. Furthermore, the recovered gradient can be used to  form an a posteriori error estimator to guide an adaptive mesh refinement.  Therefore, this method works not only for structured meshes, but also for  unstructured and adaptive meshes. Additional computational cost for this  post-processing technique is only O(N) (N is the total degrees of freedom),   comparing with O(N^2) cost for the original problem.

Using the technique of intertwining multiresolution analysis piecewise linear, continuous, orthogonal, wavelets on a regular hexagon are constructed. We will review the technique of intertwining multiresolution analysis in the one variable case then indicate the modifications necessary for the two variable construction. This is work with George Donovan and Doug Hardin.

Medical imaging is the application of mathematical and engineering models to create images of the human body for clinical purposes or medical science by using a medical device. One of the main objectives of medical imaging research is to find the boundary of the region of the interest. The procedure to find the boundary of the region of the interest is called a segmentation. The purpose of this talk is to present a variational region based algorithm that is able to deal with spatial perturbations of the image intensity directly. Image segmentation is obtained by using a Gamma-Convergence approximation for a multi-scale piecewise smooth model. This model overcomes the limitations of global region models while avoiding the high sensitivity of local approaches. The proposed model is implemented efficiently using recursive Gaussian convolutions. The model is applied to magnetic resonance (MR) images where image quality depends highly on the acquisition protocol. Numerical experiments on 2-dimensional human liver MR images show that our model compares favorably to existing methods.This work is done in collaborated with Mikael Rousson and Chenyang Xu.