Mathematics Colloquia and Seminars

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Magnetic resonance imaging (MRI) reconstruction from sparsely sampled data has been a difficult problem in medical imaging field. We approach this problem by formulating a cost functional that includes a constraint term that is imposed by the raw measurement data in k-space and the L1 norm of a sparse representation of the reconstructed image. The sparse representation is usually realized by total variational regularization and/or wavelet transform. We have applied the Bregman iteration to minimize this functional to recover finer scales in our recent work. Here we propose nonlinear inverse scale space methods in addition to the iterative refinement procedure. Numerical results from the two methods are presented and it shows that the nonlinear inverse scale space method is a more efficient algorithm than the iterated refinement method.

In this work, we specifically solve the Chan-Vese active contour model by multiphase level set methods. We first develop a fast algorithm based on calculating the variational energy of the Chan-Vese model without the length term. We check whether the energy decreases or not when we move a point to another segmented region. Then we draw a connection between this algorithm and the topological derivative, a concept emerged from the shape optimization field. Furthermore, to include the length term of the Chan-Vese model, a preprocessing step is taken by using nonlinear diffusion. Numerical experiments have demonstrated the efficiency and the robustness of our algorithm.