Computational and Applied Math Proseminar

Thursday, October 26, 2006, 3:15 p.m. GWC 604

Wolfgang Stefan

Dept. Math. & Stats.

Ill-Posed Inverse Problems in Brain Imaging and
Seismic Wave Propagation

Abstract The ever increasing computational power and recent developments in theoretical mathematics make it more and more feasible to solve large problems in image and signal reconstruction. For a long time there was, and still is, a large gap between what state of the art signal processing methods can achieve and what is used in many practical applications. This is unfortunate, because in many cases both fields, application and mathematics can greatly benefit from a more involved cooperation. Applications can benefit from sharper images and signals with a higher signal to noise ratio, thus highlighting the signal parts of interest. Mathematics benefits from new perspectives into the problems and a more intuitive point of view which often leads to the development of more efficient, robust algorithms, and provides a good starting point to understand the mathematical nature of the problem.

Only mathematics can answer the question why some methods work and others do not. In my dissertation I will focus on two fields in signal/image processing, a recently developed wavelet approach to solve an image decomposition problem f=u+v, that was initially proposed by Osher et. al. The initial problem was altered many times to fit both mathematical needs like the duality of the target spaces for u and v and different applications. The newly proposed wavelet version, which is meant to approximate the traditional method, however, is inexpensive enough to be applied to the large data sets from positron emission tomography scanners (PET), which are used to image brain activity. We can show that we get better results then the currently used traditional methods while we have to address questions of the practical discrete implementation and the connection to the theoretical continuous model.

The second problem is a deblurring problem, applied to seismic traces. In a paper to appear this year we applied total variation regularized deconvolution to clean up seismic traces used to probe the deep Earth, in particular the core mantle bounder. The next step will be to extend this method to allow more uncertainty in the assumed blurring operator leading to a total least squares problem (TLS). We will combine the the TLS approach with TV to better fit the application, and try to carry over some results that are known for Tikhonov regularization.

For further information please contact: mittelmann@asu.edu