Non-quadratic Penalization in Generalized Local Regularization for Linear and Nonlinear Inverse Problems

Michigan State University, East Lansing MI

Investigators

Abstract

Variational methods with non-quadratic penalty terms (such as those associated with total variation or sparsity constraints) are extremely effective solution methods for the resolution of sharp features of nonsmooth solutions to applied inverse problems. Unfortunately, such globally-defined methods are often associated with large computational costs, a situation made even worse by the scale of the underlying problem (e.g., large imaging data sets). Using their expertise in methods of generalized local regularization, the investigator and her colleagues develop localized non-quadratic regularization methods for ill-posed inverse problems as a way to reduce computational costs while still maintaining the sharp resolution of solutions. Applications of these new methods include problems of image deblurring, nonlinear autoconvolution, blind deconvolution, fractional integration/differentiation, and inverse and backward heat conduction. The methods developed by the investigator and her colleagues lead to faster, less computationally-intensive techniques for the accurate reconstruction of pictures in a wide number of applications, from satellite image gathering, biomedical imaging (CT scans, PET scans, etc.), to geophysical exploration. The improved methods enable the analysis of more data in a shorter amount of time, and with less overall expense. In addition to imaging applications, these new methods are also applicable to the problem of determining hidden cracks and weaknesses in structures and materials, and the detection of ozone levels in the atmostphere, as well as to problems arising in the analysis of nano-structures.