Most attempts at recovering a high quality image from a low quality one or recovering infor- mation from an image or a set of measurements, present themselves as inverse problems, and generally ill posed ones at that. This is often described (either explicitly or implicitly) as an opti- mization problem on the set of required parameters under some appropriate norm. These param- eters can consist of the restored image, a segmentation contour, object parameters, or a myriad of other questions. In order to perform such an optimization though, an appropriate model of the degradation process needs to be developed. Another issue, is that due to the generally ill posed nature of most of these problems, addi- tional regularization is required in order to recover a sane results. That is, we need to somehow impose the space of valid, or sought after, solutions onto the problem. This should however be done carefully so as not to find something that doesn’t exist just because we looked too hard. This work deals with various aspects of the optimization problem. The first half of the work deals with adaptive regularization methods while the second half deals with the direct problem arising in differential interference contrast (DIC) microscopy. The endeavor into regularization methods starts with exploring non-linear anisotropic scale spaces and their properties. These results are then extended by applying the ideas of anisotropic regularization to deconvolution, using theWiener filter as a test case. Next, coresets, an optimization methods using smart weighted random sampling is developed for the application to dictionary learning methods. As for the second half of the work, the DIC microscope is designed to look at transparent material. Roughly speaking, it can be seen as looking at the derivative of the light field rather than the intensity. This allows looking at transparent biological material without damaging it, unlike florescence based microscopy. We were interested specifically at applications for in vitro fertilization (IVF). In our specific case there is interest in assessing the viability of human embryos in order to choose the best candidates to be returned to the womb. The problem with the DIC microscope is that it provides contrast at object boundaries, but in turn looses, or more correctly augments, 3D information, which is just the type of information that doctors are looking for. This part of the work deals with building a model for DIC microscopy, and the limits on such models, namely, the ofter used first Born approximation.

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