Data, Signal, and Image Modeling: Theory and Algorithms

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To analyze and process signals, or images, a mathematical signal model is first chosen to describe the data. For example, band limited or spline models are often used for the acquisition of Diffusion Tensor Magnetic Resonance Images (DTMRI) and for tracking white matter in the brain. Sampling, data compression, or data transmission are often modeled by shift-invariant spaces such as wavelets. The goals of this research include the development of a mathematical framework for finding the "best" shift-invariant model from observed data for a given class of signals, and the development of computationally efficient algorithms for finding the so-called shift-invariant signal models.For many applications, it is important to adapt the signal model to the observed data in order to improve the quality of the analysis, or to reduce the processing time, for example. To find the shift-invariant space model that is optimally adapted to a given class of measured data, this project will develop mathematical transformations that render the search mathematically and computationally manageable. Testing and applications will focus on Diffusion Tensor Magnetic Resonance images, and data obtained from mass spectrometry for visualizing the location of proteins in tissues.