Analytic and Geometric Methods in Inverse Problems and Imaging

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This proposal seeks to answer some fundamental mathematical questions in the fields of Inverse Problems and Image Processing, as well as to contribute specific applications to Medical Imaging. The field of Inverse Problems studies novel methods to obtain images from noninvasive measurements, for next generation imaging modalities. Image Processing treats images obtained from existing modalities and the mathematics involved in denoising, segmenting, registering and extracting information from them.*** Classically, the data in Inverse Problems consists of measurements in the exterior of the object under investigation, or on its boundary. There have been considerable advances by numerous researchers in the systematic study of such problems, but several challenging questions remain open, and new ideas for addressing some of them will be sought. In addition, a new class of inverse problems considers situations where some interior information can be obtained (for instance using Magnetic Resonance Imagers). In joint work with A. Tamasan and A. Timonov, we have found a connection between one such problem arising in electric impedance imaging and the theory of minimal surfaces in non-euclidean geometries (determined by the measured data) and of related weighted least gradient problems. In recent joint work with A. Tamasan and J. Veras, the practical application has lead us to an interesting novel boundary value problem which has not been previously considered in geometric measure theory and will be investigated. We will also seek to obtain corresponding efficient and scalable numerical algorithms for its solution, and apply these to experimental data obtained in collaboration with M.Joy's group at the University of Toronto.******The study of weighted least gradient problems has also led us to several analytic and geometric questions in Image Analysis. In a seminal paper, Tadmor, Nessar and Vese introduced a multiscale decomposition of images based on a sequence of variational problems involving a similar regularization functional; they showed that this can be considered as a nonlinear harmonic decomposition, with successive terms adding finer scale details. In joint work with K. Modin and L. Rondi we have recently extended this approach to registration problems, thus obtaining a (multiplicative) harmonic decomposition of diffeomorphisms. Theoretical and practical implications of this novel decomposition will be investigated. We will also apply the multiscale approach to the inverse problem described above, as a possible tool for handling non-smooth impedances.*** Solution of some of the problems we plan to study is expected to have a significant impact in the development of next generation imaging modalities as well as to the image analysis of clinical data. **