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Efficient Algorithms for Convolutional Inverse Problems in Multidimensional Imaging

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Document pages: 17 pages

Abstract: Multidimensional imaging, capturing image data in more than two dimensions,has been an emerging field with diverse applications. Due to the limitation oftwo-dimensional detectors in obtaining the high-dimensional image data,computational imaging approaches have been developed to pass on some of theburden to a reconstruction algorithm. In various image reconstruction problemsin multidimensional imaging, the measurements are in the form of superimposedconvolutions. In this paper, we introduce a general framework for the solutionof these problems, called here convolutional inverse problems, and develop fastimage reconstruction algorithms with analysis and synthesis priors. Theseinclude sparsifying transforms, as well as convolutional or patch-baseddictionaries that can adapt to correlations in different dimensions. Theresulting optimization problems are solved via alternating direction method ofmultipliers with closed-form, efficient, and parallelizable update steps. Toillustrate their utility and versatility, the developed algorithms are appliedto three-dimensional image reconstruction problems in computational spectralimaging for cases with or without correlation along the third dimension. As theadvent of multidimensional imaging modalities expands to perform sophisticatedtasks, these algorithms are essential for fast iterative reconstruction invarious large-scale problems.

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