  Multidimensional imaging, capturing image data in more than two dimensions,
has been an emerging field with diverse applications. Due to the limitation of
two-dimensional detectors in obtaining the high-dimensional image data,
computational imaging approaches have been developed to pass on some of the
burden to a reconstruction algorithm. In various image reconstruction problems
in multidimensional imaging, the measurements are in the form of superimposed
convolutions. In this paper, we introduce a general framework for the solution
of these problems, called here convolutional inverse problems, and develop fast
image reconstruction algorithms with analysis and synthesis priors. These
include sparsifying transforms, as well as convolutional or patch-based
dictionaries that can adapt to correlations in different dimensions. The
resulting optimization problems are solved via alternating direction method of
multipliers with closed-form, efficient, and parallelizable update steps. To
illustrate their utility and versatility, the developed algorithms are applied
to three-dimensional image reconstruction problems in computational spectral
imaging for cases with or without correlation along the third dimension. As the
advent of multidimensional imaging modalities expands to perform sophisticated
tasks, these algorithms are essential for fast iterative reconstruction in
various large-scale problems.
