Efficient Algorithms for Convolutional Inverse Problems in Multidimensional Imaging

TL;DR

This paper introduces a general framework and fast algorithms for solving convolutional inverse problems in multidimensional imaging, enabling efficient reconstruction in high-dimensional data with diverse applications.

Contribution

It develops a unified approach with analysis and synthesis priors, using ADMM for efficient, parallelizable solutions to complex multidimensional image reconstruction problems.

Findings

Algorithms successfully applied to 3D spectral imaging reconstruction

Achieved fast, parallelizable solutions for large-scale inverse problems

Demonstrated versatility across different multidimensional imaging scenarios

Abstract

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.