Diffraction tomography (DT) from limited projection data has been an active research topic for over three decades. The interest has been steadily fueled due to its application in multiple disciplines including medical imaging, structural health monitoring and non-destructive evaluation to name a few. This paper explores the applicability of compressed sensing to recover complex-valued objective functions (e.g., complex permittivity in microwave tomography). Generally, compressed sensing based tomographic reconstruction has been studied under full angular access. In this paper, the effect of lowering the angular access in addition to highly limited number of projection data is explored. The effectiveness of the reconstruction methods is tested with severely limited dataset which would render reconstruction impossible by traditional iterative approximation methods. Furthermore, results show that complex-valued phantoms can be reconstructed from as few as 15 projections from 120˚ coverage, a significant finding. In this study, the Total Variation (TV) has been used as the l1 norm within the compressed sensing framework. The robustness of the algorithm in presence of noise is discussed. Use of multiple sparse domains has also been explored briefly. The results show the effectiveness of TV as a regularization parameter even for complex-valued images under the compressed sensing regime. This is a pertinent observation as TV is a simple norm to implement. For a large class of images, especially in medical imaging, this implies the availability of a steady l1 norm for easy implementation of compressed sensing reconstruction for complex-valued images.
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