Tomography is a ubiquitous imaging modality applied to numerous fields like medical imaging, geophysical imaging, structural health monitoring etc. It is a process to view cross-sectional profile of an object or region of interest through solving an inverse problem. When diffracting sources are used as the interrogating energy, the specific tomographic reconstruction is termed `Diffraction Tomography' as the algorithm accounts for diffraction effects. As a result, these algorithms are more complex than straight ray tomography algorithms viz. computed tomography which has been very successfully implemented for X-ray tomography, PET etc. Ideally, projection data covering full 360o around the region of interest is necessary for accurate reconstruction. However, in many practical applications, this is not always feasible. As a result, reconstruction is performed on limited datasets, essentially making the process a recovery from an under-determined system. This thesis focuses on development of novel and efficient methods to handle the challenges on limited data for image reconstruction under diffraction tomography. For a moderately limited coverage, some inherent redundancies in Diffraction Tomography projection data can be used to reduce the effect of limited coverage. For highly limited angular coverage, these redundancies are no longer available and cannot be exploited. Recently, however, optimization techniques involving l1-norm minimization schemes under the so called `compressed sensing' regime have shown promise. These algorithms are capable of almost exact reconstruction of the object even with highly limited number of projections. This research has explored both techniques for moderate and highly limited angular tomography. In the first part of the thesis, for moderate angular access limitations, an optimum method for exploiting redundancy within projection data has been formulated. In the second part, for highly limited coverage with further limitations on the number of available projections, reconstruction schemes under compressed sensing regime have been examined. Further, this research demonstrates reconstruction of complex-valued objective function under the regime of compressed sensing. This generalizes the application of tomographic reconstruction for newer applications such as examining new complex structures (such as metamaterial and other smart material based structures) where knowledge of complex permittivity values is essential in evaluating structural integrity or morphological aberrations. The compressed sensing method heavily relies on sparsity of the reconstructed signal in some transformation domain. In this research, the sparsity has mainly been exploited through gradient magnitude of images. In a variety of applications, gradient magnitude of images are highly sparse, even if the images themselves are not. So the gradient magnitude of images can be effectively used as the sparse domain. Further, incorporation of multiple sparse domains into the compressed sensing framework has been explored. Using Haar wavelets in addition to gradient magnitude of images as the sparse domain has successfully been employed showing potential for significant improvements in image reconstruction from highly limited data through further research.
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