In this thesis we address two inverse problems. The first is the reconstruction of sparse signals. In the second, electron paramagnetic resonance imaging (EPRI) is considered.
The problem of recovering sparse signals has recently generated much interest among researchers. The research in this area has led to development of faster algorithms with provable performance guarantees. The sparse algorithms can significantly reduce the amount of data required for extracting information of interest from observations. In this thesis we consider the sparsity pattern recovery problem under a probabilistic signal model where
the sparse support follows a Bernoulli distribution and the signal restricted to this support
follows a Gaussian distribution. For the maximum aposteriori estimate, we show that the energy in the original signal restricted to the missed support is bounded above; the bound is of the order of the energy in the noise signal projected to the subspace spanned by the active coefficients of the original signal. We also derive sufficient conditions for no missed detection and no false alarm in support recovery.
Electron Paramagnetic Resonance Imaging (EPRI) is an imaging modality which has a great potential for clinical oximetry applications for cancer treatment and wound healing. Because of several reasons including long data collection time and low signal-to-noise ratio (SNR), this modality has not yet become a clinically successful method. We address these two problems in this thesis. We use the structure present in the EPR spectrum in the form of both Lorentzian line shape and sparse nature of the spin probes implanted for oximetry. We experimentally demonstrate that this leads to two orders of magnitude reduction in data acquisition time. We also propose and build a digital data acquisition system that enhances SNR by enabling simultaneous acquisition of multiple harmonics of both absorption and dispersion components of the signal. A novel convergent iterative algorithm is proposed for processing the data in the presence of phase noise and unknown microwave phase.