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Electron paramagnetic resonance (EPR), and its molecular imaging modality, is a powerful tool to noninvasively map various biological and chemical markers within objects of interest. Reliable data acquisition is a major impeding factor for longitudinal hands-off measurements. Measurements are especially challenging in biomedical applications, as live objects are not static. Frequent changes occur that require constant fine recalibration of the EPR detection system, called the resonator. To enable longitudinal imaging, a technology permitting automatic digital control of resonator coupling, tuning, and EPR data acquisition was developed. Automation was achieved through the utilization of a microcontroller and digital peripheral components such as digitally controlled capacitors, a digital frequency source, and a printed circuit board resonator. Several applications of this technology have been suggested and tested, including in vivo EPR imaging. The first was to develop a tool for the optimization of light-based 3D printing, for which oxygen plays a major role. Towards this goal, an EPR oxygen-sensitive probe was incorporated into 3D printing resin. Oxygen depletion was measured during the 3D printing process as the polymerization front progressed. After printing, oxygen depletion was again measured during the post-curing process, proposed as a method to optimize post-curing light intensity, temperature, and duration in order to produce quality 3D printed constructs. The second application of longitudinal EPR imaging was directed toward resolving an important problem of oxygen delivery to thick (>1 cm) bioprinted models. Oxygen-sensitive EPR probes (water-soluble trityls or crystalline lithium octa-n-butoxynaphthalocyanine) were introduced into bioinks (liquid hydrogels containing cells, nutrients, and other biological factors) before printing. Bioinks become solid structures after printing due to crosslinking. EPR imaging was demonstrated to measure oxygen consumption by the cells embedded in the bioprints. As expected, an increase of oxygen depletion was observed by introducing a nutrient (pyruvate) to bioink. A numerical MATLAB simulation program was developed to predict rates of oxygen consumption by the cells in the bioprint. The input parameters for the mathematical model include the size and number of cells, the diffusion coefficient of the media, and rates of oxygen transfer through the cell membrane. The software is being further refined and optimized for computational speed. Future efforts will be aimed at improving the speed and scope of EPR automatic digital control, imaging the oxygen depletion process in commercial 3D printers, and applying EPR mapping of oxygen consumption rate to quantify the delivery of oxygen to cells deep inside bioprinted tissue models. Optimizing the delivery of oxygen to cells would overcome the challenge of the limit of diffusion, facilitating the development of larger and more complex bioprinted tissues and organs. These complex tissue and organ models are envisioned for use in drug testing, biomedical research, and, in the distant future, implants in humans.

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 wherethe sparse support follows a Bernoulli distribution and the signal restricted to this supportfollows 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.