Positron emission tomography (PET) is a nuclear medicine functional imaging modality, applicable to several clinical problems, but especially in detecting the metabolic activity (as in cancer). PET scanners use multiple rings of gamma ray detectors that surround the patient. These scanners are quite expensive (1-3 million dollars), therefore a technology that would allow the reduction in the number of detectors per ring without affecting image quality, could reduce the scanner cost, thereby making this imaging modality more accessible to patients. In this thesis , a mathematical technique known as compressive sensing is applied in an effort to decrease the number of detectors required, while maintaining good image quality. A CS model was developed based on a combination of gradient magnitude and wavelet domains to recover missing observations associated with PET data acquisition. The CS model also included a Poisson-distributed noise term. The overall model was formulated as an optimization problem wherein the cost function was a weighted sum of the total variation and the L1-norm of the wavelet coefficients. Subsequently, the cost function was minimized subject to the CS model equations, the partially observed data, and a penalty function for noise suppression (the Poisson log-likelihood function). We refer to the complete model as the WTV model. This thesis also explores an alternative reconstruction method, wherein a different CS model based on an adaptive dictionary learning (DL) technique for data recovery in PET imaging was developed. Specifically, a PET image is decomposed into small overlapped patches and the dictionary is learned from these overlapped patches. The technique has good sparsifying properties and the dictionary tends to capture local as well as structural similarities, without sacrificing resolution. Recovery is accomplished in two stages: a dictionary learning phase followed by a reconstruction step. In addition to developing optimized CS reconstruction, this thesis also investigated: (a) the limits of detector removal when using the DL CS reconstruction algorithm; and (b) the optimal detector removal configuration per ring while minimizing the impact on image quality following recovery using the CS model. Results of these investigations can serve to help make PET scanners more affordable while maintaining image quality. These results can also be used to improve patient throughput by redesigning scanners so that removed detectors can be placed in axial extent to image a larger portion of the body. This will help increase scanner throughput hence improve scanner efficiency as well as patient discomfort due to long scan time.