Tradeoffs and limitations in statistically based image reconstruction problems

Advanced nuclear medical imaging systems collect multiple attributes of a large number of photon events, resulting in extremely large datasets which present challenges to image reconstruction and assessment. This dissertation addresses several of these challenges. The image formation process in nuclear medical imaging can be posed as a parametric estimation problem where the image pixels are the parameters of interest. Since nuclear medical imaging applications are often ill-posed inverse problems, unbiased estimators result in very noisy, high-variance images. Typically, smoothness constraints and a priori information are used to reduce variance in medical imaging applications at the cost of biasing the estimator. For such problems, there exists an inherent tradeoff between the recovered spatial resolution of an estimator, overall bias, and its statistical variance; lower variance can only be bought at the price of decreased spatial resolution and/or increased overall bias. A goal of this dissertation is to relate these fundamental quantities in the analysis of imaging systems. A relationship between list-mode (single photon) measurements and binned measurements is shown. A model for the measurement statistics for a Compton Scatter Single Photon Emission Tomography (Compton SPECT) system is derived, and reconstructed images from both simulated and measured data are presented. In order to reduce the computations involved in reconstruction, we explore lossy compression of the projection data using vector quantizers. Asymptotic expressions for the loss in the Kullback-Liebler divergence due to quantization for a low contrast lesion detection task are derived. A fast and effcient method of optimizing the measurement space partitioning using a lattice vector quantizer is presented, and results for a 2D Positron Emission Tomography (PET) imaging system show an optimal bit allocation. Finally, some fundamental limitations in image reconstruction are derived. In particular, the tradeoff between bias, resolution, and estimator variance is explored using a 2D image deconvolution problem as a motivating example.