Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction

In this dissertation, we address several inverse problems associated with multi-channel sampling and reconstruction that pertain to parallel magnetic resonance imaging (pMRI). The first part of this dissertation addresses adaptive design of spatio-temporal acquisition and reconstruction in model-based pMRI wherein the signal model is a sparse support. We develop a highly-accelerated real-time dynamic MRI technique, dubbed PARADISE, which incorporates a physiologically-driven sparse support model in the joint spatial domain and temporal frequency dimension. The imaging scheme gains its acceleration from: (i) sparsity of the support model; and (ii) the redundancy in data acquired by the parallel receiver coils. The PARADISE adaptation procedure ensures that maximally compressed MR data is acquired by optimally exploiting the degrees of freedom in the joint k-t sampling space, thereby enabling high accelerations and quality in the cine reconstruction stage. We propose and verify the efficacy of a geometric multi-channel sampling design algorithm that does not require explicit knowledge of the channel characteristics. Accompanied by a customized pulse sequence, the fast semi-blind acquisition design technique enables streamlined implementation of the method in a clinical setting. Moreover, the unified multi-channel sampling framework explicitly accounts for speed limitations of gradient encoding, provides performance guarantees on achievable image quality both in terms of noise gain and aliasing distortion, and allows for analysis of the method's robustness to model mismatch. We present in-vivo results demonstrating the feasibility of the PARADISE scheme -- and its distinctive features and effectiveness -- for high resolution non-gated cardiac imaging during a short breath-hold. The second part of the dissertation addresses the problems of blind and nonblind perfect inversion of multi-channel multi-rate systems. Driven by applications in multi-sensor imaging systems such as pMRI, we focus on systems wherein each channel is subsampled relative to the Nyquist rate but the overall multi-channel system is oversampled. We address the feasibility of perfect reconstruction (PR) using short finite impulse response (FIR) synthesis filters given an oversampled but otherwise general FIR analysis filter bank (FB). We provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR and, in addition, study the requirements for achieving near-optimal noise performance. Next, we address the problem of multi-channel perfect interpolation (PI) by building upon the developed framework for the multi-channel PR problem. We present the theory and algorithms for identifying a FIR multi-input multi-output interpolation bank that achieves PI both with and without the knowledge of the channel characteristics. The theory developed for the latter case, called the blind PI problem, is in turn used to develop a self-calibrating algorithm, dubbed ACSIOM, for blind identification of the interpolation FB with limited calibration data. We also provide performance guarantees for the proposed algorithm and propose an improved iterative scheme to tackle scenarios where only very limited calibration data is available. The main practical motivation for the presented blind PI method is to tackle the image reconstruction problem in self-calibrated pMRI applications. We present in-vivo parallel MRI results that demonstrate the effectiveness of the developed method in self-calibrating MR image reconstruction with comparison to state-of-the-art -- nevertheless heuristic -- alternatives