Date of Award
Doctor of Philosophy
Ricardo F. von Borries
This dissertation introduces the theory of compressive sensing with prior information about a signal's sparse representation. We show, mathematically and in numerical simulations, that prior information improves signal reconstruction, in terms of number of required measurements, computation time and signal-to-noise ratios. Following, we present a set of methods for enhanced magnetic resonance imaging and tomography that can be combined with prior information, for enhanced image quality.
In developing the theory of compressive sensing with prior information, we provide a mathematical proof of the required condition (in terms of number of linear measurements) for reconstruction using the ideal approach of l0-minimization, as well as the necessary and sufficient conditions for reconstruction using l1-minimization. We then develop the theory for evaluating the probability of reconstruction when dealing with stochastic signals, both in the case without and with prior information. Furthermore, we compare the proposed models with the empirical probability of reconstruction, by evaluating in Monte-Carlo simulations the percentage of cases in which the conditions for reconstruction with prior information are satisfied, opposed to compressive sensing without prior information. These analyzes show that prior information about the sparse representation's support reduce the number of linear measurements required to reconstruct a signal, both in terms of the theoretical lower bound and of the l1-minimization procedure, commonly used in the compressive sensing literature.
Regarding practical reconstruction algorithms, we provide an optimization procedure for compressive sensing with prior information based on lp-minimization. The experimental results show that further improvement can be obtained by combining support prior information with the reduction of the p parameter in the lp-minimization. Furthermore, by using this method we show that partial support information also reduces the total computation time, when using a direct method to solve the inner linear systems. For a fixed number of taken linear measurements, on the other hand, the experiments show that prior information improves the signal-to-noise ratios of the reconstructed signals.
The dissertation also proposes a compressive sensing method for enhanced reconstruction of magnetic resonance (MR) images using a prefiltering strategy in the k-space domain. This method can improve the quality of the reconstruction over a standard compressive sensing approach. In particular, it leads to higher signal-to-noise ratios, and generally to lower reconstruction times. Also, it easily allows for a parallel implementation since different computation stages are independent of each other, which can further reduce the reconstruction times by a factor of up to three, in the tested reconstruction schemes.
Finally, we show how the proposed MR imaging method can be combined with the previous approach of compressive sensing with prior information. The experiments conducted over simulated and real MR images and functional MR images show a further improvement by combining prefiltering in the k-space domain with the use of prior information, in terms of visual quality and signal-to-noise ratios.
Received from ProQuest
Cristiano Jacques Miosso
Miosso, Cristiano Jacques, "Compressive Sensing with Prior Information Applied to Magnetic Resonance Imaging" (2010). Open Access Theses & Dissertations. 2542.