Approximate message passing for compressed sensing magnetic resonance imaging

Thesis

Magnetic Resonance Imaging (MRI) is a non-invasive, non-ionising imaging modality with unrivalled soft tissue contrast. A key consideration for MRI is data acquisition time, which is limited by inherent technological and physiological constraints. Compressed sensing is a relatively recent framework that can reduce the MRI acquisition time by undersampling randomly and exploiting presumed redundancies in the data.

The Approximate Message Passing (AMP) algorithm is an iterative compressed sensing method that efficiently reconstructs signals that have been sampled with i.i.d. sub-Gaussian sensing matrices. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in MRI, AMP performs poorly in practice.

In response, this thesis proposes the Variable Density Approximate Message Passing (VDAMP) algorithm for undersampled MRI data. We present three versions of VDAMP: single-coil VDAMP, where receiver coil sensitivities are ignored, Parallel-VDAMP (P-VDAMP), which includes coil sensitivities, and Denoising-P-VDAMP (D-P-VDAMP), which incorporates the statistical modelling capabilities of neural networks. Central to VDAMP is a property that we term "coloured state evolution", where the difference between the intermediate image estimate at a given iteration and the ground truth is distributed according to a zero-mean Gaussian with known covariance. We demonstrate that coloured state evolution can be leveraged to yield an algorithm that converges rapidly, and to a competitive reconstruction quality, without the need to hand-tune model parameters.

Authors: Millard, C

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: approximate message passing; magnetic resonance imaging; image reconstruction; compressed sensing
• Subjects: Magnetic resonance imaging; Signal processing; Numerical analysis; Image reconstruction; Mathematics