Compressed Sensing & Low-Rank Models
Jennifer Steeden1
1University College London, London, United Kingdom

Synopsis

Accelerated MRI techniques have transformed cardiovascular MR and have been investigated in many clinical applications during the last decade to speed up MRI scans. This talk introduces the key components of Compressed Sensing and Low-rank methods, and how these are implemented in MRI. Examples of clinical applications and current challenges of Compressed Sensing and Low-rank methods are discussed.

Background

Cardiovascular magnetic resonance (CMR) imaging is highly valuable tool for non-invasive diagnosis and monitoring of cardiovascular disease. However, MRI is a slow imaging technology, suffering from long acquisition times. This also makes it sensitive to motion: therefore it is necessary to account for cardiac and respiratory motion during the acquisition, further increasing the scan time.
There has been a huge amount of research into speeding up MRI scans over the last 20 years. This includes the use of efficient non-Cartesian trajectories, as well as data undersampling. These must be combined with suitable reconstruction technologies to enable high quality images, by exploiting prior information. Compressed sensing (CS) and low-rank reconstruction methods enable high levels of acceleration, by exploiting sparsity of the data (in some pre-defined basis). Here a non-linear reconstruction is performed to enforce the sparsity of the image and consistency with the acquired MR data.

MR Reconstruction: Inverse Problem

In general terms, the relationship between the k-space signal, $$$ s $$$, and resultant image, $$$ ρ $$$ can be expressed as a system of linear equations:
$$ s=Eρ $$
where the encoding operator, $$$ E $$$, includes the Fourier transform, the sampling mask and the coil sensitivities. Solving this inverse problem is ill-posed, and hence this is typically reformulated as a regularized optimization:
$$ \overbrace{ρ}=argmin_{ρ}⁡‖Eρ-s‖_2^2 +λ‖ϕ(ρ)‖_1 $$
where the image is recovered by balancing between the data consistency term and an additional regularization term. The weighting parameter λ controls the degree of regularization and needs to be chosen according to the noise level of the acquired data.

Compressed Sensing

In CS the regularization term works by assuming the data is sparse in some known transform domain, Φ. In cardiac MRI, the sparsifying transform can be chosen as a spatial sparsifier, e.g. spatial finite difference, or spatial wavelets, or as a temporal sparsifier, e.g. temporal finite difference, or temporal Fourier transform (figure 1).
Recommended review papers can be found at [1] and [2].

Low-Rank Reconstruction

Low-rank image reconstruction also exploits the fact that there a high degree of correlation, by representing the data in a low-dimensional subspaces. Low-rank image reconstruction may work by exploiting Globally low-rank (GLR) regularization techniques, Locally low-rank (LLR) regularization techniques, or patch-based image reconstructions exploiting local (i.e., within a patch) and non-local (i.e., between similar patches) similarities.A Casorati matrix is usually formed from the undersampled image sequence, and the missing samples are then estimated using low-rank matrix completion [3][4]. Low-rank reconstruction has been combined with CS-based techniques to further improve image quality, particularly for high acceleration factors.

Applications to CMR

Both CS and Low-rank reconstructions have been successfully applied to CMR data. Compressed Sensing reconstructions have been applied to Cardiac 2D cine MRI [5][6], 3D cine [7][8], 3D LGE [9][10] and Whole-heart MRA [11][12]. Similarly low-rank reconstructions have been applied to cine MRI [13] [14][15], 3D coronory imaging [16] and 3D MRA [17]. Combined low-rank and sparse reconstructions have been demonstrated in dynamic imaging [18][19] and for perfusion imaging [20].

These technologies have enabled significant reduction in CMR scan times, by enabling high quality reconstructions of heavily undersampled data.

Acknowledgements

I would like to thank my funders: UKRI Future Leaders Fellowship (MR/S032290/1).

References

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Proc. Intl. Soc. Mag. Reson. Med. 30 (2022)