Dynamic MRI reconstruction using low-rank plus sparse decomposition with smoothness regularization

TL;DR

Dynamic MRI reconstruction benefits from jointly modeling a slowly varying background and sparse dynamics. The SR-L+S framework introduces a temporal smoothness regularizer on the background L, in addition to the traditional low-rank plus sparse decomposition, yielding a convex objective that combines a nuclear norm on L, an $\ell_1$ sparsity term on S, and a smoothness penalty on L. An efficient proximal gradient algorithm with auxiliary variables delivers closed-form updates for L, S, and the smoothness term; evaluation on PINCAT and cardiac perfusion data shows consistent improvements over state-of-the-art L+S methods in both quantitative metrics and visual quality, as well as better L/S separation. The approach enhances robustness to noise and undersampling, with potential extensions to tensor decompositions and deep-unfolded solver architectures for further performance gains.

Abstract

The low-rank plus sparse (L+S) decomposition model has enabled better reconstruction of dynamic magnetic resonance imaging (dMRI) with separation into background (L) and dynamic (S) component. However, use of low-rank prior alone may not fully explain the slow variations or smoothness of the background part at the local scale. In this paper, we propose a smoothness-regularized L+S (SR-L+S) model for dMRI reconstruction from highly undersampled k-t-space data. We exploit joint low-rank and smooth priors on the background component of dMRI to better capture both its global and local temporal correlated structures. Extending the L+S formulation, the low-rank property is encoded by the nuclear norm, while the smoothness by a general \ell_{p}-norm penalty on the local differences of the columns of L. The additional smoothness regularizer can promote piecewise local consistency between neighboring frames. By smoothing out the noise and dynamic activities, it allows accurate recovery of the background part, and subsequently more robust dMRI reconstruction. Extensive experiments on multi-coil cardiac and synthetic data shows that the SR-L+S model outp

Figures (3)

• Figure 1: Comparison of reconstruction results of PINCAT synthetic data (14th frame) with 8-fold undersampling using different methods. First row: Original (fully-sampled) and reconstructed images. Second row: Enlarged views of yellow boxed region. Third row: Error maps (display scale of [0, 0.2]) with respective to the original image. Forth & fifth rows: y-t images at the vertical cut line (slice $x=64$) in the reconstructed images, and the corresponding error maps, respectively.
• Figure 2: Comparison of reconstruction results of multi-coil cardiac perfusion MRI with 8-fold undersampling using different methods. First row: Original (fully-sampled) and reconstructed images. Second row: Enlarged views of corresponding heart region with myocardial wall enhancement. Third row: Error maps with respective to the original image. Forth & fifth rows: y-t images at the vertical cut line (slice $x=64$) in the reconstructed images, and the corresponding error maps, respectively.
• Figure 3: The L and S component separation of cardiac perfusion dMRI sequence of a patient with coronary artery disease. (a) The x-y view of the original image (14th frame), the reconstructed image by ${\bf X} = {\bf L}+{\bf S}$, and the reconstructed ${\bf L}$ component and ${\bf S}$ component using different L+S models. (b) The y-t views of reconstructed images by different models.