L+S Reconstruction

Low-rank plus sparse (L+S) reconstruction

Ricardo Otazo (NYU), Emmanuel Candes (Stanford), Daniel K. Sodickson (NYU)

Project Summary

High spatial and temporal correlations in dynamic MRI data sets enabled the application of compressed sensing and low-rank matrix completion to accelerate data acquisition. The combination of both techniques is very attractive. Recently, a new matrix decomposition method consisting of a superposition of a low-rank matrix (L) and sparse matrix (S) has been proposed by Emmanuel Candes et al. to increase the performance of principal component analysis (PCA) in the presence of outliers. In this work, we apply the L+S decomposition to reconstruct undersampled dynamic MRI, where the low-rank component can model the correlated background and the sparse component can model the dynamic information that lies on top of the background.

The L+S approach aims to find the low-rank (L) and sparse (S) components of a matrix M with linearly dependent rows or columns by solving the following optimization problem:

min||L||* + λ ||S||1    subject to    Μ = L + S

where ||L||*  is the nuclear norm or sum of singular values of the matrix L and ||S||1 is the l1-norm or sum of absolute values of the entries of S. In order to apply this idea to dynamic MRI, the time-series of images can be reorganized into a matrix M, where each column is an image frame. The L+S reconstruction of undersampled dynamic MRI data can be formulated as follows:

min||L||* + λ ||ST||1     subject to    E(L + S) = d

where T is the sparsifying transform, E is the acquisition matrix and d is the undersampled k-t data. The reconstruction result is given by M = L + S.


Reconstruction of a 8-fold accelerated cardiac perfusion data set: (left-right) standard compressed sensing, low-rank plus sparse (L+S), low-rank component (L) and sparse component (S). The L+S technique suppresses the background before enforcing sparsity, which effectively increases sparsity (M-L is more sparse than M) and thus improves reconstruction performance.





MIP reconstruction of a 7.5-fold accelerated time-resolved peripheral MR angiography using low-rank plus sparse (L+S) reconstruction: (left-t-right) L+S, L and S. The L+S method automatically separated the background (L) from the contrast-enhancement (S) without the need of subtraction or modeling.




Principal Investigator: 
Ricardo Otazo


Philanthropic Support

We gratefully acknowledge generous support for radiology research at NYU Langone Health from:
• The Big George Foundation
• Raymond and Beverly Sackler
• Bernard and Irene Schwartz

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