Sampling and recovery of MRI data using low rank tensor models

In this paper we investigate the utility of several low-rank models for recovery of Magnetic Resonance Imaging (MRI) data from limited sampling in the k - t space for dynamic imaging. In particular, for 3D temporal (2D space + time) MRI data we employ several tensor factorization techniques and assess the degree of dimensionality reduction, or compressibility, that can be obtained. This algebraic approach is more data adaptive, in contrast to existing compressed sensing (CS) based methods that exploit sparsity in a transform domain, such as wavelets or total variation. Further, we compare these tensor factorization approaches in recovering temporal MRI data under limited sampling. Respecting traditional MRI data acquisition methods, the sampling process is restricted to be uniformly random along only one k space direction. Experimental results on synthetically sub-sampled MRI data show promise in using tensor factorization for sampling and recovery of MRI data.

[1]  N J Pelc,et al.  Unaliasing by Fourier‐encoding the overlaps using the temporal dimension (UNFOLD), applied to cardiac imaging and fMRI , 1999, Magnetic resonance in medicine.

[2]  Mathews Jacob,et al.  Accelerated Dynamic MRI Exploiting Sparsity and Low-Rank Structure: k-t SLR , 2011, IEEE Transactions on Medical Imaging.

[3]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[4]  Misha Elena Kilmer,et al.  Novel Factorization Strategies for Higher Order Tensors: Implications for Compression and Recovery of Multi-linear Data , 2013, ArXiv.

[5]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[6]  Shuchin Aeron,et al.  5D and 4D pre-stack seismic data completion using tensor nuclear norm (TNN) , 2013, SEG Technical Program Expanded Abstracts 2013.

[7]  Misha Elena Kilmer,et al.  Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging , 2013, SIAM J. Matrix Anal. Appl..

[8]  Misha Elena Kilmer,et al.  Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[9]  Justin P. Haldar,et al.  Low-Rank Modeling of Local $k$-Space Neighborhoods (LORAKS) for Constrained MRI , 2014, IEEE Transactions on Medical Imaging.

[10]  Armando Manduca,et al.  A Unified Tensor Regression Framework for Calibrationless Dynamic , Multi-Channel MRI Reconstruction , 2012 .

[11]  Prateek Jain,et al.  Provable Tensor Factorization with Missing Data , 2014, NIPS.

[12]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[13]  Bart Vandereycken,et al.  Low-rank tensor completion by Riemannian optimization , 2014 .

[14]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[15]  Stuart Crozier,et al.  Multidimensional Compressed Sensing MRI Using Tensor Decomposition-Based Sparsifying Transform , 2014, PloS one.

[16]  Robin M Heidemann,et al.  Generalized autocalibrating partially parallel acquisitions (GRAPPA) , 2002, Magnetic resonance in medicine.

[17]  M. Kilmer,et al.  Tensor-Tensor Products with Invertible Linear Transforms , 2015 .

[18]  A. Stegeman,et al.  On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition , 2007 .

[19]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.