Abstract
Recently, the sparsity which is implicit in MR images has been successfully exploited for fast MR imaging with incomplete acquisitions. In this paper, two novel algorithms are proposed to solve the sparse parallel MR imaging problem, which consists of l 1 regularization and fidelity terms. The two algorithms combine forward-backward operator splitting and Barzilai-Borwein schemes. Theoretically, the presented algorithms overcome the nondifferentiable property in l 1 regularization term. Meanwhile, they are able to treat a general matrix operator that may not be diagonalized by fast Fourier transform and to ensure that a well-conditioned optimization system of equations is simply solved. In addition, we build connections between the proposed algorithms and the state-of-the-art existing methods and prove their convergence with a constant stepsize in Appendix. Numerical results and comparisons with the advanced methods demonstrate the efficiency of proposed algorithms.
Highlights
Reducing encoding is one of the most important ways for accelerating magnetic resonance imaging (MRI)
To solve the problems existing in the algorithms mentioned above, this paper develops two fast numerical algorithms based on the operator splitting and Barzilai-Borwein techniques
The above experiments illustrate that the proposed algorithms forwardbackward operator splitting shrinkage (FBOSS) and forward-backward operator splitting projection (FBOSP) have better performance than Bregman operator splitting (BOS), splitting Barzilai-Borwein (SBB), and Alternating minimization (AM) in terms of computational efficiency, DTD generated from the TV-based SparseSENSE model can be diagonalized by fast Fourier transform (FFT)
Summary
Reducing encoding is one of the most important ways for accelerating magnetic resonance imaging (MRI). Several rapid numerical algorithms can solve the numerical difficulties, which are, for example, alternating direction method of multipliers (ADMM) [26], augmented Lagrangian method (ALM) [27], splitting Bregman algorithm (SBA) [28], splitting Barzilai-Borwein (SBB) [24], and Bregman operator splitting (BOS) [29] The efficiency of these methods largely depends on the special structure of the matrix operator DTD (such as Toeplitz matrix and orthogonal matrix) and the encoding kernel (without the sensitivity maps). They are not suitable for simultaneously dealing with general regularization operator D and the parallel encoding matrix A That is, these algorithms are not able to solve the problem (1) efficiently because the complex inversion of the large size matrix has to be computed, if DTD and/or ATA cannot be diagonalized directly by fast Fourier transform (FFT). Appendix proves the convergence of the proposed algorithms with constant stepsizes
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