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**1 - 4**of**4**### RESEARCH ARTICLE Balanced Sparse Model for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

"... Compressed sensing has shown to be promising to accelerate magnetic resonance imag-ing. In this new technology, magnetic resonance images are usually reconstructed by en-forcing its sparsity in sparse image reconstruction models, including both synthesis and analysis models. The synthesis model assu ..."

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Compressed sensing has shown to be promising to accelerate magnetic resonance imag-ing. In this new technology, magnetic resonance images are usually reconstructed by en-forcing its sparsity in sparse image reconstruction models, including both synthesis and analysis models. The synthesis model assumes that an image is a sparse combination of atom signals while the analysis model assumes that an image is sparse after the application of an analysis operator. Balanced model is a new sparse model that bridges analysis and synthesis models by introducing a penalty term on the distance of frame coefficients to the range of the analysis operator. In this paper, we study the performance of the balanced model in tight frame based compressed sensing magnetic resonance imaging and propose a new efficient numerical algorithm to solve the optimization problem. By tuning the balanc-ing parameter, the new model achieves solutions of three models. It is found that the bal-anced model has a comparable performance with the analysis model. Besides, both of them achieve better results than the synthesis model no matter what value the balancing parameter is. Experiment shows that our proposed numerical algorithm constrained split augmented Lagrangian shrinkage algorithm for balanced model (C-SALSA-B) converges faster than previously proposed algorithms accelerated proximal algorithm (APG) and alter-nating directional method of multipliers for balanced model (ADMM-B).

### A projection algorithm for gradient waveforms design in Magnetic Resonance Imaging

, 2014

"... Collecting the maximal amount of useful information in a given scanning time is a major concern in Magnetic Resonance Imaging (MRI) to speed up image acquisition. The hardware constraints (gradient magnitude, slew rate,...), physical distortions (e.g., off-resonance effects) and sampling theorems (S ..."

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Collecting the maximal amount of useful information in a given scanning time is a major concern in Magnetic Resonance Imaging (MRI) to speed up image acquisition. The hardware constraints (gradient magnitude, slew rate,...), physical distortions (e.g., off-resonance effects) and sampling theorems (Shannon, compressed sensing) must be taken into account simultaneously, which makes this problem extremely challenging. To date, the main approach to design gradient waveform has consisted of selecting an initial shape (e.g. spiral, radial lines,...) and then traversing it as fast as possible. In this paper, we propose an alternative solution: instead of reparameterizing an initial trajectory, we propose to project it onto the convex set of admissible curves. This method has various advantages. First, it better preserves the density of the input curve which is critical in sampling theory. Second, it allows to smooth high curvature areas making the acquisition time shorter in some cases. We develop an efficient iterative algorithm based on convex programming and propose comparisons between the two approaches. For piecewise linear trajectories, our approach generates a gain of scanning time ranging from 20 % (echo planar imaging) to 300% (travelling salesman problem) without degrading image quality in terms of signal-to-noise ratio (SNR). For smoother trajectories such as spirals, our method better preserves the sampling density of the input curve, making the sampling pattern relevant for compressed sensing, contrarily to the reparameterization based approaches.

### Balanced Sparse Model for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

, 2015

"... Compressed sensing has shown to be promising to accelerate magnetic resonance imaging. In this new technology, magnetic resonance images are usually reconstructed by en-forcing its sparsity in sparse image reconstruction models, including both synthesis and analysis models. The synthesis model assum ..."

Abstract
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Compressed sensing has shown to be promising to accelerate magnetic resonance imaging. In this new technology, magnetic resonance images are usually reconstructed by en-forcing its sparsity in sparse image reconstruction models, including both synthesis and analysis models. The synthesis model assumes that an image is a sparse combination of atom signals while the analysis model assumes that an image is sparse after the application of an analysis operator. Balanced model is a new sparse model that bridges analysis and synthesis models by introducing a penalty term on the distance of frame coefficients to the range of the analysis operator. In this paper, we study the performance of the balanced model in tight frame based compressed sensing magnetic resonance imaging and propose a new efficient numerical algorithm to solve the optimization problem. By tuning the balancing parameter, the new model achieves solutions of three models. It is found that the balanced model has a comparable performance with the analysis model. Besides, both of them achieve better results than the synthesis model no matter what value the balancing parameter is. Experiment shows that our proposed numerical algorithm constrained split augmented Lagrangian shrinkage algorithm for balanced model (C-SALSA-B) converges faster than previously proposed algorithms accelerated proximal algorithm (APG) and alternating directional method of multipliers for balanced model (ADMM-B).