Balanced sparse model for tight frames in compressed sensing magnetic resonance imaging

Abstract

Compressed sensing has shown to be promising to accelerate magnetic resonance imaging. In this new technology, magnetic resonance images are usually reconstructed by enforcing 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).

Fig 1. Difference between the coefficients and the canonical coefficient of a signal.

Fig 2. The relation of analysis, synthesis, and balanced models.

Fig 3. Images used in simulations.

(a) is a T2- weighted brain image, (b) is a T1- weighted brain image, (c) is a water phantom image, (d) is a k-space undersampling pattern with 40% data are sampled.

Fig 4. Reconstructed T2 weighted brain images using analysis, balanced and synthesis models.

(a) the fully sampled image; (b)-(d) are reconstructed images using analysis, balanced and synthesis models, respectively; (e)-(g) are 6 times scaled reconstruction errors for images in (b)-(d), respectively. The RLNEs for (b)-(d) are 0.114, 0.122 and 0.128.

Fig 5. Empirical convergence of C-SALSA-B solving Equation (12).

Left is the objective function, right is the value of the constrained term.

Fig 6. Reconstruction error RLNEs in the iterations using different algorithms.

Fig 7. Impact of the balancing parameter γ on reconstructed errors for datasets in Fig. 3.

Fig 8. Comparisons of three models for different percentages of acquired k-space data.

Fig 9. Comparisons on PBDW-based reconstructed images for three models.

(a) the fully sampled image; (b)-(d) are reconstructed images using analysis, balanced and synthesis models, respectively; (e)-(g) are 6 times scaled reconstruction errors for images in (b)-(d), respectively. The RLNEs for (b)-(d) are 0.085, 0.086 and 0.114.

Fig 10. Impact of the balancing parameter γ on reconstructed errors when PBDW, contourlets and TIDCT are used as tight frames.

Fig 11. Comparisons of C-SALSA-B to APG and ADMM-B for more MR images.

Fig 12. Comparison of FCSA and C-SALSA-B.

Fig 13. Impact of the balancing parameter γ on reconstructed errors when orthogonal wavelets is used.