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).

Projected Iterative Soft-Thresholding Algorithm for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

“…To handle the constraint in the synthesis-like analysis model in (14), we introduce an indicator function…”

“…As shown in last section, both pISTA and pFISTA converge to an approximate model (27) instead of the exact analysis model (9) or (14). The model (27) is not new in general image restoration and it was called the balanced sparse model that balances solutions between synthesis and analysis sparse models [33][34][35].…”

“…Compared to orthogonal systems, redundant systems, such as tight frames and dictionaries, can benefit from that redundancy in noise removal and artifacts reduction in signal processing [10][11][12]. Within the field of CS-MRI, the quality of reconstructed images is improved with redundant systems [6,[13][14][15][16]. For these redundant systems, there exists significant difference between these two models as reported in [8,14,17,18].…”

“…Within the field of CS-MRI, the quality of reconstructed images is improved with redundant systems [6,[13][14][15][16]. For these redundant systems, there exists significant difference between these two models as reported in [8,14,17,18]. Theoretically, CS theory [10] reveals that analysis models works under a wider range of dictionaries than the synthesis models, and it was proved in [19] that analysis models converge to some partial differential equation models with geometric interpretations.…”

“…Theoretically, CS theory [10] reveals that analysis models works under a wider range of dictionaries than the synthesis models, and it was proved in [19] that analysis models converge to some partial differential equation models with geometric interpretations. In practice, researchers in signal processing observed that analysis models reaches lower reconstructed error than synthesis models [8,14,17,18].…”