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Local sparsity enhanced compressed sensing magnetic resonance imaging in uniform discrete curvelet domain.

Yang B, Yuan M, Ma Y, Zhang J, Zhan K - BMC Med Imaging (2015)

Bottom Line: It updates representation coefficients of lowpass sub-band coefficients over dictionary, transform sub-bands coefficients and k-space measurements upon the ideas of constrained split augmented Lagrangian shrinkage algorithm.Experimental results on in vivo data show that the proposed method obtains high-quality reconstructed images.The proposed sparsity structure can fit and provide hierarchical sparsity for magnetic resonance images simultaneously, bridging the gap between predefined sparse representation methods and explicit dictionary.

View Article: PubMed Central - PubMed

Affiliation: School of Information Science & Engineering, Lanzhou University, Tianshui South Road No.222, Lanzhou, 730000, China. 18919849359@163.com.

ABSTRACT

Background: Compressed sensing(CS) has been well applied to speed up imaging by exploring image sparsity over predefined basis functions or learnt dictionary. Firstly, the sparse representation is generally obtained in a single transform domain by using wavelet-like methods, which cannot produce optimal sparsity considering sparsity, data adaptivity and computational complexity. Secondly, most state-of-the-art reconstruction models seldom consider composite regularization upon the various structural features of images and transform coefficients sub-bands. Therefore, these two points lead to high sampling rates for reconstructing high-quality images.

Methods: In this paper, an efficient composite sparsity structure is proposed. It learns adaptive dictionary from lowpass uniform discrete curvelet transform sub-band coefficients patches. Consistent with the sparsity structure, a novel composite regularization reconstruction model is developed to improve reconstruction results from highly undersampled k-space data. It is established via minimizing spatial image and lowpass sub-band coefficients total variation regularization, transform sub-bands coefficients l 1 sparse regularization and constraining k-space measurements fidelity. A new augmented Lagrangian method is then introduced to optimize the reconstruction model. It updates representation coefficients of lowpass sub-band coefficients over dictionary, transform sub-bands coefficients and k-space measurements upon the ideas of constrained split augmented Lagrangian shrinkage algorithm.

Results: Experimental results on in vivo data show that the proposed method obtains high-quality reconstructed images. The reconstructed images exhibit the least aliasing artifacts and reconstruction error among current CS MRI methods.

Conclusions: The proposed sparsity structure can fit and provide hierarchical sparsity for magnetic resonance images simultaneously, bridging the gap between predefined sparse representation methods and explicit dictionary. The new augmented Lagrangian method provides solutions fully complying to the composite regularization reconstruction model with fast convergence speed.

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Framework of local sparsity enhanced CS MRI reconstruction
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Fig1: Framework of local sparsity enhanced CS MRI reconstruction

Mentions: In this section, a composite regularization CS MRI method established on a novel composite sparsity structure is presented. In the sparsity structure, UDCT decomposes spatial image into one lowpass sub-band and several highpass sub-bands. Patch-based dictionary is learnt from the lowpass sub-band coefficients patches via Sparse K-SVD [25]. Then a novel composite regularization reconstruction model is thereby established and solved via VS and ADMM-2. The reconstruction model involves spatial image and transform coefficients regularization and k-space data fitting. The framework in Fig. 1 shows clearly the implementation process of the proposed method, in which the unknown MR image x is initialized with a zero-filling reconstructed image via direct inverse Fourier transform to k-space measurements, denoted as . UDCT decomposes both the real and imaginary parts of x0 into S levels, each level possessing 2κs directional sub-bands. The real and imaginary parts of complex-valued MR image are handled separately because UDCT can only perfectly deal with real-valued data. Take the real part of zero-filling reconstructed image for example, the lowpass UDCT sub-band is divided into maximum overlapped patches (for dividing method, refer to [35]) as training database for DL to enhance its sparsity. The obtained dictionary Dr (r=1(0) denotes result over real (imaginary) part) is the result of Sparse K-SVD to the training database. The sparse encodings set are referred to as the double sparse coefficients for all training lowpass UDCT sub-band patches over learnt Dr. For imaginary part, the same procedure is implemented. Let x0 be the initial intermediate image and (Dr)† the pseudo-inverse of Dr. The reconstruction step starts afterwards. All nonoverlapping vector form patches (n×1 sized) are arrayed to produce a matrix from lowpass UDCT sub-band of intermediate image. Results of (Dr)† multiplying with the above matrix are the representation coefficients (differing from double sparse coefficients) of lowpass UDCT sub-band coefficients over the dictionary. They are generally not sparse but easier to handle in our reconstruction approach. The composite regularization reconstruction formulation is solved by using VS and ADMM-2 based on C-SALSA thoughts in an iterative procedure (an updated intermediate image for once iteration), which involves modifying the representation coefficients, UDCT sub-bands coefficients and k-space measurements. The proposed method is named as local sparsity enhanced CS MRI(LSECSMRI). Formulations and implementations of the proposed sparsity structure and relevant reconstruction model are described in detail in the following content.Fig. 1


Local sparsity enhanced compressed sensing magnetic resonance imaging in uniform discrete curvelet domain.

Yang B, Yuan M, Ma Y, Zhang J, Zhan K - BMC Med Imaging (2015)

Framework of local sparsity enhanced CS MRI reconstruction
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4528851&req=5

Fig1: Framework of local sparsity enhanced CS MRI reconstruction
Mentions: In this section, a composite regularization CS MRI method established on a novel composite sparsity structure is presented. In the sparsity structure, UDCT decomposes spatial image into one lowpass sub-band and several highpass sub-bands. Patch-based dictionary is learnt from the lowpass sub-band coefficients patches via Sparse K-SVD [25]. Then a novel composite regularization reconstruction model is thereby established and solved via VS and ADMM-2. The reconstruction model involves spatial image and transform coefficients regularization and k-space data fitting. The framework in Fig. 1 shows clearly the implementation process of the proposed method, in which the unknown MR image x is initialized with a zero-filling reconstructed image via direct inverse Fourier transform to k-space measurements, denoted as . UDCT decomposes both the real and imaginary parts of x0 into S levels, each level possessing 2κs directional sub-bands. The real and imaginary parts of complex-valued MR image are handled separately because UDCT can only perfectly deal with real-valued data. Take the real part of zero-filling reconstructed image for example, the lowpass UDCT sub-band is divided into maximum overlapped patches (for dividing method, refer to [35]) as training database for DL to enhance its sparsity. The obtained dictionary Dr (r=1(0) denotes result over real (imaginary) part) is the result of Sparse K-SVD to the training database. The sparse encodings set are referred to as the double sparse coefficients for all training lowpass UDCT sub-band patches over learnt Dr. For imaginary part, the same procedure is implemented. Let x0 be the initial intermediate image and (Dr)† the pseudo-inverse of Dr. The reconstruction step starts afterwards. All nonoverlapping vector form patches (n×1 sized) are arrayed to produce a matrix from lowpass UDCT sub-band of intermediate image. Results of (Dr)† multiplying with the above matrix are the representation coefficients (differing from double sparse coefficients) of lowpass UDCT sub-band coefficients over the dictionary. They are generally not sparse but easier to handle in our reconstruction approach. The composite regularization reconstruction formulation is solved by using VS and ADMM-2 based on C-SALSA thoughts in an iterative procedure (an updated intermediate image for once iteration), which involves modifying the representation coefficients, UDCT sub-bands coefficients and k-space measurements. The proposed method is named as local sparsity enhanced CS MRI(LSECSMRI). Formulations and implementations of the proposed sparsity structure and relevant reconstruction model are described in detail in the following content.Fig. 1

Bottom Line: It updates representation coefficients of lowpass sub-band coefficients over dictionary, transform sub-bands coefficients and k-space measurements upon the ideas of constrained split augmented Lagrangian shrinkage algorithm.Experimental results on in vivo data show that the proposed method obtains high-quality reconstructed images.The proposed sparsity structure can fit and provide hierarchical sparsity for magnetic resonance images simultaneously, bridging the gap between predefined sparse representation methods and explicit dictionary.

View Article: PubMed Central - PubMed

Affiliation: School of Information Science & Engineering, Lanzhou University, Tianshui South Road No.222, Lanzhou, 730000, China. 18919849359@163.com.

ABSTRACT

Background: Compressed sensing(CS) has been well applied to speed up imaging by exploring image sparsity over predefined basis functions or learnt dictionary. Firstly, the sparse representation is generally obtained in a single transform domain by using wavelet-like methods, which cannot produce optimal sparsity considering sparsity, data adaptivity and computational complexity. Secondly, most state-of-the-art reconstruction models seldom consider composite regularization upon the various structural features of images and transform coefficients sub-bands. Therefore, these two points lead to high sampling rates for reconstructing high-quality images.

Methods: In this paper, an efficient composite sparsity structure is proposed. It learns adaptive dictionary from lowpass uniform discrete curvelet transform sub-band coefficients patches. Consistent with the sparsity structure, a novel composite regularization reconstruction model is developed to improve reconstruction results from highly undersampled k-space data. It is established via minimizing spatial image and lowpass sub-band coefficients total variation regularization, transform sub-bands coefficients l 1 sparse regularization and constraining k-space measurements fidelity. A new augmented Lagrangian method is then introduced to optimize the reconstruction model. It updates representation coefficients of lowpass sub-band coefficients over dictionary, transform sub-bands coefficients and k-space measurements upon the ideas of constrained split augmented Lagrangian shrinkage algorithm.

Results: Experimental results on in vivo data show that the proposed method obtains high-quality reconstructed images. The reconstructed images exhibit the least aliasing artifacts and reconstruction error among current CS MRI methods.

Conclusions: The proposed sparsity structure can fit and provide hierarchical sparsity for magnetic resonance images simultaneously, bridging the gap between predefined sparse representation methods and explicit dictionary. The new augmented Lagrangian method provides solutions fully complying to the composite regularization reconstruction model with fast convergence speed.

Show MeSH