Background
Compressed sensing(CS) was first presented in the literature of Information Theory as an abstract mathematical idea [
1,
2]. The fundamental idea behind CS is: rather than first sampling at a high rate and then compressing the sampled data, directly sensing the data in a compressed form (at a lower sampling rate) is preferred. CS points out that a signal can be recovered exactly from a small set of random, linear and nonadaptive measurements if it has a sparse representation. Suppose
\(\textbf {x}\in \mathbb {C}^{p}\) denotes the unknown compressible (
N-sparse) signal to be reconstructed.
Ψ denotes a tight frame sparse transform matrix. Then
x can be sparsely represented as
α=
Ψ x, where ∥
α∥
0=
N(
N≪
p). It is possible to measure a relatively small number of “random" linear combinations of signal (much smaller than the number of signal samples nominally defining it) allowing accurate reconstruction, which is comparable to that attainable with direct knowledge of the
N most important coefficients. The measurement process is denoted as
y=
Φ x, where
\(\boldsymbol {\Phi }\in \mathbb {C}^{m\times p} \left (m\ll p\right)\) denotes measurement matrix irrelevant to the sparse transform basis. Thus
$$ \textbf{y}=\boldsymbol{\Phi}\textbf{x}=\boldsymbol{\Phi}\boldsymbol{\Psi}^{-1}\boldsymbol{\alpha} $$
(1)
in which
Φ Ψ −1 is termed as the sensing matrix. The sensing matrix should satisfy three properties including the null space property, restricted isometry property and bounded coherence [
3]. Given measurements
y and the sensing matrix, the reconstruction problem turns out to be an optimization problem of the form
$$ \arg\min_{\textbf{x}, \hat{\boldsymbol{\alpha}}}\left\|\hat{\boldsymbol{\alpha}}\right\|_{0} \ \ \text{s.t.} \ \boldsymbol{\Phi}\boldsymbol{\Psi}^{-1}\hat{\boldsymbol{\alpha}}=\textbf{y} $$
(2)
(
2) can be solved by various nonlinear reconstruction approaches.
In magnetic resonance imaging(MRI), the sampled combinations are simply individual Fourier coefficients (
k-space samples). MRI is a relatively slow imaging modality at a limited data acquisition speed. Undersampling
k-space allows speeding up imaging but introduces aliasing in the reconstructed magnetic resonance(MR) images simultaneously, because it violates the Nyquist sampling theorem. Compared with that by the sinc function interpolation using sampled data restricted by Nyquist sampling theorem, CS enables MR image reconstruction with little or no visual information loss from randomly undersampled
k-space measurements. Hence, it is natural to introduce CS into undersampled MRI. The emerging method to reduce MRI scanning time via CS is termed CS MRI [
4,
5]. Three requirements for successful CS MRI are: the MR image can be sparsely represented (compressible); the aliasing artifacts brought by
k-space undersampling are incoherent (noise like) in the transform domain; then CS solves the general reconstruction formulation using nonlinear method by constraining both sparsity and
k-space measurements consistency. In CS MRI, incoherent random, radial and spiral trajectories [
4,
6,
7], etc, are used to acquire measurements from
k-space.
Sparsity is of vital importance for reducing artifacts in CS MRI reconstruction. The generally used sparse representation methods are spatial finite difference [
4,
8,
9], discrete wavelet transform(DWT) [
4,
8,
9], sharp frequency localization contourlet(SFLCT) [
10,
11], discrete curvelet transform using fast algorithm(FDCT) [
12‐
14], discrete shearlet transform(DST) [
15,
16], sparsity along temporal axis for dynamic cardiac imaging [
17‐
19] and the combination of some of these transforms [
4,
20,
21]. Dictionary has also been introduced for sparse representation and adaptive data fitting [
22,
23] and it is learnt from intermediate reconstructed or fully sampled images. Furthermore, double sparsity model has been proposed likewise. It combines adaptive dictionary learning(DL) with predefined sparse priors for signal flexible representations, stability under noise and reducing overfitting [
24,
25]. Besides, nonlocal processing [
26] methods have been explored as well based on the similarity of image patches [
27‐
30] and sparsity originated from this similarity [
31,
32]. Established on the above sparse representation approaches, various CS MRI methods have been presented for handling the ill-posed linear inverse problem, including convex, nonsmooth sparse regularization (
l 1 and total variation) based LDP [
4], TVCMRI [
33], iterative thresholding CS MRI based on SFLCT [
11], FCSA [
21], NLTV-MRI upon nonlocal total variation(TV) [
9], reconstruction under wavelet structured sparsity(WaTMRI) studied in [
34], reconstruction via using DL [
35‐
37] and PANO [
32] by using variable splitting(VS) and quadratic penalty reconstruction technique [
38] incorporated with patch-based nonlocal operator, etc. Besides, a novel individual MRI reconstruction framework of low-rank modeling for local
k-space neighborhoods(LORAKS) [
39] was also proposed. LORAKS generated support and phase constraints in a fundamentally different way from more direct regularized methods [
4,
40]. Among these reconstruction methods, DWT based MRI reconstruction gave rise to feature loss and edge blur with numerous aliasing in reconstructed images. WaTMRI provided new thought for CS MRI by making full use of the coefficients structural relationship. PANO was recently proposed to sparsify MR images by using the similarity of image patches, achieved considerably lower reconstruction error and allowed us to establish a general formulation to constrain the sparsity of similar patches and data consistency. The availability of guide image and how the gridding process affect PANO imaging with nonCartesian sampling remain to be carefully analyzed. Besides, LORAKS provided very flexible implementation and was easily incorporated with other constraints. Furthermore, 3D dynamic parallel imaging has also been presented and was of great significance for practical MRI applications. 3D dynamic parallel imaging was generally established on TV and sparsity along temporal axis [
17‐
19] and structured low-rank matrix completion [
41‐
45].
In this paper, a novel composite sparsity structure is developed, which is inspired by double sparsity model. In this composite sparsity structure, uniform discrete curvelet transform(UDCT) [
46] decomposes MR images hierarchically into one lowpass sub-band and several other highpass sub-bands. Then an adaptive dictionary is learnt from the hardly sparse lowpass sub-band coefficients patches. This adaptive DL allows a smaller amount of calculation with little (or no) decrease of efficiency compared with the double sparsity model. UDCT has quite similar properties to those of wrapping-based FDCT, such as
C(logN)
3 N −2 mean square error(MSE) decay rate for
C 2-singularities signal with
N most important transform coefficients in the curvelet expansion, tight frame property, highly directional sensitivity and anisotropy in the sense that they both employ alias free subsampling in frequency domain. Additionally, UDCT possesses some specialities making it superior over FDCT in CS MRI applications, such as a smaller redundancy of 4 and clear coefficients parent-children relationship. The goal of the proposed composite sparsity structure is to capture various directional features of images hierarchically, provide more flexible and sparse representations for lowpass sub-band adaptively, and reduce overfitting and computational complexity simultaneously. Consistent with this composite sparsity structure, one reconstruction model is provided. It involves minimizing UDCT sub-bands coefficients
l 1 regularization, image and lowpass sub-band coefficients TV penalty and constraining
k-space measurements fidelity. Then a new fast convergent augmented Lagrangian(AL) reconstruction method is presented to solve the reconstruction model. It is established on the constrained split augmented Lagrangian shrinkage algorithm(C-SALSA) [
47], translates the original formulation into another constrained one via VS and then solves the constrained one by using the variant of ADMM [
48,
49](ADMM-2 [
47]). The ADMM-2 resulting from our reconstruction problem involves quadratic problem (which can be solved exactly in closed form),
l 1 regularization, a shrinkage operation and an orthogonal projection on a ball.
The remainder of this paper is organized as follows. Section “
Methods” depicts the prior work, the proposed CS MRI method including the composite sparsity structure and corresponding reconstruction model, and its validity in detail. In section “
Results and discussions” some reconstruction results are exhibited for the proposed method and current CS MRI techniques. The ability in handling noise, convergence speed and influences of the proposed reconstruction model parameters fluctuation are analyzed, etc. Finally, conclusions and future work are explicit in section
Conclusions and future work.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BXY and MY were the principal investigators, participated in the framework design, carried out experiments and drafted the manuscript. YDM participated in the design of the framework, helped drafting and revising the manuscript. JWZ and KZ helped analyzing experimental results and revising the manuscript. All authors read and approved the final manuscript.