2. 卡尔加里大学电子与计算机工程学院 卡尔加里
2. School of Electrical and Computer Engineering, University of Calgary, Calgary AB T2N 1N4, Canada
Magnetic resonance imaging (MRI) is a crucial medical diagnostic technique which offers clinicians with significant anatomical structure for lack of ionizing. Unfortunately, although it enables highly resolution images and distinguishes depiction of soft tissues, the imaging speed is limited by physical and physiological constraints and increasing scan duration may bring in some physiological motion artifacts [1]. Therefore, it is necessary to seek for a method to decrease the acquisition time. Reducing the number of measurements mandated by Nyquist sampling theory is a way to accelerate the data acquisition at the expense of introducing aliasing artifacts in the reconstructed results. In recent years, compressed sensing (CS) theory, as a promising method, has proposed an essential theoretical foundation for improving data acquisition speed. Particularly, the application of CS to MRI is known as CSMRI [2][6].
The CS theory states that the image which has a sparse representation in certain domain can be recovered from a reduced set of measurements largely below Nyquist sampling rates [2]. The traditional CSMRI usually utilizes predefined dictionaries [1],
[7][9], which may fail to sparsely represent the reconstructed images. For instance, Lustig et al. [1] employed the total variation (TV) penalty and the Daubechies wavelet transform for MRI reconstruction. Trzasko et al. [6] proposed a homotopic minimization strategy to reconstruct the MR image. Instead, adaptive dictionary updating in CSMRI can provide less reconstruction errors due to the dictionary learned from sampled data [10], [11]. Recently, Ravishankar et al. supposed that each image patch has sparse representation, and presented a prominent twostep alternating method named dictionary learning based MRI reconstruction (DLMRI) [12]. The first step is for adaptively learning the dictionary, and the second step is for reconstructing image from undersampled
All the above methods use conventional patchbased sparse representation to reconstruct MR image, it has a fundamental disadvantage that the important spatial structures of the image of interest may be lost due to the subdivision into patches that are independent of each other. To make up for deficiencies of conventional patchbased sparse representation method, Zeiler et al. [16] proposed a convolutional implementation of sparse coding method (CSC). In the convolutional decomposition procedure, the decomposition does not need to divide the entire image into overlapped patches, and can naturally utilize the consistency prior. CSC was first introduced in the context of modeling receptive fields in human vision [17]. Recently, it has been demonstrated that CSC has important applications in a wide range of computer vision problems, like low/midlevel feature learning, lowlevel reconstruction [18], [19], networks in highlevel computer vision or hierarchical structures challenges [16], [20], [21], and in physicallymotivated computational imaging problems [22], [23]. In addition, CSC can find applications in many other reconstruction tasks and featurebased methods, including denoising, inpainting, superresolution and classification [24][30].
In this paper, we propose a new formulation of convolutional sparse coding tailored to the problem of MRI reconstruction. Moreover, due to the image gradients are a sparser representation than the image itself and therefore may have sparser representation with the CSC than the pixeldomain image, we learn CSC in gradient domain for better quality and efficient reconstruction. The present method has two benefits. First, we introduce CSC for MRI reconstruction. Second, since the image gradients are usually sparser representation than the image itself, it is demonstrated that the CSC in gradient domain could lead to sparser representation than those using the conventional sparse representation methods in the pixel domain.
The remainder of this paper is organized as follows. Section 2 states the prior work in CS and CSC. The proposed algorithm CSC in gradient domain (GradCSC) that employing the augmented Lagrangian (AL) iterative method is detailed in Section 3. Section 4 demonstrates the performance of the proposed algorithm on examples under a variety of sampling schemes and undersampling factors. Conclusions are given in Section 5.
2 BackgroundIn this section, we first review several classical models for CSMRI, and then introduce the theory of CSC. The following notational conventions are used throughout the paper. Let
The choice of sparsifying transform is an important question in CS theory. In the past several years, reconstructing unknown image from undersampled measurements was usually formulated as in (1) where assuming the image gradients are sparse
$ \min\limits_{u}\, \left\{ {{\mu }_{1}}{{\left\ u \right\}_{TV}}+\frac{1}{2}\left\ {{F}_{p}}uf \right\_{2}^{2} \right\} $  (1) 
where
Sparse and redundant representations of image patches based on learned basis has been drawing considerable attention in recent years. Specifically, Ravishankar et al. [12] presented a method named DLMRI to reconstruct MR image from highly undersampled
$ \min\limits_{u, D, \Gamma }\, \left\{ \sum\limits_{l}{\left\ D{{\alpha }_{l}}{{R}_{l}}u \right\_{2}^{2}+\nu {{\left\ {{F}_{p}}uf \right\}^{2}}} \right\}\rm{ }\\ {\rm s.t.}\quad {{\left\ {{\alpha }_{l}} \right\}_{0}}\le {{T}_{0}}~~~ \forall \, l $  (2) 
where
Although conventional patchbased sparse representation has widely applications, it has some drawbacks. First, it typically assumes that training image patches are independent from each other, hence the consistency of pixels and important spatial structures of the signal of interest may be lost. This assumption typically results in filters are simply translated versions of each other, and generates highly redundant feature representation. Second, due to the nature of the mathematical formulation that a linear combination of learned patches, these traditional patchbased representation approaches may fail to adequately represent highfrequency and highcontrast image features, thus loses some details and textures of the signal, which is important for MR images.
2.2 Convolutional Sparse CodingZeiler et al. [16] proposed an alternative to patchbased approaches named CSC, decomposing the image into spatiallyinvariant convolutional features. CSC is the sum of a set of convolutions of the feature maps by replacing the linear combination of a set of dictionary vectors. Let
$ \underset{d, z}{\mathop{\rm min}}\, \sum\limits_{i=1}^{2}{\frac{1}{2}\left\ X\sum\limits_{k=1}^{K}{{{d}_{k}}*{{z}_{k}}} \right\}_{2}^{2}+\beta \sum\limits_{k=1}^{K}{{{\left\ {{z}_{k}} \right\}_{1}}} \nonumber\\ {\rm s.t.}\quad\left\ {{d}_{k}} \right\_{2}^{2}\le 1~~~ \forall\, k\in \left\{ 1, \ldots, K \right\} $  (3) 
where the first and the second terms represent the reconstruction error and the
However, the CSC has led to some difficulties in optimization, Zeile et al. [16] used the continuation method to relax the equality constraints, and employed the conjugate gradient (CG) decent to solve the convolutional least square approximation problem. By considering the property of block circulant with circulant block (BCCB) matrix in the Fourier domain, Bristow et al. [32] presented a fast CSC method. Recently, Wohlberg [33] presented an efficient alternating direction method of multipliers (ADMM) to further improve this method.
3 Convolutional Sparse Coding in Gradient DomainThe image gradients are sparser than the image itself [13], therefore it has sparser representation in the CSC than that in the pixeldomain image. This motivates us to consider the CSC in the gradient domain. It is expected that such learning is more accurate and robust than that from pixel domain. In this work, we propose an algorithm to reconstruct the image by iteratively reconstructing the gradients via CSC and solving for the final image.
3.1 Proposed ModelTo reconstruct image from the image gradients, we propose a new model as follows:
$ \begin{align} & \underset{u, d, z}{\mathop{\min }}\, \left\{ \sum\limits_{i=1}^{2}{\frac{1}{2} \left\ {{\nabla }^{(i)}}u\sum\limits_{k=1}^{K}{{{d}_{k}}*{{z}_{k}}} \right\}_{2}^{2}\right. \nonumber\\ &\qquad \left.+~\beta \sum\limits_{k=1}^{K}{{{\left\ {{z}_{k}} \right\}_{1}}} +\frac{{{\nu }_{1}}}{2}{{\left\ {{F}_{p}}uf \right\}^{2}}\right\} \nonumber\\ & \, {\rm s.t.}\quad \left\ {{d}_{k}} \right\_{2}^{2}\le 1~~~ \forall\, k\in \left\{ 1, \ldots, K \right\} \end{align} $  (4) 
where
In order to better understand the benefit of the CSC in the gradient domain, one demonstration of visual inspection between traditional sparse coded dictionaries and GradCSC filter is shown in Fig. 1. The learned dictionaries by DLMRI and GradDL are depicted in Figs. 1(a) and (b), both of which are learned from the Lumbar spine image in Fig. 2. The learned filters by GradCSC shown in Fig. 1(c) are learned from the dataset in [16]. Compared to the traditional sparse coded dictionaries in Figs. 1(a) and (b), it can be seen from Fig. 1(c) that the convolutional filter in GradCSC shows less redundancy, crisper features, and a larger range of feature orientations.
In the regularization term of (4), the global finite difference operators
Algorithm 1. The GradCSC algorithm 
1: Initialization: 2: For 3: 4: Updating 5: Updating 6: 7: 8: End 9: Output 
Equation (4) can be rewritten as follows by introducing auxiliary variables
$ \begin{align} & \nonumber\underset{u, {{w}^{(i)}}, d, z}{\mathop{\min }}\, \left\{ \sum\limits_{i=1}^{2}{\frac{1}{2}\left\ {{w}^{(i)}}\sum\limits_{k=1}^{K}{{{d}_{k}}*{{z}_{k}}} \right\_{2}^{2}} \right. \\ &\nonumber \qquad \left. +~\beta \sum\limits_{k=1}^{K}{{{\left\ {{z}_{k}} \right\}_{1}}} +\frac{{{\nu }_{1}}}{2}{{\left\ {{F}_{p}}uf \right\}^{2}}+\sum\limits_{k=1}^{K}{in{{d}_{C}}({{d}_{k}})} \right\} \\ & \, {\rm s.t.}\quad {{w}^{(i)}}={{\nabla }^{(i)}} u~~~\forall \, i \end{align} $  (5) 
then, by employing the AL technique and denoting
$ \begin{align} & \left\{{{u}^{j+1}}, {{w}^{j+1}}, {{d}^{j+1}}, {{z}^{j+1}}\right\} \nonumber\\ & \qquad={\arg} \underset{u, {{w}^{(i)}}, d, z}{\mathop{\rm min}}\, \sum\limits_{i=1}^{2}{\frac{1}{2}\left\ {{w}^{(i)}}\sum\limits_{k=1}^{K}{{{d}_{k}}* {{z}_{k}}} \right\_{2}^{2}}\nonumber\\ &\qquad\quad+\frac{{{\nu }_{1}}}{2}\left\ {{F}_{p}}uf \right\_{2}^{2} +\beta \sum\limits_{k=1}^{K}{{{\left\ {{z}_{k}} \right\}_{p}}}+ \sum\limits_{k=1}^{K}{in{{d}_{C}}({{d}_{k}})}\nonumber\\ &\qquad\quad+\frac{{{\nu }_{2}}}{2}\left\ {{\left({{b}^{(i)}}\right)}^{j}}+\nabla u{{w}^{(i)}} \right\_{2}^{2} \end{align} $  (6) 
$ {{\left({{b}^{(i)}}\right)}^{j+1}}={{\left({{b}^{(i)}}\right)}^{j}}+\nabla {{u}^{j+1}}{{\left({{w}^{(i)}}\right)}^{j+1}} $  (7) 
where
1) Updating the Solution
$ \begin{align} \nonumber{{u}^{j+1}}=\, & \arg \underset{u}{\mathop{\min }}\, \bigg\{ {{\nu }_{1}}\left\ {{F}_{p}}uf \right\_{2}^{2} \\ & +~{{\nu }_{2}}\left\ {{({{b}^{(i)}})}^{j}}+\nabla u{{({{w}^{(i)}})}^{j}} \right\_{2}^{2} \bigg\}. \end{align} $  (8) 
Recognizing that (8) is a simple least squares problem admitting an analytical solution. The least squares solution satisfies the normal equation
$ \left( {{\nu }_{1}}F_{p}^{T}{{F}_{p}}+{{\nu }_{2}}{{\nabla }^{T}}\nabla \right){{u}^{j+1}}={{\nu }_{1}}F_{p}^{T}f+{{\nu }_{2}}{{\nabla }^{T}}({{w}^{j}}{{b}^{j}}). $  (9) 
However, directly solving the equation can be tedious due to (9) has a high computation complexity (
$ {{u}^{j+1}}={{F}^{1}}\left( \frac{F\left[{{\nu }_{1}}F_{p}^{T}f+{{\nu }_{2}}{{\nabla }^{T}}({{w}^{j}}{{b}^{j}})\right]}{{{\nu }_{1}}FF_{p}^{T}{{F}_{p}}{{F}^{T}}+{{\nu }_{2}}F{{\nabla }^{T}}{{F}^{T}}F\nabla {{F}^{T}}} \right) $  (10) 
similarly as described in DLMRI and GradDL method [12], the matrix
2) Updating the Gradient Image Variables
$ \begin{align} \nonumber{{({{w}^{(i)}})}^{j+1}}=&\ \arg ~\underset{{{w}^{(i)}}}{\mathop{\min }}\, \Bigg\{ \frac{1}{2}\left\ \sum\limits_{k=1}^{K}{d_{_{k}}^{j}*z_{_{k}}^{j}}{{w}^{(i)}} \right\_{2}^{2} \\ & \ +\frac{{{\nu }_{2}}}{2}\left\ {{\left({{b}^{(i)}}\right)}^{j}}+{{ \left({{\nabla }^{(i)}}u\right)}^{j+1}}{{w}^{(i)}} \right\_{2}^{2} \Bigg\}. \end{align} $  (11) 
The least squares solution satisfies the normal equation, and the solution of (11) is as follow:
$ {{\left({{w}^{(i)}}\right)}^{j+1}}=\frac{{{\nu }_{2}}\left[{{\left({{b}^{(i)}}\right)}^{j}}+{{\left({{\nabla }^{(i)}}u\right)}^{j+1}}\right]+\sum\limits_{k=1}^{K}{d_{_{k}}^{j}*z_{_{k}}^{j}}}{{{\nu }_{2}}+1}. $  (12) 
3) Updating the Coefficients
$ \begin{align} &\underset{{{d}^{j+1}}, {{z}^{j+1}}}{\mathop{\min }}\, \frac{1}{2}\left\ {{w}^{(i)}}\sum\limits_{k=1}^{K}{{{d}_{k}}*{{z}_{k}}} \right\_{2}^{2}\nonumber\\ &\qquad +\beta \sum\limits_{k=1}^{K}{{{\left\ {{z}_{k}} \right\}_{1}}}+\sum\limits_{k=1}^{K}{in{{d}_{C}}({{d}_{k}})}. \end{align} $  (13) 
The problem in (13) can be solved by employing an AL algorithm like mentioned above, (13) needs to introduce auxiliary variables
$ \begin{align} & \nonumber\left\{ d_{k}^{j+1, \ell +1}, z_{k}^{j+1, \ell +1}, r_{1}^ {^{\ell +1}}, r_{2}^{^{\ell +1}}, r_{3}^{^{\ell +1}} \right\} \\ \nonumber&\qquad ={\rm arg }\ \underset{{{d}^{j+1}}, {{z}^{j+1}}, {{r}_{1}}, {{r}_{2}}, {{r}_{3}}}{\mathop{\min }}\, \Bigg\{ \frac{1}{2}\left\ {{\left({{w}^{(i)}}\right)}^ {j+1}}{{r}_{1}} \right\_{2}^{2} \\ \nonumber&\qquad\quad +\sum\limits_{k=1}^{ K}{in{{d}_{C}}({{r}_{2, k}})}\ +\beta \sum\limits_{k=1}^{K}{ {{\left\ {{r}_{3, k}} \right\}_{1}}}\\ \nonumber&\qquad\quad +\frac{{{\mu }_{1}}}{2}\left\ {{r}_{1}}\sum\limits_{k=1}^{K}{d_{_{k}}^{j+1}*z_{_{k}}^{j+1}}+\lambda _{_{1}}^{j} \right\_{2}^{2}\ \\ &\qquad\quad +\frac{{{\mu }_{2}}}{2}\left\ {{r}_{2}}d_{_{k}}^{j+1}+ \lambda _{_{2}}^{j} \right\_{2}^{2}+\frac{{{\mu }_{3}}}{2}\left\ {{r}_{3}}z_{_{k}}^{j+1}+\lambda _{3}^{\ell } \right\_{2}^{2} \Bigg\} \end{align} $  (14) 
at the
$ \begin{align} &\nonumber \lambda _{_{1}}^{\ell +1}=\lambda _{_{1}}^{\ell }+r_{_{1}}^{\ell +1}\sum\limits_{k=1}^{K}{d_{_{k}}^{j+1, \ell +1}*z_{_{k}}^{j+1, \ell +1}} \\ \nonumber& \lambda _{2}^{\ell +1}=\lambda _{_{2}}^{\ell }+r_{_{2}}^{\ell +1}d_{_{k}}^{j+1, \ell +1} \\ & \lambda _{3}^{\ell +1}=\lambda _{3}^{\ell }+r_{3}^{\ell +1}d_{_{k}}^{j+1, \ell +1}. \end{align} $  (15) 
ADMM is chosen to solve the (14). The corresponding five subproblems can be solved as follows:
$ \begin{align} \nonumber d_{k}^{j+1, \ell +1}=&\ {{\left( {{\left(z_{k}^{j+1, \ell }\right)}^{T}}z_{k}^{j+1, \ell }+\frac{{{\mu }_{2}}}{{{\mu }_{1}}}I \right)}^{1}} \\ \nonumber &\ \times\left( {{\left(z_{k}^{j+1, \ell }\right)}^{T}}\left( r_{_{1}}^{\ell }+\lambda _{_{1}}^{\ell } \right)+\frac{{{\mu }_{2}}}{{{\mu }_{1}}}\left( r_{2}^{\ell }+\lambda _{_{2}}^{\ell } \right) \right) \\ \end{align} $  (16) 
$ \begin{align} \nonumber z_{k}^{j+1, \ell +1}=&\ {{\left( {{\left(d_{k}^{j+1, \ell +1}\right)}^{T}}d_{k}^{j+1, \ell +1}+\frac{{{\mu }_{4}}}{{{\mu }_{3}}}I \right)}^{1}} \\ &\ \times\left( {{\left(d_{k}^{j+1, \ell +1}\right)}^{T}}\left(r_{_{3}}^{\ell }+\lambda _{_{3}}^{\ell }\right) +\frac{{{\mu }_{4}}}{{{\mu }_{3}}}\left(r_{_{4}}^{\ell }+\lambda _{_{4}}^{\ell }\right) \right) \end{align} $  (17) 
$ \begin{align} r_{1}^{^{\ell +1}}=&\ \left({{\mu }_{1}}+1\right){{I}^{1}}\nonumber\\ &\ \times\left[\left({{\mu }_{1}}\sum\limits_{k=1}^{K}{z_{_{k}}^{j+1, \ell }*d_{_{k}}^{j+1, \ell +1}}\lambda _{_{1}}^{\ell }\right)+{{\left({{w}^{(i)}}\right)}^{j+1}}\right]\\ \end{align} $  (18) 
$ \begin{align} r_{2, k}^{^{l+1}}=&\ \begin{cases} \frac{d_{_{k}}^{j+1, \ell +1}\lambda _{_{2}}^{\ell }}{{{\left\ d_{_{k}}^{j+1}\lambda _{_{2}}^{j} \right\}_{2}}}, &d_{_{k}}^{j+1, \ell +1}\lambda _{_{2}}^{\ell }\ge 1\\[4mm] d_{_{k}}^{j+1, \ell +1}\lambda _{_{2}}^{\ell }, & {\rm else} \end{cases} \end{align} $  (19) 
where
$ \begin{align} r_{3}^{^{\ell +1}}=&\ \max \left\{ 1\frac{\beta }{{{\mu }_{3}}{{\left\ z_{_{k}}^{j+1, \ell +1}\lambda _{3}^{\ell } \right\}_{2}}}, 0 \right\}\nonumber\\&\ \circ \left( z_{_{k}}^{j+1, \ell +1}\lambda _{3}^{\ell } \right) \end{align} $  (20) 
where
In this section, we evaluate the performance of proposed method at a variety of sampling schemes and undersampling factors. The sampling schemes employed in the experiments contain the 2D random sampling, pseudo radial sampling, and Cartesian sampling with random phase encoding (1D random). The size of images we use in the synthetic experiments are
In this subsection, we evaluate the performance of GradCSC under different undersampling factors with same sampling scheme. Fig. 2 illustrates the reconstruction results at a range of undersampling factors with 2.5, 4, 6, 8, 10 and 20. We added the zeromean complex white Gaussian noise with
In this subsection, we evaluate the performance of GradCSC at various sampling schemes. The results are presented in Fig. 3 which reconstructed an axial T2weighted brain image at 8fold undersampling factors by applying three different sampling schemes, including 2D random sampling, 1D Cartesian sampling, and pseudo radial sampling. The PSNR and HFEN indexes versus iterative number for method DLMRI, GradDL and GradCSC are plotted in Figs. 3(b) and (c). Particularly, we present the reconstructions of DLMRI, GradDL and GradCSC with 2D random sampling in Figs. 3(d), (e), and (f), respectively. In order to facilitate the observation, the difference image between reconstruction results are shown in Figs. 3(g), (h), and (i). As can be expected, the convolution operator enables CSC outperform DLMRI and GradDL methods for most of specified undersampling ratios and trajectories. This exhibits the benefits of employing the convolutional filter for sparse coding. In particular, in Fig. 3(d), (e), and (f) the skeletons in the top half part of the GradCSC reconstruction appear less obscured than those in the DLMRI and GradDL results. the proposed method GradCSC reconstructs the images more accurately with larger PSNR and lower HFEN values than the GradDL approach. Particularly when sampled at 2D random trajectory, our method GradCSC outperforms others with a remarkable improvement from 0.7 dB to 5.8 dB.
We also conduct experiments to investigate the sensitivity of GradCSC to different levels of complex white Gaussian noise. DLMRI, GradDL and GradCSC are applied to reconstruct a noncontrast MRA of the circle of Willis under pseudo radial sampling at 6.67fold acceleration. Fig. 4 presents the reconstruction results of three methods at different levels of complex white Gaussian noise, which are added to the
In this work, a novel CSC method in gradient domain for MR image reconstruction was proposed. The important spatial structures of the signal were preserved by convolutional sparse coding. For the new derived model, we utilized the AL method to implement the algorithm in a few number of iterations. A variety of experimental results demonstrated the superior performance of the method under various sampling trajectories and
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