Elsevier

Signal Processing

Volume 170, May 2020, 107444
Signal Processing

Compressed sensing MRI based on the hybrid regularization by denoising and the epigraph projection

https://doi.org/10.1016/j.sigpro.2019.107444Get rights and content

Highlights

  • This paper proposes a hybrid regularization by denoising (HRED) constraint.

  • HRED constraint exploits the weighting of BM3D denoiser and FFDNet denoiser.

  • HRED constraint is employed to CSMRI model to construct HRED-MRI algorithm.

  • HRED-MRI is tackled via the alternating projection implemented by the epigraph method.

  • HRED-MRI only has one parameter and can achieve more excellent image reconstruction performance.

Abstract

Signal models play a paramount role in compressed sensing magnetic resonance imaging (CSMRI), which aims to accurately recover magnetic resonance (MR) images from highly undersampled measurements. In recent decade, lots of works exploit the sparsity and the low rank for CSMRI. However, there are some defects involving many finely-tuned parameters and pending to further improve the image reconstruction quality. In order to address the above issues, this paper proposes a hybrid regularization by denoising (HRED) constraint, in which we employ the weighting of two types of denoisers such as the BM3D and a fast flexible denoising convolutional neural network (FFDNet). Essentially, the HRED constraint exploits the complementarity of multifarious priors, including the non-local similarity and sparsity induced by BM3D, and the learning deep priors induced by FFDNet. We plug the proposed HRED constraint into CSMRI framework to construct a CSMRI algorithm dubbed HRED-MRI. Concretely, we leverage the HRED constraint to formulate a CSMRI problem, and then tackle the formulated CSMRI problem via the alternating projection implemented by the epigraph method. The epigraph method is equivalent to the gradient descent method whose step sizes are selected in an adaptive manner. Thereinto, epigraph sets of the HRED constraint and the data fidelity term are defined, and the two epigraph sets are all convex. The HRED-MRI relied on the HRED constraint and epigraph method, only has one parameter which needs to be tuned. Compared with the state-of-the-art CSMRI approaches, experiments validate that the HRED-MRI can achieve more excellent image reconstruction performance and better robustness to noise.

Introduction

Magnetic resonance imaging (MRI) has been succeeded in applying to the science research and medical practical diagnosis both at home and abroad, owing to its excellent properties [1,2]. Normally, the visually legible MR images are required to offer applications in the imaging experiment and clinical disease state inspection. Hence, how to yield accurate MR images from K-space measurements is a key issue to be addressed. The compressed sensing (CS) method proposed by Donoho [3] and Candes et al. [4] has been employed to recover MR images from highly undersampled measurements [5], [6], [7], [8], due to the sparsity of MR images in certain transform domain. Recent years, a multitude of researchers have presented numerous outstanding CSMRI algorithms to recover MR images accurately.

The field of CSMRI methods has seen roughly two types of algorithms. The first type tends to be non-adaptive CSMRI algorithms applied for MR image reconstruction and sparse representations of MR images in certain transform domain. Lusting et al. [9] explored the total variation (TV) regularization to construct a CSMRI algorithm in 2007, and the algorithm accomplished to recover MR images in the spatial domain. Kim et al. [10] utilized the global prior built on the dual-tree complex wavelet transform to propose a CSMRI algorithm. Qu et al. [11] employed the sparsity-induced regularization based on the contourlet transform to present an algorithm for acquiring the high quality of the reconstructed MR images. Zhang et al. incorporated the region of support and phase correction matrix to a two-level iterative CSMRI algorithm for guaranteeing image reconstruction accurately [12]. However, the aforementioned CSMRI algorithms just contribute to accurately reconstructing some MR images which take on the specific features, since these algorithms by their owns possess certain limitations to choose priors. Therefore, researchers have blended a number of different sparsity-induced regularizations into the CSMRI framework, for instance, the wavelet transform and contourlet transform regularizations proposed by Qu et al. [13]; the wavelet transform and TV regularizations presented by Huang et al. [14]; the wavelet transform, TV and nonlocal self-similarity regularizations offered by Huang et al. [15]. The combinatorial regularization-based CSMRI algorithms can promote the image reconstruction precision, and MR images do not need to take on some specific features. The sparse representations of MR images applying the above-mentioned CSMRI algorithms are non-adaptive, which is prone to exert a negative impact on reconstructing MR images. The CSMRI algorithms possessing the self-adapted ability should be developed for adaptive sparse representing and reconstructing MR images.

A second type of CSMRI algorithms has been designed to accommodate the learning methods, including the database-free learning models (such as the dictionary, transform and patch-based nonlocal operator learning models) and the database learning models (such as the deep learning frameworks). The database-free learning methods typically involve mathematical image models referring to the sparsity assumption. Usually, the database-free learning methods can be divided into two types, that is, the partially-adaptive learning models (such as the PBDW-based CSMRI reconstruction [16], MR image reconstruction with FDLCP [17]) and the fully-adaptive learning models (such as DLMRI [1] and TLMRI [18]). Ravishankar and Bresler [1] introduced a fully-adaptive CSMRI algorithm by employing the adaptive synthesis dictionary learning (DLMRI). In order to speed up the image reconstruction process of DLMRI, an adaptive CSMRI algorithm on the basis of the adaptive sparsifying transform learning (TLMRI) [18] was proposed. In [19], the data-driven tight frame CSMRI algorithm was presented via exploiting the tight frame learning. Moreover, some CSMRI algorithms built on the combination of adaptive learning methods including the adaptive synthesis dictionary learning and the adaptive tight frame learning, also have been developed, such as GradDLRec [20], TRIMS [21] and TFG-MRI [22]. In addition, Zhang et al. [23] presented an exponential wavelet iterative shrinkage thresholding CSMRI algorithm (EWISTARS) for reconstructing more accurate MR images. Liu et al. [24] proposed a projected fast iterative soft-thresholding CSMRI algorithm (pFISTA) to rapidly recover MR images. More recently, to avoid losing the consistency of pixels and the crucial spatial structures for MR images, some CSMRI algorithms exploited the convolutional sparse coding method to act as the regularization, such as GradCSC [25] and FoECSC [26]. Simulation experiments manifested that adaptive CSMRI algorithms are capable of outperforming non-adaptive CSMRI algorithms in terms of the image reconstruction quality. Besides, an alternative CSMRI algorithm by using BM3D denoising algorithm (BM3D-MRI) [27], solved the CSMRI problem as the denoising problem and utilized the non-local similarity to determine the sparsifying transform [28]. BM3D-MRI can acquire better performance for recovering MR images. The aforesaid CSMRI algorithms using the normal sparsity do not take into account the group sparsity. The group sparsity can explore more accurate sparse priors and increase the degree of sparsity. Accordingly, some CSMRI algorithms based on group sparsity-induced regularizations have been proposed, for example, GSTV [29] proposed by Jiang et al., GSNR [30] presented by Liu et al. and SP-GS [31] proposed by Yu et al.. The group sparsity-induced CSMRI algorithms can yield promising image reconstruction performance as well. However, the algorithms in the light of the group sparsity exist in the drawbacks of several finely-tuned parameters and the high computation complexity.

The database learning methods replace mathematically designed models of signals with data-driven or adaptive models motivated by the field of deep learning techniques [32]. Recently, deep learning (DL) has been poured great attention into image processing studies, and has generally returned dividends in image processing performance. Since DL affords many powerful frameworks for extracting deep priors from MR image training datasets, and has the lower computation complexity in the test process, some researchers explored DL-based approaches to recover MR images. The literatures [33,34] presented the feed-forward networks which belong to deep neural networks for MR image reconstruction. The feed-forward networks learn the mappings from undersampled MR images to fully sampled versions, and can utilize the undersampled test images to reconstruct MR images. However, the feed-forward network strategy does not guarantee the data consistency in the test process [28]. To overcome this problem, Schlemper et al. [35] proposed the DL approach based on the convolutional neural network (CNN) by cascading a mapping block, a data consistency block and the dual block structure. This approach adding the explicit data consistency, pledged the consistency with measured image data in the test process. Alternatively, another class of the DL framework is that the networks are used to promote the image reconstruction quality of the previous iterative reconstruction algorithms [28]. For example, the alternating direction method of multipliers algorithm can be implemented by a multi-layer CNN (DeepADMM) [36], and the existing iterative algorithm was expressed as a convolutional network [37]. More elaborate architectures can be seen in [38], [39], [40]. The advantages of this class of the DL approach are that the nonlinear functions and kernels are learned and parameterized by the networks in place of being determined in original algorithm [28]. Although the above DL-based CSMRI approaches can dramatically reduce reconstruction time, they received similar reconstruction results as the conventional CSMRI algorithms. In order to achieve superior reconstruction quality, the generative adversarial network (GAN) framework consisting of a generator network and a discriminator network is a choice, as an example in [41], Yang et al. presented a conditional GAN-based DL approach (DAGAN) for dealiasing and fast CSMRI. In conclusion, DAGAN can improve MR image reconstruction accuracy and speed up the image reconstruction time.

On the basis of the existing CSMRI approaches, the performance of the MR image reconstruction has made favorably progresses in boosting the reconstruction accuracy, intensifying the robustness to noise or speeding up the reconstruction process. However, needing many finely-tuned parameters and pending to further enhancement of the image reconstruction quality, are still the pivotal issues for some conventional CSMRI algorithms. Although the DL-based CSMRI approaches have provided the great potential in reconstructing MR images with much faster reconstruction speed, improving image reconstruction quality is not achieved prominently different from what the conventional CSMRI algorithms can obtain.

In this work, motivated by well-designed existing CSMRI algorithms, we propose the HRED-MRI algorithm to address the above-mentioned issues. Our HRED-MRI coalesces the data fidelity term with the HRED constraint to formulate the CSMRI problem for MR image reconstruction. Thereinto, the HRED constraint is designed by the proposed HRED model consisting of the hybrid BM3D and FFDNet. Then, the formulated HRED-MRI problem is tackled by the alternating projection implemented by the epigraph method. Simulation experiments verify that HRED-MRI can reconstruct MR images more accurately, compared to the state-of-the-art CSMRI approaches including the conventional and DL-based CSMRI approaches. Our contributions are described as follows:

  • (1)

    Inspired by the RED paradigm, we present a novel HRED model by integrating the weighting of the BM3D denoising algorithm and FFDNet denoising network into the image denoising engine of the RED model. The advantages of the HRED model are that: i) BM3D is a block-matching and 3D filtering denoiser, which can leverage the non-local similarity and sparsity of MR images; ii) FFDNet is a fast and flexible CNN-based denoising network with a tunable noise level map as the input, which can explore the deep priors from MR image training datasets and can establish a tradeoff between the inference speed and the denoising performance; and iii) More than one denoising algorithm can be selected as the denoising engine to plug into the HRED model relying on its flexibility. In this work, we choose the fusion of the two denoising algorithms as the denoising engine. Then, the proposed HRED model is employed to constitute the HRED constraint. The advantage of the HRED constraint is that it can comprehensively exploit the complementarity of the priors excavated by BM3D and FFDNet, respectively.

  • (2)

    A CSMRI algorithm termed HRED-MRI, is proposed to accurately reconstruct MR images. Virtually, the HRED-MRI amalgamates the data fidelity term with the HRED constraint together to formulate a CSMRI problem. The HRED constraint is used to deal with the corresponding MR image denoising problem under the weighting of BM3D denoiser and FFDNet denoiser, and can explore the complementarity of the priors of MR images. It is worth noticing that we exploit FFDNet to learn the image deep priors in HRED-MRI similar in essence to the non-DL algorithms introduced in CSMRI communities. But, the main difference of HRED-MRI is that the powerful FFDNet can capture the deep prior and statistical distribution to complement the non-local similarity of MR images mined by BM3D, compared with non-DL CSMRI algorithms. The fundamental difference of HRED-MRI is the decoupling of the prior from the data consistency term, compared with DL-based CSMRI approaches.

  • (3)

    We exploit the alternating projection optimization strategy of the epigraph method to solve the formulated CSMRI problem. The epigraph method is equivalent to the gradient descent method whose step sizes are selected in an adaptive manner. The epigraph sets of the HRED constraint and the data fidelity term are defined, and the two epigraph sets are all convex. The epigraph projection can be acquired via the equivalent gradient descent method. The virtue of the epigraph projection is that it can avert many finely-tuned parameters. As well, in the projection process, one of our advantages is that the step sizes of the equivalent gradient descent method are selected in an adaptive manner, i.e. the step sizes gradually decrease along with the projected value gradually approximating the optimum value; another of our advantages is that the noise intensity of BM3D and FFDNet gradually decreases along with the projection distance descending. These advantages are benefit for improving the performance of HRED-MRI in image reconstruction aspect. HRED-MRI can achieve the superior reconstruction performance with preserved perceptual image details from highly undersampled measurements, compared with the state-of-the-art DL-based approaches and conventional CSMRI algorithms. Furthermore, HRED-MRI is suitable for the real-time processing.

The remainder structure of the paper is arranged as follows: Section 2 reviews the preliminary knowledge about the CSMRI algorithm and the RED model; Section 3 introduces HRED-MRI and the proposed algorithm for MR image reconstruction; Section 4 performs various simulation experiments, and the results confirm that our proposed HRED-MRI can achieve superior performance of the real-valued and complex-valued MR image reconstruction; finally, Section 5 gives the concluding remarks and expectations of the future work.

Section snippets

Preliminaries

In this part, we briefly give an overview of the CSMRI theoretical framework in MR image processing realm, and then describe the implementation of the RED model. The following notational conventions are used for the full paper. We assume that xCN represents an MR image vector to be reconstructed, yCM indicates the undersampled K-space measurement vector of an MR image, FuCM×N denotes the undersampled Fourier coding matrix, and λ is a positive penalty parameter.

The HRED constraint and HRED-MRI algorithm

In this work, to further promote the quality of the reconstructed MR image and do not need to adjust many parameters for the CSMRI algorithm, we propose an explicit image-adaptive regularization functional based on the weighting of BM3D and FFDNet for a hybrid RED constraint. The newly proposed HRED constraint is utilized to formulate the CSMRI problem solved by the epigraph projection method, and to reconstruct MR images with highly quality from the undersampled measurements.

Experiment settings

In this section, diverse simulation experiments were performed to test the validity of the proposed HRED-MRI with the real-valued and complex-valued MR images. In the experiments, different types of sampling schemes were selected, such as the pseudo radial, 1D Gaussian, 2D Gaussian, 2D Poisson disc, Cartesian and 2D variable density random sampling modes. For the pseudo radial and 1D Gaussian sampling modes, 10%, 20%, 30% and 40% sampling rates retained raw K-space data were simulated

Conclusion

In this work, we proposed a new HRED-MRI algorithm by integrating the HRED constraint into the CSMRI problem. We explored the hybrid denoising engine to define the HRED constraint, and the hybrid denoising engine leveraged the weighting of BM3D and FFDNet. That is because there exists in the sparsity and the non-local similarity induced by BM3D, and the learning deep priors induced by FFDNet. The HRED-MRI can exploit the complementarity of the different priors to reconstruct MR images.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We sincerely acknowledge the editors and reviewers for their hard-working efforts on the paper. We would like to gratefully acknowledge the authors for sharing their source codes. The project was funded in part by the China NSFC 61471313 and 61901406, the Natural Science Foundation of Hebei Province F2019203318, the University Natural Science Research Project of Anhui province KJ2019A0805, and the Science and Technology Major Special Project of Anhui Province 18030901022. We would like to

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