Compressed sensing (CS) has been applied to accelerate magnetic resonance imaging (MRI) for many years. Due to the lack of translation invariance of the wavelet basis, undersampled MRI reconstruction based on discrete wavelet transform may result in serious artifacts. In this paper, we propose a CSbased reconstruction scheme, which combines complex doubledensity dualtree discrete wavelet transform (CDDDTDWT) with fast iterative shrinkage/soft thresholding algorithm (FISTA) to efficiently reduce such visual artifacts. The CDDDTDWT has the characteristics of shift invariance, high degree, and a good directional selectivity. In addition, FISTA has an excellent convergence rate, and the design of FISTA is simple. Compared with conventional CSbased reconstruction methods, the experimental results demonstrate that this novel approach achieves higher peak signaltonoise ratio (PSNR), larger signaltonoise ratio (SNR), better structural similarity index (SSIM), and lower relative error.
Magnetic resonance imaging (MRI) is a powerful noninvasive imaging modality, which is ubiquitously used in modern medical diagnosis [
In recent years, a variety of techniques have been proposed to enhance the quality of MRI, which can be roughly classified into three categories [
To enhance the image reconstruction quality and reduce the reconstruction artifacts, in this paper, we propose a novel reconstruction scheme, which combines CDDDTDWT with FISTA. Although dualtree complex wavelet transform has also been exploited in the literature [
The remainder of this paper is organized as follows. Section
The CDDDTDWT is an overcompleted discrete wavelet transform that combines doubledensity DWT [
Twodimensional (2D) doubledensity dualtree DWT includes 2D real doubledensity dualtree DWT and 2D complex doubledensity dualtree DWT. The former is constructed from two oversampled 2D doubledensity DWT in parallel, which is redundant by a factor of two. Figure
The filter bank structure for real 2D doubledensity dualtree DWT.
The latter is formed by utilizing four oversampled 2D doubledensity DWT in parallel to the input image. The filter bank structure of this transform can be obtained by extending the one illustrated in Figure
The two levels of 2D complex doubledensity dualtree DWT.
The CSMRI image reconstruction problem is defined as follows:
As an alternative formulation, applying
Since
The FISTA algorithm is applied to solve the optimization problem of (
Applying the sparsity transform CDDDTDWT to a local optimal image
The recovered image
In (
The proposed algorithm combining the complex doubledensity dualtree and fast iterative shrinkage thresholding algorithm (FISTACDDDT) for solving (
To evaluate the performance of the proposed reconstruction algorithm, we implement the complex doubledensity dualtree wavelet and conventional wavelet using the software in [
The MR images. (a) SheppLogan phantom, (b) axial brain, and (c) spine.
Gaussian random
The undersampling patterns. (a) Gaussian random sampling at a sampling ratio of 20% and (b) radial sampling at a sampling ratio of 9%.
In this work, FISTA based on three different sparsity transforms is utilized to solve the optimization problem of (
We first conduct the experiment on the SheppLogan phantom image shown in Figure
The peak signaltonoise ratio (PSNR), signaltonoise ratio (SNR), structural similarity (SSIM) index, and relative error (Rel.Err) are used to evaluate the FISTACDDDT recovery performance. The PSNR is calculated using the following equation:
The SNR is defined as
The definition of the SSIM index is given by
The Rel.Err is defined as
Note that, for all the figures in this part, various approaches are labeled by different colors below the images. The green dotted lines mean FISTADWT, the pink dotted lines denote FISTACDT, and the blue lines represent the proposed FISTACDDDT.
Figure
Reconstructed results of SheppLogan phantom images using Gaussian random mask (a–d) and radial mask (e–h) with a 20% sampling ratio among different approaches. (a) Gaussian random sampling mask, (b) and (f) FISTADWT, (c) and (g) FISTACDT, (d) and (h) FISTACDDDT, and (e) radial sampling mask.
Figure
The comparison results among three different MR reconstruction algorithms using Gaussian random mask (a and b) and radial mask (c and d) at a 20% sampling ratio using SheppLogan phantom image. (a) and (c) PSNR versus iterations; (b) and (d) Rel.Err versus iterations.
For further analysis, the PSNRs of the reconstructed images using different methods are plotted at various sampling ratios. It is clear from Figure
Comparisons among different approaches at different sampling ratios. (a) Gaussian random sampling and (b) radial sampling using a SheppLogan phantom image.
Figures
Reconstructed images using 20% radial sampling. (a) An axial brain image, (b) FISTADWT, (c) FISTACDT, (d) FISTACDDDT, and (e)–(h) magnified images of the regions marked by white rectangles in (a)–(d), respectively.
Reconstructed images using 20% Gaussian random sampling. (a) A spine image, (b) FISTADWT, (c) FISTACDT, (d) FISTACDDDT, and (e)–(h) magnified images of the regions marked by white rectangles in (a)–(d), respectively.
Note that the PSNRs of reconstructed axial brain images using radial sampling mask by FISTADWT, FISTACDT, and FISTACDDDT are 33.99 dB, 37.58 dB, and 38.87 dB, respectively. The magnified images are shown in Figures
Table
Numerical results for an axial brain MR image by different reconstructed methods using radial sampling mask with
Sampling ratio  Algorithms  SNR (dB)  Rel.Err (%)  SSIM 

15%  FISTADWT  14.51  4.99  0.7461 
FISTACDT  17.26  3.76  0.8393  
FISTACDDDT  20.26  3.19  0.9328  


18%  FISTADWT  17.68  3.28  0.8409 
FISTACDT  21.15  2.41  0.9246  
FISTACDDDT  22.74  2.17  0.9558  


20%  FISTADWT  19.84  2.61  0.8703 
FISTACDT  22.53  1.88  0.9427  
FISTACDDDT  23.82  1.76  0.9610  


25%  FISTADWT  20.94  1.76  0.9067 
FISTACDT  24.37  1.30  0.9544  
FISTACDDDT  25.17  1.25  0.9651  


28%  FISTADWT  22.35  1.36  0.9242 
FISTACDT  25.42  1.00  0.9600  
FISTACDDDT  26.01  0.99  0.9675 
Figures
Numerical results for a spine MR image using different reconstructed methods employing Gaussian random sampling mask with
Sampling ratio  Algorithms  SNR (dB)  Rel.Err (%)  SSIM 

15%  FISTADWT  13.19  11.77  0.7555 
FISTACDT  14.35  11.55  0.7955  
FISTACDDDT  17.32  8.34  0.8694  


18%  FISTADWT  16.08  7.48  0.8349 
FISTACDT  17.68  7.26  0.8656  
FISTACDDDT  19.51  3.57  0.9170  


20%  FISTADWT  18.28  3.07  0.8937 
FISTACDT  20.48  2.43  0.9301  
FISTACDDDT  20.78  2.40  0.9435  


25%  FISTADWT  19.49  2.30  0.9196 
FISTACDT  21.78  1.88  0.9520  
FISTACDDDT  21.80  1.81  0.9599  


28%  FISTADWT  19.05  2.21  0.9262 
FISTACDT  22.22  1.90  0.9551  
FISTACDDDT  22.27  1.76  0.9262 
The comparison of PSNR versus sampling ratio among three different MR reconstruction algorithms using (a) Gaussian random mask and (b) radial mask on an axial brain image.
The comparison of PSNR versus sampling ratio among three different MR reconstruction algorithms using (a) Gaussian random mask and (b) radial mask on a spine image.
Considering the superiority of CDDDTDWT in preserving edges and maintaining higher directional selectivity, the proposed reconstruction approach combines the CDDDTDWT with FISTA to produce better recovery results with a faster convergence rate. Although the ISTA and TwIST can be integrated with CDDDTDWT as well, both of them were designed for simple regularization problems. Besides, they have some drawbacks that cannot be ignored. ISTA based on the operatorsplitting strategy is a promising method, which has been successfully used in signal recovery. However, it belongs to the firstorder algorithm that converges quite slow. As a variant of ISTA, TwIST is also an iterative thresholding algorithm, which is not guaranteed to converge globally. In contrast, FISTA has a faster convergence rate and better reconstruction accuracy, as proved in [
It is worth noting that Zhu et al. [
In this paper, we develop a new image reconstruction method for CSMRI based on complex doubledensity dualtree wavelet transform. The filter bank structure of the CDDDTDWT is explored. This novel approach has been applied to SheppLogan phantom and axial brain and spine image reconstruction and compared with two popular methods, namely, FISTADWT and FISTACDT. The reconstructed results demonstrate that our scheme improves the PSNR and SNR as well as SSIM index and reduces the reconstructed artifacts significantly. In both simulation and experiments on in vivo data, we use the FISTA as the reconstruction algorithm. However, it can only solve the unconstrained minimization problems. An algorithm that can solve both unconstrained and constrained convex optimization problems will be studied in the future work.
The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.
This work was supported by National Science Foundation of China (Grants nos. 81371537, 91432301, and 81527802), Major State Basic Research Development Program of China (973 Program) (Grant no. 2013CB733803), and Fundamental Research Funds for the Central Universities of China (Grant no. WK2070000033). The authors also thank the two healthy volunteers from whom they obtained the original MR datasets.