We demonstrate the performance of our method by comparing it with state-of-the-art methods on 15 single-spool heart, 7 single-spool DCE, and a multi-spool brain MRI datasets at different sampling rates and 50%). The results show that our method significantly outperforms the other state-of-the-art methods in reconstruction quality with a comparable running time and is robust to noise. In this thesis, we introduce a new dynamic CS-MRI reconstruction method that extends the state-of-the-art CSC-based reconstruction methods [18,19].
We demonstrate the effectiveness of our reconstruction method and compare it with state-of-the-art methods in Chapter V.
The Compressed Sensing In Dynamic MRI Problem
In our experiment on 15 cardiac and 7 dynamic contrast-enhanced (DCE) MRI datasets at three different Cartesian sampling rates and 50%), the proposed reconstruction method produces significantly better image quality compared to several state-of-the-art methods (e.g., k-t FOCUSS [ 9 , 10 ], single-scale 3D CSC [ 19 ], blind compressive detection [ 13 ], patch-based dictionary learning [ 14 , 15 ], and FTVNNR [ 23 ]) and runs at efficient speed with GPE acceleration. To overcome this drawback, a good reconstruction method using compressed sensing MRI [3] is used, which produces high-quality images and fast acquisition time.
Dictionary Learning Algorithms
Since its inception in the seminal work of [3], CS-MRI has been actively studied to speed up the time-consuming MRI imaging process. Conventional CS-MRI algorithms are mainly based on promoting data reduction using regularization on universal sparse transform models. CS-MRI for dynamic data is also proposed by implementing spatial and temporal coherence (i.e., k-t FOCUSS [9, 10] and [28]).
These conventional CS-MRI methods suffer from computational overhead due to solving expensive nonlinear l1 minimization problems. Thus, conventional DL in CS-MRI [11-15] approaches have been successfully applied to improve MRI reconstruction quality. 18 , 19 ] first used CSC to solve CS-MRI problems, greatly improving the running time and reconstruction quality by building more compact and expressive shear invariance convolution filters.
In this approach, zero-fill (Fig. 5b, zero-fill reconstruction) and randomly initialized filters (e.g., in Fig. 5d, the different filter sizes) are iteratively updated until they converge, as in Figs. to our knowledge, our approach is the first attempt to use GA for automatic parameter selection in the CS-MRI reconstruction problem. Then, the proposed method consists of two components: the reconstruction process that takes a zero-fill reconstruction as an initial guess to improve image quality (i.e., removing undersampling artifact, Fig. 1 yellow box), and the parameter search process by using a GA ( Fig. 1 blue box).
Reconstruction Process
Moreover, our method iteratively updates the result to enforce measurement consistency (see Algorithm 1 and the constraint term in Equation 7), while the other methods perform one-time reconstruction for each frequency data and then combine them. The fourth term✓krtslk1 is the total variation energy that forces the temporal coherence of the low-frequency image. The rest of this equation is the collection of constraints: the first constraint preserves the connection between the undersampled measurement m and the undersampled reconstructed image using the k-space mask RwithF2 operator; the second constraint limits the Frobenius norm for each atom dn,k within a unit length.
The problem (7) can be divided into two sub-optimization problems as follows, which can be iteratively updated to the global minimum solution. 7 with respect to sl, which contains the total variation along the time axis and the measurement limitation term. 2kRF2(sh+sl) mk22 (9) This problem can be solved efficiently by an iterative clipping algorithm using primal-dual.
The variables g and d are related by a projection operator as a combination of a truncated matrix with the corresponding dictionary size followed by a zero-padding in oder to make the dimension of g the same as that of x, and the variable g must also be padded with zeros to make its size similar to gf3 and xf3, so we can use the Fourier transform to solve this problem. 13 can be unbounded using the double variable, h, and further regulates measurement consistency and double differences with . The update rule for u can be defined as a fixed-point iteration with the difference between x and y (u converges when x and y converge to each other).
Complexity analysis of proposed reconstruction algorithm
2kRshf2 mhk22 (29) Previously, df3 andxf3 were obtained in the 3D Fourier domain, we need to bring it to the same space by applying an inverse Fourier transform along the time axis F2H. After the iteration process, sh will be the result of applying a 2D inverse Fourier transform F2H to shf2.
Parameter Searching Process
Moreover, we also set possible lower and upper bounds for each parameter to limit the search space of the GA, as illustrated in Table 2. In particular, our experiment shows that a GA can be applied only once to find the optimal parameters used for similar types work. of data (i.e. cardiac or DCE MRI data). To assess the performance of our method, we performed experiments on 15 cardiac MRI datasets from The Data Science Bowl [41] (30 frames of a 256⇥256 image over the cardiac cycle of a heart for each dataset), 7 tumor DCE MRI datasets (128⇥128 images with 128 frames of each dataset) and a multi-coil MRI brain dataset with 12 frames in which each frame consists of 12 parallel acquisitions) in three different rates of Cartesian space undersampling masks and 50% sampling masks), as shown in Fig.
GA is also set to run only once to search parameters for five generations with L= 200 and ⌧ = 20. More specifically, for each type of data set, we arbitrarily choose one full-sampled data and the sampling mask to optimize parameters and reuse these found parameters to reconstruct the remaining MRI data sets. We also compare with the intermediate versions of our method (i.e. incremental addition of new features, such as multi-scale expansion of CSC, elastic net regularization and spectral decomposition, to the base version of 3D CSC [19] to assess how each addition function affects the overall performance of the method (see Table 3).
Multi-scale 3D CSC withl1, l2 regularization Elastic multi-scale Elastic multi-scale and temporal TV minimization Elastic multi-scale TV.
Reconstruction quality evaluation
Our method is shown to generate less visual artifact compared to other methods (see the region of interests and pixel-wise error maps), even under an extremely low sampling rate (12.5%). It is also worth noting that our method can reconstruct temporal changes in DCE data more accurately compared to tok-t FOCUSS and DLTG at a very low sampling rate (see Fig. 9 12.5%). The Akep value, derived from the [43] model, is also a commonly used quality metric for dynamic MRI (especially DCE) to assess the consistency of the reconstructed image sequence because this value reflects the degree of MRI signal enhancement and the exchange . rate in terms of brightness and contrast.
Therefore, the Akep value characterizes the rate of MRI signal change in the tumor region of interest (ROI), which has been shown to provide relevant information regarding tumor perfusion and permeability. To assess how the proposed reconstruction method performs, we measure the Akep on each reconstructed image (actually Akep is generated per pixel) and generate a least squares fitting curve. Moreover, our method effectively reduces the intensity variation (or noise) of the reconstructed images (see the variation of red circles is much smaller than that of other results) due to enforcing temporal coherence using TV energy.
Overall, the proposed method achieves a better reconstruction quality than other state-of-the-art methods. Shift-invariant convolution filters can represent both spatial and temporal features well, while multi-scale 3D CSC with l1 and l2 regularization shows its performance with better M SE, P SN R and SSIM. Moreover, our frequency-divided reconstruction approach, using temporal TV minimization for low frequency and multi-scale 3D CSC with elastic net regularization for high frequency, can significantly improve the image quality as well as the convergence speed, which will be discussed in Section 5.4.
Extension to multi-coil parallel MR
We tested the parallel version of our method on a 12-spool MRI brain dataset and compared with FTVNNR, BCS and 3D-CSC methods for image quality assessment. 14 also shows that our method is less prone to pixel-wise errors and generates images closer to the full reconstruction than FTVNNR, BCS and 3D-CSC.
Robustness to noise
Convergence evaluation
The proposed method converges faster (i.e. requires less number of epochs) than 3D-CSC, multi-scale and multi-scale elastic. 19 and 20 show the adequate convergence curves of the CSC methods with a large P SN R improvement and no divergence. In our observation, when we used the required parameters for all data sets in the same type of MRI, the efficiency of the convergence rate still remains, as shown in Fig.
Running time evaluation
We found that the frequency-dependent reconstruction, that is, using temporal total variation for low-frequency reconstruction and multi-scale 3D CSC with elastic mesh regularization for high-frequency reconstruction, played an important role in improving the overall reconstruction quality and convergence speed. . -scale Elastic multi-scale Elastic multi-scale TV. a) Akepmaps of 25% sampling reconstruction of Tumor 1 dataset. 3D-CSC Multi-scale Elastic multi-scale Elastic multi-scale TV Our working method. a) Convergence curves from two different Cardiac 3 and 4 datasets. 3D-CSC Multi-scale Elastic multi-scale Elastic multi-scale TV Our working method.
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