Global and local constrained parallel MRI reconstruction by exploiting dual sparsity and self-consistency
Introduction
Imaging speed is one of the main limitations of magnetic resonance imaging (MRI), and it leads to motion-related artifacts during acquisition. Although fast MR pulse sequences enable accelerated acquisition, the realization of high-resolution 3D imaging is still challenging owing to both physical (e.g., gradient slew rate) and physiological (e.g., nerve stimulation) constraints. Instead, parallel MRI is a well-established acceleration technique that acquires a fraction of phase encodings (PEs) based on multiple receiver coils. Parallel MRI techniques are generally classified based on SENSE [1], [2], [3] and GRAPPA [4], [5], [6]. Specifically, SENSE-like methods employ coil sensitivity directly in the image domain, and are very difficult to obtain accurate sensitivity information, which in turn can potentially lead to ill-conditioning problems during the inverse problem. Conversely, GRAPPA-like methods operate in k-space by simultaneously enforcing calibration and self-consistencies over the entire k-space. Given that GRAPPA-like methods exploit it implicitly in k-space, they tend to be more robust to errors in coil calibration. In particular, in volumetric imaging applications, controlled aliasing, termed as CAIPIRINHA [7], [8], substantially reduces the g-factor noise penalty by evenly distributing the undersampling between the two PE axes. Nevertheless, the acceleration is theoretically limited by the number of receiver coils.
The application of compressed sensing (CS) and matrix completion (MC) to parallel MRI has been extensively explored as a signal recovery method from incomplete measurements [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. For successful applications, the following two conditions should be satisfied: 1) MR signals are sparse or rank-deficient in a certain domain (e.g., wavelets and k-space,) and 2) Fourier encoding is incoherent. Images are then reconstructed by solving nonlinear problems that employ soft- or hard-thresholding shrinkage operations in the corresponding domain with data fidelity. During the reconstruction, the signal priors (sparsity or low rank) play an important role in reducing artifacts and noise, while the data fidelity term is used to preserve the spatial resolution and image contrast. However, given that the two constraints are competing, images can suffer from a loss of image details with high acceleration. Unlike the aforementioned approaches that utilize global priors (sparsity and low rank) in a transform domain, multi-scale approaches (e.g., wavelets) were introduced as a local constraint in [21]. It was observed that the wavelet transform yields three directional high subbands (horizontal, vertical, and diagonal), and each subband is ideally represented in compact and separable high-frequency k-space partitions. The reconstruction is then locally performed in the wavelet and k-space domains. Despite the improved reconstruction accuracy, this approach is suboptimal because of the following reasons: 1) it does not consider the actual frequency response of each high subband, 2) it potentially leads to signal discontinuity between local k-spaces, and 3) it also suffers from image blurring due to the direct trade-off between the sparsity constraint and k-space data fidelity with increasing acceleration.
Given the considerations above, the signal priors (sparsity and low rank) are effective in suppressing artifacts and noise but lead to image blurring. Conversely, the self-consistency of parallel MRI is important in preserving image accuracy but yields amplified noise. In this study, we propose global and local constrained parallel MRI with multi-level variable density (VD) sampling, which combines the two competing constraints but effectively decouples them under the joint estimation framework. Furthermore, simulations and experiments are performed to demonstrate the effectiveness of the proposed method when compared to parallel MRI, CS, and the combined method.
The main contribution of this study corresponds to the formulation of an algorithm that locally and globally estimates the missing signals under dual sparsity and self-consistency constraints. Ideally, each wavelet subband can be represented in a compact and separable k-space partition based on the fundamental duality between the transform domain sparsity and weighted k-space data. Interestingly, the fundamental duality converts a large-scale sparse signal recovery to a regionally independent small-scale problem [16], which enables local k-space reconstruction depending on the wavelet filter direction. Although conventional high-subband CS utilizes spectral-weighted k-space as data fidelity, each subband signal is not perfectly represented within the localized k-space data. This in turn leads to signal loss of image features, and this results in signal discrepancy between the localized k-space data. To the best of our knowledge, the proposed model is the first method that addresses the aforementioned potential issues by 1) introducing a complementary local k-space model with dual sparsity and 2) enforcing self-consistency over the entire k-space. This proposed algorithm is of significant value in addressing the problems related to wavelet-filtered signal loss and signal discontinuity in k-space.
Section snippets
Controlled aliasing in volumetric parallel MRI
Controlled aliasing employs a periodically shifted undersampling in PE axes [7], [8], in which the sampling points in the axis are successively shifted with respect to each other using an additional gradient offset in the axis. The sampling pattern increases the distance between aliasing voxels, thereby making them easier to separate because of the increased sensitivity variations. Once the data are Fourier transformed in the frequency encoding (FE) direction, missing signals in k-space
Multi-level VD sampling
In the proposed method, multi-level VD sampling in k-space is employed, wherein the sampling density becomes sparser from the low to high frequency region as follows: 1) Nyquist sampling in the central region of k-space (low frequency), 2) periodically shifted lattice undersampling [7], [8] between the two regions (intermediate frequency), and 3) incoherent pseudo-random undersampling in the peripheral region of k-space (high frequency) (Fig. 1). The sampling pattern is based on the hypothesis
Experimental setup
In our experiments, we used 3D in vivo human brain datasets, which were acquired using 3D magnetization prepared rapid acquisition with gradient echo (MP-RAGE) sequence with the following parameters: flip angle = 9, time-of-repetition = 1900 ms, time-of-echo = 2.52 ms, time-of-inversion = 900 ms, field-of-view = mm3, and matrix size = in , , and directions, respectively. A 12-channel head coil was used for signal reception. The fully acquired 3D brain data
Discussions
One of the key requirements in CS involves realizing incoherent aliasing artifacts in the transform domain. As shown in [9], this criterion is satisfied using pseudo-random measurements if either the wavelet transform or discrete cosine transform is used. However, parallel MRI is preferred because image pixels that are spatially far apart are aliased together. This desirable property is further exploited in CAIPIRINHA with periodically shifted undersampling. Given the considerations above, it
Conclusion
In this study, we introduced an efficient, global, and local constrained reconstruction that employs dual sparsity and self-consistency. This effectively decoupled the object function in Eq. (13) into CS in Eqs. (14), (15) and parallel MRI in Eq. (16), thereby yielding no direct trade-off between image accuracy and noise. With respect to CS reconstruction, we introduced a complementary local k-space model as a union of locally decomposed and leaked residual signals. The high-frequency signals
CRediT authorship contribution statement
Suhyung Park: Conceptualization, Methodology, Software, Writing - original draft, Writing - review & editing. Jaeseok Park: Writing - original draft, Supervision.
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
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1013603).
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