The primary design choices left to the user involve selecting an appropriate sparsifying transform and choosing the value of the tuning parameter. Throughout this paper, we achieve reasonable results using a four-level ‘9-7’ DWT; although not specifically tuned for brain images, a different transform may be necessary to apply DESIGN successfully to other types of data. The first experiment depicts the effect of choosing different values for the tuning parameter λ on DESIGN. As the primary design choice left to the user, the tuning parameter may be selected via either a parameter sweep, as is done in this paper, or a method like cross-validation.
The images depicting the results of reconstruction using DESIGN demonstrate that the proposed method mitigates noise amplification due to both undersampling and GRAPPA, improving PSNR and supporting the notion of sparsity-enforcing regularization as an effective denoising method. However, improvements in PSNR, like mean-squared error, are not indicative of improved diagnostic quality, and care must be taken to avoid oversmoothing. The included visual comparisons suggest that the DESIGN denoising method may be beneficial at moderately high accelerations, especially at a field strength of 3 T with a receive coil array with 32 channels. In addition, despite the sparsity regularization component of the joint optimization method, the algorithm functions properly with uniform undersampling, relying on GRAPPA to mitigate the aliasing; this feature obviates the need to dramatically redesign pulse sequences used for acquisition; existing noisy GRAPPA reconstructions may be improved by post-processing with this method.
However, the oversmoothing observed at very high accelerations suggests that this method may not be suitable for generating diagnostically useful images with extreme undersampling. The results from simulations involving synthetic 1.5 T data (with SNR degraded by added noise and gray/white matter contrast reduced according to T1
values from (34
)) are depicted in supplemental Figure S1
. The reduction in image quality across all methods suggests that the feasible range of acceleration with this method is not as great as for 3 T; similar degradation is probable when far fewer coil channels are available. In addition, fewer channels reduces the ability of GRAPPA to resolve aliasing at high accelerations, and residual aliasing may remain in images denoised using DESIGN, reducing the proposed method’s utility at high accelerations. Further study is required to understand the limits of DESIGN in terms of aliasing and SNR loss from using 12- or 16-channel systems, and lower acceleration factors may be necessary with such systems.
The analysis using a synthetic contrast phantom supports that although DESIGN blurs low-contrast elements slightly, the degradation is not significant. Analyses of effective resolution conducted using both synthetic (based on the phantom described in (33
)) and real Siemens multipurpose resolution phantoms are depicted in supplemental Figures S2
. Although the reconstructed image quality varies significantly among methods, the effective resolutions estimated using the full width at half maximum (FWHM) of horizontal and vertical cuts of the 2-D point-spread function (PSF) are all on the order of one voxel. However, this analysis is limited to simple features and relatively low noise levels and does not predict the contrast or texture degradation or loss of resolution that would result with applying DESIGN to denoise human images acquired at high accelerations; such oversmoothing may hinder the isolation of small features or low-contrast regions. Further tests on images with pathologies are necessary before the effects of oversmoothing can be ascertained definitively.
DESIGN denoising exhibits several distinctive characteristics when compared to L1 SPIR-iT, the state-of-the-art method for combining compressed sensing and parallel imaging. First, when random undersampling is not possible, GRAPPA, and hence DESIGN, is far more effective at mitigating coherent aliasing than the underlying SPIR-iT approach; these artifacts are clear in several instances at high accelerations. Furthermore, according to the retained SNR maps, DESIGN is much more effective at denoising than L1 SPIR-iT, ignoring the uniform sampling constraint. Moreover, not having to convolve the SPIR-iT (or GRAPPA) kernel with the k-space data in every iteration simplifies the implementation of the algorithm. In situations where random undersampling is used, DESIGN can denoise the L1 SPIR-iT result, mitigating SNR loss at high accelerations.
In conclusion, the proposed method successfully combines GRAPPA and sparsity-based denoising. Adjusting the framework to accommodate non-uniform or non-Cartesian sampling patterns, using SPIR-iT or another non-uniform GRAPPA-like operator, and/or gridding techniques, would enable applicability to a greater number of acquisition schemes, including radial and spiral trajectories. In addition, the proposed algorithm can benefit greatly from implementation on a GPU, since the dominant computational operations (FFT and DWT) are all highly parallelizable. Such an implementation would enable cost-effective real-world application on clinical scanners. Further work includes testing DESIGN denoising on a variety of other types of MR images, carefully examining the effect of denoising on low-contrast anomalies like small tumors or lesions, and exploring more sophisticated sparsifying transforms or penalty functions suggested by a Bayesian analysis of the image denoising problem.