NOise reduction with DIstribution Corrected (NORDIC) PCA in dMRI with complex-valued parameter-free locally low-rank processing - PubMed (original) (raw)

NOise reduction with DIstribution Corrected (NORDIC) PCA in dMRI with complex-valued parameter-free locally low-rank processing

Steen Moeller et al. Neuroimage. 2021.

Abstract

Diffusion-weighted magnetic resonance imaging (dMRI) has found great utility for a wide range of neuroscientific and clinical applications. However, high-resolution dMRI, which is required for improved delineation of fine brain structures and connectomics, is hampered by its low signal-to-noise ratio (SNR). Since dMRI relies on the acquisition of multiple different diffusion weighted images of the same anatomy, it is well-suited for denoising methods that utilize correlations across the image series to improve the apparent SNR and the subsequent data analysis. In this work, we introduce and quantitatively evaluate a comprehensive framework, NOise Reduction with DIstribution Corrected (NORDIC) PCA method for processing dMRI. NORDIC uses low-rank modeling of g-factor-corrected complex dMRI reconstruction and non-asymptotic random matrix distributions to remove signal components which cannot be distinguished from thermal noise. The utility of the proposed framework for denoising dMRI is demonstrated on both simulations and experimental data obtained at 3 Tesla with different resolutions using human connectome project style acquisitions. The proposed framework leads to substantially enhanced quantitative performance for estimating diffusion tractography related measures and for resolving crossing fibers as compared to a conventional/state-of-the-art dMRI denoising method.

Keywords: Brain imaging; Denoising; Diffusion MRI; Human connectome project; Multiband; Simultaneous multi-slice; Singular value decomposition.

Copyright © 2020. Published by Elsevier Inc.

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Figures

Fig. 1.

Fig. 1.

Flowchart of the NORDIC algorithm for a series m(r, τ). Firstly the series is normalized with the calculated g-factor kernels as m(r, τ)/g(r). From the normalized series the Cassorati matrix Y = [y1, ⋯ , y j, ⋯ , y N] is established and the low-rank representation of Y is calculated as Y L = U · Sλthr · VH, where λ(i) = 0 for λ(i) < λ thr. After reforming the series m LLR(r, τ) the normalization with the calculated g-factor is reversed as m NORDIC(r, τ) = m LLR(r, τ) · g(r).

Fig. 2.

Fig. 2.

The interaction between the spectra of the underlying model, additive noise and the observed noise perturbed measurements, and the threshold estimated with asymptotic properties and hard thresholding based on the maximal singular value. The four spectra shown in each of the four columns are generated by Eqs. (5)-(8), respectively, given in section titled Numerical Evaluation of Threshold Choice; they represent an asymptotic model (λ Model), an asymptotic model with low-rank (λ LR Model), a spectra from dMRI which is full rank and falls below the maximal singular value of the noise (λ dMRI) and a low-rank signal which does not fall below the maximal singular value of the noise (λ LR dMRI). The MPPCA technique uses the asymptotic properties of the noise spectra to infer a threshold, and the NORDIC uses the prior knowledge of the noise-level for this.

Fig. 3.

Fig. 3.

Real valued simulation of the quality of the “optimal” signal recovery with NORDIC. Fig. A shows for a single slice the quality of the images reconstructed with denoising using NORDIC and MPPCA subsequent to SNR degradation with the addition of noise, and Fig. B shows analogous images for q_-space contrast, Δ_q_1_q_2m_, as the difference between volumes with different q_-vectors and same b_-value. The reference images are shown in Fig. 3A.i, and 3Bi before addition of noise and in Fig. 3A.ii, and 3Bii after addition of noise. The NORDIC methods are compared for patch-sizes of 53, 73, and 113 (middle row), and the MPPCA method (bottom row) are compared using the Gaussian noise and Rician noise. Panel C shows the structural similarity index (SSIM) restricted to the brain between m without noise, and NORDIC with patch sizes 53, 73, 113, 153, and 193. Panel D shows the structural similarity index (SSIM) restricted to the brain between Δ_q_1_q_2m without added noise, and NORDIC processing with patch sizes 53, 73, 113, 153, and 193. The SSIM for Panels C and D are averaged over all slices and diffusion direction for the different patch sizes. The SSIM from D is 0.54 for NORDIC using a patch size of 113, 0.49 for MPPCA with Gaussian noise, and 0.50 for MPPCA with Rician noise.

Fig. 4.

Fig. 4.

Effect of phase-stabilization and patch averaging for the NORDIC processing of complex data using an 113 patch. Reformatting to oblique coronal slices of the 0.9 mm isotropic data, acquired with oblique axial acquisition, are shown. The left figures are without phase-removal of the slice-specific smooth diffusion phase, and the right figures are with removal of the diffusion phase. For both approaches, the impact of patch-averaging with all 113 patches are shown.

Fig. 5.

Fig. 5.

Top row shows the effect of NORDIC across different resolutions on FA maps for a single slice. Bottom row presents corresponding diffusion weighted image (b = 3000 s/mm2) for the same slice. The FA maps are obtained after EDDY processing, and the images in the bottom row are before EDDY processing. From left to right, in groupings of 4, the three resolutions of 1.5 mm, 1.17 mm and 0.9 mm are shown. For each grouping, the images with the standard processing are shown adjacent to the images with the NORDIC processing. Supplemental Fig. S3, illustrates the same slices and also includes reconstruction by MPPCA for comparison.

Fig. 6.

Fig. 6.

Comparison of NORDIC processing with averaging of repetitive acquisitions to increase SNR. The left two columns are for a single acquisition across 3 different resolutions and the right two columns are for the averaging of the repetitive acquisitions. In each case, EDDY processing was applied. Supplemental Fig. S4 illustrates the same data and also includes reconstruction by MPPCA for comparison.

Fig. 7.

Fig. 7.

Effect of patch size in NORDIC processed dMRI data on fiber detection rate and accuracy for the three different resolutions. The 5 different patch sizes compared are with _n_3 for n = {3, 7, 11, 15, 19}. The top row, shows the detection rate of voxels which supports a second and third fiber, and the bottom row shows the gain in fiber orientation accuracy after NORDIC relative to the standard processing as a ratio between the dispersion determined for the standard and NORDIC processing. For MPPCA, the fiber detection rates were [94%, 77% and 38%] for the 2nd fiber, and [73%, 31% and 8%] for the 3rd fiber, with gains in Fiber orientation accuracy of [2.2, 3.4, and 3.0] for the 2nd fiber and [2.2, 3.3 and 1.9] for the third fiber.

Fig. 8.

Fig. 8.

Quantitative metric in brain regions SLF and PCR for standard and NORDIC processed data for the 5 subjects scanned at different resolutions and the 3 subjects each scanned at a single resolution with multiple repetitions. The top row illustrates fiber orientation dispersion (reflecting the uncertainty in the fiber orientation estimation) for voxels within a VOI supporting a second fiber (left), and supporting a third fiber (right); for this metric, lower height of the bar indicates better performance (lower uncertainty). The gain in fiber orientation accuracy (i.e. a decrease in dispersion reported as the ratio of the dispersions calculated with standard to that calculated with NORDIC processing) is shown in the lower row for the voxels supporting second (left) a third fiber (right); for this metric, the higher bar indicates better performance for NORDIC. The rightmost column in Fig. 7 shows the segmentation of the SLF and PCR used for quantification of crossing fibers. The error bars for the multi-resolution single repetition data represents the variability between subjects, and the error bars for the single-resolution multiple repetitions shows the variability within subjects but over different acquisitions. In case of MPPCA for the single-resolution multiple repetitions, gains in Fiber orientation accuracy were [2.2, 3.4 and 3.0] for the 2nd fiber for the 1.5, 1.17 and 0.9 mm resolution data, respectively; the corresponding numbers were [2.2, 3.3 and 1.9] for the third fiber with MPPCA processing.

Fig. 9.

Fig. 9.

Scatter plot of the detection rate of voxels with second (top row) and third (bottom row) fibers against the fiber orientation dispersion, reflecting the uncertainty in the fiber orientation estimate for the fibers in the brain regions PCR and SLF after bedpost processing. The VOI is determined from the subject independent JHU-ICBM atlas, and resampled to the data-space for each subject. The vertical axis (detection rate) is expressed as% of voxels in the VOI that contain two or three fiber crossings (second and third fibers, respectively).

Fig. 10.

Fig. 10.

Comparison of connectivity distributions from the probabilistic tractography results for 0.9 mm data (A), and 1.17 mm ((B) upper panel) and 1.5 mm ((B) lower panel) data representing connectivity of the entire subject specific PCR.

Fig. 11.

Fig. 11.

Comparison of tractography streamlines without and with NORDIC processing. The top row shows a comparison for the 0.9 mm isotropic resolution data analyzed in Figs. 5-10 using a single repetition with standard processing (A), and with NORDIC processing (B). The bottom row, likewise shows a comparison for the 0.7 mm isotropic resolution data with standard processing (C) and with NORDIC processing (D). This improvement in the tractography streamlines are fully consistent with the probabilistic tractography results given in Fig. 10. The 0.7 mm data set has different acquisition parameters and as such cannot be directly compared to the 0.9 mm data per se; it is included here only to show a more dramatic improvement possible with lower SNR data.

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