Gregory Lemberskiy1,2, Steven Baete1, Jelle Veraart1, Timothy M. Shepherd1, Els Fieremans1, and Dmitry S Novikov1
1New York University School of Medicine, New York, NY, United States, 2Microstructure Imaging INC, New York, NY, United States
Synopsis
We develop random matrix theory (RMT)-based MRI image reconstruction able to increase SNR by up to 10-fold, and to radically increase resolution for routine clinical acquisitions. RMT offers an objective criterion for separating signal from noise across all coils, voxels and MRI contrasts, by utilizing the redundancy in MRI measurements. We demonstrate RMT on a 0.8x0.8x0.8 mm3 neuro exam that includes a series of
multiple T2w, T1w, diffusion, and fMRI images on a 3T clinical scanner. RMT can serve as a paradigm for reconstructing multiple contrasts, enhancing image quality for low-field scanners, increasing MR value, and improving biomarker precision.
Introduction
The signal-to-noise ratio (SNR) defines MR image quality and fundamentally limits the MRI resolution. Here, we unlock an untapped reserve for an order-of-magnitude SNR
increase. We employ random matrix theory (RMT)1-5 to
develop a noise-removing image reconstruction methodology able to recover the signal from below the noise floor, by utilizing the information across multiple coils, voxels and MRI
contrasts. The resulting clinical 3T neuro protocol achieves $$$0.8\times0.8\times0.8$$$mm resolution for fMRI, advanced diffusion (dMRI), $$$T_1$$$ and $$$T_2$$$.
Methods
We propose to utilize the redundancy in measuring the tissue of interest from multiple vantage points, Fig.1. Including different
rf coils, MRI
measurements (e.g., anatomical, diffusion and functional), and adjacent
voxels. This redundancy allows us to gain SNR in each of $$$M\sim100$$$
distinct measurements almost by the same $$$\sqrt{M}$$$ as if we were averaging over the
same measurement $$$M$$$ times.
RMT identifies
pure-noise principal components $$$\lambda$$$ in a redundant measurement, i.e., when it has a small number $$$P$$$ of significant components relative to the size of the $$$N\,\times\,M$$$ data-matrix $$$X$$$.
While thermal noise makes $$$X$$$ full-rank, in the limit $$$N,M\to\infty$$$ the pure-noise components are distributed according to the Marchenko-Pastur (MP) distribution
3$$p_\gamma (\lambda)=\frac{\sqrt{(\lambda_+-\lambda)(\lambda-\lambda_-)}}{2\pi\gamma\sigma^2\lambda},\,\,\,\lambda_-<\lambda<\lambda_+\,\,\,\,\,(1)$$Here $$$\sigma$$$ is the noise level, $$$\gamma=M/N$$$, and $$$\lambda_\pm=\sigma^2(1\pm\sqrt{\gamma})^2$$$ are the boundaries of MP distribution. We identify and remove the contribution (1) to the PCA spectrum of $$$X$$$.
Contrary to previous approaches
6,7, we start from a
rank-3 data matrix$$$\,\mathcal{X}$$$:
coils$$$\,\times\,$$$
measurements ($$$T_2w,\,T_1w$$$, dMRI, fMRI, etc)$$$\,\times\,$$$
voxels, size $$$N_c\times N_m\times N_x$$$, which we reshape to size $$$M\,\times\,N$$$ to make it closest to a square, $$$M\,\sim\,N$$$. Redundancy in coils and measurements enables a sharp separation between pure-noise and signal, and lets us minimize the size $$$N_x\sim27-45$$$ of the local spatial
patch.
The distribution (1) relies on uncorrelated noise. Hence, we apply RMT after coil noise decorrelation and phase-unwinding, and before k-space filling, since standard k-space filling techniques
8-12 make noise
correlated
13.
Experiments were performed on a 3T Siemens
Prisma system with a 20-channel head/neck coil,
of which 16 elements were enabled. After RMT denoising, standard reconstruction steps were applied, including adaptive-combine
14.
- Diffusion at different
resolutions (Fig.1-3): A 3-slice brain slab from a 31 y/o female volunteer was acquired with different isotropic resolutions [0.8,0.9,1.0,1.5,2.0] mm
(original SNR = [2.5,3.1,4.0,7.9,14.9] at $$$b=0$$$), $$$T_R=3$$$s,$$$\,T_E=132$$$ms, with 8 $$$b$$$-shells [0,250,500,1000,2000,3000,4000,5000] along [1,6,6,12,20,30,30,40] directions, respectively. Scan time was 7min per resolution.
- Multimodal protocol at $$$0.8\times0.8\times0.8$$$mm resolution (Fig.4-5): A 5-slice segment of the brain from a 28 y/o
male volunteer with various contrasts was acquired through the hand knob of the
precentral gyrus, using GRAPPA10 $$$R=2$$$, pFourier 6/8 with POCS (original SNR
3.31 at $$$b=0$$$). Total scan time was 12:30. There were $$$N_m=256$$$ total measurements
[70 SE diffusion ($$$10 b = 0, 20 b = 1000, 40 b = 2000$$$), 13 SE IR images for $$$T_1$$$ mapping ($$$T_R=4000\,$$$ms, $$$T_E=77$$$, $$$BW = 1125.2$$$ Hz/pix, $$$T_I=[33-2000]$$$), 13 SE images for $$$T_2$$$ mapping ($$$T_R=4000$$$,$$$\,T_E=[80-200]$$$), and 160 GE images for a
right index finger tapping fMRI ($$$T_R=1000$$$ms, $$$T_E=39$$$ms, FA$$$=62^\circ$$$, using 8 on/off 20s epochs). The standard protocol at 2.4 mm resolution was generated by using 1/3 of the k-space in each dimension.
Processing
All images were aligned using TOPUP
15+EDDY
16,17. The z-scores to
measure cortical activation were determined after motion correction, slice
timing correction, brain extraction, high-pass temporal filtering, and BOLD modeling using FSL-FEAT
18 [Fig.4,5]. 36 SE images for $$$b=0$$$ were used to calculate proton density $$$S_0$$$, $$$T_1$$$ and $$$T_2$$$ times, via:$$S(T_I,T_R,T_E)=S_0[1-2e^{-T_I/T_1}+e^{-T_R/T_1}]e^{-T_E/T_2}\,\,\,\,\,\, (2)$$Diffusion images were processed for WLLS diffusion tensor/kurtosis
19 estimation. Fiber orientation distributions were estimated using mrtrix3
20 for up to $$$l=6$$$ spherical-harmonics order.
Results & Discussion
In Fig.1, we summarize RMT-reconstruction, and
demonstrate perfectly white-noise residuals down to $$$5\sigma$$$ (Fig.1G).
Fig.2: RMT recovers high-$$$b$$$ diffusion images from under the noise floor, down to $$$0.8\times0.8\times0.8$$$ voxels, yielding at least $$$5\times$$$ noise reduction.
Fig.3: RMT provides up to $$$10\times$$$ precision increase for mean diffusivity. Moreover, RMT successfully removes the noise-floor bias in both MD and RK, where mean values at high and low match.
Fig.4: RMT increases temporal SNR over $$$5\times$$$ for task-fMRI, enabling unprecedented $$$0.8\times0.8\times0.8$$$ voxels at 3T. The fMRI time-courses
(normalized by their mean values) show clear noise reduction, with the full-protocol RMT reconstruction yielding the highest gain in
precision.
Fig.5: RMT reaches its fullest potential on a multimodal neuro protocol, where information is shared between contrasts. Indeed, the number $$$P$$$ of information-carrying components outside of the distribution (1) after full-protocol-RMT is notably smaller than the sum of $$$P$$$ from RMT on individual modalities (fMRI, diffusion, etc). Conclusions & Outlook
- By objectively identifying the low-rank structure12,21-25 across all MRI contrasts/voxels/coils, RMT reconstruction unlocks an untapped $$$\sim10\times$$$ SNR reserve, essentially for free.
- RMT prompts acquiring all clinically necessary contrasts with the same highest possible resolution and jointly denoise
all of them, whereby each measurement alone would drown in noise.
- RMT is complementary to hardware improvements (high field, multi-coils, strong gradients). It can further improve image quality for low-field scanners reducing cost and increasing MRI value.
- RMT has potential to enhance the performance of modern acquisitions and reconstruction algorithms, and increase accuracy and precision of imaging biomarkers lowering sample sizes for human studies.
- RMT reveals information content of MRI, and serves as a paradigm to combine multiple contrasts, without relying on priors or training data.
Acknowledgements
We thank Daniel Sodickson and Kawin Setsompop for discussions. This work was
supported by the NIH under awards number R01NS088040
(NINDS) and R01EB027075 (NIBIB), and by the Center
of Advanced Imaging Innovation and Research (CAI2R,
www.cai2r.net), a NIBIB Biomedical Technology Resource Center: P41 EB017183. References
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