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MRI Below the Noise Floor
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) distribution3$$p_\gamma (\lambda)=\frac{\sqrt{(\lambda_+-\lambda)(\lambda-\lambda_-)}}{2\pi\gamma\sigma^2\lambda},\,\,\,\lambda_-<\lambda<\lambda_+\,\,\,\,\,(1)$$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 approaches6,7, 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 techniques8-12 make noise correlated13.

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-combine14.

  • 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 TOPUP15+EDDY16,17. 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-FEAT18 [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/kurtosis19 estimation. Fiber orientation distributions were estimated using mrtrix320 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.

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Figures

Fig.1 RMT Reconstruction. (A) From the rank-3 measurement $$$\mathcal{X}$$$, to locating and removing MP distribution within the spectrum of covariance matrix $$$C=XX'/N$$$, to further reconstruction steps. (B) coil noise decorrelation, (C) phase removal, (D) noise removal by removing MP principal components. Residuals r normalized by σ display no anatomy. (E-G) principal component distributions for each step. (G) Coil decorrelation enables perfect Gaussian residuals: $$$p(r)\sim e^{-r^2}$$$ down to $$$r^2=25$$$ or 1 ppm in the tail, i.e. 5σ precision in identifying noise.

Fig.2 RMT recovers signal from under the noise floor. Diffusion images acquired from 5 resolutions (2.0-0.8 mm isotropic voxels, rows) and for 8 diffusion weightings (b = 0 – 5000 s/mm2, columns), are reconstructed (A) in a standard way (individually), and (B) with the RMT approach. (C) Direction-averaged $$$S(b)$$$ over white matter, normalized by $$$S(b=0)$$$, get "squashed" by the relatively increasing noise floor (shaded) for the standard (blue) images. RMT-reconstructed $$$S(b)$$$ (red) pierce through the noise floor, yielding at least a 5x SNR gain at the highest resolution.

Fig.3 Accuracy and precision for diffusion MRI. (A) Mean diffusivity (MD) and radial kurtosis (RK) are shown for standard and RMT reconstructions across different resolutions. (B) Boxplots of MD and RK for each resolution are shown for each resolution across white matter. A red dashed line across MD and RK boxplots represents the median of RMT at the lowest resolution, which serves as a ground truth. (C) Table displays the relative error (coefficient of variation, CV) of MD calculated from $$$b≤1000$$$ images. RMT decreases the CV (increases precision) by up to 10x in a uniform ROI (ALIC).

Fig.4 fMRI temporal SNR gain. tSNR=mean/(standard deviation) over all t-points. (A) tSNR histograms and (B) maps of tSNR gain ratio. The activated hand knob region (red box) is further enlarged in Fig.5B. fMRI signals are shown for a voxel in white matter (C, no activation), and for the highest z-score voxel in the hand-knob region (D). Full-protocol RMT achieves the highest tSNR gain, matching tSNR at 0.8 mm isotropic resolution to that of the 2.4 mm isotropic (clinical resolution for pre-operative fMRI) reconstructed via standard method (A).

Fig.5 Joint RMT reconstruction of a multimodal exam at 0.8mm. (A) Maps comparing low-res (2.4mm), standard, and full-protocol RMT for an axial slice through the sensorimotor cortex. (B-E) Full protocol RMT achieves notably better noise removal compared to RMT on individual contrasts (cf. Fig.4). Moreover, joint RMT yields more coherent fiber orientations (E) for the white-dashed box outlined in the DEC-FA map. (F) Information is shared among contrasts: the sum of the number of significant components for each contrast is notably greater than P for the joint RMT on the full protocol.

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)
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