Uncertainty in denoising of MRSI using low-rank methods

Purpose Low-rank denoising of MRSI data results in an apparent increase in spectral SNR. However, it is not clear if this translates to a lower uncertainty in metabolite concentrations after spectroscopic fitting. Estimation of the true uncertainty after denoising is desirable for downstream analysis in spectroscopy. In this work, the uncertainty reduction from low-rank denoising methods based on spatiotemporal separability and linear predictability in MRSI are assessed. A new method for estimating metabolite concentration uncertainty after denoising is proposed. Automatic rank threshold selection methods are also assessed in simulated low SNR regimes. Methods Assessment of denoising methods is conducted using Monte Carlo simulation of proton MRSI data and by reproducibility of repeated in vivo acquisitions in 5 subjects. Results In simulated and in vivo data, spatiotemporal based denoising is shown to reduce the concentration uncertainty, but linear prediction denoising increases uncertainty. Uncertainty estimates provided by fitting algorithms after denoising consistently underestimate actual metabolite uncertainty. However, the proposed uncertainty estimation, based on an analytical expression for entry-wise variance after denoising, is more accurate. It is also shown automated rank threshold selection using Marchenko-Pastur distribution can bias the data in low SNR conditions. An alternative soft-thresholding function is proposed. Conclusion Low-rank denoising methods based on spatiotemporal separability do reduce uncertainty in MRS(I) data. However, thorough assessment is needed as assessment by SNR measured from residual baseline noise is insufficient given the presence of non-uniform variance. It is also important to select the right rank thresholding method in low SNR cases.


Stein's Unbiased Risk Estimate (SURE)
The SURE equations for SVT,from Candès et al (17), and SVHT, adapted from Ulfarsson and Solo (18), are reproduced in full here.
)*+(-,/) (12 (,412 where m and n are the dimensions of matrix M, is the threshold, ( is the i th singular value of M, is the standard deviation of the i.i.d. noise, and 3 = max ( , 0). SURE accurately predicts the MSE of the denoised spectral data from the singular values, matrix dimensions and i.i.d. noise variance of the input noisy data. An example of the SURE-predicted MSE in data identical to that shown in Supporting Figure S1 is shown below. Figure S1.

Supporting Figures
Supporting Figure S1: a 'Metabolite' maps for the three peak explicit rank-3 simulation data. b Spectra from three voxels of the same simulation data showing different relative 'metabolite' concentrations.
Supporting Figure S2: Underestimation of the rank of the data by the MP algorithm in noisy data. The solid horizontal lines show the estimated MP threshold in the noisy (blue) and very noisy (red) case. In the very noisy case, the rank is underestimated as 2 in the global case (a) and as 1 in the local case (b). Panel c shows example spectra and the relative noise levels.
Supporting figure S3: Monte Carlo covariance and estimated covariance for the global and local ST denoised single peak data. There is close agreement between them. All plots are shown with the same arbitrary colour scale.
Supporting Figure S4: Uncertainty (standard deviation) of the single peak amplitude as a function of noise standard deviation. This plot shows the same data as Figure 6 but resolved across all input noise levels. Solid lines show the uncertainty estimated by Monte Carlo simulation, dashed lines by the fitting algorithm, and dotted lines by the proposed bootstrap process. The line 'averaged data' shows the uncertainty that would be achieved if all untreated (noisy) voxels were averaged before fitting.
Supporting Figure S5: Equivalent to Figure 7 for the 'all unique' group of metabolites. a Mean uncertainty of the fitted concentrations of the 'all unique' metabolites expressed as a ratio to the original noisy data. Uncertainty measured by MC simulation is compared with that estimated by the FSL-MRS fitting algorithm. b Normalized RMSE of the noisy and denoised data comparing the fitted concentrations of the 'high signal' metabolites to that of the noiseless synthetic data.
Supporting Figure S6: Estimated and MC measured variance for a single voxel of the simulated 1H-MRSI data at each of the five different noise levels. Panel a shows the variance for the global ST method and b shows that for the local ST method. The estimated variance captures the structure of the true (MC) variance, but globally underestimates the magnitude.
Supporting Figure S7: Covariance estimated using the proposed method and MC measured covariance for a single voxel of the simulated 1H-MRSI data at a single noise level (9-minute equivalent). Panel a shows the covariance for the global ST method and b shows that for the local ST method. The main diagonal slightly underestimates the variance (as in Figure S6), nor does the estimated covariance fully capture the complex covariance structure of the actual (MC) covariance in either case.
Supporting Figure S8: Estimated variance at different noise levels by Monte Carlo simulation, FSL-MRS fitting and bootstrap fitting for a single combined resonance (a NAA+NAAG) and 'all unique' metabolites (b). This plot is equivalent to Figure 8 but for the different grouping of metabolites.