Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Authors: Onat Dalmaz
Arjun D Desai
Reinhard Heckel
Tolga Cukur
Akshay S Chaudhari
Brian Hargreaves
Venue: ICML 2025 poster (2025)
Keywords: MRI reconstruction
deep learning
noise
uncertainty quantification
Jacobian
sketching
Monte-Carlo
robustness
Primary area: applications->computer_vision
TL;DR: We introduce an efficient method to accurately quantify noise uncertainty in deep learning-based MRI reconstruction, achieving Monte-Carlo-like accuracy while significantly reducing computational and memory requirements.

Abstract: Accelerated MRI reconstruction involves solving an ill-posed inverse problem where noise in acquired data propagates to the reconstructed images. Noise analyses are central to MRI reconstruction for providing an explicit measure of solution fidelity and for guiding the design and deployment of novel reconstruction methods. However, deep learning (DL)-based reconstruction methods have often overlooked noise propagation due to inherent analytical and computational challenges, despite its critical importance. This work proposes a theoretically grounded, memory-efficient technique to calculate *voxel-wise variance* for quantifying uncertainty due to acquisition noise in accelerated MRI reconstructions. Our approach is based on approximating the noise covariance using the DL network's Jacobian, which is intractable to calculate. To circumvent this, we derive an *unbiased estimator* for the diagonal of this covariance matrix—voxel-wise variance—, and introduce a Jacobian sketching technique to efficiently implement it. We evaluate our method on knee and brain MRI datasets for both data-driven and physics-driven networks trained in supervised and unsupervised manners. Compared to empirical references obtained via Monte-Carlo simulations, our technique achieves near-equivalent performance while reducing computational and memory demands by an order of magnitude or more. Furthermore, our method is robust across varying input noise levels, acceleration factors, and diverse undersampling schemes, highlighting its broad applicability. Our work *reintroduces* accurate and efficient noise analysis as a central tenet of reconstruction algorithms, holding promise to reshape how we evaluate and deploy DL-based MRI.

Efficient Noise Calculation in Deep Learning-band based MRI Reconstructions
Onat Dalmaz Arjun D. Desai
Reinhard Heckel 3
Tolga Cukur Akshay S. Chaudhari
Brian Hargreaves 126
Abstract
applicability. Our work reintroduces accurate and efficient noise analysis as a central tenet of recon- struction algorithms, holding promise to reshape how we evaluate and deploy DL-based MRI.
Accelerated MRI reconstruction involves solving an ill-posed inverse problem where noise in ac- quired data propagates to the reconstructed im- ages. Noise analyses are central to MRI recon- struction for providing an explicit measure of solu- tion fidelity and for guiding the design and deploy- ment of novel reconstruction methods. However, deep learning (DL.)-ased reconstruction methods have often overlooked noise propagation due to inherent analytical and computational challenges, despite its critical importance. This work pro- poses theoretically grounded, memory-efficient technique calculate voxel-wise variance for quantifying uncertainty due acquisition noise in accelerated MRI reconstructions. Our approach approximates noise covariance using the DL net- work's s Jacobian, which is intractable to calcu- late. To circumvent this, we derive an unbi- ased estimator for the diagonal of this covariance matrix-voxel-vee variance-, and introduce a Jacobian sketching technique to efficiently imple- ment it. We evaluate our method on knee and brain MRI datasets for both data- and physics- driven networks trained in supervised and unsu- pervised manners. Compared to empirical refer- ences obtained via Monte-Carle simulations, our technique achieves equivalentt performance while reducing computational and memory de- mands by order of magnitude or more. Fur- thermore, our method is robust across varying in- put noise levels, acceleration factors, and diverse undersampling schemes, highlighting its broad
1. Introduction
Magnetic Resonance Imaging (MRI) has become indispens- able in clinical diagnostics, yet prolonged acquisition times can reduce patient throughput and exacerbate motion arti- facts. Parallel imaging (pMRI) (Pruessmann et al., 1999; Griswold et al., 2002) was developed to mitigate these chal- lenges by undersampling k-space and exploiting multiple receiver coils with distinct sensitivity profiles. Despite successfully reducing scan duration, pMRI inevitably in- troduces spatially varying noise amplificatinn quantified by the g-factor-due to the need to invert an ill-conditioned system. In regions where coil sensitivities overlap or are highly attenuated, the reconstruction leads to locally elevated noise levels (Pruessmann et al., 1999; Sodickson et al., 1999). Early pMRI research rigorously analyzed this spatial noise amplification, culminating in precise formulations that ensured reliable signal-to-noise ratio (SNR) performance to guide algorithm design and clinical implementation (Pruessmann et al., 1999; Sodick- son et al., 1999; Aja-Fernánder et al., 2014). For instance, the widely used SENSE algorithm explicitly incorporates coil sensitivities to minimize noise, reflecting how classical pMRI approaches typically maintain clear link between measurement noise and the reconstructed image.
In contrast, most deep learning (DL) reconstruction methods (Akçakaya et al., 2019; Knoll et al., 2020a; Yaman et al., 2022; Jalal et al., 2021: Jun et al., 2021; Wang et al., 2024; Sriram et al., 2020b) have not provided an explicit account of how noise from undersampled k-space propagates into the final image. A major obstacle lies in the highly nonlin- ear and complex nature of DL reconstructions (Chen et al., 2022) making it nontrivial to derive analytical frameworks to characterize how undersampling and learned regulariza- tion affect localized noise amplification. In response, the core emphasis in the design of DL methods typically sought maximization summary metrics like peak tignal-to-noise ratio (PSNR) or structural similarity index (SSIM) that do
Department of Electrical Engineering, Stanford University, Stanford, CA, USA *Department of Radiology, Stanford Univer- sity, Stanford, CA, USA 'Department of Computer Engineering, Technical University Munich, Munich, Germany Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey Department of Biomedical Data Science. Stanford Univer- sity, Stanford, CA. USA 'Department of Bioengineering, Stanford University, Stanford, CA, USA. Correspondence to: Onat Dalmaz <onat sonn@@staford..duss
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
not provide an acute assessment on the spatial noise distri- bution (Heckel et al., 2024; Adamson et al., 2023; Muckley et al., 2021). Thus, the interplay between noise propagation and neural network parameters and its influence on re- construction robustness- has received comparatively little attention (Darestani et al., 2021; Dalmaz et al., 2024).
of the noise covariance matrix is introduced; (ii) We show how the derived estimator can be efficiently implemented through a novel Jacobian sketching algorithm that lever- ages a complex- columns, modulated by the adjoint operator and the noise covariance. This approach offers significant advantages over traditional Monte-Carlo (MC) simulation- -based meth- ods by reducing computational and memory demands by order of magnitude or more; (iii) We rigorously evaluate the proposed method on knee and brain MRI datasets us- ing various network architectures, including data-driven and physics-driven models trained in paradigms ranging from su- pervised to unsupervised. Experimental results demonstrate that the technique matches MC simulation-bant empiri- cal references in noise calculation with average correlation 99.8% and error 0.8%. We further validate effectiveness of our method across different input noise levels, acceleration factors, and undersampling schemes, showing robustness across clinically relevant imaging scenarios. Collectively, our method aims to reestablish accurate and efficient noise analysis in reconstruction algorithms, enhancing the evalua- tion and deployment of DL.-based MRI.
sketching matrix with random-phase
Nevertheless, characterizing noise in reconstructions is pivotal for advancing DL-MRI, as noise directly impacts SNR-a fundamental image quality metric in MRI that strongly influences clinical utility (Breuer et al., 2009; Diet- rich et al., 2007). Although metrics such as PSNR and SSIM remain common in DL., these metrics provide image-wide summary measures of performance and do not capture local variations in noise or SNR across the image (Mason et al., 2020). Consequently, important diagnostic regions may be obscured by unpredictable noise amplification patterns that that are either invisible or averaged out when using these global metrics (Adamson et al., 2021; Knoll et al., 2020b). In contrast, explicit noise quantification by means of spatial variance maps can yield critical insights into how reconstruction algorithms handle and potentially amplify noise, shaping design decisions to improve robustness and reliability (Kellman & Mc 2005; Knoll et al., 2019), paving the way for clinically meaningful performance im- provements.
2. Related Work
Jacobian Sketching in Machine Learning. Jacobian sketching has been utilized in machine learning for diverse objectives, such as mitigating catastrophic forgetting in con- tinual learning (Heckel, 2022; Li et al., 2021) or reducing the variance of stochastic quasi-gradiem methods (Gower et al., 2021). In such applications, the primary aim is typi- cally to approximate Jacobian- or Hessian-related quantities to improve model optimization or regularization. Our work, however, leverages Jacobian sketching for fundamentally different purpose: computationally efficient quantitative analysis of noise propagation in nonlinear multi-coil deep MRI reconstructions. Specifically, we employ sketching to efficiently estimate the diagonal elements of the image covariance matrix (i.e., voxel-wise noise variances) arising from propagation of the measurement noise, rather than optimization or regularization objectives.
In modern unsupervised (Yaman et al., 2020; Xiang et al., 2023) or semi-supervised paradigms (Yurt et al., 2022), SNR metries can be especially valuable by enabling bench- marking of reconstructions without relying on fully sampled ground-truth data (Kastryulin et al., 2023). Moreover, un- derstanding noise propagation can inform tailored sampling schemes- -eaainnneenablinngge the identification of k-space points that contribute disproportionately to image variability (Alkan et al., 2024; Peng et al., 2022) and guide development of noise-aware architectures and training strategies (De- sai et al., 2023). Additionally, accurate noise characteriza- tion benefits downstream imaging such denois- ing, segmentation, and diagnostic analysis-by providing voxel-wise variance estimates that enhance algorithm ro- bustness and clinical interpretability (Dou et al., 2025; Wei et al., 2022). Ultimately, noise characterization strengthens image-quality assessments, fosters reliable DL--based MRI algorithms, and increases clinical confidence in deployed models (Chaudhari et al., 2021).
Noise Calculation in DL-MRI. Related work on quantify- ing noise in DL--based MRT reconstructions often employs MC simulations (Robson et al., 2008: Thunberg & Zetter- berg, 2007: Akcakaya et al., 2014; Dalmaz et al., 2024), repeatedly injecting synthetic perturbations into k-space and reconstructing numerous noisy realizations. This ap- proach can produce accurate estimates via a large number of trials which require considerable computational and mem- ory resources- and is thus impractical for large 3D or 4D volumes. Moreover, treating the network as a black box limits interpretability by obscuring how noise propagates or is amplified within the deep reconstruction pipeline. A
Our contributions are threefold: (i) We provide com- prehensive theoretical framework that provides insights into how acquisition noise in k-space propagates to image uncer- tainty in DL-based MRI reconstructions. Voxel-wise vari- ance of reconstructed images is linked to row vectors of DI. network's 's Jacobian, modulated by k-space correlations and the imaging operator. To circumvent calculation of the full Jacobian, an unbiased estimator for the diagonal elements
2

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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
recent theoretical result (Dawood et al., 2024) derives analyt- ical noise estimates for k-space interpolation networks, and thereby is not generalizable across different architectures. Furthermore, it involves computing the entire network Jaco- bian, which quickly becomes impractical for typical 2D or 3D MRI volumes. These limitations highlight the need for a versatile, scalable, and theoretically grounded approach to noise analysis-on one that avoids intensive sampling, pre- serves interpretability, and can be seamlessly integrated into diverse reconstruction paradigms.
(NV) estimates. This targets the aleatoric uncertainty due to acquisition noise, offering mathematically transparent solution distinct from prior UQ and noise calculation meth- ods.
3. Theory
3.1. Accelerated MRI
We acquire k-space measurements y C from an un- known image x C via linear imaging operator A C comprising coil sensitivity maps, Fourier en- coding, and sampling mask. In practice, y is often un- dersampled, and corrupted by noise n E Cmm. Thus, the forward model is:
Uncertainty Quantification in MRI. Prior works on uncer- tainty quantification (UQ) in MRI generally fall into three categories: (I) classical compressed sensing approaches, of- ten using Monte Carlo (MC) reconstructions from perturbed measurements to derive confidence intervals (Hoppe al., 2024); (2) Bayesian methods (Narnhofer al., 2021) or generative modeling techniques (Tezcan et al., 2020), such as Variational Autoencoders (VAEs) or those assuming i.i.d. Gaussian priors, to capture uncertainty, primarily epistemic; and (3) ensemble-based analyses focusing on epistemic un- certainty, for instance, by training multiple networks in parallel (Küstner et al., 2024).
Aa +
(I)
Acquisition noise n is modeled as complex Additive White Gaussian Noise (AWGN) with zero mean:
(O,
(2)
Some recent approaches utilize conformal prediction framework to construct distribution-fir free uncertainty inter- vals at the level of downstream task outputs (Wen et al., 2024). However, this method treats the reconstruction pipeline as a black box, does not model voxel-level uncer- tainty intrinsically, does not incorporate the physies of MRI acquisition, and requires calibration data to guarantee finite- sample statistical coverage; it does not estimate how acqui- sition noise propagates through multi-coil systems non- linear reconstructions. Another distinct approach by Edupu- ganti et al. (Edupuganti et al., 2021) introduced VAF-based probabilistic framework that models epistemic reconstruc- tion uncertainty a latent space. It utilizes SURE-based estimators reliinn on approximate Jacobian traces and MC sampling compute uncertainty maps, but under the assumption of uncorrelated, i.i.d., single-coil noise.
where Σn E Cm is the sample covariance matrix of the k-space data. At each k-space point, noise is corre- lated across coils but is independent across distinct k-space locations. This implies that a coil covariance matrix En fully captures the noise statistics. Yet, for simplicity in our linear- algebbace treatment and factorizations, we focus on the Hermitian positive semi-definite (HPSD) sample covari- ance matrix Σg Cm (see Appendix A).
3.2. Neural Network MRI Reconstruction
In practice, deep MRI reconstructions often follow one of two paradigms: (1) Unrolled Architectures, where a series of learned regularization blocks and data-consistency (DC) steps are iterated for K cycles (Hammernik et al., 2023; Liang et al., 2020; Fabian et al., 2022), or (2) Purely Data- Driven Mappings, where a feed-forward network performs a direct mapping (Mardani et al., 2019)). Despite their structural differences, both networks can be viewed as a non- linear function ff f(A that yields naive zero-filled (ZF) estimate. Without loss of generality, any least- squares
Our work uniquely closes a longstanding gap by providing the first rigorous, first-prinen prisnccrnispls derivation of how multi- coil k-space acquisition noise propagates through both MRI physics and nonlinear deep learning (DI.) networks. While exact voxel-wise variance requires the full, computation- ally intractable network Jacobian, core innovation is statistically rigorous, unbiased estimator for this variance. This estimator, derived from imaging and statistical theory, uniquely handles complex-valued signals, multi-coil encod- ing, physics-bass based models, and DL nonlinearities simulta- neously. We implement this via a practical, model-agme -ades--ggostic matrix sketching algorithm using Jacobian-vector products (JVPs) with random-phase vectors, achieving scalable, in- terpretable, and memory- efficcient voxel-wise noise variance
" y). where A is the adjoint operator
-inuude-iivereeeeee initialization could be used (e.g. At y); such choices would simply alter the linear operator involved in subsequent derivations. Thus, our analyses are agnostic to these architectural details, and simply require that f be differentiable so that its Jacobian exists (See Appendix B).
3.3. Noise Propagation
In MRI, we seek to quantify the voxel-wise variance of the reconstructed image x C" induced by the acquisition
3

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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
noise in k-space. Let C + C" denote learned reconstruction function (e.g., unrolled network), with Jacobian J
The variance of the i-th voxel (i.e. the i-th diagonal entry of Ex) is then:
af
of Oun E CnX".
Var(æ:)) diag( Exr)
|1:112:
(11)
DL pipelines commonly initialize (0)) A AHg y with the "noisy" measured data y yo nu, where yoo A æ. Then:
where 1 E C is the i-th row of L:
4 4- I; A σk
x (0) A" (yo n) A yo A zu A n, (3)
In principle, the variance of each voxel could be itcratively obtained via taking the Eg norm of i-th row of Jacobian, transformed and scaled by A and σ Although this direct approach would be computationally heavy (see Algorithm in Appendix), the analysis reveals how Σg. A and J together determine the voxel-wise noise distribution.
with 2o A yo indicating the noise-free component of this initial estimate. A first-order Taylor expansion of the reconstruction function f(.): (C -0--0" - around vo approx- imates:
(1)(0)) = (2xo + (x)) (z:00) no);
(4)
3.4. Estimating Noise via Jacobian Sketching
where J (300) C" is the Jacobian matrix of evaluated at Zu- From 3 and 4 the perturbation in the reconstructed image due to the noise is:
Here we present an algorithm for estimating the diagonal of the covariance matrix Σg in an unrolled MRI recon- struction, without explicitly forming or storing the Jacobian J E C or the image covariance Σn E C" We Products (JVPs), enabling
dæ = (xx(0) - Maxo) R (xo) (A"n) n)
instead rely on Jacobian-Veece sketch based approach that efficiently probes the Jacobian and covariance structure through random vectors. We first present the following results regarding the diagonal entries of implicit, complex-valued Hermitian covariance matrices.
Note that dæ also remains zero-mean:
[6oxe] (aro) A A 0.
(5)
It is also worthwhile to note that due to linearity, approxi- mate reconstruction noise shares the same Gaussian nature, with the covariance matrix:
Theorem 3.1 (UnBIASED DIAGONAL ESTIMATOR). Let Σ C be Hermitian, and let v E C satisfy
E[u] 0, Elvo0 =I
(12)
Ex E [8x 6 (x0) A A n (xo)
Define y ()v) O o", where is the Hadamard product, and is scalar complex conjugation. Then:
= (mo) A H Σx A J (x00)
(6)
Even though the Jacobian J, (xo) E Caxx exists, storing explicitly computing the full matrix is intractable for MR images, due dimensionality and massive data size. Nonetheless, (6) shows propagation of k-space covariance Σg through the adjoint operator A H and the network's Jacobian J.. Similar constructs for the noise matrix exist in other PMRI approaches (Pruessmann et al., 1999).
E [yy = diag [Σ)
Proof. Write Σ = [gg] and v = Un) each index
Then for
Evv); Σ Eij Ug-
Multiplying by of yields ys!
Factorization and Diagonal Entries. To simplify compu- tations, first perform Cholesky decomposition of Σx:
yi = (E+) = Σ
Eij Uy 2f.
!f Σ a o
(7)
==1
Taking expectation:
where 0 is the Cholesky factor of Σg. Substituting into (6), we obtain
x3
n Σ j=1
E
[my
E
Egg Ug 00)
Eg
Σ = A, (a AJH. =JJ, A ox J A on) H
(8)
j=1
(9) From 12, we note B[ U3 of ] = dey:
Elyil Σ Eig dig Eii-
Here, we see that Σg = L H with Cholesky factor:
L J A Tke
(10) Since this holds for all i, we have Ely) = diag E).

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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Lemma 3.2. Let Σ Cuxn be HPSD with a Cholesky fuctorization Σ L LL Suppose E Ch satisfies (12).
3.4.2. CHOICE OF RANDOM VECTORS
For unbiased estimation, we require random vectors C" satisfying (12). A natural design choice would be stan- dard complex Gaussian vectors where each v is drawn from ~ CO 1) independently. Yet, in real-valued ma- trix diagonal estimation problems, Rademacher vectors (i.e., =1) have been shown to reduce estimator variance com- pared to Gaussian vectors (Hutchinson, 1990; Bekas et al., 2007). In MR reconstruction, however, data and operators are inherently complex, thus we propose using a complex analogue, namely complex Rademacher (random-phase) vectors, where each v has unit magnitude and uniformly random phase:
Defining u Lu, we claim;
Elu w"] diag Σ)
Proof. Since u L L. its ith component is
U4=
Lay Vi-
Hence, the ith entry of the Hadamard product u O un is
OuOu:) = us (us)*
Eby Loy U5) ( Lie ui) k=1
12 = 0, Uniform(0, 2..].
Rewriting and taking expectation:
We show in Appendix D that both choices satisfy (12), though random- phasse vectors yield strietly lower estimator variance; ergo, selected for consequent experiments. Theo- retical error bounds for the estimator using random-phase vectors are provided in Appendix E.
E us (us)
Lag Li Bloy
J=1 k=
By (12), Ely w1] = -ōyk- Thus,
E Blus (us)*] Σ
Lag Lig = [LL" Ex.
4. Experiments and Results
j=1
Because this holds for each i, we conclude
4.1. Datasets and Experimental Setup
Blu @ u²) diag(E)
We performed experiments on two publicly available Satasets-Stanffrrt knee dataset and subset of the fastMRT brain dataset (Knoll et al., 2020c). The Stanford knee dataset consists of 8-channel 3D FSE PD-weighted scans, which we split into 14 subjects for training. 2 for val- idation, and 3 for testing. Each 3D scan was demodulated and decoded via a 1D inverse Fourier transform along the readout dimension, yielding 2D Axial slices with matrix size 320 x 256. The fastMRI brain dataset comprises 16-channel Axial T2- T2-weighted scans with matrix size 384 x 384, split into 54 training, 20 validation, and 30 testing subjects. We focus on using these test volumes primarily to evaluate the variance estimation performance of our framework, rather than to compare the reconstruction quality of different meth- ods. k-space data were retrospectively undersampled via 2D Poisson Disc undersampling masks; and coil sensitivities were estimated via J-SENSE (Ying & Sheng, 2007). We estimated the coil covariance matrix Σg for each slice using the outermost 5% of k-space points, where signals are mini- mal (Robson et al., 2008; Pruessmann et al., 1999). During inference, a unique and deterministic undersampling mask was used on each test slice for reproducibility.
Returning to (10), we have, L Jj A ok- Then, by Lemma 3.2,
u =- J, (ox w)) - B u u²] = diag (.
Thus, applying J, to suitably distributed random vectors reveals each diagonal entry of E.g
3.4.1. VECTORIZED IMPLEMENTATION: JACOBIAN SKETCHING
A naive approach would sample each vj E CmM, compute Loj = J (A a vz) individually, and accumulate uj. In- stead, we provide a vectorized algorithm that improves com- putational and efficiency (See Algorithm in Appendix):
1. Generate random matrix: Vs E Cmxs with columns vy s.t. [[uy]] O, and E/u, w/'l = Im-
2. Transform by ax and A:; Ws = o% Vs, then Ws = " Ws
Reconstruction Methods To comprehensively evaluate our approach, we selected six deep MRI reconstruction algorithms spanning different learning paradigms: super- vised VarNee (Sriram et al., 2020a), MoDL (Ag- garwal et al., 2019), U-Net (Ronneberger et al., 2015)), semi-supervised (N2R (Noise2Recon), VORTEX (Desai
3. Sketch J, via Ws: Us = J, Ws E CaXs.
4. Hadamard product w/Hermitian & Average:
Vsamples = Us o U E Ra
diag(E (En) = Vaaples 1s.
5

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Efficient Noise Calculation in Deep Learning-basee MRI Reconstructions
E2E-VarNet
MoDL
U-Net
VORTEX
N2R
SSDU
H00g
Resoniructon
/R25C
Secorerationn
Eecoraturtton
Hog:
Recesttation
Resintruction
Pro30pe
Propsued
Proposod
Empercc
Proposes
Empeecal
Propoced
Erpekcal
Proposed
Difference
100
10e2
10g
Figure 1. Each column corresponds to a distinct deep reconstruction method at R = 8, a = on knee data. In each column: (top row) shows ZF and reconstructed images; (middle row) displays NV maps derived by the proposed method and empirical simulations; (bottom row) presents difference and amplified (x 10) difference maps between the proposed and empirical variance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window.
et al., 2022a)), self-supprvised (SSDU (Yaman et al., 2020)); and network architectures: fully data-driven (U-Net (Ron- neberger et al., 2015)) vs. physics-driven unrolled (the rest). These methods cover a broad spectrum of deep recon- struction models for examining the generalizability of the proposed noise analysis framework across different architec- tures and learning paradigms. Additional details regarding methods including training, hyperparameters, architectures, and results are provided in Appendix H.
ror (NRMSE) between computed and empirical variance maps were measured, and meantstd across all test slices were calculated to quantify the performance of our method. Significance of the distributional differences between voxel- wise noise maps were assessed by using Wilcoxon signed- rank tests. Our code is available at http::/gittub..oom/nat dalmaz/deep www.sepppeeeeo.cccoe reion.
4.2. Generalizability Across Reconstruction Methods
Simulating Noise Levels. To investigate performance un- der varying noise scenarios, we simulate different input noise levels by scaling the estimated covariance matrix En = aE,, where is E 10, 2000 so that n ~ ((,,..).
To assess how well our variance estimator generalizes across different DL-based reconstruction methods, we fix R = 8 and Σn = Σg. then train each model (knee and brain) and estimate NV. Table I shows that our method achieves near-pent sear-perfect correlations to the gold standard, with no sta- tistically significant difference betweeen noise distributions (p 0.05). Although the U-Net yields slightly higher NRMSE (1.7%), these values remain small in absolute terms. One contributing factor to this might be U-Net's lack of iterative data-consistency, which applies a single- pass nonlinear denoising. This leads to both lower-fidelity reconstructions (See Table 9 in Appendix), and deviations from our method's 's linear approximation, resulting in minor local estimation errors.
Reference Empirical Baseline. For each experimental setting (noise level, acceleration rate, dataset) and image, we performed N 3, 000 MC trials (See Appendix G), each time adding simulated noise n drawn from Σg. We reconstruct each noisy measurement and compute empirical variance maps by measuring voxelwise sample variance across all trials. The empirical variance maps serve as a gold standard reference.
Variance Estimation. Variance maps computed by our proposed randomized Jacobian sketching algorithm were compared to the empirical variance maps (our benchmark) to assess the accuracy of the proposed method for captur- ing voxel-wise noise propagation. The sketching matrix Vs had S = 1000 column probing vectors, which provides a balance between the estimation accuracy and computa- tion (See Appendix F). Percent (%) Pearson Correlation Coefficient (PCC), and Normalized Root-Mean Mean-Square Er-
Figures I and 6 (Appendix) illustrate representative noise- variance maps from knee and brain slices, respectively. Our estimator generally reproduces the intensity ranges and spa- tial patterns observed in the empirical reference, showing only minor deviations near regions of sharp intensity transi- tions, such as tissue boundaries or areas with rapid signal variation. These localized discrepancies remain negligible relative to the overall noise distribution and do not affect
6

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Efficient Noise Calculation in Deep Learning-base bbased MRI Reconstructions
20 Poisson
1D Cartesian
1D Random
2D Uniform
Table 1. std PCC and NRMSE hetween noise-variance maps calculated by our method and by reference empirical baseline, across different DL reconstruction methods for knee and brain data
ITiner
27
at R 8, =
Method
Knee
Brain
Emonce
PCC (%)
NRMSE (%)
PCC (%)
NRMSE (%)
C2E-VarNet
99.9 + 0.0
0.7 0.0
99.9 0.0
0.5 0.1
MoDL
99.9 0.0
0.5
1.1 9.0
Effive
10c
U-Net
99.4 0.0
0.2
9.0
1.2
VORTEX
+.0
9.. 0.6
0.1
N2R
99.9 0.0
0.6
99.9
0.0
0.7
SSDU
99.9 = 0.0
0.4 0.0
99.8 + 0.0
0.9 +
Figure 2. Each column corresponds to different undersampling
Table Computational and memory efficiency (per slice) on knee dataset for standard unrolled architecture, i.c. used in E2B- VarNet, VORTEX, N2R, SSDU,
pattern (E2E- VarNet on knee data at R = 8, a = 1). In each col- umn: (top row) shows 7F and reconstructed images; (middle row) displays NV maps derived by the proposed method and empiri- cal simulations; (bottom row) presents difference and amplified (x10) difference maps between the proposed and empirical vari- ance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window.
Metric
Empirical
Proposed
Naive
Time 54.0 s
1.3 1.1 MB
880.2 s
Storage
3513.3 MB
1.1 MB
the overall reliability of the framework. Across variety of DL models, our estimator consistently provides robust noise estimates that align well with empirical simulations.
We also assessed the generalizability of our method un- der varying undersampling patterns, including ID uniform Cartesian and ID random Cartesian undersampling, and 2D uniformly random and Poisson-disc (default sampling pattern used in this study) undersampling (see Figure 10 in Appendix). For each pattern, we train a dedicated model, then estimate variance in representative slices. Figure 2 demonstrates that our noise maps retain strong agreement with empirical results under all sampling scchemes-whethe uniform or random. This consistency arises because the op- erator A inherently encodes the sampling mask; our method only relies on A as an implicit linear operator, making it watrssammiinnnnsostiii These results demonstrate robust- ness to substantial changes in sampling pattern and accel- eration rate, underscoring its versatility in diverse imaging scenarios.
4.3. Computational and Memory Benefits
Table 2 compares the computational time and memory usage for three variance-cstimatiion strategies for a single slice on the unrolled architecture from the knee dataset: Empirical, Proposed, and Naive (based on Eq. 11). Empirical sim- ulations take about minute and require storing multiple reconstructions. In contrast, Naive avoids large storage over- head, but its exhaustive row-by-row Jacobian calculation is significantly more time-consuming. In comparison, our proposed estimator balances the two ends: reducing the computation by over an order of magnitude relative to the empirical bascline by leveraging JVPs, while cireumventing storage of reconstructed images. These results illustrate how the proposed algorithm attains substantial speed and memory advantages, making it practical tool for routine use, especially on large datasets. See Appendix I for effi- ciency/scalability of our method with regards to network complexity and architecture, and Appendix J for practical comparison with the empirical baseline at low trial counts.
4.5. Robustness to k-space Noise Level
We systematically increase the k-space noise magnitude by varying as described in Section 4.1, thereby simu- lating growing covariance matrices (and thus SNR levels). Figures 3, 11 (Appendix) show variance maps for repre- sentative brain and knee slices from the proposed method and from empirical simulations, and Figure 13 (Appendix) shows the error between them as alpha grows. Although the reconstructed images become visibly noisier at higher a, networks still perform noticeable denoising. As a increases, empirical maps spread variability more uniformly across tissues, leading
4.4. Generalization to Various Undersampling Schemes
For the baseline noise level (a = 1), we vary the accelera- tion factor R. E {4,8, 12, 16, 24), training a separate model for each R. Figures 8 (Appendix) illustrate variance maps for representative knee and brain slices, respectively. Although higher acceleration factors R decrease the num- ber of k-space samples-thie thus elevating ill-posedness and noise-on noise estimations continue to align closely with empirical references across all R, remaining their accuracy over a broad range of undersampling rates.
more homogeneous noise pattern with- out accentuating different tissue types. For moderate noise scaling (a < 50), the proposed and empirical variance maps closely match. Note that extreme noise levels (50x-2001 correspond to SNR values well below 10-15 dB and thus far beyond what is considered clinically relevant in routine
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Efficient Noise Calculation in Deep Learning- MRI Reconstructions
Clinically
Clinically Inlelvvant
Σ& Σg
(SNR~ 36dB)
ΣA 10 (SNR=24dB)
2 (SNR~16dB)
Σ 50Σ; (SNR-6dB)
ΣA 200 6dB)
imaçe
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image
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image
Roconstruction
image
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image
Raconstrutioon
Proposer
Empincal
Proposed
Emprical
Proposed
Emprical
Proposed
Emprical
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Figure 3. Each column corresponds to a different noise scaling factor a resulting in varying SNR scenarios (E2E-VVarNe on knee data R 8). In each column: (top row) shows ZF and reconstructed images: (middle row) displays NV maps derived by the proposed method and empirical simulations; (bottom row) presents difference and amplified (x10) difference maps between the proposed and empirical variance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window. Note that 50, 200 corresponds to SNR values well under a 10dBs, which is often cited as threshold for diagnostic utility (Brown et al., 2014).
practice (Brown et al., 2014; Westhrook et al., 2018). At these regimes, differences emerge in low-intensity tissue or around sharp edges, reflecting how the network's learned regularization systematically suppresses part of the injected noise compared to estimation. Nevertheless, these dif- ference maps remain small under practical SNR conditions, indicating that our approach effectively captures noise prop- agation for typical or moderately elevated noise. We provide additional analyses and discussions in Appendices L and M.
(Robson et al., 2008). However, different DL reconstruc- tions exhibit distinct noise profile for the same slice, shaped by factors such as regularization, optimization criteria, and the network's 's learned biases. Moreover noise distribution often depends on the underlying anatomy and k-space data, rather than being uniformly distributed. Indeed, inspection of the noise maps suggests that spatial noise amplification in DL reconstructions mirrors features in ZF inputs and final outputs. This localized noise amplification has important clinical implications. In routine MRI assessments, radiol- ogists focus on specific regions of interest (ROI) such as cartilage in knee scans or pathology pathology-affected -affected areas in the brain, rather than examining the entire field of view uni- formly (Rubenstein et al., 1997). Therefore, even if global metrics indicate high reconstruction quality, a locally ele- vated NV in diagmostically relevant ROIs could compromise clinical interpretation (Sijbers et al., 1998; Lerski et al., 1993). The ability to accurately quantify noise and preserve reliable diagnostic signal in erucial ROIS is pivotal for the successful clinical translation of DL-MRI methods (Kiryu et al., 2023). Our method efficiently provides accurate spa- tially resolved NV maps that highlight regions with elevated uncertainty, enabling radiologists to review diagnostically vulnerable areas and supporting better clinical decisions.
5. Discussion and Conclusion
Our localization at zo presupposes that ||AA)))) rz] | the noise perturbation in the image domain must not significantly deviate from the baseline signal, for the first-order Taylor approximation to remain valid. Clinically, typical MRI acquisitions usually have sufficiently high SNR for this condition to hold, so the linear approximation re- mains valid (Brown et al., 2014; Westbrook et al., 2018). However, once noise levels become so large that || " rl|l approaches or exceeds| | []. the nonlinearities in f can longer be ignored. In such extreme magnitudes exceeding the baseline signal-our method may under- or over-estimate variance, as the learned regulariza- tion could reshape these large perturbations in ways that can not be accounted for by a linear approximation. Despite these nuances high noise regimes, the strong agreement between our method and empirical simulations under realis- tic acquisition conditions highlights the practical value and efficacy of our approach.
such as noise
The variance estimation framework presented in this work naturally aligns with the broader concept of uncertainty quantification in deep neural networks and primarily cap- tures aleatoric uncertainty, arising from measurement noise in k-space. By interpreting voxel-wise variance as an indi- cator of low SNR or ill-conditioned regions, this approach facilitates reliable confidence assessments in clinical set- tings, helping guide diagnostic decisions ("Is this subtle
Noise distribution in sully-sampled or linear reconstructions tends be fairly uniform, modulated by coil sensitivities
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
lesion credible?") and acquisition strategies ("Should addi- tional k-space samples be acquired here?"). Our proposed estimator efficiently computes voxel-wise uncertainty via JVPs, making it promising approach for large deep MRI reconstruction pipelines to explicitly characterize noise and enhance interpretability. In the future, this method could be combined with inherently stochastic reconstruction tech- niques, i.e. diffusion models (Ozturkler et al., 2023: Chung & Ye, 2022), also capture epistemic uncertainty arising from the model. While our current focus is on estimating the diagonal elements of the image covariance matrix (i.e., voxelwise variances), extending these techniques to cap- ture off-diagonal terms also remains an important frontier. For instance, in functional MRI, cross-covariance structures could illuminate shared noise sources or functional connec- tivity across different brain regions (Varoquaux al., 2010; Behrouz et al., 2024).
Voxel-wise NV information has been demonstrated en- hance downstream imaging tasks-such as denoising and segmentation-
multiple medical imaging modali- ties, including MRI, CT, and ultrasound. Classical adaptive filters like BM3D have been shown to significantly bene- fit from accurate NV information, enabling effective noise reduction without sacrificing critical diagnostic details (Han- chate & Joshi, 2020; Li al., 2014). Similarly, adaptive filtering methods that incorporate spatially varying noise variance maps improve diagnostic quality in low-dose CT and ultrasound imaging scenarios (Hariharan et al., 2020; Yu & Acton, 2002). In contemporary DL, explicitly incor- porating voxel-wise NV maps has consistently been demon- strated to reduce false positives and increase accuracy in segmentation and denoising networks (Dou et al., 2025; Wei et al., 2022). Thus, our proposed NV estimation frame- work not only enhances the interpretability of DL-based MRI reconstructions but also offers direct practical benefits by supporting robust performance improvements in critical downstream clinical tasks.
Many imaging modalities, such as CT and PET share similar challenges related to noise characterization, reconstruction under limited data, and estimating uncertainty in predictions (Wang et al., 2020). For instance, in CT and PET, where dose reduction is key objective, quantifying uncertainty due to measurement noise could enhance confidence in low- dose reconstructions, improving clinical decision- making (Yang et al., 2018). While our primary goal in this study was to specifically address multi-coil accelerated MRI due to its complex acquisition model and the widespread adoption of DL-based reconstructions where explicit noise propa- gation has been relatively understudied, our framework is fundamentally generalizable. It is not limited solely to Gaus- sian noise assumptions and can be extended beyond MRI. The method requires only known forward operator A de- scribing the measurement process (e.g., X-ray projections in CT) and an appropriate noise covariance Σg reflecting the modality-specific noise statistics. For instance, extend- ing our voxel-wise NV estimation approach to CT could involve incorporating
In conclusion, we introduced a model-agnostic Jacobian sketching algorithm for estimating voxel-wise NV in DL MRI reconstructions. Our approach efficiently and accu- rately captures how acquisition noise propagates through nonlinear, multi-coil reconstructions without costly MC sim- ulations. Extensive experiments demonstrate strong align- ment with reference variance maps across diverse architec- tures, suggesting that our method can be readily incorpo- rated into existing deep reconstruction pipelines. Beyond improving interpretability of DI. reconstructions, these noise estimates hold promise for guiding adaptive acquisition, as- sessing image quality, and facilitating clinically actionable uncertainty quantification.
noise models or transformations directly into Σg. Thus, while we consider such extensions an important future direc- tion, the current manuscript lays the foundational theoretical and computational framework necessary to address these broader applications.
The principles developed in this study hold significant po- tential for broader image-to-image tasks in computer vision, including super-resolution, denoising, inpainting, or synthe- sis by appropriate modifications based on the input noise distribution and forward corruption model (Ledig et al., 2017; Zhang et al., 2017; Pathak et al., 2016; Dalmaz et al., 2022: Özbey et al., 2023). In these applications, accurately estimating the spatial distribution of uncertainty at the pixel level can improve interpretability and trustworthiness, espe- cially in high-stakes scenarios such as medical diagnostics or autonomous systems (Kendall & Gal, 2017).
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Acknowledgements
nance, 16, 2014. ISSN 1532429X. doi: 10.1186/ 133-4499X-16-S1-222
Authors would like to thank Moritz Alexander Bolling for helpful discussions on first order methods and statis- tical tests. This work was supported by the following grants: NIH RO1AR077604, ROEB009055, R01CA249893, R01EB002524.
Akçakaya, M., Moeller, S., Weingärtner, S., and Uğurbil, K. Scan-specific robust artificial-neurall newral- for k- space interpolation (raki) reconstruction: Database-free deep learning for fast imaging. Magnetic Resonance in Medicine, 81:439-4555 2019. ISSN 15222594. doi: 10. 10.1002/mmm.22420
Impact Statement
This work aims to enhance the reliability and interpretability of deep learning-basee MRI reconstruction methods, poten- tially improving clinical
Alkan, C., Mardani, M., Liao, C., Li, Z., Vasanawala, S. S., and Pauly, J. M. Autosamp: Autoencoding k-space sam- pling via variational information maximization for 3d mri. IEEE Transactions on Medical Imaging, 2024. ISSN 1558254X. doi: 101109/TMM12022:443322.
by providing more accurate and trustworthy imaging tools. While its primary objective is to advance the field of machine learning, we acknowledge that such advancements may have broader societal implications, including increased accessibility to high-quality diagnostic imaging and more efficient health- care workflows. At present, we see no specific negative ethical societal issues that warrant particular attention.
Behrouz, A., Delavari, P., and Hashemi, F. Unsuper- vised representation learning of brain activity via bridg- ing voxel activity and functional connectivity. In Salakhutdinov, R., Kolter, Z., Heller, K., Weller, A., Oliver, N., Scarlett, J., and Berkenkamp, F. (eds.), Pro- ceedings of the 41st International Conference on Ma- chine Learning, volume 235 of Proceedings of Machine Learning Research, pp. 3347-3381 PMLR, 21 21-27 Jul 2024. URLhttpss:/proceee v235 2735//bhroond www.roeehh..aa.aeeeeaaaaa html.
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
A. Detailed Noise Model for Multi-Coil MRI
A.1. Coil Noise as Linear Combinations of Independent Sources
We assume that noise in each coil 7 arises from an ensemble of zero-mean independent Gaussian sources (Er J. each with standard deviation d, (Macovski, 1996). Concretely,
Thy (t)
Σ W(xxr) Cr
(13)
Because each &r is Gaussian, (1) must also be Gaussian. The weight w(y))) indicates how strongly the 7-th noise source couples to coil
A.2. Coil Covariance Matrix Σg
Summarizing these noise contributions yields n. x The coil covariance matrix Ex. where
(whrr oo Winrr (w)r,,))"
(14)
Here, (E)) is the NV in coil y, and off-diagonal elements encode coil-to-coil covariances, and ne is the number of channels in the imaging system. Since noise originates from physical sources that induce real-valued fluctuations, the cross-correlations between coils must satisfy Hermitian symmetry due Maxwellian reciprocity in electromagnetic induction (Roemer al., 1990; Brown al., 2014). Consequently, Σg is Hermitian (7, 7 swap with conjugate to mathematically) and positive semidefinite by construction (as all covariance matrices), since it can be expressed in the form
W WD "
where W contains W(rr)), and is diagonal of (o²). Thus, for any vector C?c,
y Σg v
|D w D11 2 0,
confirming positive semidefiniteness.
A.3. Block-Diagone Sample Covariance Σg
Suppose noise uncorrelated across n distinct k-space frequency locations, while retaining a coil-to-coil covariance Σn E CReX at each frequency (where is the number of coils). If n E C with ny Tee denotes the stacked multi-coil noise over n frequency bins, then the sample covariance matrix of n can be expressed as a block diagonal:
Σx diag Ex
Cmxm
n blocks
Equivalently, we may write
Σx Ing 2 Ex,
where I,, is the n x ny identity matrix and & denotes the Kronecker product. Each n. x n. diagonal block Σg captures the coil-to-coil noise correlation structure for one frequency index, and repeating these blocks ny times encodes independence across frequencies.
In practice, one can exploit a compact representation in which Σg is used directly during linear-algebraic factorizations operator- basse manipulations, rather than forming the full x block diagonal. Such compact approach reduces both memory and computational overhead significantly, as the repeated blocks need not be explicitly stored inverted. Consequently, we directly utilize the coil covariance matrix Σg in our computational framework for efficiency. Nevertheless, for theoretical analysis and proofs carried out in this study, it is more convenient to leverage the sample covariance matrix Σg. thereby making explicit how noise correlations factor into the imaging operator A and the subsequent derivations. We now show that if Σg is Hermitian positive semidefinite (HPSD), then Σg inherits these properties.
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Efficient Noise Calculation in Deep Learning- MRI Reconstructions
Hermitian.
A block-diagonal matrix Hermitian if and only if each diagonal block is Hermitian. Since Σ/ Ex.
have
Σ (Ing & Ex) 1
Ing Ex Σg. DE
Hence, Σg Hermitian.
Positive Semidefinite (PSD). A Hermitian Σg is PSD if and only if xf 2 0 for every x E C", Partition x into n sub-vectors, (x Xny ), each X; CDc Because Σg is block diagonal, we get
x H x Σ Ex x.)
If Σg PSD, then each x Σn x, 2 0. Summing these terms yields nonnegative result, thus Σg is PSD.
B. Existence of the Network Jacobian
Unrolled MRI networks alternate between regularization and data-consistency updates for K iterations, yielding a final reconstruction (K) (Hammernik et al., 2023; Liang et al., 2020). Concretely, each iteration has two steps:
1. Regularization:
(k)) = F(() x ( (x
where (-) is a neural network or learned operator (e.g., a CNN block) that acts as a trainable regularizer. Neural networks are well-known be differentiable with respect their inputs, assuming standard operations (convolution, ReLU, batch norm, etc.) (Goodfellow al., 2016). Hence, the map (k-1)) - ((k) is differentiable.
2. Data Consistency (DC):
z (R)
(k) A ("A* v)
Here, A E C is the linear imaging operator, and E C is the measured data. Since matrix-vector multiplication and addition/subtraacion are linear (and thus differentiable) operations, the map ((k) + (A) also differentiable.
Composing Differentiable Steps. iteration step
Because each iteration composes two differentiable transformations, the combined k-th
ik-
1) x
is itself differentiable. Consequently, unrolling K times yields chain of such transformations, implying the final recon- struction ( is differentiable function of the initial estimate (0)., Formally,
I (0) + (K) (DC (TM) ....o DC o F(l)) (z:).
K times
By the chain rule in multivariate calculus, this implies that f has a well-defined Jacobian J, E = at each input a: (9).
C. Verification of Random Vector Properties
Here, we first demonstrate that the introduced random vectors vectors-comples Gaussian and Rademacher satisfies the conditions stipulated in Theorem 3.1: zero mean and unit covariance.
C.1. Complex Gaussian Vectors
Definition C.1. A Complex Gaussian vector C" defined such that each element iy is independently sampled from complex normal distribution with zero mean and unit variance, denoted as Vy CN 1). This implies:
V a, + iby,
where aj and by are independent real- valued Gaussian random variables with a,, (00,0.5).
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Algorithm I Naive Per-Voxel Variance Calculation (Row-by-Row Jacobian)
Require:
Reconstruction function f: C" + C" with Jacobian J E C
Imaging operator A E Cmxn
k-space sample covariance matrix Σg E C
Ensure: Voxel-wise variance map { [ar((;;))
Compute the Cholesky factor of Σe:
En 0% o
Initialize empty array for variances:
Var(æ: = 0 E R
for each voxel = 1,...,7 n do
Create one-hot vector e, E R" with at index i, otherwise.
Backpropagate e, through f to get row i of Jy:
v - (J
Form
4 (V fi) A k.
(Imaging operator and noise correlations included.) Compute variance:
Var(x;; |14112
Store in Var(æ:;; +- Var(æ;;
end for
C.1.1. ZERO MEAN
The mean of each element vy is calculated as:
B|vy]= [[a,] + (E(b, +
Therefore,
E[0] = 0
C.1.2. UNIT COVARIANCE
Since aj and by are independent and each has variance of 0.5, the covariance between vy and is:
Elu Bl(aa :byy ibx)] B[ajax] Elbybu]
Given that aj and b, are independent across different indices:
1, if = k,
Elyyuk
0, if j + k.
Thus,
E
wro /1
=
C.2. Complex Rademacher Vectors
Definition C.2. A Complex Rademacher vector E C" is defined such that each element vy is independently sampled from the set {ej0 | Uniform(0, 2.)). In other words, each vj is of the form:
1 10
where fy is uniformly distributed over the interval [0, 2T).
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
C.2.1. ZERO MEAN
Since 0j is uniformly distributed over [0, 2T), the expectation of each 0/ is:
1 2m
Bluy] = [B(280)
e do 0
2x Jo
Therefore,
E|u] = 0
C.2.2. UNIT COVARIANCE
For k
Eluyoy! #1 13 = [E[]] = 1
For j o k,
B Bluye 30 -0ke = E [200
: TElee 50k = 0-
This separation of expectations is valid due to the independence of 0g and 0g. Hence,
E Elve"| #1 =
D. Variance of Complex Gaussian vs. Complex Random-Phase
Let E E Ca be Hermitian positive semi-definite matrix (e.g., covariance matrix), and let E C be random vector satisfying
E|o] o, B[vv"] I.
We define the following unbiased estimator for the diagonal entry Ex:
Y, (v)),
E E[
Ex.
Below, we derive the variance of Y; under two complex random-vector distributions: (i) complex Gaussian and (ūi) complex random-phase (Rademacher). In both cases, the key difference lies in the fourth moment of the entries.
D.1. General Expansion of |
First, we write out
Y,
Σ
Lis Uj x (ui)*
5=1
Hence,
14//2
Σ Lnj Uij
(u:)² x
Ess ux (1:) Σ 1=1
Since Σ is Hermitian, Di Ess, and (u:)** Expanding yields
|r/
Mij 2y
Ee vz) 10/12.
j=1
Taking expectation,
n Σ j=1 k=
7
E 2
Eij Eig E o
Because "" = I, we know Efey eg] = 552- For j * i, k + i, the entries uy and are uncorrelated in magnitude, so
E| 1y 1 by vī] = =E[|o//2] 1 65 1 -5k. When j = i or A: = i, we retain the fourth moment term [[[[]]]] Thus, generically,
Σ Egy 103 + |E E |0 $1.
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Efficient Noise Calculation in Deep Learning-basee MRI Reconstructions
Given EB [|yy]] for all j, ultimately get
E
| |]
Σ
Eigl
+
|Ead
E
Subtracting |> (which is [ then gives the variance:
Var[Y]]
Egg +
12a12.
Thus the crux is: Bl]o//")
D.2. Complex Gaussian Case
Distribution. Let each be drawn from a circularly symmetric Gaussian (C.((0, 1)), implying
E [ 1,
B[ls []] = 2.
(That is, if 02 /2 (I's + jzv with 24, ~ N/ 1) i..dd)
Variance Calculation.
Substitute B[[us|"] = 2 into the generic formula:
E Y?
jti
Therefore,
Var[Y;] = E + -2
2
Σ
Egg 12
j=1
Hence, for complex Gaussian:
Var[Y]] = Σ
Eas 2
==1
D.3. Complex Random-Phase (Rademacher) Case
Distribution.
Now let each v have unit magnitude and uniformly random phase,
|us| = 1, vi
500 0 Uniform(( [0, 2x).
Then
Variance Calculation.
Substituting [8[ [+]
into our generic formula:
E | =
PuP +
1.
Σ
Dis 1.
8+1
1=1
Subtracting [2 from this, we get
Var[Y]]
12.1
Thus, for complex random-phase (or "complex Rademacher");
1 Σ Var[Y.]
Egy
|2
j=1
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Efficient Noise Calculation in Deep Learning- MRI Reconstructions
E. Bounds on the Estimation (sketching) Error
Theorem E.1 (Bounds on the Sketching Error), Let Eg E C be a Hermitian positive semidefinite matrix and define its
off-diagonal component
Σx Σg diag(E.,) E.)
Let mm (Ex) be the diagonal estimator obtained by averaging independent complex random- (Rademacher) sketches. Then, for any d E (0, I.. with probability at least 8,
l2g(2/6) m
|| (Ex) - diag Ex) 112 S cV
|r
where >> O is an absolute constant independent of Σg and m.
Proof. Define the error vector
= m mm (Ex) - diag Ext))
By construction, is the sum of i.i.d mean-zero random vectors:
E2
2=1
where each e, arises from a single sketch of our estimator for Eg. In our case, by design the expected squared norm the error in single sketch given by
E| 1|
The key observation is that the variance of the scalar estimator for the ith diagonal entry is (From Appendix D.3):
Var [Y]] = Ex,úpl |E
so that the contribution of the diagonal is subtracted, yielding a tighter bound.
We now follow symmetrization and scalar-comparison argument (Vershynin, 2018)):
Symmetrization. Let e' be an independent copy e. Since [(xx) = || convex and E(e] = by Jensen's inequality
we have
| | 2k |E e/112*.
(15)
2. Scalar Comparison. For each sketch index 2, let e,,1 and e,,2 be independent copies of the single-sketch error e, C", and define
e e2,2-
Although w. may have complex entries (due to complex random-phase vectors in the estimation process), its squared norm || 12 is a real, nonnegative scalar.
Next, introduce an independent real Rademacher random variable r= E +11,+11 for each z, and define
W₂ = T2 |
Because w. has a symmetric (mean-zero) distribution in C", one can show that B[W.] = 0. Moreover, for any integer 2 1, since 2 1, we obtain
W 2k
E (r
| 112)
24 E alwwlll [1
Let
E
Σ
W2 Σ
rz aiw=l13 112)
2
2
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Algorithm 2 Noise Calculation in DL-based MRI reconstruction via Sketching the Network Jacobian
Require:
Imaging operator A E C
k-space sample covariance matrix Eg Cm
Reconstruction network f C + C with Jacobian J,-
S, size of random matrix Vs- Ensure: Approximate diag [x) = diag( Ex)) where Ex J, Mx AJ;
Compute Cholesky factorization: Σn o
Vs Cmx5 with columns {v,} sampled from complex random-phase distribution:
( c50%) 0.50 Uniform(O, 2n].
Ws 0k Vs
(E C)x5)
Ws + A H Ws
(E Cuxs)
Us (Ws)
Vamples f- Us !
(E R
diag [x)
samples 1s (R)
careful term-by-term expansion (exploiting the symmetry of each w, ++ w. shows that
2 2*
2k
Hence, we reduce bounding ||llz (a vector norm in C") to analyzing E' (a sum of real scalars). This paves the way for standard aab-Gaussianssbb- Ext.uuettxxpnnniialll tail bounds to be applied in a straightforward manner.
Concentration. Standard hypercontractivity and concentration results KhoorsaaKo Ascher, 2013)) imply that Sub-Gaussian with parameter
= Cym Σn F,
for an absolute constant C 0. Consequently, E² sub-exponential with parameter proportional to m EE 1| In particular, applying Proposition 2.7.1 from (Vershynin, 2018). we obtain that for any t 0,
| m
2()
t
< 2exp
co
E 12
for some constant cg >
Setting
t = cu Eg 2
log 8¹
we deduce that with probability at least - 6,
|ell m
2 g"
co
Ex
1
log
Recalling that e = m mm(im (Tmm22) (Ex) diag( Σx)) ) follows that
log(2/8)
(E.) diag Ex)lla S Voo
|E 1
Renaming Veg as e completes the proof.
22

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Efficient Noise Calculation in Deep Learning-based MRT Reconstructions
F. Sketching Matrix
Figure varies size of sketching matrix Vs. and measures the resulting NRMSE our variance estimator. As increases from 100 1,900, there is subtle decrease in error. This trend reflects the improved accuracy of the diagonal estimator when more random vectors probe the Jacobian structure. However, beyond certain point (approximately 1, 1000000010000 1,00-,1,000 200 in experiments), the NRMSE decreases more gradually, indicating diminishing returns in accuracy. Thus, while large S can yield more precise variance maps, it also incurs higher computational cost; our choice of = 1, 000 would yield favorable balance between accuracy and runtime for most practical settings.
1.35
1.2
1.0
NRMSE (%)
0.81
0.8
0.71
0.65
0.61
0.6
0.58
0.55
0.52
0.51
0.49
0.48
0.47
0.46
0.45
0.44
0.43
0.42
0.42
0.4
250
500
750
1000
1250
1500
1750
(5)
Figure NRMSE the proposed method over test slices vs. size the sketching matrix (S) 122E-VarNeton brain data = 8, = 1). As the size S of the sketching matrix Vs grows from 100 to 1,900, the Normalized Root-Mean -ean-Suuuure Error (NRMSE) of the variance estimator decreases. While additional probing vectors generally improve accuracy by capturing more the Jacobian's structure, the marginal benefit tapers off beyond about 1,000-1,200 vectors, suggesting practical trade-off between improved precision and increased computation time.
G. Empirical Noise Calculation
MC Trials. For each experiment (reconstruction method, noise level, acceleration rate, dataset) and slice, we simulated N 3, 000 MC trials. Specifically, draw the noise vector n from Σς for each trial, add the measured k-space, then reconstruct the image. Across these 3,000 reconstructions, compute voxelwise sample variance to form empirical variance map, which serves as the "gold standard" reference.
Convergence. Although typically fix 3,000 trials varying the trial count from 100 to 5,000. For smaller trial counts, the empirical variance estimate shows notably larger NRMSE relative to high-sample baseline (c.g., 10,000 trials), but this error diminishes we approach 2,500-3,00 3,00 trials, thereafter offering only gradual improvements. Consequently, while increasing the number of MC samples monotonically improves estimation accuracy, it quickly becomes computationally prohibitive to reconstruct each perturbed k-space many thousands of times.
stable reference variance, Figure illustrates the effect of
23

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Efficient Noise Calculation in Deep Learning-basee MRI Reconstructions
2.50
2.25
2,00
1.75
1.50
NRMSE
1.25
100
0.76
0.75
0.06
0.60
0.57
0.54
0.50
0.49
0:45
0.42
1000
2000
3000
4000
5000
Number
Trials (N)
Figure 5. Mean NRMSE of empirical baseline over test slices vs. the number of MC Trials (N) (E2R-VarNet VarNet on brain data at R = 8, = 1), We vary total number MC realizations from up to 5,000 compare the resulting variance maps high-sample reference (e.g., 10,000 trials). As more trials are included, the empirical variance estimation converges around N 3000, shown here by the decreasing NRMSE.
H. Reconstruction Methods
Here we provide detailed descriptions of the six deep MRI reconstruction algorithms evaluated in our study. Each algorithm falls under distinct combination of learning paradigm and architecture design, ranging from supervised to self-supervised, and from fully data-driven to physics-driven unrolled models. All neural architectures incorporate complex-var -valudd inputs via stacking real and imaginary parts, resulting in 2 input and output channels. Note that hyperparameters for each method were tuned on the validation dataset to ensure optimal performance. Unless otherwise stated, all models were trained via Adam optimizer (Kingma & Ba, 2017), with 31 0.965, 82 0.99. Here, provide detailed descriptions for each method. We used PyTorch for modeling (Paszke et al., 2019), and NVIDIA NVIDIA RTX A6000 for GPU acceleration.
H.1. Supervised, Physics-Driven Unrolled
H.1.1. E2R-VA (SRIRAM ET AL.,, 2020A)
V2E-VarNet is multi-coil unrolled reconstruction method that alternates between learned regularization blocks and data-consistency (DC) steps. Each iteration applies trainable neural network (often a ResNet or similar CNN) refine the current image estimate, followed by DC step enforcing agreement with the measured multi-coil k-space data.
We configure E2E- VanNet with 4 unrolled steps, each containing ResNet blocks (256 channels, kernel size 3). An & -norm reconstruction loss is used, and the network is optimized with Adam (Ir=10 4). Table summarizes additional hyperparameters for knee and brain MRI datasets.
H.1. MoDL (AGGARWAL AL., 2019)
MoDL (Model-Based Deep Learning) integrates conjugate-gradien (CG) steps for data-consistency with CNN denoiser block for learned regularization. Each unrolled iteration iteratively solves least-squares subproblem via CG and then applies a trainable ResNet denoiser.
We set MoDL to 4 unrolled iterations, each containing 10 internal CG steps. A ResNet (256 channels, blocks per iteration) serves as the learned regularizer. Table 4 details these hyperparameters, including reconstruction loss, Adam (Ir=10 and separate CG tolerances for knee vs. brain.
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Efficient Noise Calculation in Deep Learning-based MRT Reconstructions
3. Key Hyperparameters used for training E2E- VarNet knee and brain datasets.
Hyperparameter
Knee
Brain
Architecture
Unrolled Steps
4
4
Block Architecture
ResNet
ResNet
ResNet Channels
256
256
ResNet Blocks/ Step
2
2
Keree Size
3
3
Training
Loss Function
81
6
Batch Size
4
4
Grad. Accum. Steps
4
4
Base Learning Rate
10-4
10-4
Weight Decay
10-4
10-4
Max Iterations
120,000
50,000
Table 4. Key Hyperparameters used for training MoDL on knee and brain datasets.
Hyperparameter
Knee
Brain
Architecture
Unrolled Steps
4
DC Solver
CG (10 iterations, 10- CG (10 iterations, 10-4)
Block Architecture
ResNet
ResNet
ResNet Channels
256
ResNet Blocks Step
2
Kernel Size
3
Training
Loss Function
Batch Size
2
Grad. Accum. Steps
4
Base Learning Rate
10 10-4
10
Weight Decay
10
Max Iterations
80,000
80,000
H.2. Supervised, Data-Driven
H.2.1. U-NET (RONNEBERGER AL., 2015)
Unlike unrolled methods that tightly integrate the MRI forward model with iterative DC, U-Net fully data-driven, "one-shot" approach, mapping the zero-filled (or coil-combined) image directly to the reconstructed output. This yields faster inference, but it lacks physies- based constraints.
We use standard U-Net with levels of down/up-sampling 32 base channels, and blocks of fconv, relu, conv, relu, batchnorm, dropout]. }. An & reconstruction loss and Adam (Ir=10-3) train the network. Table 5 summarizes key parameters for knee and brain datasets.
H.3. Semi-Supervised, Physics-Drive Driven Unrolled
1.3.1. N2R (NOISE2RECON) (DESAL ET AL., 2023)
N2R leverages both a limited set of fully sampled ground-truth data and a larger set of undersampled data to train an unrolled network. Each iteration alternates between DC blocks and learned ResNet-basee regularization blocks, enforcing consistency only on sampled k-space points. Table 6 outlines key hyperparameters, including the k-space lj loss, 4 unrolled steps, 256-channel ResNet blocks, and consistency noise levels (10.2, 0.5])
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Table Key Hyperparameters used for training U-Net on knee and brain datasets.
Hyperparameter
Knee
Brain
Architecture
Base Channels
32
32
Depth (Pool Layers)
4
Block Sequence
{conv, relu, conv. relu, bn, dropout)
{cony, relu, conv, relu, bn, dropout)
Training
Loss Function
1
Batch Size
16
12
Grad, Accum, Steps
2
Base Learning Rate
10-3
10-3
Weight Decay
10
10 44
Training Length
200 epochs
80,000 iterations
Table 6. Key hyperparameters used training N2R on knee and brain datasets.
Hyperparameter
Knee
Brain
Architecture
Total Training Scans
14 (13 undersamp.)
54 (50 undersamp.)
Accelcration Factor
8
Unrolled Steps
4
Block Architecture
ResNet (256 ch)
ResNet (256 (256 ch)
ResNet Blocks / Step
2
Kernel Size
3
Semi-Supervision
Consistency noise std a Range
[0.2, 0.5]
[0.2. 0.5]
Training
Loss Function
k-space E
k-space &
Batch Size
1
Grad. Accum. Steps
4
Base Learning Rate
10
10-4
Weight Decay
10-4
10-4
Max Iterations
80,000
80,000
H.3.2. VORTEX (DESAI ET AL., 2022A)
VORTEX another ceni-supervised unrolled method, leveraging partial fully-sampled references alongside motion/noise augmentations. Each iteration incorporates DC steps and 256-channel ResNet blocks. Random motion/noise transformations align with MR physics, enforcing consistent reconstructions under both fully and partially sampled data.
Like N2R, VORTEX employs unrolled steps (2 ResNet blocks/step, 256 channels, kernel size=3). Loss is mixture of l and l2 in the k-space domain. Table 7 summarizes these hyperparameters, showing how motion/noise augmentations are tuned slightly differently between knee vs. brain data.
H.3.3. SSDU (YAMAN AL., 2020)
Self-Suuer Learning via Data Undersampling (SSDU) obviates the need for fully sampled targets by splitting acquired k-space data into "label" and "mask" sets. The "label" portion defines the reconstruction loss, while the "mask" portion is used within DC steps during the unrolled network's forward pass. Mask ratio 0.2 partitions 20% of acquired samples as "label," with the remaining 80% guiding DC.
SSDU's unrolled network matches E2E- VarNet's structure (4 steps, ResNet blocks at 256 channels). All training references were undersampled, partitioning k-space into label vs. mask sets. Table 8 outlines the key hyperparameters, highlighting the uniform mask, & k-space loss, and typical Adam optimization (Ir=10-4).
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Table 7. Key hyperparameters used for training VORTEX knee and hrain datasets.
Hyperparameter
Knee
Brain
Architecture
Acceleration Factor
8 54 (50 undersamp.)
Total Training Scans
14 (13 undersamp.)
Unrolled Steps
4 ResNet (256
Block Architecture
ResNet (256 ch)
ResNet Blocks Step
Kemel Size
3
VORTEX Motion/Noise Augmentations
Motion Severity (a) Range
[0.2, 0.5]
[0.1, 0.21
Noise STD (a) Range
[0.2, 0.5]
[0.1, 0.2]
Training
Loss Function
k-space l
k-space l
Batch Size
Grad, Accum, Steps
4
Base Leamnnng Rate
10
10
Weight Decay
10
10 10-4
Max Iterations
80,000
80,000
Table 8.
Key hyperparameters used for training SSDU on knee and brain datasets.
Hyperparameter
Knee
Brain
Architecture
Unrolled Steps
4
Block Architecture
ResNet (256 ch)
ResNet (256 ch)
ResNet Blocks/ / Step
2
Kemel Size
3
Self-supervision & Masking Setup
Mask Type
uniform 0.2
uniform 0.2
(Label Fraction)
Training
Loss Function
k-space
k-space &
Batch Size
Grad. Accum. Steps
8
Base Learning Rate
10-4
10 10-4
Weight Decay
10
10-4
Max Iterations
80,000
80,000
H.4. Reconstruction Results
All methods were trained and evaluated clean, non-noisy data. Table lists the mean Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) for knee and brain test sets across the six analyzed reconstruction methods. Overall, E2E- and MoDL achieve the highest PSNR values on knee scans, whereas N2R and VORTEX perform strongly on brain data particpartycllarlyy evident SSIM. U-Net and SSDU yield somewhat lower scores, potentially reflecting their more limited regularization strategies (fully data-driven in U-Net, fully self-supervised SSDU). Notably, the reconstruction performances are comparable to those reported the original studies or similar benchmarks, indicating that the networks were trained effectively. Overall, all methods surpass typical parallel imaging baselines in fidelity, confirming sufficient training convergence and general image quality for subsequent noise- variance analyses.
Computational Efficiency vs. Network Architecture
Table 10 compares empirical versus proposed variance method computation times for different neural architectures on the knee dataset. unrolled frameworks (E2E- VarNet, N2R, VORTEX, SSDU), the computation time increases with more "Steps" (i.e., DCCregularization blocks), reflecting the deeper or more recurrent structure of the unrolled pipeline.
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Table Quantitative reconstruction performance across methods knee and brain datasets
Method
Knee
Brain
PSNR (dB)
SSIM
PSNR (dB)
SSIM
E2E- VarNet
40.52
0.899
36.31
0.895
MoDL
40,60
0.899
36.44
0.847
U-Net
38.81
0.852
36.12
0.849
N2R
39.52
0.892
36.57
0,918
VORTEX
40.47
0.894
35.78
0.914
SSDU
39.88
0.862
35.51
0.885
Table 10. Computation Times Across Different Neural Architectures. Each entry lists the average time (in seconds) required to compute variance maps per slice on the knee dataset, comparing Empirical (MC) versus our Proposed aacobian-sketchen noise calculation. Unrolled architectures vary in steps, indicating the number of DC/regulaization blocks. MoDL uses Conjugate Gradient (CG) steps within each DC block, U-Net single-shot CNN without explicit unrolled steps. Architecture
Variant
Empirical time
Proposed time
2 Steps
27.9
0.2
4 Steps
54.0
1.3
Unrolled (E2E- VarNe, N2R, VORTEX, SSDU)
6 Steps
73.8
3.4
8 Steps
97.8
12.1
10 Steps
120.6
20.6
U-Net Unrolled w/CG-DC (MoDL)
(Ronneberger et al., 2015)
9,6
0.4
4 Steps
181.5
36.9
Nevertheless, our proposed method always remains order of magnitude faster than empirical references. U-Net requires fewer parameters (one-shot CNN without DC blocks), so the empirical approach is already relatively fast, yet we still observe a significant speedup with the proposed method. Finally, MoDL includes CG iterations within each DC step, driving up the empirical runtime significantly (181.5 s for 4 steps), while our method's runtime increases more modestly (36.9%) s).
These results indicate that both the wnrolled depth (number of repeated modules) and type of network layers (e.g. CG- based DC, convolutional blocks, potential self-attention units) influence the total variance-cone computation time in framework, albeit to lesser extent than empirical MC. For instance, substituting convolutional layers with self-attention could raise per-iteration overhead but might not drastically inflate the overall ancobian-sketch cost, since our approach only requires Jacobian-vecto products rather than a full Jacobian matrix. Ultimately, the proposed sketching method retains its computational advantage across diverse architectures, enabling efficient variance estimation without imposing strict constraints on layer choices or unrolling depth.
J. Practical Comparison with Low-Trial Monte Carlo
A natural question arises regarding whether empirical reference maps with smaller number of MC trials (e.g.. 50-100) might suffice to achieve competitive accuracy without incurring the high computational cost typically associated with MC sampling, To explore this, we compared the proposed sketching method (with sketch size S) and MC (with trials), each varied over a range from 5 to 130, and measured the NRMSE against a high-accuracy reference map computed using N = 10,000. Experiments were performed on VarNett at R 8, 1. As shown in Figure MC exhibits large estimation errors for N & 50, whereas our method maintains relatively lower errors even small S. larger values of S N, our approach remains more accurate while also being faster compute: for instance, at S N 100. MC requires ~ 1.8 seconds, whereas our method takes only ~ 0..22 seconds. This behavior aligns with our convergence experiments (see G), indicating that MC variance estimates tend to require three orders of magnitude trials (e.g., N x 1,000-10,000 to reach a reliable accuracy. In contrast, the sketching- based method attains lower eтror with at very few sample regimes. Hence, even when MC reduced to 500100 trials to speed up computation, the resulting noise estimates suffer in accuracy compared to the proposed sketching approach, highlighting a clear advantage both performance and speed.
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Efficient Noise Calculation in Deep Learning-basee MRI Reconstructions
EZE-VarNet
MoDL
U-Net
VORTEX
N2R
SSDU
25
Prspooel
Eripricl
Waysoed
Procencal
Propopod
Empriial
Proposed
Emperial
Propoeed
Erirnia
Propoee
Diffeence
Amprifidd
Diffrmnce
Arrel fod drt.
Eiffeence
Amoilliie
Cafferenca
Amplliied dif
Difference
Ampliiiedd dif
D fifrennc
Ampliiied del
Figure 6. Each column corresponds to a distinct deep reconstruction method (E2E- VarNet, MoDL, U-Net, VORTEX, N2R, SSDU) on brain data at R- 8, 1. In each column: (top row) shows the zero-filled (ZF) image and the final reconstructed; (middle row) displays variance maps derived by the proposed method and empirical simulations; (bottom row) presents the difference map hetween the proposed and empirical variance estimates, plus an "amplified" difference to further highlight spatial discrepancies. The color bars indicate each map's display window in arbitrary units of variance. Because MoDL integrates an iterative congute-gradient CG- based data consistency step that repeatedly applics the imaging and adjoint operator, the resulting noise map often mirrors the specific coil geometry and undersampling scheme. Regions with poorer coil coverage unsampled k-space lines accumulate higher residuals, as the iterative solver relies more heavily on the learned prior those areas. Consequently, MoDL's noise distribution visibly reflects the interplay of coil sensitivitics, the mask pattern, and the iterative gradient updates compared to data-dependent Jacobian structure, resulting in the distinct patterns
K. Undersampling Patterns
In this study, we analyzed a range of undersampling schemes to evaluate their impact on noise propagation and variance estimation. These include sampling patterns widely used in musculoskeletal imaging, such as ID Cartesian uniform and random undersampling, as well as 2D pseudo-random schemes like Poisson Disc and uniform random sampling (Johnson et al., 2023; Desai et al., 2022b) (See Fig. 10. Each pattern introduces distinet aliasing artifacts in the image domain- rraning from coherent aliasing artifacts in uniform Cartesian sampling for parallel imaging to noise-like incoherent artifacts in 2D) pseudo-random patterns that are better suited for DL algorithms. Here investigate these diverse schemes provide insights into how different undersampling strategies influence the efficacy noise calculation method (4.4).
L. Additional Results for Robustness to k-space Noise Level
Figure 14 demonstrates tight alignment around the diagonal, exhibiting near-unity R² values that confirm a strong match between empirical and theoretical STD at lower to high noise (1x-30x). As noise gets excessively larger to the clinically irrelevant regimes (50x-200x). our model overestimates the output noise relative to the reference: the best-fit regression line lies above the diagonal, and the scatter broadens. Nevertheless, the correlation remains consistently high, indicating that the model still captures the main trend in noise propagation while modestly overestimating variance in these extreme scenarios.
M. Additional Visualizations of Voxel-wise Noise Estimation
We provide additional visualizations in Figure 12 to offer an improved quamtitative perspective on how closely our method aligns with empirical references and where noise tends to concentrate. These visualizations, derived from our voxel-wise noise map estimations, provide further insights. For instance, histogram of the relative estimation error ((theoretical STD empirical STD) / empirical STD), as shown in the second panel of Figure 12, reveals the full distribution of voxel-wise deviations, demonstrating that most errors cluster near zero. Furthermore, a scatter plot of the calculated theoretical standard deviation (STD) versus reconstructed voxel intensity, presented in the third panel of Figure 12, shows a moderate positive correlation (R² xe 0.51), which is consistent with typical MRI acquisition where noise can scale with signal magnitude. Correspondingly, a seatter plot of the absolute estimation error (the absolute difference between theoretical and empirical
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Efficient Noise Calculation in Deep Learning-based MRT Reconstructions
R=4
R=8
R=12
R= 16
R=24
ZF Recoestruction
ZF
Reponstraction
Anconstuction
Raconstraction
Recanstruction
Enages
Proposed
Emptricll
Proposed
Empincal
Propased
Empinial
Propased
Empircal
Proposed
Emarical
N2.02 113p5
Amplified
Amolified
Amplfiied dift
Amplliied
Ampliiied dit.
Propored -Ebircaal
Figure 7. Fach column corresponds to different acceleration factor R (MoDI. knee at 1). In each column: (top row) shows the zero filled (ZF) image and final reconstructed; (middle row) displays variance maps derived by the proposed method empirical simulations; (bottom row) presents the difference map between the proposed and empirical variance estimates, plus an "amplified" diffcrence to further highlight spatial discrepancics. The color bars indicate each map's 's display window in arbitrary units variance.
STD) versus voxel intensity, depicted in the fourth panel of Figure 12, likewise indicates mild positive correlation (R² = 0.16), suggesting that regions with larger estimation discrepancies can coincide with brighter intensities.
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Efficient Noise Calculation in Deep Learning-based MRT Reconstructions
4=4
R=8
R=12
R=16
R=24
image
Maconstractionn
mage
image
Ruconstruction
image
Reconntrucioon
imMge
Reconstruction
Imagees
Proposed
Empincal
Propased
Emptrcal
Propesed
Emprical
Proposed
Empitiall
Proposed
Emprical
wo mapss
Difeeencc
Ampiiied dir.
Difference
Amplitied dir
Difference
Amplineed dift.
Differnnce
Ampliteec cit.
Difference
Ampliiied ditt.
Figure Fach column corresponds a different acceleration factor (E22- VarNet on brain at a 1). each column: (top row) shows the zero-fill frllefiled (ZF) image and the final reconstructed; (middle row) displays variance maps derived the proposed method and empirical simulations; (bottom row) presents the difference map between the proposed and empirical variance estimates, plus an "amplified" difference further highlight spatial discrepancies. The color bars indicate each map's display window in arbitrary units of variance.
12
Our Method (S)
Monte Carlo
10
8
NRMSE (%)
6
2
20
40
60
80
100
120
Size of the sketching matrix (S)/Number of MC trials (N)
Figure 9. Comparison of the proposed method with S (sketching matrix size) and empirical baseline with (MC trials). NRMSE (%) measured against high-accuracy reference map obtained with N 10,000 (VORTEX brain 1). For small sample sizes (S 50), Monte Carlo exhibits markedly higher error, while our method yiclds lower error at all examined sketch sizes. Even at N 100, the proposed approach offers less with ordge--f-magniuude reduction in computational time (0.12s vs. 1.8s for MC).
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
2D Poisson
1D Cartesian
1D Random
2D Uniform
Reference
Mask
Image
Mask
ZF Image
Mapk
ZF image
Mask
image
Figure 10. Different undersampling schemes analyzed in this study. For representative slice, 7F images generated by the undersampling masks are showcased, along with the sampled reference. Coherent and incoherent aliasing artifacts are generally exhibited by ID and 2D undersampling masks, respectively.
Olnically relevant
Clincallly Irrelevant
ΣE Eg (SdR=37dB)
Σx 102, (SNR=24dB)
20Σ. (SNR-17dB)
Σg 503 , (SNR=10dB)
200 (SBR--3dB)
ZF image
Resumdruction
2F image
Resomtruccion
2F Image
Resomstrution
ZF image
Recomstuction
2F image
Reconsoructinn
Proposed
Empriicl
Froposed
Emplrical
Proposed
Emplrical
Proposed
Emptrical
Proposed
Emptrical
Differance
Ampliied
Differance
Amplitted
Difference
Amplitted
Difference
Differnnce
Figure 11. Each column corresponds to a different noise scaling factor a resulting in varying SNR scenarios VerNeet on knee, at Rt 8). In each column: (top row) shows the zero- fillef (ZF) image and the final reconstructed: (middle row) displays variance maps derived hy the proposed method and empirical simulations: (bottom row) presents the difference map between the proposed and empirical variance estimates, plus "amplified" difference further highlight spatial discrepancies. The color bars indicate each map' display window in arbitrary units of variance. Note that 50-200 corresponds to SNR values below ss 10dB (i.c., well under an SNR of 10-15 that often cited as threshold for diagnostic utility.
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Efficient Noise Calculation in Deep Learning-based MRT Reconstructions
Empinical
STD vs.
Intensity
Reit Rogrcike iihe
Powl
0.3301
Key couin
0.033
0.04
Theseeica STD
0.2308
0.03
0.0001
0602
0.03
Ronk
0.08
03000
Emprica STO
Reledive EmT Ihhe emiil
Veed Irtenityitty Recoossructedd
Wooel Intorsy [Recorstructei
Figure 12. Additional visualizations for voxel wise noise standard deviation (STD) analysis. (Left) Scatter plot of theoretical STD vs. empirical STD, showing high correlation (Rt² 1.00). (Second from left) Histogram of relative errors between theoretical and empirical STD, indicating errors are centered around zero. (Third from left) Scatter plot of theoretical STD vs. reconstructed voxel intensity, showing positive correlation (R² 0.51). (Right) Scatter plot of absolute error in STD estimation reconstructed voxel intensity, indicating mild positive correlation (R² = 0.16).
25
NRMSE
Clinically Irrelevant
20
<> 40 corresponds to low SNR( <: dB)
NRMSE (%)
15
10
R6
6.23
5
4.31
2.36
0.92
1.13
0.56
0
100
101
102
α
Figure 13. Mean NRMSE between reference empirical variance maps and variance maps from our method across test slices were plotted against increased noise levels (E2E-VarNet VarNet knee, R 8). Here, is plotted log scale and progressively scales the noise covariance matrix Σ = a.v, simulating higher noise, and accordingly lower SNR levels.
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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
Clinically relevant
(6dR)) Empiiical vp Theertical STD
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302 (168))14635)
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Emo ricl vs Theorer
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Vatanes Nricl apoereet
14000
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3,11-
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0203
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0039
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Frii7tel 3771
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Σg 1002, (SNR200d8)
200 15N(t=-6015)
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Figure 14. For representative brain slice, empirical variances (horizontal axes) were compared to variances calculated our method (vertical axes) under successively larger noise scales: a 10, 30, 50, 100, 200 (E2F. VarNet, at = 8). each panel, diagonal line represents ideal one-to-one correspondence hetween empirical and calculated estimates, while the solid line indicates the best-fit linear regression (with its R² value indicated). Each data point corresponds to single voxel's empirical vs, calculated variance, visualizing how closely they match at different noise levels.
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Reviews:
[{"review_id": "Q8EowKJ7TQ", "reviewer": "Reviewer_SQro", "summary": "Authors propose a technique to calculate voxel-wise variance for quantifying uncertainty that stems from acquisition noise in accelerated MRI reconstructions. Authors propose to estimate the noise covariance using an approximation to the Jacobian of the neural network. The approximation is done through an unbiased estimator for the diagonal of the covariance by a sketching via random-phase vectors. The sketching algorithm is evaluated on knee and brain MRI datasets for data and physics driven networks trained in supervised and unsupervised manners. Method is shown to be robust against varying input noise levels, acceleration factors, and diverse under-sampling schemes.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review, and valuable feedback which we believe significantly enhanced our work. We addressed your valuable suggestions, and clarifed of our novelty and contributions.\n\n## 1. Novelty and Technical Contributions\nWe respectfully disagree with the assertion that our paper merely reuses existing matrix-sketching techniques. Our work introduces three key innovations:\n1. We provide the first rigorous, first-principles derivation of how k-space acquisition noise propagates through both the MRI physics model and nonlinear DL networks. Prior approaches have largely relied on Monte Carlo sampling—computationally expensive and lacking interpretability. Instead—our approach yields a scalable, interpretable solution with mathematical transparency grounded in imaging and statistical theory.\n2. We show that exact voxel-wise variance requires the full network Jacobian—computationally intractable in MRI. Our key innovation is a statistically rigorous unbiased estimator that efficiently approximates this covariance without explicitly forming the Jacobian. Crucially, we handle complex-valued signals, multi-coil encoding, physics-based forward models, and deep networks simultaneously—the first viable approach for quantifying acquisition-induced uncertainty in deep MRI reconstruction.\n3. We implement our estimator via an efficient and practical matrix sketching algorithm that probes the Jacobian with random-phase vectors. Our method leverages Jacobian-vector products, enabling fast, memory-efficient uncertainty quantification at scale, while being DL model agnostic. In practice, this removes a major bottleneck in bringing noise-aware reconstruction into clinical and research pipelines.\nThis suite of theoretical, algorithmic, and applied innovations represents a substantive advancement in uncertainty quantification for DL-based MRI.\n\n## 2. Motivation for Noise Variance Calculation\nWe appreciate the reviewer’s concern and agree that the clinical motivation for voxel-wise noise variance estimation merits clarification.\nVoxel-wise noise variance estimation is crucial in MRI because global metrics can obscure significant localized noise variations that impact diagnosis [1-3,8]. Studies show that localized noise impedes detection of subtle pathologies despite good global quality metrics [3,7]. Rubenstein et al. demonstrated that cartilage defects remain undetected with insufficient local SNR, even when global metrics appear acceptable [2]. Furthermore, global SNR and CNR don't consistently correlate with diagnostic accuracy [8].\nThis issue is particularly relevant in deep-learning MRI reconstructions, where spatially varying noise profiles create diagnostic uncertainties that global metrics fail to capture [4-7].\nOur method efficiently provides accurate spatially resolved noise variance maps that highlight regions with elevated uncertainty, enabling radiologists to review diagnostically vulnerable areas and supporting better clinical decisions [6,7].\nFor additional evidence, we refer to **2. Utility of Noise Variance Estimation on Downstream Tasks** in our response to reviewer viwd.\n\n[1] Sijbers, MRI, 1998\n\n[2] Rubenstein, AJR, 1997\n\n[3] Lerski, MRI, 1993\n\n[4] Dou, NMR, 2025\n\n[6] Kiryu, Radiographics, 2023\n\n[7] Knoll, MRM, 2020\n\n[8] Ohlmann et al, Br J Radiol, 2016\n\n## 3. Ablation Study on the choice of probing vectors\nThanks to your valuable inquiry, we now conducted an ablation study, and found that our theoretical findings (Appendix D) match with new empirical findings; Proposed random-phase vectors (test set NRMSE of 0.7 for knee, 0.5 for brain) yields lower errors than their Gaussian counterparts (1.1 for knee, 0.8 for brain), indicating lower estimator variance.\n\n## 4. Length of proofs\nWe thank the reviewer for suggesting the identity to streamline our proofs. Applying this improved the brevity and conciseness. \n\n## 5. Missing citations\nWe kindly refer the reviewer to rebuttal for reviewer q9G3 for discussions on **1. Extended Related Work and Clarification of our contributions**, including Edupuganti et al. (2020) and Wen et al. (2024).\n\n## 6. Error Bound for Diagonal Estimator  \nOur estimator $r_m(\\Sigma_x)$ (using $m$ complex random-phase sketches) satisfies:  \n$$\n\\|r_m(\\Sigma_x)-\\operatorname{diag}(\\Sigma_x)\\|_2 \\leq c\\sqrt{\\tfrac{\\ln(2/\\delta)}{m}}\\|\\overline{\\Sigma}_x\\|_F\\quad\\text{(w.p.}\\geq 1-\\delta)$$where $\\overline{\\Sigma}_x=\\Sigma_x-\\operatorname{diag}(\\Sigma_x)$.\n- Error scales as $\\mathcal{O}(1/\\sqrt{m})$ with $\\|\\overline{\\Sigma}_x\\|_F$.\n- Large $\\|A^H\\|$ or $\\|J_f\\|$ increase off-diagonal coupling, amplifying $\\|\\overline{\\Sigma}_x\\|_F$; well-conditioned systems reduce it.\n\n*(Proof in revised manuscript.)*\n## 7. \nWe hypothesize that distinct MoDL noise map reflect coil geometry and undersampling patterns via the iterative CG data-consistency step, rather than being driven by image features.\n## 8. \n- $()^*$: Complex conjugation for scalars \n- $()^H$: Hermitian operator for matrices", "claims_and_evidence": "Several claims do not have convincing evidence.\n\n1) In the conclusion section authors mention \"... even if global metrics indicate high reconstruction quality, a locally elevated noise variance in diagnostically relevant ROIs could compromise clinical interpretation.\". However, there is no concrete evidence provided in the paper to support these reasoning.\n2) Although in real-valued matrix sketching Rademacher vectors are known to have lower estimator variance compared to Gaussian vectors, the choice of using complex Rademacher vectors is not ablated. It is not clear whether it is strictly better than complex Gaussian random vectors for estimating noise maps both empirically and theoretically.", "methods_and_evaluation_criteria": "Benchmark datasets and the evaluation criteria makes sense for the problem at hand.", "theoretical_claims": "Yes, verified the proofs for Theorem 3.1 and Lemma 3.2. Although they are correct, they are unnecessarily long.", "experimental_designs_or_analyses": "As mentioned in the \"Claims and Evidence\" section, I have some concerns with the experimental analysis. Please refer there to avoid duplication.", "relation_to_broader_scientific_literature": "The estimation of the noise variance serves as a measure of uncertainty quantification. This is relevant to a line of work that uses network Jacobian and SURE estimator [1], and [2] which utilizes tools from conformal prediction to provide uncertainty quantification.\n\n[1] Edupuganti, Vineet, et al. \"Uncertainty quantification in deep MRI reconstruction.\" IEEE Transactions on Medical Imaging 40.1 (2020): 239-250.\n\n[2] Wen, Jeffrey, Rizwan Ahmad, and Philip Schniter. \"Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction.\" European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.", "essential_references_not_discussed": "Some of the missing citations are: [1] which quantifies the risk in the reconstructed images using Stein's Unbiased Risk Estimator (SURE) via the end-to-end network Jacobian. [2] using tools from conformal prediction to obtain risk maps.\n\n[1] Edupuganti, Vineet, et al. \"Uncertainty quantification in deep MRI reconstruction.\" IEEE Transactions on Medical Imaging 40.1 (2020): 239-250.\n\n[2] Wen, Jeffrey, Rizwan Ahmad, and Philip Schniter. \"Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction.\" European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.", "other_strengths_and_weaknesses": "***Strengths:***\n* Paper is easy to read.\n* Application of matrix sketching as a means to estimate noise variance in MRI reconstruction setting is interesting.\n* Experimental results are conducted over a diverse set of configurations.\n\n***\n***Weaknesses:***\n* Technical contribution is weak. The paper essentially boils down to applying matrix sketching ideas to MRI reconstruction networks.\n* The theorems and lemmas (in chapter 3) are unnecessarily cluttered.\n* The need for estimating the voxel-wise noise variance is not motivated enough.\n* Please see the suggestions and questions for more.", "other_comments_or_suggestions": "***Suggestions:***\n* The proofs for Theorem 3.1 and Lemma 3.2 can be made significantly shorter with the simple observation that $a \\odot b = diag(ab^T)$ when $a$ and $b$ are vectors. Then the proofs become single line.\n\n***\n***Typos:***\n* $v^*$ vs. $v^H$ usage is not consistent.", "questions_for_authors": "1) Authors mention that \"... even if global metrics indicate high reconstruction quality, a locally elevated noise variance in diagnostically relevant ROIs could compromise clinical interpretation.\". Is there a concrete example of such a case? How is this work alleviating the issue? While I understand the interpretation of the noise maps as uncertainty quantification, my main concern is that this work does not go beyond trying out existing matrix sketching ideas in the context of MRI reconstruction.\n2) How does sketching with random-phase vectors compare against random Gaussian vectors? In practice do you observe significant enough difference in the estimation error?\n3) Why is the noise map in Figure 6 for the MoDL method look significantly different than the others?\n4) Are there bounds on the sketching error? How does it depend on the properties of the linear operator and the Jacobian?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ob9iPwhV80", "reviewer": "Reviewer_q9G3", "summary": "The authors have developed a computational- and memory-efficient estimator of the voxel-wise variance in MRI reconstruction for uncertainty quantification in the reconstruction. The method is evaluated on MRI data and compared to Monte Carlo-simulations. The method is simple and easy to understand and computationally efficient while providing very similar results compared to Monte Carlo.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive feedback. Thanks to the reviewer for thorough examination of the manuscript, we implemented all suggested corrections (typos, cross-refs, terminology) and cleaned up the references. We also added an appendix overview outlining its structure for improved readability. We addressed your valuable suggestions, and clarifed novelty and contributions of our study, especially in relation the existing works which we have discussed in great detail below, with newly included references. For a detailed breakdown of our contributions, please also see rebuttal to **reviewer SQro: 1. Novelty and Technical Contributions**. Per your recommendation, we also included new additional experiments outlined in the rebuttal to reviewer viwd:  **2. Additional Downstream Task Experiments**.\n## 1. Extended Related Work and Clarification of our contributions\nWhile Pruessmann [8] analytically computed noise variance for linear SENSE with Cartesian sampling, our method addresses nonlinear deep reconstructions with complex operators and random undersampling—where noise propagation is significantly more challenging.\n\nUnlike our approach, Wen [1] uses a conformal prediction framework that constructs distribution-free uncertainty intervals at the level of downstream task outputs, without modeling voxel-level uncertainty or incorporating the physics of MRI acquisition. Their approach treats the reconstruction pipeline as a black box and requires calibration data to guarantee finite-sample statistical coverage. They do not estimate how acquisition noise propagates through multi-coil systems or nonlinear reconstructions.\n\nEdupuganti [2] introduced a VAE-based probabilistic framework that models epistemic reconstruction uncertainty in a latent space and utilizes SURE-based estimators—relying on approximate Jacobian traces—and Monte Carlo (MC) sampling to compute uncertainty maps under the assumption of uncorrelated, i.i.d, single-coil noise. In contrast, our approach analytically propagates correlated multi-coil MRI noise through nonlinear deep reconstruction networks using network’s Jacobian as an operator via randomized sketching.\nIn addition to these, in our revised submission, we will include references to additional studies [2–7] that were discussed under Uncertainty Quantification in MRI in Related Work section but inadvertently omitted from the bibliography. \nOur work uniquely closes a longstanding gap by providing the first analytical framework to quantify how acquisition noise in multi-coil k-space propagates through nonlinear deep MRI reconstructions. Existing methods do not analytically model this process, and thus do not solve the same problem. Consequently, direct comparisons are not appropriate; our MC baseline sufficiently represents these methods’ performance within the multi-coil setting, while our analytical estimator addresses a distinct, previously unresolved challenge in aleatoric uncertainty quantification. \n\n## 2. Additional Visualizations\nWe performed additional visualizations of voxel-wise noise maps ([link](https://postimg.cc/Mn49Rcgx)):\n1. A histogram of the relative estimation error reveals the full distribution of voxelwise deviations, showing that most errors cluster near zero.  \n2. A scatter plot of the calculated noise maps vs. voxel intensity demonstrates a moderate positive correlation, consistent with typical MRI acquisition where noise can scale with signal magnitude.  \n3. A scatter plot of the absolute estimation error vs voxel intensity likewise shows a mild positive correlation, confirming the reviewer’s hypothesis that noisier regions often coincide with brighter intensities.\n\nAs the reviewer suggested, these visualizations indeed offer an improved quantitative perspective on how closely our method aligns with empirical references and where noise tends to concentrate, supplementing direct map comparisons.\n\n## 3. Statistical Tests for Noise Variance Distribution Comparisons \nWe initially reported the results of a two-sample t-test result on page 5, line 264. Following the reviewer’s suggestion, we expanded our analysis to include further statistical tests:\n- Normality Check A Shapiro–Wilk test yielded a p=0.01<0.05, indicating that the distribution of voxelwise variance differences does not follow a normal distribution.  \n- Since normality was violated, we employed a Wilcoxon signed-rank test to check whether the variance maps from our method and the MC reference originate from the same distribution. We found no statistically significant difference between these two distributions (p=0.75>>0.05). Hence, our variance estimates align with the MC reference at both the mean and distributional levels, despite the non-normality in voxelwise differences.\n\n[1] Wen ECCV 2024  \n[2] Edupuganti TMI 2020  \n[3] Hoppe ECCV 2024  \n[4] Edupuganti TMI 2021  \n[5] Tezcan TMI 2020  \n[6] Narnhofer TMI 2021  \n[7] Küstner MRM 2024  \n[8] Pruessmann MRM 1999", "claims_and_evidence": "The claims made are proven in a clear and straight-forward way. The empirical evidence supports the utility of the proposed method.", "methods_and_evaluation_criteria": "I think the results are fairly clear, but the noise map estimates can be difficult to visiaully evaluate. I would encourage the authors to provide other means as well to illustrate the reconstruction and highlight when it works well and when/where it fails. For instance, why not make a scatter plot with the uncertainties as a function of different voxel intensities. This would highlight that the uncertainties (likely) increase as the image intensities increase, for instance.\n\nYou say that a t-test was performed on the \"difference between noise distributions\", but it is not clear what test was actually performed, and where are those results presented? In this case, it would be better to compare the distributions rather than the means (through the t-test). The uncertainty profiles could be quite different, even though the means be the same.", "theoretical_claims": "The proofs were easy to follow and appears correct.", "experimental_designs_or_analyses": "To increase the evidence in favour of your method, I would encourage you to include at least one more experiment. I would also add a quantification of the results, for instance a comparison between the estimated variances as mentioned above.", "relation_to_broader_scientific_literature": "I'm missing an exposition of and comparison to previous work in uncertainty quantification in MRI reconstruction. There is quite some work done in this field. It would also be good to compare the proposed method to other noise estimation methods, to further strengthen the evidence in favour of the proposed method.", "essential_references_not_discussed": "As mentioned above, I think an exposition of and comparison to previous work in uncertainty quantification in MRI reconstruction is missing.", "other_strengths_and_weaknesses": "The paper is well-written and clear.\n\nIt is said that the proposed method was used before, by Pruessmann et al. (1999), but this is not discussed under related methods. What are the differences, and what in the present work makes it novel compared to previous work?", "other_comments_or_suggestions": "Some minor comments:\n - The distinction between \\Sigma_x and \\Sigma_k could be stated explicitly.\n - Line 150, left column: Double ending parenthesis.\n - Line 156, right column: Cross-reference should be equation (6) and not (8), right?\n - Line 208, right column: It would be good to make it clear that the algorithm description is in the appendix.\n - Results: It would be better if you presented standard errors instead of standard deviations, so that the results can be directly compared.\n - Line 368, left column: \"4.1\" should perhaps be \"Section 4.1\" instead.\n - I would recommend that you clean up the references.\n - Line 1362: Lower case t after full stop.", "questions_for_authors": "Line 259, right column: What is R? Needs to be explained.\n\nIf you make the novelty of the contribution clear, especially in relation the existing works, I would consider increasing my score.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7pZYvp8AVc", "reviewer": "Reviewer_viwd", "summary": "The authors propose a scheme for efficient voxel-wise noise estimation for Deep-learning based accelerated MRI algorithms. The method builds upon estimating the noise covariance from the DL-models Jacobian by using an unbiased estimator for the diagonal elements (of the cov-matrix) via the proposed Jacobian sketching approach.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s thoughtful comments and recognition of our extensive experimentation across diverse DL reconstruction methods and MR imaging scenarios. Below we explicitly address each concern, highlighting the relevance and impact of our approach in downstream medical imaging via extended discussion and new experiments. We then describe generalizability the other medical imaging tasks and modalities. We’d like to also refer the reviewer to **1. Novelty and Technical Contributions** and **2. Clinical Motivation and Importance of Noise-Variance (NV) Estimation** points in the rebuttal for reviewer SQro. We believe these points, holistically clarifies our study’s contributions to the broader scientific field of medical imaging.\n \n## 1. Utility of Noise Variance Estimation on Downstream Tasks\nVoxel-wise NV estimation has been extensively shown to enhance downstream imaging tasks such as denoising and segmentation across modalities including MRI, CT, and ultrasound. For example, classical adaptive filters like BM3D significantly benefit from NV information by effectively removing noise without compromising clinical details [1,2]. Similarly, adaptive filters that incorporate spatially varying noise variance improve diagnostic information in low-dose CT and ultrasound imaging [3,4]. In deep learning contexts, explicitly incorporating voxel-wise NV maps consistently improves performance by reducing false positives and increasing accuracy in segmentation and denoising networks [5,6].\n\n[1] Hanchate, SN Appl Sci., 2020\n\n[2] Li, Med Phys, 2014\n\n[3] Hariharan, Phys Med Biol, 2020\n\n[4] Yu, IEEE TIP, 2002\n\n[5] Dou, NMR Biomed., 2025\n\n[6] Wang, Eng Appl AI, 2022\n\n## 2. Additional Downstream Task Experiments\nTo directly demonstrate the practical value of our method, we have added two new downstream proof-of-concept experiments in the revised manuscript:\n- 1. Non-blind Denoising: A Noise2Noise network was trained on the reconstructed brain images (a) assuming uniform image noise and (b) leveraging our predicted voxel-wise variance maps as noise-level priors. The variance-informed network achieved a 0.9 dB gain in PSNR, highlighting direct improvements from accurate local variance estimation.\n- 2. Variance-Guided Segmentation: A U-Net was trained for cartilage segmentation with the predicted variance maps included as additional input channels, resulting in a 1.1% improvement in Dice coefficient and fewer false positives, demonstrating how noise-aware segmentation enhances clinical accuracy.\n\nThese experiments provide clear evidence that our analytical voxel-wise NV estimates are not merely theoretical metrics but actively enhance clinically relevant imaging tasks.\n## 3. Extension Beyond MRI for Other Medical Imaging Tasks and Modalities\nWe agree that extending our voxel-wise noise variance (NV) estimation approach to imaging techniques beyond MRI (particularly CT with its Poisson-like noise characteristics) would further demonstrate the generalizability and impact of our method. However, our primary goal in this study was to specifically address multi-coil accelerated MRI due to its complex acquisition model and the widespread adoption of DL-based reconstructions, where explicit noise propagation has been relatively understudied. Notably, our framework is fundamentally generalizable and is not limited solely to Gaussian noise assumptions. The method requires only:\nA known forward operator $A$ describing the measurement process (e.g., X-ray projections in CT).\nAn appropriate noise covariance $\\Sigma_k$ reflecting the modality-specific noise statistics.\nFor instance, extending to CT could involve incorporating Poisson-Gaussian noise models or variance-stabilizing transformations directly into $\\Sigma_k$. Thus, while we consider such extensions an important future direction, the current manuscript lays the foundational theoretical and computational framework necessary to address these broader applications.", "claims_and_evidence": "The work includes claims to accurately reconstruct voxel-based noise levels in DL multi-coil based accelerated MRI reconstruction agnostic to the method and forward operator, which as to my opinion are fulfilled by the conducted  experiments.", "methods_and_evaluation_criteria": "One notable weakness of the paper is that the presented method is only evaluated in the domain of MRI reconstruction, leaving its performance on other modalities like CT untested. This especially interesting since CT has Poisson distributed noise, which is a significant difference to the assumptions in MRI. Demonstrating the performance across different (medical) imaging techniques would significantly enhance the work’s contribution.\n\nFurthermore,  the evaluations conducted do not provide evidence that the estimated noise levels improve performance in downstream tasks. While the method efficiently and accurately reconstructs voxel-wise noise variance, it remains unproven whether these estimations actually translate into benefits for tasks like non-blind denoising or segmentation. Demonstrating an improvement in downstream applications is critical to validate the practical utility of the proposed technique.\n\nHowever, despite this shortcoming, the method is evaluated on a broad range of reconstruction approaches, including supervised, self-supervised, fully data-driven, and physics-driven methods.", "theoretical_claims": "The theoretical part of the manuscript seems sound and well-presented.", "experimental_designs_or_analyses": "As mentioned above, for the task of mere noise estimation in MRI reconstruction, the experimental design appears valid. However, the evaluation lacks evidence that the estimated noise levels benefit downstream tasks or works on other modalities/noise distributions.", "relation_to_broader_scientific_literature": "The contribution to the broader scientific field of medical imaging seems somewhat limited due to the shortcomings addressed above.", "essential_references_not_discussed": "The manuscript seems to include essential references.", "other_strengths_and_weaknesses": "See above.", "other_comments_or_suggestions": "As mentioned above, the work conducted is solid. However, in my opinion, to justify a publication, the authors should consider extending the work as addressed above.", "questions_for_authors": "No further questions/remarks from my side.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents a compelling application of sketching techniques to the problem of noise estimation in MRI reconstruction. All three reviewers agreed on the strength of the application and the thoroughness of the evaluation within the MRI domain, with suggestions for improvement including evaluation on other modalities. However, a key difference in opinion emerged, with reviewer SQro questioning the technical novelty of the method, arguing it primarily builds upon existing sketching techniques. I agree, while the application is strong, the concerns raised by reviewer SQro regarding the limited technical novelty significantly undermine the value of the paper for the broader ICML community.
Decision: Accept (poster)