Self-Supervised Diffusion MRI Denoising via Iterative and Stable Refinement
Authors: Chenxu Wu
Qingpeng Kong
Zihang Jiang
S Kevin Zhou
Venue: ICLR 2025 Poster (2025)
Keywords: Diffusion based models
Self-supervised MRI denoising
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Magnetic Resonance Imaging (MRI), including diffusion MRI (dMRI), serves as a ``microscope'' for anatomical structures and routinely mitigates the influence of low signal-to-noise ratio scans by compromising temporal or spatial resolution. However, these compromises fail to meet clinical demands for both efficiency and precision. Consequently, denoising is a vital preprocessing step, particularly for dMRI, where clean data is unavailable. In this paper, we introduce Di-Fusion, a fully self-supervised denoising method that leverages the latter diffusion steps and an adaptive sampling process. Unlike previous approaches, our single-stage framework achieves efficient and stable training without extra noise model training and offers adaptive and controllable results in the sampling process. Our thorough experiments on real and simulated data demonstrate that Di-Fusion achieves state-of-the-art performance in microstructure modeling, tractography tracking, and other downstream tasks. Code is available at https://github.com/FouierL/Di-Fusion.

Published as a conference paper at ICLR 2025
SELF-SUPERVISEE DIFFUSION MRI DENOISING VIA ITERATIVE AND STABLE REFINEMENT
Chenxu Wu! Qingpeng Kong!? Zihang Jiang S. Kevin Zhou Zhou 1,2.A,
School of Biomedical Engineering, Division of Life Sciences and Medicine, USTC MIIRCCEE Center, Suzhou Institute for Advance Research, USTC
State Key Laboratory of Precision and Intelligent Chemistry, USTC KKey Laboratory of Intelligent Information Processing of CAS, Institute of Computing Technology, CAS wucchexx,, qpkong27 @mail. wutuctt... edu. en, { |jhh103,, s. kevin. zhou) @ @mmail com
ABSTRACT
Magnetic Resonance Imaging (MRI, including diffusion MRI (dMRT), serves as "microscope"" for anatomical structures and routinely mitigates the influence of low signal-to-noise ratio scans by compromising temporal or spatial resolution. However, these compromises fail meet clinical demands for both efficiency and precision. Consequently, denoising is a vital preprocessing step, particularly for dMRI, where clean data unavailable. In this paper, we introduce Di-Fusion, fully self-supervised denoising method that leverages the latter diffusion steps and an adaptive sampling process. Unlike previous approaches, our single-stage framework achieves efficient and stable training without extra noise model training and offers adaptive and controllable results in the sampling process. Our thorough experiments real and simulated data demonstrate that Di-Fusion achieves state- -f-the-art performance microstructure modeling, tractography tracking, and other downstream tasks. Code is available at http://github..oo/Fouirrr DDii-uuiio.
INTRODUCTION
Characterizing real-world noise using data distributions is difficult (Huang et al. 2021), particularly in non-invann non-invasive imaging modalities such as Magnetic Resonance Imaging (MRI), where the noise predominantly originates from numerous factors including thermal fluctuations (Fadnavis al. 2020a). MRI, including its subtype Diffusion-weighted Magnetic Resonance Imaging (dMRI) (Basser al. 1994), serves vital tool for observing inferred structures (Le Bihan 2003 Le Bihan et al. 2006 Schilling et al. 2019) and necessitates a high Signal-to-Noise Ratio (SNR) for better clinical decision making. While is possible to improve the SNR by increasing the acquisition time or reducing the image resolution, either way hinders the clinical application of MRI. Therefore, much research has focused on processing techniques like denoising for dMRI to improve its SNR and reduce acquisition time, which holds great significance for clinical efficiency and accuracy.
The dMRI typically consists of 4D data (X E Rwxhxdxl), including 3D spatial coordinates (w.h and d) and 1D diffusion vectors (?), in which diffusion is measured along different gradient direc- tions (Westin et al. 2016). Different clinical applications require varying numbers of diffuusion vectors and acquisition strategies, leading to diverse noise sources and distributions, which complicates noise modeling and denoising implementation. For supervised methods Gibbonn et al. 2019 Kaye et al.. 2020), not only is it non con-practical to obtain paired data with high SNR and low SNR. but the diversity of dMRI also leads to distributional shifts among different datasets, resulting in a fundamental drop in their performances (Darestani et al.) 2021). Different from these approaches, self-supervised solution for dMRI denoising through a single-stage construction and an efficient adaptive sampling process. Without the need for paired training data or clean data, our method is capable of removing the noise from dMRI with a denoising diffusion model. To mitigate the drift problem, Fusion process is proposed align the forward process. Moreover, as real-workh noise is difficult to characterize, a "Di-" process is introduced to represent the noise distribution in a more
method offers
"Corresponding authors

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effective manner. Consequently, our method Di-Fusion is able to achieve better denoising results while preserving the desired anatomical structures.
The main contributions of our work three-fold: (i) We propose Di-Fusion, stable and self- supervised dMRI denoising method leveraging the latter diffusion steps (Section 3.2p. Di-Fusion
integrates the statistical self-supervised denoising techniques (Batson & Royer 2019) into the dif- fusion models through the Fusion process and "Di-" process (Section 3.1 (ii) Di-Fusion enables
iterative refinement through an adaptive sampling process (Section 3.3 (iii) With thorough compar-
isons on real and simulated data, Di-Fusion demonstrates state-of-the-ar denoising performance in microstructure modeling. tractography, and other downstream tasks (Section
BACKGROUND AND RELATED WORKS
2.1 STATISTICAL SELF- SUPERVISEE IMAGE DENOISING
Built upon the assumption that additive noise is pixel-wise independent, Noise2Noise (Lehtinen et al. 2018) learns the process of image restoration solely by observing corrupted measurements:
arg min Ell fe (x') ==|²)
arg min fll (x') vl E v1()
(1)
where x and s' are independent corrupted measurements of the clean ground truth y and fa is denoising function which is parameterized by 0. Due to the assumption of independent noise, E|| is usually a constant. Furthermore, Noise2Self (Batson & Royer 2019) proposes the J-invariant theory, using only the same corrupted measurement to perform denoising. Following this theory. Noise2 (Krull et al.) 2019), Laine al. (Laine al. 2019) and Noise2Same (Xie cet al. (2000) focus on how to construct unorganized collections of corrupted images by masked-based blind spot networks. Noisier2Noise (Moran et al. 2020) and Noisy-As-Clean (Xu al. 2020) add additional noise to the original noisy image to generate training image pairs. Nevertheless, these methods exhibit a significant drop in performance when confronted with real-world noisy images. particularly when the explicit noise model is unknown (Huang et al.
2021 Mansour & Heckel 2023).
2.2 DIFFUSION MODELS
Denoising Diffusion Probabilistic Model (DDPM) (Ho et al. 2020 Sohl-Dickstein et al. 2015) emerges as a powerful generative model, which is composed of parameterized Markov chain with 7 diffusion steps to fit given data distribution. The forward process (z, (x=(x serves to perturb the data by gradually adding Gaussian noise based on a pre-defined noise schedule $1,... ,r (Following (Ho et al., 2020), of := 3,, a := - 3, and ã₂ := I are sets of predetermined constants in this paper) until the data distribution approaches a standard Gaussian distribution:
q (Mur) (2)) = II (m. a (x)) == N' (me v - 1,00-1,
The reverse process starts from Gaussian distribution ~ N (0, and uses a parameterized Gaussian transformation kernel Fg to learn the btep-by-step restoration of the original data distribution:
pr (FOTT) = pxpp() II pr (1==1 pr (2))) = N (x--1; Fo(
Recently, there has been a large interest in exploring ways to enhance the extensibility and sam- pling efficiency of DDPM. For enhancing extensibility, (Song & Ermon 2019) uses gradient of the log density as a force to pull a random sample across the data space towards regions with a high data density characterized by p (x) (Croitoru et al. 2023) by adopting Langevin dynamics algorithm (Hyvärinen & Dayan 2005). (Song et al. (020b) further extend the score function as solutions to reverse-time Stochastic Differential Equation (SDE) and extends DDPM to continuous states. Cold diffusion (Bansal et al. 2024) investigates the necessity of Gaussian noise any form of randomness for diffusion models to work effectively in practical scenarios. (Zhou et al. 2024) introduces family of processes that interpolate between two paired distributions given endpoints. For accelerating sampling speed, (Song et al. 2020a) and (Watson et al., (202I) generalize DDPM by introducing class of Mon-Markovian diffusion processes that achieves the same sampling objective.
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Previous works have demonstrated that diffusion models can be effectively applied image restora-
tion tasks Kawar et al. 2022 Xia et al. 2023 Ozdenizei & Legenstein 2023 Chung et al. 2022b
Saharia et 2022a Fei et al. 2023). Conditioned
low-resolution input image, (Saharia et al. 2022bb performs image super-resolution via repeated refinement. (Chung et al.] (022a) (Song et al.
2021), (Song al., 2024) and (Gao et al.] 2023) extend diffusion models inverse problems in
medical imaging. However, these models require clean data (e.g., normal-dose CT) to capture their prior data distribution, which makes direct application of these methods dMRI data impractical
because no clean data is available in dMRI itself. Our method does not require extra noise model training or clean ground truth y and can be applied to the aforementioned scenarios.
2.3 RELATED WORKS
The initial denoising methods employed for dMRI are adaptations techniques developed for natural images, like non-local means (NL-means (Coupé et al. (008) and its variants (Chen et al. 2016) Coupé et al. 2012)). Under the assumption that small spatial structures exhibit relative consistency across varied dMRI measurements, Local Principal Component Analysis (LPCA) (Manjón et al.) 2013) and its Marchenko- rank approximation. Training the Noise2Noise (Lehtinen et al. 2018) model directly using the same slices from different volumes can result in excessively smooth outcomes (Shown in the experiments of (Xiang et al. (023)). So, utilizing the entire volumes, Patch2Self (Fadnavis al. (2020a) trains a full-rank locally y linear denoiser to perform volume-wise denoising. Patch2Self2 (Fadnavis et al.)
extension (MPPCA) (Veraart et al. 2016) project dMRI to a local low-
2024) further enhances the computational efficiency of Patch2Self. Recently, Corruption2Self' (Tu et al. 2025) extends denoising score matching to accommodate noisy observations and provides a framework for denoising MRI. A state-of wef-the-art art sulf-suupe method DDM2 (Xiang et al. 2023) is
proposed for denoising dMRI, which incorporates statistical image denoising into the diffusion model three-stage framework. However, the results obtained by DDM2 are prone to over-denoising as its performances in downstream tasks are not satisfactory (Section 4.2
3 METHODS
4D aMRI consists of independent noisy samples acquired at different gradient directions. Considering x = X. X.,..i. (i: slice index, j: volume index) as the target slice to denoise, x' -i. and x are independent corrupted measurements of the clean ground truth y. In this section, demonstrate how to decompose the single-step mapping from x' to ø into T steps using a parameterized Markov chain (We denote Fa as the parameterized transformation kernel in our method). We provide complete definition of the entire Di-Fusion in Appendix B
There five questions be answered in our method. Q1: Since x' and x still different, how can we obtain the forward process to construct the multi-step mapping between two endpoints? Q2: How can we represent the noise distribution without extra noise model training? Q3: How can training be conducted with only noisy data? Q4: Why does Di-Fusion only leverage the latter diffusion steps? Q5: How does the reverse process enable iterative refinement?
3.1 MODIFICATIONS OF FORWARD PROCESS
Q1 We use Fo to map from x' to x. considering x' as FT and x as No. Fo should take x, and as input and output Yout close to .::
x + & = Tont = Fo (z,, t), | < E,
where represents small positive value, is perturbation term that depends on f, and decays as +-+ 0. Performing the reverse process of DDPM, we find that æ;-1 should be a linear interpolation between Fout and a; plus noise instead of 5T-1, the major difference is introduced by Fout (See Appendix C.T for detailed derivations):
vā,- (I + Ex)
Va: (1 - āt-1)
It-1 1 Ōi
+ 0,2 + 8T-1 Var-la' + VI 1-11a- a.-12, (5)
1 1- Õt.
miferrdifeeence
where 2~ N { are obtained by directly performing the forward process in DDPM and {Mt)² are obtained from the reverse process of DDPM. Since the component ét 0 as t - 0,
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Final nC
Pasion Process
AY
Target
Ra
(a) Diffusion
Drift
Diffunion Steps
X1
Xp
Diffusion Stops
(b) Training
Sampling
Figure 1: (a) Fusion process (Section 3.1 aligns {ā,}" to { 1 and avoids drift ("Drift" means drifted results, "Final" means the denoised version of "Target"); (b) Training the latter diffusion steps (Section 3.2) imposes restrictions on the generation ability of diffusion models and decreases uncertainty; (c) Sun-Walls accelerated sampling (Section 3.3) accelerates the entire sampling process.
larger proportion of æt-1 aligns closer to æ, rather than merely being a noisy version of x'. If we still feed Fz-1 and 1 into Fe. will cause output deviations, which accumulate in the sampling chain and ultimately lead to the drift problem (Fig. [ (a)).
F Fusion process (Q1) Since Fg learns the mapping from x' to {xt); should be combinations of r and x', augmented with a sampled noise 2 ~ N (0, I). These combinations can be approximated by utilizing the reverse process in DDPM to compute the linear interpolation between x' to x:
x = Afx + Naa',
(6)
9 (Fr|=]) == N (xei Vana (1 (1-);)) 1))
(7)
V V and X/ var(1-ar-- for simplification. As I decreases, x 1-a. becomes closer to x since Xj has a higher value. By substituting zi for x' in Eq. 5 the Fusion process be achieved, which obtain 2t with different
where rewrite Af
1-a,
shown in Fig. S19 Intuitively, the
Fusion process gradually introduces the target denoising slice x to the model, guiding the model to optimize in a fixed direction, thereby mitigating the drift. We thereby address Q1 by defining the forward process (Ir (x:))
Q2 Approximating noise as is definitely feasible approach. However, the noise distribution in the real world often exhibits complex statistical properties, and thus cannot be easily captured mathematically (Section 2.I. Similar challenges also exist in dMRI.
"Di"" process (Q2) To better characterize real-world noise, we represent the noise distribution involving the input noisy data. Since x and x' are independent corrupted measurements of the redun- dant part and have independent noise, directly calculating x - leaves some linear combinations of noise (xr = y + n1, y + 122, x x' ni 722, here n and nı represent the noise in x and x', respectively), we perform a zero-mean operation on these linear combinations of noise to comply with the zero-mean constraint of 2:
Ln=l (Xmm x' mn
Ex-x! mess ((x -1) -z-=22 Me-2'
m==
(
W
where mess (-) means spatial shuffling operation originated from DDM2 (Xiang et al. 2023 Mar-r' the mean x - x'. Ea-z' theoretically preserves the variance information of the noise (See Appendix C.2 for proof) and will serve as the noise distribution employed in both (zx and Pyr (xt-1 In this case, the forward process and reverse process no longer follow a Gaussian distribution, but they can be represented as Eq. (18) and Eq. (IO), respectively. In Fig. S20 demonstrate through experiments that E has different statistical properties from 2. In Fig. S27 show the impact of Ex-z² and on the reverse process.
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Training Process
Sampling Process
Volume
Xo Xosz
Ebu
Volume
dd
Yes
CS
No Adaptive Termination
Figure 2: Overview of single-stage Di-Fusion. The training process does involve any extra model training apart from Fø, and the sampling process offers adaptive and controllable results.
3.2 TRAINING PROCESS
J-Invariance optimization (Q3) When training Fø, first consider æ and x' as J {x, x') Assuming that the noise distributions of r and x' are mutually independent, the model with as input and the optimization target satisfies the property of input-output independence. According the Proposition declared in Noise2Self (Batson & Royer 2019), the loss between Fo (ar') and x will in expectation equal to the loss between Fo (a') and clean ground truth y. plus constant E|[æ - y (Bq. (1). Therefore, minimizing Fe (z') z]| is equivalent to minimizing BB (x') yll with respect to the clean ground truth and our simplified training objective is:
Iiimple (0) Ex Co-n
[| FolVaiat + vI aiE. -', |]
Intuition of training the latter diffusion steps (Q4) DDPM, shown that when conditioned on the same latent, the samples share high-level attributes (when conditioned on say T250, the samples are close to each other) (Ho et al. 2020). because the thorough training in the former diffusion steps (xy + ær; that DDPM possesses diverse generative capabilities. Since we perform image denoising task with such a strong prior (from one noisy volume to another noisy volume), training only the latter diffusion steps possible reduce the diverse generative capabilities of Fe. More precisely, only the last T, steps in the Markov chain (IT, - xn) are trained. In this way, a generative training task is simplified into a conditional generation task (ZT. Fo. Te < T)), with more zo information provided in { To (Fig. (b))
There are two main reasons for adopting this strategy. Firstly, training the latter diffusion steps weakens the generation capacity the diffusion model, reducing its diversity. This, in turn, lowers the uncertainty in denoising results for task. Secondly, with the same training time, obtaining a more stable Fo is possible. By training only the latter diffusion steps, each step receives more training iterations, resulting in improved stability for the model performance. Algorithm outlines the training process, and Fig. (left) provides an overview of the entire training process.
Algorithm Training process
Initialize Fg randomly; input 4D data: X E IR wxxxdxxx repeat
Uniform (f1, ,Te)) x X.,.,..j X X.,,,.--1
training the latter diffusion steps in Section 3.2 slice index, j: volume index
Ex- mess ((x x') pl=xz))
Eq.
x = Na.a"'
Eq.
take gradient descent step on: Valxx- F (Vonsz VI- - 11]1 resample and until converged
Eq. (9
3.3 SAMPLING PROCESS
We make two specific modifications pur (at-1 [2;) to achieve adaptive sampling process and directly begin the sampling process at ær.- See Fig. 2 (right) for an overview of the entire sampling process and Algorithm for detailed description of the complete sampling process.
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Run- Walk accelerated sampling After substituting the standard normal distribution in Eq. B with Su-z', a typical reverse process p, (It-1 (zt) could be formulated as:
PF (1))) Iz-1 Xj Fo (Iz, t) + X2 + (ax - 11) En-a':
(10)
where is a constant. DDIM (Song et al. 2020a) notes special case when = 0 for all t + the forward process is deterministic given 7t-1 and x except for process, the coefficient before the noise &x-x' becomes zero, resulting in an implicit probabilistic model (Mohamed & Lakshninarayanan) 2016). However, we do not follow the uniform step strategy of DDIM in the sampling process; instead, we use Run-Walk accelerated sampling. Consider a DDPM sampling process from IT, to zo. when t is large (t > T,.1 S T, S Tc). the speed during each reverse process is slow; thus, acceleration can be applied (Run). Conversely, when is small (t T,), the speed is fast, and deceleration required (Walk). In equation form, the difference between 24-1 and 2t can be represented as (See Appendix C.3 for additional derivations):
1; in the sampling
Ft-1 - Jt
X1
m1) +
(11)
speed
perturbation
When / is large (e.g. 1> T,), X1 approaches zero and the speed (Af (ær xx)) towards is relatively slow. This is when we perform accelerated sampling. When reaching the latter sampling process, X progressively increases and the speed towards zo quite fast. This is when stop accelcrating. When T, 1, Run-Walk accelerated sampling degenerates into DDIM sampling. When T, Te. Run- Walk Run-Walk accelerated sampling degenerates into DDPM sampling.
Now let consider the forward process as defined not all ( but on subset ( Tres h. where is an increasing sub-sequence of [l...., Tc] of length S. In particular, we define the sequential forward process over Zr,...., Trs (I,, V (A x + A x') + vI- a 1 < k < S). The sampling process now samples according to reversed(T) (In practice, 7= (1, ................................ T, 1, T,, T, + p.... Te p, Te). where p an integer representing the accelera- tion factor). This can be more intuitively understood in Fig, (c).
Algorithm 2 Sampling process
Load pre-trained Fe; input: X E Ruxhxdxl, i, and CSNR X. +. x X
i: slice index, j: volume index
xx-z² = mess (x x.) -z-22'
> Eq.
8
Fr. Var, (A 7 Xi x²)
+ v1 σT. Eu-2'
Eq.
2ms I
2 x
by = 2 E4 ((mmn01)) >01)
+
>> Eq. 12
2. m=1 n= (ism P22)
for Th- = reversed (1,2, ....
Tr 1, Tr, + D.
Te - p. Te} do
Ex-Z' mess (5 (6x-2')
Shuffle
& again
Fout = Fg (ITz: TT))
> Eq. (4
d || - Tout 1| x bz
> Eq.
13
if d2 > CSNR then
Zo out break
b In Section 3.3
else
Iry X7t Tout + A3 Ira + (orn 7)) Ex-a'
> Eq.
10
end if
end for
return 20
Towards iterative and controllable refinement (Q5) During experiments, observe that the intermediate outputs, Fout, obtained during the sampling process demonstrate a substantial success in denoising. Therefore, we explore the feasibility of adaptive termination stop sampling prematurely. More specifically, the degree of denoising in Tous can be characterized by its distance from x. Nevertheless, directly computing this distance ||æ Iouell presents problem. When the slice index is located the edges, the resulting distance tends to be smaller due to the reduced
We do this by multiplying ae with 77, where 77 0 if special instructions are provided.

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low
Fiber to Bundie Coherence (FBC)
high
Nolsy_ (1120)))) (095)
ASCM(
DIP[1930)
Nr2N|2550]
P25(1989)
DDM2[681)
(URS(1862)
Figure Density map of FBC projected on the streamlines of the OR bundles. The numbers
in parentheses represent the number of streamlines. Di-Fusion generates the minimal number of streamlines while maintaining high FBCs (consider "Noisy_filtering as references for high FBCs).
amount of brain tissue in these edge slices. Hence, it would be preferable to calculate coefficient b. that accounts for the ratio brain tissue to the entire image. Here, we adopt a simple definition:
Em=l En=1
w
Em=1 m=
w
dr 2.
(12)
>m=1
201
m==1 (p))
2. --E=> 2n=1 (p))
where P1 and P2 are constants depending on the data normalization methods employed and I (() an indicator function. b, can be used to correct d,;
is
=1= Zout 112 by-
(13)
Since dz has been corrected, we can pre-define a universal value CSN to perform the iterative and controllable refinement on each slice. During pr (24) we first get Tout Fo (1x, and compute dN (Eq. [33). Then if dæ is greater than CSNRR xo Tout and the refinement iteration breaks. In contrast, the refinement iteration continues if de is smaller than CSNRR. In extreme cases, when CSN'R = O, the reverse process will immediately terminate and output Tg. When CSNR = 1, the complete reversed(T) will be executed until completion.
EXPERIMENTS
4.1
DATASETS AND COMPETING METHODS
Datasets To thoroughly evaluate Di-Fusion, we perform experiments three publicly available
brain dMRI datasets acquired using different, commonly-uset acquisition schemes: (i) High-Angular
Resolution Diffusion Imaging (Stanford HARDI, X IR 10 X0oxxxx16xx110 76x150 (Rokem) 2016)); (ii) Multi-
Shell (Sherbrooke 3-Shell dataset, X E R 1883x128xxxxxX
Garyfallidis et al. 2014)); (tii) Single-
Shell (Parkinson's 's Progression Markers Initiative (PPMI) dataset, X E R 72x64 (Marek
20 Simulated experiments are carried out on the fastMRI datasets (Tibrewala et al. 2023 Zbontar al. 2018). We simulate noisy data with five different noise intensities.
ct al.
Competing methods We compare Di-Fusion with five competing methods in the main paper (all experimental details are provided in Appendix D.I): (i) Adaptive Soft Coefficient Matching (ASCM), an improved extension of non-local means denoising (Coupé et al. 2012). (ū) Deep Image Prior (DIP), self-supervised denoising method (Ulannov et al. 2018). (iū) Noisier2Noise (Nr2N),
statistic-based denoising method (Moran et al. 2020). (iv) Patch2Self (P2S), multi-volume denoising method (Fadnavis et al. 2020a)). (v) DDM2, state-of-the-art denoising method (Xiang el al.. 2023). More comparisons with other denoising methods, including MPPCA (Veraart al. (0166), (Pang et al. 2021), and Patch2Self2
Noise2Score (Kim Ye, 2021), Recorrupte22Rcoouuute (Fadnavis et al. 2024), can be found in Appendix E
4.2
IMPACTS ON DOWNSTREAM CLINICAL TASKS
Effect on tractography The noise in dMRI can impact tractography results, potentially causing the generation of spurious streamlines by the tracking algorithm (Fadnavis al.) [220a]] Garylalifdis et al. 2014 Schilling et al. 2019). We explore the effect of denoising on probabilistic tracking (Girard et al. 2014) by employing the Fiber Bundle Coherency (FBC) metric (Portegies et al. 2015) and reconstruct the optic radiation (OR) bundles (See Appendix D.2 for details). Since low FBCs indicate which fibers are poorly aligned with their neighbors, we further clean the tractography results of noisy data (captioned by pooisyfilterrnn"" using a stopping criterion (Meesters et al. 2016). In Fig.
In our experiments, P1 -0.93 and P2 = -0.95, changing their values has little impact on the results.
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Corpus Callosum (CC)
Centrum Semiovale (C50)
Corpus Callosum
Centrum Semiovale
Constralned Spherical Doconvoluttin
(CSO)
Diffusion Tonsor Model
MOD AARO EDEEP LUN
MEST'S L.00 recohaoa1 drtal
(DT.0
Figure 4: Scatter plots of the microstructure model predictions against input data. The top-left of each plot shows the quantitative It metric computed from each model fit on the corresponding data. Our data points are more concentrated (higher It²).
show the effect on the tractography of OR. Although DDM2 yields the fewest streamlines, noticeably, it misses the high FBCs indicated by the white arrow in Fig. 3 Di-Fusion generates the minimal number of streamlines while maintaining high FBCs, which indicates that our method maximizes the denoising performance while preserving fiber bundle information.
Effect on microstructure model fitting Denoising methods can be compared based on their accuracy in fitting the diffusion signal (Ades-Aron et al. 2018). We apply two commonly used mi- crostructure fitting models, namely diffusion tensor model (DTT) (Basser et al. 1994) and Constrained Spherical Deconvolution (CSD) (Tournier et al., 2007), ) on noisy and denoised data (Appendix [D.2 for details). We show the quantitative Rt metric of microstructure predictions against the original data for Corpus Callosum (CC) and Centrum SemiOvale (CSO) in Table S2 and the corresponding scatter plots are in Fig. 4 As measured by R², Di-Fusion achieves the best results across all four different settings. This means that Di-Fusion aids in the characterization of the microstructure.
Effect on diffusion signal estimates We further examine how the denoising quality translates creating quantitative and clinically-relevant DTI (Basser al. 1994) diffusion signal estimates (Details are in Appendix D.2). In Fig. S9] we show fractional anisotropy, axial diffusivity, mean dif- fusivity, and radial diffusivity comparisons. Our method effectively suppresses noise and reconstructs fiber tracts.
4.3 QUANTITATIVE AND QUALITATIVE RESULTS ON in-vivo DATA
Quantitative results on SNR/CNR metrics Considering the infeasibility of using metrics that need clean ground truth and their limited correlation with clinical utility (Mason et al. 2019), computing metrics in downstream clinical regions of interest is more reasonable (Adam son et al., 2021). To quantify the denoising performance, we employ Signal-to-Noise Ratio (SNR) and Contrast-to-Noise Ratio (CNR) metrics (Details in Appendix D.3). The quan- titative denoising results are reported as mean and standard deviation scores for the complete 4D volumes in Fig S12 Di-Fusion indicates better performance in terms of SNR/CNR metrics.
Residual
N/A
Qualitative results In Fig. 5 we
visualize the denoising results and residuals on axial slices for Stanford
HARDI (Fig. [S13] S14 and Fig. ST. for more qualitative results). From the residuals of Fig. 5 the area indi- cated by the red arrow does not appear in "OURS", indicating that Di-Fusion does not remove any anatomical struc- ture during denoising. All pictures are best viewed when zoomed in.
Output
Noisy
ASCM
DIP
N22N
P2S
DDM2
OURS
Figure 5: Qualitative results. "OURS" results are obtained when CSNR 0.040. The area indicated by the red arrow does not appear in "OURS", indicating that Di-Fusion does not remove structural information during denoising.
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4.4 QUANTITATIVE RESULTS ON SIMULATED DATA
We show the PSNR and SSIM metrics F in Table I (Implementation details are in Appendix D.4 see Fig. S16 qualitative results). For Nr2N, DDM2, and Di-Fusion, three rounds of experiments are conducted to provide eπror bars. As P2S utilizes linear regressors, each round's 's results are consistently similar. Hence, P2S's error bars are not provided. When the noise intensity is high, our method performs the best and shows stable performance. Di-Fusion holds a tremendous potential for generalization and applicability for its stable performance and better performance under high noise intensity. Moreover, these results provide evidence that Di-Fusion could be extended to self-supervised MRI denoising without relying on any clean data.
Table I: Quantitative results on simulated data (Noisy means simulated data). PSNR (dB) and SSIM (%) are reported. Numbers are presented as mean value with standard deviation. Di-Fusion exhibits more stable performance with smaller standard deviation.
Simulation
Simulation 2
Simulation 3
Simulation
Simulation 5
Method
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
Noisy
11.38
13.72
20.65
16.09
37.41
20.38
$2.02
23.76
64.62
26.63
P2S
24.53
11.20
43.94
17.63
65.34
24.91
78.61
30.26
86.13
33,87
Nr2N
23.720.3
16.78o.2
50.832m
22.13004
71.77691
26.801 69
70.327
26.40s.17
87.314.0
32.82om
DDM2
26.02263
17.92..06
49.5715.
21.761.61
59.64s.47
23.254.05
77.962.16
29.14..1
81.43223
31.35221
OURS
39.051.42
19.91e.32
62.021.11
23.60us
77.360.1
26.960.7
83.43139
28.69b67
89.52a.x
30.630.7
4.5
ABLATION STUDIES
Without the Fusion process, the results in the early sampling phase do not deviate significantly (Fig. S24). but when we do not perform an adaptive termination, the results show areas that are absent in the noisy data (Fig. S25 indicating that they have indeed drifted; Without the "Di-" process, the results lack some high-frequency information, and the overall gray value of the denoised images has also changed (Case in Fig. (S24). In Fig. [20 we demonstrate through experiments that the noise computed by "Di-" process has different statistical properties from 2; Without training the latter diffusion steps, the denoising results are noticeably smoother and have more hallucinations (Fig.
S24 and Fig. $25). The details of the above ablation studies can be found in Appendix G.I
Furthermore, we balance the training epochs for different T:4 and show R of microstructure model fitting results in Fig. [6] The results show that choos- ing any Tr within reasonable range (T, < 500) will not have significant influence on the denoising results and the training difficulty of every step is relatively consistent.
0.90
0.90
cc (CED)
EC CSO (ST-
Moan
2000
During the reverse process, Run- Walk accelerated sampling not only enables accelerated sampling, but also sures that the sampling quality remains mainly unchanged (Fig. (S26). In Table the sampling time indicates that the adaptive termination and Run-Walk accelerated sampling (Section 3.3) together greatly reduce the sampling time. The details of the above ablation studies can be found in Appendix G.2
Figure 6: R² of microstructure model fitting on CSD & DTI obtained when T. is different. When To 500, the performance is consistent.
DISCUSSIONS
On comparisons with related work In Appendix A we discuss the differences between Di-Fusion and Patch2Self, as well DDM2. In essence, Di-Fusion surpasses Patch2Self by significantly
Wee have clean ground truth in simulation settings.
#1 x 10 epochs for T. 300, 2 10 epachs for T. 600, etc.
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reducing the dependence the number of input volumes, thereby expanding its scope of application. In comparison to DDM2, Di-Fusion not only simplifies the method to single-stage framework but also implements iterative and controllable refinement through methods mentioned in Section 3 Moreover, in Section 4 Appendix a Di-Fusion achieves state-of-the-ar performance in the conducted experiments, showeasing its superior and stable results.
On limitations (i) Possible longer inference duration. The inference time of diffusion models is already relatively long, and there are concerns that the additional computation introduced by adaptive termination in Section 3.3 may further increase the inference duration. In Appendix G.2 we discuss the sampling burden associated with Di-Fusion. (ii) Possible hallucinations. As a class of generative models, diffusion models inevitably raise concerns regarding generating fake anatomical details (hallucinations). However, we find that with the methods introduced in Section 3 particularly the training the latter diffusion steps mentioned in Section [.2] the generative capacity of diffusion models can be restricted (Appendix G.I), which helps reduce hallucinations. (iii) Noise-artifact conflation and additive noise assumptions. Our method explicitly targets thermal noise modeled as additive Gaussian processes, consistent with common denoising frameworks (Chen et al. 2019 Ramos-Llorden al. 2021; Cordero-Grande
al., 2019). However, this intentionally excludes spatially varying distortions (c.g., cardiac pulsation) often classified as "physiological noise" (Chang al., 2005 2012: Walker al. (20TT) but better characterized as artifacts. While our assumption enables tractable solutions, it may limit effectiveness for signal-dependen noise.
6 CONCLUSION
This paper proposes Di-Fusion, end-to-end self-supervised MRI denoising method that achieves iterative and stable refinement without relying on extra noise model training or clean data. The Fusion process aligns the trajectory of the forward process and avoids drifted results. The "Di-" process characterizes real-world noise, enabling the model to capture statistical properties of the real-world noise. By training the latter diffusion steps, our model achieves enhanced stability and performance. During the inference stage, Di-Fusion offers controllable results through an adaptive sampling process. Comprehensive experiments on real and simulated data demonstrate that Di-Fusion achieves state-of-the-ar performance in microstructure modeling, tractography tracking, and other downstream tasks.
ACKNOWLEDGMENTS
This research was supported by Natural Science Foundation of China under Grant 6227 1465, Suzhou Basic Research Program under Grant SYG202338, and Open Fund Project of Guangdong Academy of Medical Sciences, China (No. YKY-KF202206).
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A COMPARISONS WITH RELATED WORKS
Comparison with Patch2Self Patch2Self (Fadnavis et al. 2020a) requires a minimum of ten additional diffusion vector volumes to denoise single diffusion vector volume. Instead, our work only needs one additional volume, which is clinically meaningful as common clinical dMRI often scans fewer than ten diffusion vector volumes (Karayumak et al. 2019 Xiang et al., 2002). Moreover, model does not require repetitive training, whereas Patch2Self necessitates training multiple regressors to perform voxel-by-voxel denoising. However, despite achieving better results, Di-Fusion takes relatively longer time than Patch2Self, which is due to the training time of the diffusion models.
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Comparison with DDM2 Our work only requires single stage for denoising, whereas DDM2 typically involves three stages. Furthermore, it's worth noting that the noise model in the first stage of DDM2 critically influences the ultimate denoising results, and finding optimal solution that simultaneously maximizes evaluation metrics scores and minimizes training time can be challenging (See Fig. S32 details).
B DI-FUSION
B.I FORWARD PROCESS
Consider x = X., (i: slice index, j: volume index) as the target slice to denoise, x' = Nx, X...i.I- PI,....7 is peeppeeeeeee rre-neffned noise schedule, of := B,, := 1 3r and ar = II Ot.. We rewrite X C 18t l- and X1 ya.(l-at-1) for simplification. 1-a:
Perform the Fusion process:
z X/x + Aga'.
(14)
Then we get linear interpolation between x and x', we compute z, based on q (z, (x,]):)):
It Var + v1 0z
(15)
The forward process can be defined if using N IO. for perturbing data distribution:
q (m+|=²) == = N (o: Vorai Tz)
(16)
However, we use "Di-" process to compute noise Ex-xx to substitute for 2:
Ex-x' = mess ((x - x') pz-22), He-x' Em= En
(Imn x' mn)
(17)
W
So the forward process can't be represented as N (xt; Varai, (1 ai I), but could be computed using the following formula:
q (xg mii + T Vanat + VT ciEo-r'
(18)
We leverage dynamic combination (the Fusion process) and continuously varying noise (the "Di-" process) to provide the model with more augmented training data, thereby enhancing its robustness. This idea is similar to those in Noise2Void (Krull et al. 2019). Noisier2Noise (Moran et al. 2020). and Noisy-as- as-Cle (Xu et al. 2020), which also utilize data augmentation construct training data.
B.2 TRAINING PROCESS
Our simplified training objective is:
|| - Folvan + /I a.c 0)]l
Lsimple (0) = Er,
(19)
We perform training the latter diffusion steps by sample Uniform ((1, Te)))
B.3 REVERSE PROCESS
The details of how to perform the reverse process in DDPM if a data predictor is used are in Appendix If a data predictor F& that directly predict To. the reverse process for DDPM becomes:
Fo (Iz, t) + Xbas (or 17) Ex-i'
(20)
And (x1-1 (xr)) can be defined as:
pr (xt- (x)) 2t-1 FoFo (Iz, Xb.r (or 11) Ex-st-
(21)
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Now let consider the forward process defined not on all {xx)1 but subset {Irz- Trg where is an increasing sub-sequence of [I...., Tel of length S. In particular, we define the sequential forward process over ........ ................................ 2ry (2T, van, x X x') + vI 5 & S S).
The Run- Walk accelerated sampling now sample according to reversed(7) (In practice, T (1,2, T, 1, T,, T, p.... Te p, )). then the reverse process become:
PJe (Tru (TTT)
- X1 Fo (TTu Tk) + x Ir + (0r. 77) Ex-x'-
(22)
Before sampling, we define an universal value CSNR and compute by:
b = 2m=1 Ex=1
Em=1 Em=1?
(23)
>m== Ex= (amm))))
2. >m= E 2 ((mm)88))
During every Pr. (Iru-1 (TTr)) we first get Tout = Fo (ZiT, Tk), and dz = | Foust 11 x bz.
Then if dz is greater than CSNR, the output To Dout and the refinement iteration breaks. In contrast, the refinement iteration continues if d. is smaller than CSN'R.
C ADDITIONAL DERIVATIONS
C. THE DIFFERENCE BETWEEN THE TWO TRAJECTORIES
The original sampling process in the Algorithm 2 of DDPM (Ho et al. 2020 is:
1
1 04 VI
It-1
2t
co( Oz
+ 022,
(24)
where eg a noise predictor. However, we use a data predictor Fø to directly predict xo in our paper. We will demonstrate how perform the reverse process in DDPM if data predictor is used.
Given a data point sampled from a real data distribution zp ~> ((x), let us define forward diffusion process in which add small amount of Gaussian noise to the sample in T steps, producing a sequence of noisy samples 2 21...., Dr. The step sizes are governed by a variance schedule (3. E (0, 1)
T
a(0,1==-1) = N(2a vI B, ((F1r110) = II ((
(25)
As the step increases, the data sample zo gradually loses its distinguishable features. Ultimately, when't T - 20, IT converges to an isotropic Gaussian distribution.
Let at = 1 - 3r and a. II as nice property the aforementioned process that we can sample x, at any arbitrary time step / in closed form using the reparameterization trick:
Vouta-1 vI 01
where 2t-1, 21-2, NO I).
V0:01-150022 + VI 11-;00-122-
where 2,-2 merges two Gaussians.
(26)
Vorao v1 0,2 0,2,
where we merge two Gaussians with different variances, (00 ofl) and N(,,of1) resulting in a new distribution (0,, (oî + o)) Here, the merged standard deviation is given by (1 - 0z) + a:(1 a=11) V1 040:-1 We can then derive:
(x NN vonso, (1 0. I).
(27)
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Consider reverse process, it is noteworthy that the reverse conditional probability is tractable when conditioned on Zy:
(28)
Using Bayes' rule, we then have (Ho 2020):
(01,-1/01,200 vo)
g(0 (T0)
20)
q (20)
(x0- va
(Iz-1 /0-1100)
(xt v
exp
2
31
1 04-1
a2
1
zt
+
1i-1 -2
2
22/00 1
+ 04 12
(3a
varuo) )
=exp
31
ar-1
exp
Olt
2/00
1t
2/or-1
Fn) 1
3e 0t-1
31
ōt-1
(29)
where C(r,, vo) ) is some function that does not involve -;-1, and the details are omitted. Following the standard Gaussian density function, the mean and variance can be parameterized as follows (recall that 1 - 3r and ā₂ = =I[==1 a:):
o, 1/( =1(
+
1/(
01 āこ
+
1 -
āt-1
Bc.
(30)
Be
3e(1
att
(1 v0) =
Dt
+
v
k0)l
at D1
+
Be
1 - āx-1
1 ā 1
va
v V
at-1
Be T1
3e
(31)
á1-1
1 Ōz
Varl
(1
Ō(-1)
1.
14.
Ütt
Thus, if it a data predictor Fg that directly predict wo, based on q( [r, 1x No), the reverse process for DDPM becomes:
Iz- = (e( 2o) + 0²²
Vai (1 ai-
24 + V Fo (It, f) + 0Z
(32)
Cr
1- Br had similar results through DDPM (Ho et al. 2020 found that both of = 3g and of T-o. experiments. We set of B1,...,7 , and hold B1.... r as hyperparameters.
Now we know how to perform the reverse process a data predictor is used. According to Eq. we know x + Fout = Fo (xz, t), then we can get:
04 (1 a.-1) It
vo--18t (x (x+
It-1
of
(33)
1 āc
1-
Now let us consider directly performing the forward process (g(2:(20)) on x' = Xepajj X (x.1) without the Fusion process (Eq. (6):
_1- Var-1z' v1 ā
(34)
thus the trajectory { obtained by directly performing the forward process in DDPM and the trajectory {x,)² obtained from the reverse process of DDPM are different, and the major difference brought by (x e)):
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00-1 (x ++)
Vai Ō4-1) + 0,2 Tt-1 Vāi- + v1 /1- 01-12. (35)
dt-1 1 dq
1 1- -q
major difference
This is because component 6 decays - 0, then larger proportion of components in 2;-1 becomes closer to æ.
If directly feed øt-1 and 1 into Fu. the output would deviate slightly further from I. This occurs because during training, Fo is optimized only with the objective: || - Fo(Voo-1z' + 1 1)ll (the training objective without the Fusion process). Im- portantly, 2 is one step closer to æ. (Øt-1 = you 19 than simply being a noisy version of æ'. This drift accumulates over the sampling chain, ultimately leading the result drift toward another slice.
I-a; (r + +x) y F2 + o22), rather l-a.
C.2 VARIANCE INFORMATION OF NOISE "D1-" PROCESS
Ex-z/ theoretically preserves the variance information of the noise:
Var (æ x²) Var (y + (v + n2))
Var (n, 112))
(36)
Var (ni) Var (n2) 2Cov (mi 12)
=
Var (n1) - Var (12),
where Cov (-) is the covariance, Var (-) is the variance, Cov (M, n22 = 0 since n and my are independent. Assuming that n1 and ng follow the same distribution, the variance information of this distribution is retained.
In Fig. [20)] show that the noise in "Di-" process has different statistical properties compared to Gaussian noise.
C.3 SPEED TOWARDS THE TARGET
The difference between _z-1 and F, can formulated as:
F--1 Dt.
Fo (x,, t) + No + (a, 11) Ex-o' - 34
X Fout + (1 X)) x4 + (a: 1) Ex-x' It.
(37)
According to Eq.
know El Touts then
substitute Fout into the Eq. 37 and get:
It-1 Ft
X1 Woust + (1 M) xz + (a, 17) Ex-z' Ft.
X (I 1) (1 X1 2t + (0, 1)) Ex-x' 2t
A (x- - xe) + + m) E Ex- + 19
(38)
speed
noise
perturbation
In DDIM (Song al. 2020a rj 0, so typically, the term "noise" disappears, leading to the following expression:
Rt-1 Zt
(1 xz) +
M
(39)
speed
perturbation
We know that e; is perturbation term that depends on t 0. c, - 0, X 1 at the same time. So the value of "perturbation" item does not change significantly when decreases; thus difference between #t-1 and n, mainly caused by the "speed" item.
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D EXPERIMENTAL DETAILS
D.1 EXPERIMENT AND REPRODUCIBILITY DETAILS
Noise schedule A typical noise schedule (Ho et al. 2020 Saharia et al. 2022b) follows "warm-
up" scheduling strategy. Inspired by DDM2, we implement a reverse "warm-up" strategy where & remains at 5e-5 for the first 300 iterations and then linearly increases to 1e? between (300, 1000] iterations (Xiang et al. 2023).
Training details Following DDPM (Ho et al. 2020), we set of = 31,... and hold 31,... as hyperparameters. Since are performing deterministic sampling process, 1j in Eq. (10) is set to 0 (we talk about how 7) influences the final results in Appendix F.2). We implement denoising functions Fo via U-Net (Ronneberger et al. 2015) with modifications suggested in (Saharia et al.) [2022b Song et al. 2020b). Inspired by (Chen et al. 2020 Song & Ermon 2019), we train Fo to condition on Ce nilorrm ([1,... T. Ta 300. Adam optimizer was used optimize 0 with fixed learning rate of LE and a batch size of 32, and Fe was trained 1e steps from scratch. All experiments were
performed on RTX GeForee 3090 GPUs in PyTorch (Paszke al. 2019). The training duration for one Fg is approximately five hours on a single RTX GeForce 3090 GPU with 5578MB of VRAM.
Sampling details During sampling, T, 50. 31 = -0.93 and 32 -0.95 and changing their values has little impact on the results (We set these two factors as a simple way to extract the brain mask). r = 0 and p = 10 if no special instructions are provided. CSN'R are provided in the figure caption.
Competing methods details In the main paper, Di-Fusion is compared against four state-of-the-art self-supervised deep learning-based denoising methods (ASCM isn't deep bearinggbaseed For fair comparisons DIP, Nr2N, and DDM2 adopt the architecture used in Di-Fusion. We follow the official repository and use the parameters that should give the optimal denoising performance for P2S (Fadnavis et al. 2020a). (i) Deep Image Prior (Ulyanov et al. 2018) trains network on random input to target noisy image. Thus, network parameter optimization must be performed for each image. In our experiments, we use their official repository to identify the optimal training iterations on a single image and then apply the same number of iterations for denoising the entire volume. (ii) Noisier2Noise (Moran et al. 2020) trains network our experiments, we add additional randomly sampled noise to x', and the training is performed to reconstruct the noisy image (Noise2Noise wasn't 't used due its pronounced over-smoothing denoising effect in the DDM2 experiments (Xiang et al., (023). We want to evaluate our method further by using advanced version, Noiseer (ii) Patch2Self (Fadnavis et al., 2020a) generalizes Noise2Noise (Lehtinen et al. 2018) and Noise2Self (Batson & Royer (2019) for voxel- by-voxel dMRI denoising. In our experiments. we followed their official implementation without adjusting their hyperparameters. (iv) DDM2 (Xiang et al. (223) proposes three-stage framework that integrates based denoising theory into diffusion models and performs denoising through conditional generation. In our experiments, we follow their official repository without adjusting their hyperparameters.
noisier input to target noisy image. In
Additionally, more comparisons with other denoising methods, including MPPCA (Veraart et al.. 2016). Noise22core (Kim & Ye, (02I). Recorupted22ecoomupee (Pang et al. 202T), and Patch2Self2 (Fadnavis et al. (024), are provided in Appendix E We implemented MPPCA using the code from DIPY (Garyfaliidis et al. 2014). For Noise2Score (N2S), we utilized their official repository
Recorrpted2Recoruupe (R2R) was implemented using its repository") For Patch2Self2 (P2S2), directly used the denoised data provided in their supplementary material FFadaavis et al. 2024].
htwpy//ytttttnooooognggoooooooee insuiidii Deep- Deaird--eee----ee-----eeee wawming-foo--mmaeee
Denoising
The denoised data is shared http::/ffihhaaae..com//fffffffft
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D.2 DOWNSTREAM CLINICAL TASKS IMPLEMENTATION DETAILS
On tractography To reconstruct white- pathways in the brain, one integrates orientation information of the underlying axonal bundles (streamlines) obtained by decomposing the signal in each voxel using a microstructure model (Behrens et al. 2014) Fadnavis et al., 2020a Garyfallidis et al. (2144). Noise that corrupts the acquired DWI may impact the tractography results, leading to spurious streamlines generated by the tracking algorithm. We explore the effect of denoising on probabilistic tracking (Girard et al., 2014) by employing the Fiber Bundle Coherency (FBC) metric (Portegies et al. 2015). We first fit the data to the Constrained Spherical Deconvolution (CSD) model (Tournier et al., 2007). The fiber orientation distribution information required to perform the tracking is obtained from the Constrained Spherical Deconvolution (CSD) model fitted to the same data. The Optic Radiation (OR) is reconstructed by tracking fibers (3x3x3 voxels ROI cube, and the seed density is 6) from the calcarine sulcus (visual cortex VI) to the lateral geniculate nucleus (LGN), After the streamlines are generated, their coherency is measured with the local FBC algorithm (Portegies et al. 2015 Duits & Franken 2011), with yellow-orange representing spurious/incoherent fibers and red-blue representing valid/coherent fibers. Since low FBCs indicate which fibers are isolated and poorly aligned with their neighbors, we further clean the results of tractography algorithms by using the stopping criterion outlined in (Meesters et al. 2016) (the stopping criterion was only performed on noisy data's density map of FBC; thus, its results are captioned by "Noisy_ filtering" and can be considered as the reference for high FBCs).
On microstructure model fitting Microstructure modeling poses complicated inverse problem and often leads to degraded parameter estimates due to the low SNR of dMRI (Novikov et al. 2018). Different denoising methods can be compared based on their accuracy in fitting the diffusion signal. We apply two commonly used diffusion microstructure models, the diffusion tensor model (DTI) (Basser et al. 1994) and Constrained Spherical Deconvolution (CSD) (Tournier et al.) (007)
(DIPY (Garyfalidii et al. 2014) has made available of these two models), on raw and denoised data. DTI is simple model that captures the local diffusion information within each voxel by modeling it in the form of a 6- 6-parameter tensor. CSD is more complex model using a spherical harmonic representation of the underlying fiber orientation distributions. In order to quantify the results, we perform a 3-fold cross- validation (Hastie et al. 2009) at two exemplary voxel locations, corpus callosum (CC), a single-fiber structure, and centrum semiovale (CSO), a crossing- fiber structure. The data is divided into three different subsets for the selected voxels, and data from two folds are used to fit the model, which predicts the data on the held-out fold. We quantify the goodness of fit of the models by calculating the R2 score (I2 metric is computed from each model fit on the corresponding data) FFddnaii et al. 2020a).
On diffusion signal estimates We examine how the denoising quality translates to downstream clinical tasks such as creating DTI (Basser et al.) 1994) diffusion signal estimates using the various denoising methods. To do the comparisons, we use the volumes acquired by the first ten diffusion directions and the ten b-value=0) volumes. Before fitting the data, we perform data pre-processing. We first use the method in (Ostul 1979) to compute a brain mask to avoid unnecessary calculations on the background of the image. Now that we have loaded and pre-processed the data we can go forward with DTI (Basser et al. 1994) fitting. We can extract the fractional anisotropy (FA), the mean diffusivity (MD), the axial diffusivity (AD) and the radial diffusivity (RD) from the DTI model.
D.3 SNR AND CNR IMPLEMENTATION DETAILS
To quantify the denoising performance, we employ Signal-to-Noise Ratio (SNR) and Contrast-to- Noise Ratio (CNR) metrics, which are also used in DDM2 (Xiang et al., 2023 We differentiate foreground and background signals following Patch2Self (Fadnavis et | 2020a): 1. Perform uniform normalization on all the data; 2. Use the method in (Ostu 1979) to compute a brain mask; 3. fit DTI (Basser et al., 1994) model to perform corpus callosum segementation; 4. signal corpus callosum signal, background is the signal out of the brain mask. 5. Compute SNR and CNR using:
Mean (signal)
Mean (signal) - Mean (background)
SNR-
CNR
(40)
Var (background)
Var (backgroun))
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Table $2: R of microstructure model fitting on CSD DTI. Bold and Underline fonts denote the best and the second-best performance, respectively. As measured by R²², Di-Fusion achieves the best results across all four different settings.
CSD
DTI
Method
CC
CSO
CC
CSO
Noisy
0.797
0.614 0.789 0.484
ASCM
0.934
0.844 0,942 0.789
DIP
0.868 0.477 0.875 0.381
Nr2N
0.959 0.908 0,961 0.872
P2S
0.927 0.754 0,725 0.675
DDM2
0.863
0.810 0.845 0.790
OURS 0.967 0.939 0.976 0.876
where Mean (.) is the mean, Var (-) is the variance; 6. Statistics are performed on all computed SNR and CNR values, and box plot like Fig. S12 is created.
D.4 SIMULATED DATA IMPLEMENTATION DETAILS
Details on making simulated data Apart from the experiments on in-vivo datasets, we further simulate noisy k-space data to demonstrate that Di-Fusion can still be used for denoising tasks on simulated noisy MRI data, which is done on fastMRI datasets (fastMRI provides raw, complex, multi-echo, and multi-coil k-space MRI data) (Tibrewala et al., 2023 Zbontar al.] 2018). To simulate the effects of adding additional complex noise to the k-space data, we employ k-space noise addition strategies that have been previously validated in prior work (Desai et al. 2021a Xiang et al., 2023). Specifically, we start by sampling Gaussian noise with different standard deviations (to simulate different noise intensities) and add to the real and imaginary parts of each coil's k-space data. Next, we utilize the inverse transformation function implemented in fastMRI (Tibrewala et al.) 2023] Zbontar et al.] 2018) to convert the k-space data into simulated noisy datasets with varying degrees of noise. We simulate five datasets with different noise intensities.
Declaration DIP not considered as a comparison method due to its long computational time (the need for retraining each image) and the mild blurring shown in Fig. Fig. S13 S14 and Fig. S11 The original Patch2Self is not included as a comparison method because it typically requires at least ten 3D volumes to achieve good results (Fadnavis al. 20206 Garyfallidis al. 2014). In contrast, the input 3D volumes in the simulated experiments are limited to a maximum of two (two for DDM2, one for Di-Fusion and Noisier2Noisee. Directly comparing it with other methods on simulated data would be unfair. However, we still develop a reimplementation of Patch2Self, with modifications to the volume extraction part to limit the input volumes (two in our modified Patch2Self). should be noted that our modified Patch2Self is solely utilized in the simulated experiments, where the limited input of two volumes does not yield optimal results. The original Patch2Self still used in the remaining experiments carried out in this paper. Nevertheless, Patch2Self remains useful approach when applied to larger number of volumes (c.g., ten).
E SUPPLEMENTARY EXPERIMENTAL RESULTS
In this section, we include comparisons with additional denoising methods, including MPPCA (Veraart et al., 2016), Noise2Score (N2S) (Kim & Ye 2021). Recorupped2Recorppee (R2R) (Pang (tt al.] D, and Patch2Self2 (P2S2) (Fadnavis et al. 2024) (reproduction details are provided in Appendix D.I.
E. MICROSTRUCTURE MODEL FITTING AND DATA DISTRIBUTION PLOTS
In Table S2 as measured by R², our Di-Fusion achieves the best results across all four different settings. An intriguing observation is that the denoised data from DDM2 exhibits a relatively higher
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6081
Density
6083
2001
0.05
6000
Gray value
Figure S7: Data distribution plots on raw and denoised data. distribution has shifted from the noisy data.
Note that DDM2 denoised data
Table S3: R of microstructure model fitting on CSD DTL Bold and Underline fonts denote the best and the second-best performance, respectively. As measured by R², Di-Fusion achieves the best results across all four different settings.
Method
Noisy ASCM MPPCA
DIP Nr2N N2S
R2R
P2S
DDM2
P2S2
OURS
CSD-CC
0.797
0.934
0.884
0.868 0.959 0.823 0.879 0.927
0.863
0.957
0.967
CSD-CSO
0.614
0.844
0.750
0.477
0.908 0.468 0.731 0.754
0.810
0.934
0.939
DTI-CC
0.789
0.942
0.881
0.875
0.961 0.831
0.872
0.725
0.845
0.973
0.976
DTI-CSO
0.484
0.789
0.614
0.381
0.872 0.348
0.677
0.675
0.790
0.867
0.876
An observation is that the data from DDM2 exhibits higher distribution than other methods. This may explain the improvement of DDM2 in CNR/SNR metrics in Fig. S12 the foreground signals have higher values. Our experiments on downstream clinical tasks in Section 4.2 have shown no correlation between high or low scores CNR/SNR metrics and the performance of the downstream clinical tasks.
distribution (higher mean value) than other methods in Fig. S7
In Table S3 we summarize the quantitative R² metrics of all comparison methods. As measured by RR, our DusFon still achieves the best results across all four different settings.
E.2 ADDITIONAL COMPARISONS ON TRACTOGRAPIYY
In Fig. E.2 we illustrate the effect on the tractography of OR using additional denoising methods. Di- serving
Fusion effectively performs denoising while maintaining high FBCs, with as the reference for high FBCs.
E.3 DTI DIFFUSION SIGNAL ESTIMATES COMPARISONS
We further examine how the denoising quality translates to downstream clinical tasks such as creating DTI (Basser et al.
1994) diffusion signal estimates using the various denoising methods. Details in Appendix D.2 In Fig. S9 we show fractional anisotropy. axial diffusivity, mean diffusivity, and radial diffusivity comparisons. Apart from the poor performance of ASCM, observe that other methods performed well on diffusion signal estimates.
(Fo
Nih
Malex Staring
F252
Figure S8: Density map of FBC projected on the streamlines of the OR bundles. The numbers in parentheses represent the number of streamlines. Di-Fusion maintains high FBCs (consider "Noisy filtering as references for high FBCs).
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Noisy
ASCM
DIP
OURS
Fractional Anisotropy
Axial Diffusivity
Difuuivity
Radial Diffusivity
Figure S9: Fractional anisotropy, axial diffusivity, mean diffusivity, and radial diffusivity comparisons. The main differences are highlighted within the red dashed box. Our method effectively suppresses noise and reconstructs fiber tracts while maintaining a grayscale consistency with the original data (No overall brightening of diffusion signal estimates, especially on axial diffusivity)
Notay
MPPCA
N2S
R2R
OURS
Fractional Anteotropy
Axial Diffusivity
Mean Diffusivity
Redial Diffusivity
Figure $10: Fractional anisotropy, axial diffusivity, parisons. The main differences are highlighted within the red dashed box. Our method effectively suppresses noise and reconstructs fiber tracts while maintaining grayscale consistency with the original data (No overall brightening of diffusion signal estimates, especially on axial diffusivity)
diffusivity, and radial diffusivity
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Noisy Image
ASCM
Noisy image
ASCM
[all volumes used] volumes used)
Fractional Anicotropy
Axial Diflusivity
Meen Diffusiity
Redial Diftusivlty
Figure S11: Fractional anisotropy, axial diffusivity, mean diffusivity and radial diffusivity compar- isons of previous version results and all volumes used results.
On radial diffusivity, all methods exhibited an improved and less noisy representation of the diffusion directions of fiber tracts. However, on fractional anisotropy. and axial diffusivity, DDM2 shows inconsistencies with the noisy image at specific locations (highlighted by the red dashed box), indicating excessive denoising. Our method effectively suppresses noise and reconstructs fiber tracts while maintaining a grayscale consistency with the original data (no overall brightening of diffusion signal estimates, especially on axial diffusivity).
In Fig. E,3 we further compare DTI diffusion signal estimates with those obtained using additional denoising methods. We follow the same steps in Appendix to estimate fractional anisotropy (FA), axial diffusivity (AD), mean diffusivity (MD), and radial diffusivity (RD). Additionally, we computed the reference for FA, AD, MD, and RD using all available dMRI data. Based on this, we performed the calculation of PSNR and SSIM for all slices. The quantitative results are summarized in Table E.3
Questions on ASCM diffusion signal estimates. Fig. S9 minimal signals are observed in the ASCM axial and mean diffusivity. We further utilize all volumes perform diffusion signal estimates and show results in Fig. [SI]] A noticeable signal should be revealed all volumes are used in the diffusion signal estimates. This finding demonstrates that the denoising results of ASCM could significantly hinder the DTI diffusion signal estimates.
E.4 QUANTITATIVE RESULTS ON in-vivo DATA
Given the absence of a consensus on image quality metrics, particularly in unsupervised reference- free settings (Chaudhari et al.) 2020; Woodard & Carley-Spencer (2066). the task of assessing perceptual MRI quality becomes a challenging research problem (Mittal et al. 201T). Considering the infeasibility of using PSNR and SSIM metrics (no ground truth reference images) and their limited correlation with clinical utility (Mason et al., 2019). computing metrics in downstream clinical regions of interest is more reasonable (Adamson et al. 2021). We follow the procedure outlined in (Xiang et al. 2023) to calculate SNR/CNR metrics (Details are in D.3). The quantitative denoising results were reported as mean and standard deviation scores for the complete 4D volumes in Fig. S12
Di-Fusion indicates better performance against competing methods. Our experiments in Section 4.2
have shown no correlation between high or low scores on CNR/SNR metrics and the performance of the downstream clinical tasks.
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Method
Metric
FA
MD
RD
AD
PSNR
23.47
31.79
33.56
26.39
Noisy
SSIM
0.8760
0.9635
0.9637
0.9247
PSNR
22.63
21.81
24.70
20.89
ASCM
SSIM
0.8897
0.8188
0.8921
0,8102
PSNR
24.76
33.01
32.42
30.21
DIP
SSIM
0.8894
0.9675
0.9630
0.9453
PSNR
26.37
35.28
36.35
29.63
MPPCA
SSIM
0.9048
0.9751
0.9780
0,9534
PSNR
29.47
38.42
38.48
34.87
Nr2N
SSIM
0.9354
0.9851
0.9865
0.9712
PSNR
23.13
29.71
31.08
25.84
N2S
SSIM
0.8691
0.9469
0.9491
0.8973
PSNR
25.93
34.54
33.82
30.21
R2R
SSIM
0.8809
0.9708
0.9678
0.9394
PSNR
24.18
32.16
30.62
33.27
P2S
SSIM
0.9090
0.9708
0.9632
0.9677
PSNR
26.77
37.53
37.39
33.21
DDM2
SSIM
0.9041
0.9872
0.9848
0.9610
PSNR
30.08
39.28
40.23
34.06
P2S2
SSIM
0.9432
0.9894
0.9921
0,9701
PSNR
30.79
40.26
40.35
35.30
OURS
SSIM
0.9450
0.9931
0.9923
0.9763
Table S4: PSNR and SSIM comparisons for DTI diffusion signal estimates. Here "AD" represents axial diffusivity, "RD" represents radial diffusivity, "MD" represents mean diffusivity and "FA" represents fractional anisotropy.
SMR
CMR
SNR
ONR
S4R
CNR
SMI
ONR
SMR
CNR
(.640)
(6080]
[8075]
(0S75)
(B285)
(0.08))
36R
CKR
S4R
CNR
SXI
CNR
Figure S12: Box plots of Quantitative SNR/CNR metrics scores. The numbers within parentheses under OURS represent the value of CSAR (Section 3.3). Di-Fusion indicates better performance in terms of SNR/CNR metrics.
We summarize the CNR and SNR metrics of all comparison methods in Table S5 achieves better results in both CNR and SNR metrics.
where our method
Table S5: Comparison of SNR and CNR. Bold and Underline fonts denote the best and the second- best performance, respectively.
Method ASCM MPPCA
DIP
R2R
N2S
Nr2N
P2S
P2S2
DDM2 OURS
SNR -0.7251
0.2372
0.1035
-0.0099
0.2266
-0.1598
-0.1616
0.1526
1.3040
1.5735
CNR
-1.0513 0.2191
0.0567
-0.1161
+0.3004
-0.2004
-0.3694
0.1177
1.2725
1.5687
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Noisy
ASCM
DIP
Nr2N
P2S
DDM2
OURS
Residual
N/A
Stanford- Harddiariii
Output
Residual
Sherbrooke- -shell
N/A
Output
Residual
N/A
PPMI
Output
Figure S13: More qualitative results on Stanford- Hardi. "OURS" results are obtained when CSAR = 0.040 (Section 3.3). Notice that Di-Fusion suppresses noise and does not show any anatomical structure in the residual plots.
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Noisy
ASCM
DIP
Nr2N
P2S
DDM2
OURS
Residual
N/A
Output
Residual
N/A
Output
Figure S14: More qualitative results on Sherbrooke 3-Shell. "OURS" results are obtained when CSNR- = 0.040 (Section 3.3). Notice that Di-Fusson suppresses noise and does not show any anatomical structure in the residual plots.
Noisy
ASCM
DIP
Nr2N
P2S
DDM2
OURS
Residusl
N/A
Output
Residual
N/A
Output
Figure S15: More qualitative results on PPMI. "OURS" results are obtained when CSNR = 0.040
(Section 3.3 Notice that Di-Fusion suppresses noise and does not show any anatomical structure in the residual plots.
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E.5 MORE QUALITATIVE RESULTS
In Fig. [ST] Fig. S14 and Fig. STS we show more qualitative results. For each of the datasets, we show the axial slice of randomly chosen 3D volume and the corresponding residuals (squared differences between the noisy data and the denoised output). We can observe that the results are generally consistent with those presented in Section 4.3) From the residuals "DDM2", it can be observed that particular regions are suppressed (especially in Fig. [ST3] the DDM2 results for the second slice show that the residuals contain a significant amount of anatomical information). Notice that Di-Fusion suppresses noise and does not show any anatomical structure in the residual plots.
E.6 QUANTITATIVE AND QUALITATIVE RESULTS ON SIMULATED DATA
We show the quantitative and qualitative results in Fig. ST6 When the noise intensity is high (left three columns), our method performs the best. When the noise intensity is low (right two columns), denoising results comparable to other methods. Considering the high PSNR and SSIM in the right two columns, it suggests that in real-world scenarios, such data may not require denoising and can still enable effective clinical use. Di-Fusion has more potential for generalization and applicability as it performs better under high noise intensity.
E.7 COMPARE UNDER MIXED B- V-LLEE IMAGES
In Fig. S17 we show additional qualitative results when training mixed b-value images (Sher- brooke 3-Shell has 1000, 2000, and 3500 b-values). Nr2N, P2S, DDM2, and our method both show minimal sensitivity to mixed b-values training data. Minor brightness variations in DDM2 and P2S for multiple b-values have negligible impact on the overall results. Our method primarily learns the mapping from one volume to another, making it less affected by varying b-values in different volumes (P2S uses all the other different volumes, DDM2 uses two different input volumes at Stage 1, and Di-Fusion only uses one different volume). This suggests that the performance of Di-Fusion is relatively robust and not reliant on specific b-vuuue configurations.
E.8 QUALITATIVE RESULTS WHEN USING FEWER DMRI VOLUMES
As shown in Fig. E.8 when using fewer dMRI volumes (20% of original dMRI volumes), Di-Fusion still demonstrates effective denoising capabilities. Please pay special attention that the 30 dMRI volumes here refer to the total training data. Additionally, using a portion of dMRI data from different individuals for model training is more clinically feasible approach.
F VISUALIZATION OF DI-FUSION
F.I FUSION PROCESS: LINEAR INTERPOLATION BETWEEN THE TWO ENDPOINTS
The noise schedule can be found in Appendix D.I In Fig. ST9 we provide a visual demonstration of of (Eq. (6)) Without the Fusion process, the model output would deviate from Tous, resulting in drifted results. By incorporating the Fusion process, where each linear interpolation 2; from x' has as the target, the inference process avoids drifted results (Fig. (a)). We further conducted ablation studies to demonstrate the significance of the Fusion process Appendix G.1
F.2 "DI"" PROCESS: DIFFERENT NOISE DISTRIBUTION
Experiment details: We computed all the E in Stanford HARDI dataset (meaning a total of 76 * 150 11400 noisy images), calculated the grayscale histogram, mean and variance of these noisy images, and presented the calculated mean and variance in the form of a histogram. At the same time, randomly sampled 11400 Gaussian noisy images and performed the same statistical operation.
Statistical properties of Ex- From Fig. $20 the noise caleulated by the "Di-" process has significantly different statistical properties from Gaussian noise. This is reflected in the fact that: 1. the variance of the calculated noise is relatively small and does not follow normal distribution 2.
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incressing
SNR
High SNR
Ground Truth
Naley Imagge
5OM
FSAR
PAAR
Ours
Our Roc
COOK
POWR 16:07
FORR
POLR
PONP
PAAR
DOM2
DD42 Peeakuu
DONR
POUR
PIUR Mi
MrN
NZZN Heee Reeetual
POAR TEA
PINE: AH
POWR
POAH
P2S
P225 Reeedaa
Figure S16: Quantitative and qualitative results on simulated data. In our experiments, CSN'R = 0.040. The red color represents the highest value for the metrie, while the blue color represents the second-highest value. Please note that these are the results of single round of simulated experiments and their corresponding PSNR and SSIM metries scores.
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Nony
(mult-shell)
(2000)
(muuttlsshll)
(2000)
DDME (multi-ssell)
DOME (2000)
(multi-shel))
(2000)
Figure S17: Additional results when training on mixed b-value images (All our results are obtained when CSWR 0.040). "(2000)" indicates using data with only b-value of 2000. "(multi-shell)" represents using data with mixed b-values, including 1000, 2000, and 3500. The performance of Di-Fusion is relatively robust and reliant specific b-value configurations
Noisy
OURS-30
OURS-150
Noisy
OURS-30
OURS-150
Noisy
OURS-30
OURS-150
Residual
N/A
N/A
N/A
Output
Figure S18: Qualitative results when using fewer dMRI volumes. OURS-30 indicates using 30 dMRI volumes, while OURS- 150 represents using 150 dMRI volumes.
t=1
t=5
t=10
t=20
t=100
t=300
x²
(xi) Eq. 6
Figure S19: Visual demonstration of x obtained by different t.
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Grayscale histogram
Mean
Variance
Gaussian
300
009
"D)-" process
D5
Figure S20: Grayscale histogram, mean and variance of these noisy images. We computed all the Ex-2' in Stanford HARDI dataset (meaning a total of 76 150 11400 noisy images), calculated the grayscale histogram, mean and variance of these noisy images, and presented the calculated mean and variance in the form of histogram. the same time, we randomly sampled 11400 Gaussian noisy images and performed the same statistical operation. The noise calculated by the "Di-" process has significantly different statistical properties from Gaussian noise. This is reflected in the fact that: 1. the variance of the calculated noise is relatively small and does not follow normal distribution 2. the counts of each pixel value on the grayscale histogram of Ex-w' are similar, rather than a normal distribution in Gaussian noise.
slice
b
2.37
1.80
1.26
1.24
1.83
4.81
17.5
Figure S21: Slices and their corresponding bz. The by values of the edge slices are relatively larger.
the counts of each pixel value on the grayscale histogram of Ea-a' are similar, rather than normal distribution in Gaussian noise. Different noise distribution makes Fg more capable of modeling real-world noise.
F.3 SAMPLING PROCESS: ITERATIVE AND STABLE REFINEMENT
Value of by In Section 3.3
we adopt a simple definition (Eq. 12 to calculate coefficient by that accounts for the ratio brain tissue the entire image. Fig. S21 displays the slices accompanied by their corresponding by. It can be observed that Eq. (12 simple method for evaluating the proportion of the brain tissue and by can used to correct d, in Eq. (3).
propose adaptive termination during the sampling process. This allows us to control the sampling process by setting the value of CSN'R. RR. In general, setting lower CSNR will preserve more anatomical details. On the other hand, setting higher CSNR will remove more noise at the cost of losing some anatomical details (see Fig. S22 for visual demonstrations).
Iterative and controllable refinement In Section 3.3
d, plots In Section 3.3 we calculate d, (Eq. (13) to represent the degree of denoising in Tout- In Fig. S23 we illustrate the variation d, during the reverse sampling process and present the adaptive termination. It can be observed that with such an adaptive
results when implementing termination, it possible to quickly obtain denoised results (low CSNR results) or further remove noise effectively (high CSA'R results).
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Increasing
Low
High
Target denoise slice
Stanford-Hardi
Figure S22: The results of sampling process obtained by different CSNR.
d
280
X
OURS (0.040)
OURS (0.075)
Ng
Figure S23: Variation of d. during the sampling process. The numbers within parentheses below OUR represent the value CSNR (Section 3.3
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Noisy
OURS Final
w/o Fusion process
w/o "Di-" process
w/o Training in T.
frlet 19
2
3
Figure S24: Qualitative results of ablation studies (Implement an adaptive termination mentioned in Section 3.3 during the sampling process and all the experiments CSVR 0.040). Hcadings distinguish results obtained using different ablation settings.
G ABLATION STUDIES
G.1 ON TRAINING IN DI-FUSION
utilize Eq. (6) to compute linear interpolation from x' to aiming to reduce drift in final results. We disable the Fusion process by substituting for Φ;. Fig. $24 when going through several reverse steps (low SCNR), the results without the Fusion process do not exhibit significant deviations. However, when the adaptive termination is not implemented (which means completing all the sampling steps), noticeable slice misalignment occurs in Fig. $25 (highlighted by the red box).
On Fusion process In Section
On "Di-" process In Section 3.1 we utilize Eq. (8) compute noise distribution Eu-ar and
use it in (x)l=?) and (#t-1 ).)) We directly replace &a-x' calculated in the "Di-" process with Gaussian noise. Without the "Di-" process, results lack some high-frequency information, and the overall gray value of the denoised images has also changed (Case 1 in Fig. S24). Some may consider using Ex-x' only during the diffusion process q (2a( and Gaussian noise during the sampling process pF (#;-1/<2)) We present the results of this setting in Fig. S27 where it can be observed that artifacts occur along the edge slices.
preform training the latter diffusion steps by optimizing Fo to condition on oz,, f ~ Uniform (1 T2)), T, 300. We disable training the latter diffusion steps by optimizing Fø condition T4e t Uniform (1..... 1000)) and balance the training iterations (training the latter diffusion steps iterations: 1e training all diffusion steps: 3) Without training the latter diffusion steps, the denoising results are noticeably smoother and have more hallucinations (Fig. and and Fig. S25). We recommend training the latter diffusion steps based on its potential advantages, which include (i) mitigating hallucinations and (ii) reducing training time with improved stability.
On training the latter diffusion steps In Section 3.2
G.2 ON SAMPLING IN DI-FUSION
On adaptive termination In Section
3.3 we introduce an adaptive termination to enable iterative
and adjustable refinement. In Fig. S21
show slices and their corresponding by. the by values of
the edge slices are relatively larger. In Fig. S22 we illustrate the impact of CSNR on the sampling results. In Fig. SS23 show variation of dz during the sampling process.
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Noisy
OURS Final
w/o Fusion process
w/o *Di- process
w/o Training in T,
Case
finet Ap
Case
Case
Figure S25: Qualitative results of ablation studies (Didn't implement an adaptive termination men- tioned in Section 3.3 during the sampling process). Headings distinguish results obtained using different ablation settings. The red box highlights the main differences.
DDPM Sampling
Run-Walt Sampling
DDIM
Sampling
PSNR: 54.27
PSNR: 38.28
SSIM: 0.993
SSIM: 0.917
Figure S26: DDPM sampling v.s. Run- Walk accelerated sampling (didn't implement an adaptive termination in Section 3.33 v.s. DDIM sampling. For all results, 7) 0. PSNR, SSIM are calculated using DDPM sampling results as references. This indicates that the sampling results from Run Mall sampling are closer the sampling results when accelerated sampling is not used.
Run- Walk accelerated sampling maintains the sampling quality In Fig. S26 we show results obtained by different sampling strategies and metrics (averaged PSNR and SSIM on all volumes) calculated using DDPM sampling results as references. Directly performing DDIM sampling on a pre-trained model may lead to biased results (use DDPM sampling results as references). Run- Walk accelerated sampling significantly improves sampling speed and reduces inference time while maintaining the sampling quality relatively unchanged.
About sampling time We do experiments to demonstrate that the additional computations in Section 3.3 do not impact the sampling speed. Firstly, set CSA'R to 1, which means that all
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Noisy
Use in 1/a x
Use 1 x in Py
(x (
(x)
Use 0 5 in
p, [ (
Figure S27: Using during p (2:-1 [x,)) Using 1- -a during p= (I,-1/æ;) Using 0 O.E during pr (1a-1 (2,) (all results didn't 't implement an adaptive termination mentioned in Section 3.3
during the sampling process).
slices undergo the extra computations and the whole sampling process since CSNR is sufficiently large. Subsequently, we remove the extra computational operations and perform sampling again, The sampling time the first experiment was 1.19 seconds per slice. In contrast, in the second experiment, it was 1.18 seconds per slice, which indicates that the additional operations have minimal impact on the sampling speed. Table S6 the sampling time per individual slice is presented for different CSNR. We find that When CSNR is low (CSNR = 0.040), the sampling time is 0.0395 seconds per slice. This indicates that our adaptive termination and Run-Walk accelerated sampling greatly reduce the sampling time.
Using Ex-ar in reverse process In Fig. $27
we demonstrate the importance of using Eu- and setting 77 in the sampling process. It can be observed from the final results that the central slices (with more brain tissue), using = ~ N' (0, during the sampling process pF (#t-1 only leads to subtle differences in the denoised results. However, the edge slices (with less brain tissue). using significantly impacts the sampling results, resulting in additional regions that appear inexplicably (highlighted by the red box, and these additional regions don't 't appear in noisy data). During the sampling process, DDM2 uses Because our sampling process is deterministic, according to the experiments DDIM (Song al. 2020b), we set = 0. We further demonstrated the sampling results in the figure with 17 and O. When 7 although the presence of unexpected regions reduced, some still remain. However, when 7 = 0, such issues don't 't arise
Table S6: Sampling time per slice for different CSN'R (Stanford HARDI). We set different CSNR parameters for Walk accelerated sampling and DDPM sampling to perform adaptive termination.
CSNR Time (s) for Run-wall
Time (s) for DDPM
0.04
0.0395
0.327
0.045
0.115
0.739
0.05
0.626
2.01
0.055
1.08
5.48
0.06
1.11
6.97
1.18
11.5
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Table S7: R of microstructure model fitting on CSD DTI. Bold and Underline fonts denote the best and the second-best performance, respectively.
CSD
DTI
Method
CC
CSO
CC
CSO
Noisy
0.797
0.614
0.789
0.484
ASCM
0.934
0.844
0.942
0.789
DIP
0.868
0.477
0.875
0.381
Nr2N
0.959
0.908
0.961
0.872
P2S (OLS)
0.927
0.754
0.725
0.675
P2S (Ridge)
0.927
0.757
0.927
0.673
P2S (Lasso)
0.824
0.471
0.816
0.429
P2S (OLS, r=1)
0.934
0.832
0.950
0.735
DDM2
0.863
0.810
0.845
0.790
OURS
0.967
0.939
0.976
0,876
Noisy Image
P2S (OLS)
P2S (Lasso)
P2S (Ridge)
P2S (OLS, r=1)
OURS
Figure $28: Comparisons with different Patch2Self experimental settings. obtained when CSNR 0.040.
"OURS" results are
H MORE COMPARISONS WITH COMPETING METHODS
H.1 COMPARE WITH DIFFERENT PATCH2SELF SETTINGS
In Fig. S28 we show additional results on comparisons with different Patch2Self experimental settings. Our modifications are limited to the denoiser type (OLS, Lasso, Ridge) and patch radius, following the official repository of Patch2Self (Fadnavis et al. 2020b Garyfallidis et al. 2014). The term "(r=1)" indicates changing the patch radius to 1, while the patch radius is assumed to be 0 if not specified. Modifying the denoiser type and patch radius in Patch2Self does not yield substantial improvements in the results. Altering the denoiser type does not impact the overall denoising time, whereas changing the patch radius significantly increases the overall denoising time. In our experiments, employing the OL.S denoiser required 4 hours, while utilizing OLS with a patch radius of took 26 hours.
show additional tractography results on comparisons with different Patch2Self experimental settings. Although the number of streamlines is the lowest in the "(OLS, r=1)" experi- mental setting, still misses the high FBCs indicated by the white arrows. There are no significant differences in the results in the remaining experimental settings. Considering the computational burden when setting the patch radius to we suggest setting the patch radius of Patch2Self to to improve efficiency.
In Fig. $29
In Table S7 we show quantitative results (on microstructure model fitting) on comparisons with different Patch2Self experimental settings. Varied experimental settings influence the performance of microstructure model fitting. Nonetheless, these modifications do not change the rankings of the best and second-hest results.
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(045,m))
(134)
(2003]
(260]]
(2228)
(1877)
1999
(1983]
Figure S29: Density map of FBC projected the streamlines of the OR bundles. The numbers in parentheses represent the number streamlines.
Noisy Image
Nei2Nei
OURS
Stanford-Hardi
Sherbrooke 3shell
PPMI
Figure $30: Qualitative comparisons with Neighbor2Neighbor. "OURS" results are obtained when
CSNR 0.040.
H.2 COMPARE WITH NEIGHBBOR2NEIGHBBOR
the mentioned methods exhibit significant drop in performance when confronted with real- world noisy images. particularly when the explicit noise model is unknown. To make up for this, Neighbor2Neighbor (Nei2Nei) (Huang et al., 202T) and Zero-shot Noise2Noise (Mansour & Heckel 2023) sub-sample individual noisy images to create training pairs and are more robust against.
In Section 2.1
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Fiber to Bundle Coherence (FBC)
low
high
Noisy filtering (1524)
Noisy (3091)
Nei2Nei (486)
OURS (1862)
Figure S31: Density map of FBC projected on the streamlines of the OR bundles. The numbers in parentheses represent the number of streamlines.
real-world noise. We compare our method with Nei2Nei using the same model in Di-Fusion. We implement Nei2Nei with parameters set to the default values specified in their official repository
The qualitative results are in Fig. $30 It can be observed that Nei2Nei does not perform well in denoising, there partial jagged artifacts in the image and significant changes in grayscale. This may be the reason that Nei2Nei denoising relies on the structural similarity of the neighboring regions in the image. This can also be seen the qualitative results of Nei2Nei. where the images maintain structural similarity in sub-sampled noisy images, leading to better denoising results. In Fig. S31 it can observed that the density map of FBC projected on the streamlines of the OR bundles is missing significant number of FBCs; thus the denoising results of Nei2Nei are unsuitable for modeling tasks.
I DDM2: STAGE HAS A HUGE IMPACT ON FINAL RESULTS
we present the results of DDM2's first stage and corresponding third stage on the Stanford HARDI dataset. By utilizing the hyperparameters from the DDM2 official repository and conducting experiments (only the training iteration in stage 1 was modified, the official training iteration is set 10c*), we have observed that coarser outcomes in the first stage yield more striking yet less stable denoising results in the final stage. Conversely, deterministic outcomes in the first stage result in more stable but uninteresting denoising results in the final stage. Different first stage results lead to drastically distinct outcomes in the third stage. The CNR and SNR scores show significant differences between different first stage results (Fig. S32 (right below)). Please note that the subsequent experiments we conduct on DDM2 are using their best results.
Fig. S32
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Iter: 100000
Target denoise slice
Iter: 10000 1
Corresponding stage 3
Iter: 20000
stagen 1
Iter: : 10000 2
Iter: 12000
Corresponding
stage 1
stages 3
Iter: 10000 3
4
2
Iter: 10000 best
0
-2
-4
ASCM
DOM2
(100)
(12)
(30.3)
(10.30
(10.3)
bea))
Figure S32: DDM2 unstable model outcomes. (right below) Show the CNR metric of different experiment settings; CNR/SNR metrics show the same trend. The red box highlights the best results obtained using the parameters from the official code repository. Having stable Stage I often leads to poor performance in CNR/SNR metrics, the red color within the parentheses represents the settings corresponding each experiment.
40


Reviews:
[{"review_id": "nYGcBeNDTm", "reviewer": "Reviewer_FDwC", "summary": "The paper proposes a novel self-supervised denoising method Di-Fusion that leverages the latter diffusion steps and an adaptive sampling process.  Di-Fusion outperforms two slightly older methods and a state-of-the-art approach on   and on downstream processes like tractography.", "questions": "Please see weaknesses above.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Paper is easy to follow.\nResults across multiple real and simulated datasets, suggesting generalizability of approach.\nBaseline is a recent state-of-the-art.\nThe authors released their code to the reviewers, which is well-written, informative and aides reproducibility.", "weaknesses": "Did the authors consider using: https://arxiv.org/pdf/2305.00042  and    https://arxiv.org/pdf/2309.05794  as baselines?\nBetter signpost the extensive results in the supplementary materials.\nSome parts of the paper read a bit odd and should be checked for oddities e.g. from the introduction 'The MRI, including ...', 'Consequently, the denoising technique plays a crucial role..'", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "1Ka1g8dTtt", "reviewer": "Reviewer_TzM3", "summary": "This paper introduces Di-Fusion, a fully self-supervised diffusion MRI (dMRI) denoising method designed to enhance the signal-to-noise ratio (SNR) of MRI data without requiring clean reference data. The authors leverage novel late diffusion steps and an adaptive sampling process to create a single-stage framework that operates without an explicit noise model. Di-Fusion demonstrates superior performance over state-of-the-art denoising methods in tasks such as microstructure modeling and tractography. The method’s efficacy is validated through extensive quantitative and qualitative evaluations on real and simulated data.", "questions": "- To my understanding, the primary goal of dMRI denoising is to reduce the number of gradients required during acquisition, thus accelerating DWI scanning. In downstream tasks based on DTI, the authors compare DTI metrics computed from noisy images with those from denoised images. Why did the authors not use more DWI data to compute a clean DTI metric as a reference for comparison?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Flexibility with data and noise models**: Instead of relying on explicit noise models or clean training data, the method relies on an N2N training strategy and pixel shuffling to reorganize the noise, providing strong generalization potential across different noise distributions. This suggests that the method has the potential to be applied to a wider range of denoising scenarios, such as cryo-EM.\n- Compared to the current state-of-the-art method, DDM^2, this approach demonstrates comprehensive improvements. Not only does it outperform in terms of performance, but it is also simpler to implement. Notably, this method does not require additional denoiser training, significantly enhancing its practical usability.\n- As a study on dMRI denoising, this paper conducts thorough and comprehensive experiments, including extensive comparisons and analyses on downstream task performance. This renders the work methodologically and experimentally well-rounded.", "weaknesses": "Please refer to the **Questions** section for details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "5HIUhft58i", "reviewer": "Reviewer_Dn7R", "summary": "This paper presents a new self-supervised learning-based denoising method for diffusion MRI (dMRI). The proposed method leveraged the diffusion modeling concept, but instead of training a diffusion model with “clean images” as x_0 and noise as x_T, it utilized two diffusion weighted images (DWIs) with different diffusion encodings at both ends of a “diffusion-like” process. A denoising network was trained by predicting one DWI using a linear combination of two DWIs and an added noise term. The linear combination coefficients are time-dependent and determined via a scheduling strategy similar to training a diffusion model. The network was then used for a conditional sampling step for generating the final denoised images. The idea to utilize images acquired with different diffusion encodings to denoise one of them is interesting and the training strategy is an interesting approach to leverage the diffusion modeling concept, especially with training only latter diffusion steps to reduce hallucinations. However, several key assumptions made are questionable and the overall methodology and presentation lacks clarity. Evaluation using only dMRI signal model goodness of fit is limited and can be biased. There are a few overstatements that can mislead the readers. Detailed comments can be found below.", "questions": "In Eq. (5) on Line 155, the authors highlighted a specific term as the ”major difference” between xt−1 and x^bar_t−1. Could the authors clarify why this particular term is considered the primary source of difference? Furthermore, can the authors elaborate on the underlying reason(s) for the “drift” in the model and how it emerges during the reverse diffusion process?\n\nAccording to the definition of the Fusion process in Eq. (6)  and the “forward process” in Eq. (7), it appears that the starting point for the forward process changes based on t, as x_t* is dependent on t. This dependence implies that the Fusion process dynamically adjusts the starting point of the forward process at each step, which is unconventional compared to typical diffusion models. Could the authors clarify the rationale behind this design?\n\nOther more recent self-supervised denoising methods should be compared, if not for all, e.g., Noise2Score and Recorrupted2Recorrupted etc.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "A diffusion-like modeling that learns the relationship between two DWI volumes with different diffusion encodings to denoise one or each other.\n\nTraining only later step diffusion to avoid hallucination\n\nA fusion strategy that exploits linear combination of two DWIs with different contrasts with time-dependent coefficients and iterative refinement.\n\nExtensive evaluations using both simulations that exactly followed the assumptions for the proposed methodology and practical magnitude DWI data.", "weaknesses": "There are statements that can be misleading in the context of MR physics (aka domain knowledge). For example, \"the noise predominantly originates from physical interferences (Fadnavis et al., 2020a)\". This statement about physical interferences is  both vague and inaccurate. This work is dealing with thermal noise or noise resulting from thermal noise in the measurements, which is not really physical interferences depending on how ones interpret them. Another example, \"Different clinical applications require varying numbers of diffusion vectors and acquisition strategies, which makes modeling the noise distribution and further implementing denoising techniques challenging\". Acquiring DWIs with varying numbers of diffusion vectors had nothing to do with the difficulty of  modeling noise distribution.\n\nMany key assumptions for the proposed method was built on do not hold which made the theoretical/mathematical foundations questionable, e.g.,\na) It seems that the authors assumed DWIs acquired with different diffusion encodings had the same underlying “clean” image and were corrupted by independent noise. This is inaccurate. In fact, two DWIs can have rather different contrasts due to the diffusion encoding effects, e.g., different diffusion encoding directions. More specifically, x and x’ cannot be simply modeled as the same y plus different noise. What are the implications of this assumption not met?\n\nb) Line 111: The authors claimed that that the proposed method does not require an explicit noise model. This is an overstatement. The J-invariance assumption, which formed the basis of the training objective in Eq. (9) implicitly requires that the noise distribution be zero-means and conditionally independent. Furthermore, additive noise model was assumed, x = y + n1 (Line 200). In dMRI, the magnitude images with higher b-values (stronger diffusion weightings) can have lower SNR for which additive noise may not hold. These need to be clarified.\n\n-  Overall, the presentation lacks clarity and there seem to be some concerning inaccuracies.\na) The linear combination relationship claimed in Section 3.1 does not seem accurate. I checked the derivation. Eq. 31 is correct which is known (so this is not a contribution of the authors), but I'm not sure about going from Eq. 31 to 32 as F_theta predicts x_0, but they are not equal, and there is also an additional term of sigma_t^2*z. Therefore, I don't think it's a correct statement to say x_(t-1) is a linear interpolation between x_out and x_t. But is this really needed for the proposed method? I really don’t see a connection between what’s argued theoretically and what’s actually being implemented.\n\nb) There are a few other inaccurate mathematical statements and notations which are confusing. For example, Eq. 7, the left side has q(x1:T |xt*) which is a joint distribution for x1 to xT, and the right side is a Gaussian distribution for xt. \nOn Line 160: {xt}1:T was described as”obtained from the reverse process.” However, in\nFigure 1 and on the right side of Equation (7) on Line 186, it appears that xt is a corrupted version of xt*.  This interpretation, along with the notation in the Fig. 1, implies that {xt}1:T would represent a forward process. It appears to this reviewer the authors had not been using a consistent definition of forward and reverse diffusion which made the overall description rather confusing. These are just examples of inconsistencies found.\n\nc) According to the J-invariance property, the noise should ideally have zero mean\nand be conditionally independent of the target output. This requirement is necessary to ensure that the expected loss for self-supervised training asymptotical approaching the supervised loss. However, the input to F(.) in Eq. (9) includes xt*, which is a linear combination of x and x’ (Eq. (6)). Given that x serves as the supervision signal for the loss, this implies a correlation between the input x∗t and the target x, which would violate the conditional independence requirement for J-invariance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "ykQJf3GIbG", "reviewer": "Reviewer_K1kH", "summary": "This is a new denoising method for dMRI data. Combines DL-based diffusion models with a bit of fusion. The fusion process stabilizes issues that DDM2 has.\n\nI suggest that the authors refrain from large claims. For example, it says that it outperforms the other methods. But I do not see any speed or memory comparisons. \n\nIn the comparisons I would also add MPPCA. I would also cite Patch2Self2 (CVPR 24). Patch2Self has clearly outperformed MPPCA however still many people use MPPCA.\n\nThe paper does a great job on the methodological sections. \nIn providing code and using open source standards.\n\nHowever, at least a thorough review of language is required.\n\nQualitatively it is hard to see large advantages over Patch2Self but nonetheless the method is useful. \n\nIn the revision please report time and memory usage. I would also compare against Patch2Self2 if possible.\n\nAlso it would be important to explain the setup. What GPUs were used for training?", "questions": "What is the actual minimum number of B0 and DWI volumes required ?\n\nCan this work with data that have a single B0? Would that be denoised?", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "Great way to stabilize the diffusion process.", "weaknesses": "Refrain from large claims. \n\nReport time and memory usage.\n\nReport setup (GPU types, numbers and VRAM).\n\nCheck statements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "GpYzolf7Hp", "reviewer": "Reviewer_9LpF", "summary": "The paper proposes a method for denoising diffusion MRI data sets.\n\nThis is a well-studied problem with many solutions in the literature. It is an important problem, as diffusion MRI is widely used for neuroscience and for clinical medicine. Recent years have seen a trend towards using self-supervised approaches to characterise the noise distribution and separate noise from the underlying signal.  This submission falls very much in this category, but proposes a different algorithm to those that are popular in the literature.\n\nExperiments compare against five baselines and results appear competitive with other methods, sometimes surpassing them.", "questions": "Corresponding to weaknesses listed above.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The algorithm appears novel, although I found it hard to tell from the literature review how novel it is - whether it takes ideas from other areas and repurposes them for this problem, or if this is an algorithm specifically designed for diffusion MRI.\n\nThe problem is an important one with widespread application.\n\nResults appear competitive on a few example images shown in the figures.", "weaknesses": "The baselines chosen do not include the most widely used denoising methods.  A clear omission is the random-matrix theory approaches proposed by Veraart et al in a series of very highly cited papers starting with Neuroimage 2016.\n\nThe only quantitative results use simulations, which seem likely to be skewed towards to capabilities of the proposed algorithm.\n\nThe qualitative results on actual human data are questionable as to whether they show improvement over baselines.  Even if they do, these are single cherry-picked examples and it is not clear whether these are advantages that manifest over large collections of images/scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)