DGSolver: Diffusion Generalist Solver with Universal Posterior Sampling for Image Restoration
Authors: Hebaixu Wang
Jing Zhang
Haonan Guo
Di Wang
Jiayi Ma
Bo Du
Venue: NeurIPS 2025 poster ()
Keywords: Image restoration
diffusion generalist solver
universal posterior sampling
deep learning
Primary area: deep_learning
TL;DR: Fast high-order generalist diffusion solvers with universal compensation mechanisms for high-quality image restoration in a training-free manner

Abstract: Diffusion models have achieved remarkable progress in universal image restoration. However, existing methods perform naive inference in the reverse process, which leads to cumulative errors under limited sampling steps and large step intervals. Moreover, they struggle to balance the commonality of degradation representations with restoration quality, often depending on complex compensation mechanisms that enhance fidelity at the expense of efficiency. To address these challenges, we introduce \textbf{DGSolver}, a diffusion generalist solver with universal posterior sampling. We first derive the exact ordinary differential equations for generalist diffusion models to unify degradation representations and design tailored high-order solvers with a queue-based accelerated sampling strategy to improve both accuracy and efficiency. We then integrate universal posterior sampling to better approximate manifold-constrained gradients, yielding a more accurate noise estimation and correcting errors in inverse inference. Extensive experiments demonstrate that DGSolver outperforms state-of-the-art methods in restoration accuracy, stability, and scalability, both qualitatively and quantitatively. Code and models are publicly available at https://github.com/MiliLab/DGSolver.

DGSolver: Diffusion Generalist Solver with Universal Posterior Sampling for Image Restoration
Hebaixu Wang 1.3 Jing Zhang" Haonan Guo 4, Di Wang2.3 JJayi Ma and Bo 2 School of Electronic Information, Wuhan University, Wuhan, China Schooo of Computer Science, Wuhan University, Wuhan, China $7hongguaneun Academy, Beijing, China State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, China
banplbabaxxxjjjjgzheeeee cv jymm20000)0mmail. com; (haonan. guo, d_wang, dubo Aubo)2hhu. edu.
Abstract
Diffusion models have achieved remarkable progress in universal image restoration. However, existing methods perform naive inference in the reverse process, which leads to cumulative errors under limited sampling steps and large step intervals. Moreover, they struggle to balance the commonality of degradation representations with restoration quality, often depending on complex compensation mechanisms that enhance fidelity at the expense of efficiency. To address these challenges, we introduce DGSolver, a diffusion generalist solver with universal posterior sampling. We first derive the exact ordinary differential equations for generalist diffusion models unify degradation representations and design tailored high-order solvers with bheue- accelerated sampling strategy to improve both accuracy and efficiency. We then integrate universal posterior sampling to better approximate manifold-constrained gradients, yielding more accurate noise estimation and correcting errors in inverse inference. Extensive experiments demonstrate that DGSolver outperforms state-of-the-art methods in restoration accuracy, stability, and scalability, both qualitatively and quantitatively, Code and models are publicly available at httpp:://github. http://hhbbb.oom///lll/////c////
Introduction
Universal image restoration (UIR) strives to achieve versatile model capable of discerning and mitigating various degradation types. Capitalizing on the superior attributes of the restored details with high fidelity, it has emerged as promising approach for enabling applications such as autonomous
driving 377, remote sensing 44 and other related fields 773 26
58 59
68 54 55 56].
Owing to the substantial advancements in deep learning, a surge of universal image restoration frameworks has emerged. Typically, the existing methods can be delineated into categories based on the dependency of representations across varied degradation distributions [64 74 9 namely blend-repressenttiin--er
methods and distinct-representation based methods, displayed in Fig. The former approaches promote the commonality among degradation representations by projecting diverse degradation distributions into a shared representation space, where the commonality mea- sures the petpddattionnnnostti extent of representations within the shared representation space (34). However, the complexity of restoration escalates as the commonality of degradation representations intensifies [00 28 29]. DiffUIR [75] reaches compromise by projecting the disparate distributions
onto an impure Gaussian distribution, retaining partial degradation conditions. FoundIR
23 further
Cooresponning author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Micutber
Generic Encoder
Generic Decoder
Text Exports
Light Esports
Espiris Pool
Degradution apnossi Represmtation
Qualiyy Refinmment
Tank-speeffic pilend-rpresentation-neess image restoration
rain
light
blar hase
Generic Docoder
Conditional Decoder
Foundation Model
SHON
light
Tesk-specific Pronpe Eiwbedding
Representation Space
(b) Distinat-eprreenntrion---- image restoration methods
Figure I: Overview of mainstream UIR methods. (a) These methods typically train generic models learn shared distribution with compensation mechanisms for quality refinement. (b) Conversely, distinetreprrssentatin---e methods integrate task-specific modules into generic models to handle diverse degradations, forcing the generic model to learn different distributions.
implements a mixture-of-exppeee system to mitigate the diminished performance resulting from the pursuit of commonality. Consequently, the compensation systems that are incurred significantly re- duce efficiency. The latter approaches utilize multi-encoder architectures prompts from foundation models to acquire distinct representations tailored for specific degradation distributions (27 42 13]. Afterwards, customized conditional guidance is incorporated into a shared decoder reconstruct the distorted information [T] Nevertheless, their performance constrained by the limited explo- ration of degradation correlations, as different tasks may potentially share complementary information that could augment the efficacy individual tasks (25]. For instance, low light conditions and noise frequently coexist, while rain and raindrops often interlinked.
In this paper, we aim to develop precise and efficient method for universal image restoration. is observed that the existing methods are plagued by two major drawbacks: One is the dilemma between the commonality of degradation representations and restoration quality, and the other is the lack of a versatile and effective compensatory mechanism for quality refinement. To this end, we propose DGSolver, a diffusion generalist solver with universal posterior sampling for image restoration, as presented in Fig. It amalgamates the merits of ordinary differential equation (ODEs) inference and leverages diffusion posterior sampling as universal accuracy
solvers for compensation for refinement under high commonality conditions. Specifically, we adopt the diffusion generalist models (DGMs) [75] 29 to unify diverse degradation types into a purely degradation- agnostic latent distribution. Subsequently,
tackle the alternative sampling from DGMs by solving the corresponding semi-linear ODEs to alleviate the discretization error accumulated in the multi- step inference process. Benefiting from the analytical structure, the solutions can be equivalently simplified to a linearly weighted integral of the neural networks with an accelerated sampling strategy. Furthermore, seamlessly integrate the underlying mechanism of generalist modeling with Bayesian posterior sampling, thereby universally serving as manifold-constraal guidance without incurring computational expenditures. Consequently, high-order ODEs numerical approximation and universal posterior sampling are combined to effectively improve the solution accuracy and stability training-free manner. Extensive experiments on natural and remote sensing scenes are conducted to verify the superior performance of our method, showcasing its remarkable restoration capability across different application scenes. Our key contributions
gradients for restoration
follows:
1. We reformulate the generalist diffusion process using ODEs and derive its analytical solution form. Besides, high-order solvers equipped with an accelerated sampling strategy are customized to mitigate accumulated discretization errors and boost efficiency during inverse inference.
2. To further enhance the accuracy and stability of restoration, we introduce universal posterior sampling strategy to offer manifold-constrained gradient guidance for noise estimation. It seamlessly integrated into generalist diffusion solvers requiring no training.

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Forvard
Inference
Correvtion-fer
Untversal Pesterior Correction
High-arder Solver Correction
Dien + W.
(00)
(00
12 (0)
(0)
(00)
th .order Solver
Degralationggeenn
(()
Representation
100
1
7
7(K)
7(
1
(K)
00]
(()
Diffasion Generalies Model
T-1
-order Solver
1-1
My-1
Generative Munifold M2
(1)
(1)
Cumullatvv Extor
M
M
urddre Solver
1 orrde Estimation
_order Estimation
.order Currected Estimation
Ground-truth
Figure 2: Illustration of the DGSolver. In the forward process, we utilize diffusion generalist model to uniformly represent diverse degradation categories into degradation-agnonn distributions. In the inverse inference, we employ a -order solver to alleviate the discretization enror accumulated in the multi-step sampling process. Simultancously, universal posterior sampling is used to stabilize the solution and further minimize the discrepancies with the ideal one generated by -order solver.
2 Background
2.1 Diffusion-based Image Restoration Model
Grounded in solid mathematical foundations, diffusion models [0 51] have emerged as powerful frameworks for image restoration (see Appendix AA). Initially, lots of works focus on structural improvements to facilitate the restoration of details [62 40 49 by simply treating degraded images as conditional inputs of the denoising network. Afterwards, InDI
[0]] and 12SB (28 pioneer pro-
found theoretical and practical improvements in the reverse process, opting for estimating the target image or its linear transformation term to replace the noise estimation. RDDM (29] extends beyond the single diffusion paradigm to independent double diffusion processes corresponding to residual and noise estimations, respectively. Consequently, the residual prioritizes certainty to restore the major structures while the noise emphasizes diversity to retrieve the dismissed details. Furthermore, DifFUIR (75) incorporates partially degraded information into the forward process and constructs a non-pure Gaussian distribution representation of different degradation types. Nevertheless, the com- monality of degradation representations and the performance of restoration are mutually constrained, with versatile and effective approach available to simultaneously enhance both.
2.2 Diffusion Posterior Sampling
To tackle the ill-posed attributes of linear inverse problem [19], diffusion posterior sampling has been developed as a novel approximation tool to reduce the solution uncertainty [1 45 Specifically, leverages the Bayesian framework to accurately estimate the clear image To from its degraded
observation in = ATo + n, where A is measurement operator and is usually the Gaussian noise.
The stochastic differential equations (SDEs) of diffusion probabilistic models [5] with diffusion term fle, t) and drift term gt) can be formulated as:
dI. = [T,, tt + g(t)dw:,
where W, is the standard Wiener process. The reverse SDE of this process can be:
dl. [S(,,, 1) (²) log (1)]d g())ww, where VI, log qı (1t) defines the predicted score functions. the case of inverse problems, we want to generate the posterior distribution qe (L A). Leveraging Bayesian rule, Eq. ( becomes:
(2)
d1, [f t) - 3² (I) log ge (1.) + V,, log (m (m + gt)
Nevertheless, V,, log ( (I)) can be computationally intractable. Chung et al. 7 first approxi-
mate gradient of the log likelihood by assuming ((4m)4:) q(4nn = E[4]/4]) NNinn
3

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A Σ = o7, I). Essentially, they substitute the unknown clean image To with its condi- tional expectation [e] and known operator A, making the term pl Lin 44) tractable. Afterwards, PSLD [8] extends pixel-level posterior sampling into latent space, showing provable sample recovery in low-dimensional subspace. Furthermore, BlindDPS [6] introduces multiple diffusion models to construct parallel reverse procedure to jointly estimate the operator kernel and clear image. In summary. existing posterior sampling relies on known or estimated degradation operators. The former still suffers from ill-posedness in the practical scene due to the availability of measurement operators, while the latter only fits well in kernel- sernel-based degradation modeling with extensive computational overhead and cannot be generalized more complex degradation scenarios.
Methodology
3.1 Diffusion Generalist Formulations
Assume that a clear image In IR& is obtained by sampling from an unknown distribution go(lo). We define forward process {4 0.27 starting with lo, such that for any / E [0, T the distribution of I is conditioned on Io, Lin, and Iree. which satisfies:
yor Co Tess lin) (Ne Jo + a(l) - 8(1)/in m (1)I),
(4)
where (tt), (t) R are differentiable functions of t with bounded derivatives, and Irea is the residual items equal to Jn To. When 1. ~ T, (T)) - 1,5(7) + 1, the endpoint 7, = (t) is merely represented by a pure Gaussian distribution with a (tt) (toouulleee variance. Therefore, we refer to the linear noise schedule [4] to generate parameter series and modify the value of ox(7), 5(7)
to 1. Consequently, these variables are monotonic, and we denote them as ae o, B, for simplicity. We prove that the following stochastic differential equation (SDE) possesses an identical transition distribution gor(11)4 (o,//o, Lres lin) as in Eq. 4 for any t E [0,T] (proof in Appendix A.15):
dl = raa) + h(() dl + g(L)dwrr,
Io ~ go))))
(5)
dæ
dō.
d3²
f(t)
h(t)
g²
(6)
dt
dt
dt
To accelerate the sampling process, we consider the associated probability flow ODE with the same marginal distribution each time as that of the SDE, which can be formulated as:
dI, dt = f(t)&res + h(t) 7 (t)V,, log (1))). 4~~9r(11
1
Apparently, the rh.s of Eq. (7) comprises two indeterminate variables, i.e. residual components Iree and score functions V log qe (Le), necessitating estimation via neural networks conditioned on the temporal variable t. Pursuant to the Tweedie formula, the score functions V,, log qe (1e) can be equivalent to noise prediction, implying that e = -B, V log qe (It). Hence, the optimal sample can be obtained by solving the following ODE from T to O:
dī, dt
g²(1) 23, cT Iin, I I i qr(Ir),
f(t) (Te, Iim f) h(t) +
(8)
where 19 (It, Lin, 1) and
t) are the residual estimator and noise predictor, respectively. In Eq. 8), Tit is linear function of Iin, while the other two parts I (T,, Tin, t) and (1) typically 23,
exhibit nonlinear dependencies on I, because of ncural networks. Previous works are ignorant of this semi-linear structure by decorating the entire Eq. (8) into black-box ODE solver, which not only increases the complexity of the solution but also exacerbates the discretization errors of both linear and nonlinear terms. Given initial value I, at time > the solution I, time <t < T can be precisely formulated as follows:
I = Ig + S. dii, (To (Jr,Isn Jsn, 7)dr - Iin J db, dr + [' dB, - dr
dr co(Tr, Iiny r)dr.
dr
Such taxonomy decouples the origin of discrete error, where the linear part be accu- rately computed, but the integrals of the nonlinear parts are still complicated. As a2 de, Br are

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strictly monotonic function of t, there exist inverse functions tz tz(-), and tz(()) satisfying t - (x((()), A { 6, 3), respectively. We change the subscripts of I and eg from t to À and denote Ires Lin = Ires Lin, a) c0 Am. T) = Eo Eo Im, 3). For each X, Eq. can be reformulated as:
Proposition 1 (Exact solution of ODEs for diffission generalist models, proof in Appendix B Given an initial value Ig time 8 > O, the solution It time < <t< T of that ODEs Eq. (8)
3. aa T Poe Jmn aidd {a co(T3, Jinn B)dB.
I, (ō, (2.-5,)[e +
(10)
In consequence, Eq. 10p provides a new perspective for approximating the optimal solution, where estimating the solution at time i is equivalent to directly approximating the integral of 7 (TT,, Liny a) from ag to at and éo (Tg, Am. 3) from 3, to 3,. In light of such formulation, we are capable of customizing diffusion generalist solvers to improve the convergence and accuracy of solutions.
3.2 Diffusion Generalist Solvers
To enhance the precision in solving Eq. 8
we propose high-order solvers for the ODEs with a convergence order guarantee. Specifically, given initial point Ir at time T and total M + 1 time steps [t decreasing from to = T to tM = O. Let {Te /== be the sequence iteratively computed at time steps {4))- using the presented solvers. To mitigate the cumulative effect significant errors, endeavor minimize the inaccuracy for the solution I,, each step. Starting with the previous solution I,
at time fç-1,
the exact solution -1, at target time t, is:
("
I1---- = -e- (ōu. 8,,,, Jin
Po P Imn a)da Jo
C0(TT, Iin, Bd. (11)
We utilize the Taylor expansion to make & orders approximation to calculate the integral of both 7o (Ta, Tin, from 0 oz; and Co Jin, B) from B,,,, to Beg res
(a Cit, ) Taa Lin, a...)) + -(..)))..))))
Pes Lin
(12)
n!.
Be, )m EO Eo B Lins B,, (B,,,...)"
Eo (T Lnn B)
(13)
n!
n=0
Substituting the above Taylor expansions into Eq. yields:
Cr Tre n+1
11 = = - (ōc. -
7())) res
(n 1)!
n=0
(14)
(Br, - Br, jut1 (n 1)!
By discarding the approximation emrors terms O(-), we approximate the ni-th order total derivatives 70(m) (.) and <(m) (-) for n
1 with finite difference [88] In the case 4. - - (ō.. - An + (āe. - -._.) - + - -e, Isn (41). (15)
1, Eq. 14 becomes:
Iin,
Notably, prior works 75 2x are a special case of the first-order solver within our methods by omitting the discrete enror. For l = 2, the calculation of the first derivative requires additional intermediate time points t. = rt, + (1 r)tz2 in the range of [t, t--1] with controllable parameters r, and we have:
(Ti, (
la = lem -6e, - -dem Men (oa, Tee M (Ke Lmn + -Bo,, )eo((en Lins tra) +
2r
(B
-Bap)) (co(Jeu Lin, tw) eo co
(Tre lin, tw) 1hen Uee lin.
2r
(16)
For > 2, the technical derivation details and algorithms are deferred to Appendix B
5

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sampling stop
stort point
intermediate point
and point
inferennce provess
Timelas)
Exe-
(4)
(()
12
(8)
(e) Naive sampling
(B)
(B)
(8)
(A)
(8)
(8)
(b) Quene-bared sampling
(c) Tlme-comsuming comparsons
Figure 3: Different sampling strategies of diffusion generalist solvers (k- = 2).
3.3 Universal Posterior Sampling
DPS circumvents the intractability of posterior sampling in diffusion models via a novel ap- proximation, which is generally applicable to inverse problems with accessible kernels. Inspired by this, we propose universal posterior sampling from the perspective of residual modeling. Under well-defined conditions, Eq. 4 adequately unifies diverse degradations into a degradation-agnoste representation, facilitating the perception of commonalities while exacerbating the restoration bur- den for discriminability reduction. Eq. (3) offers insights that V,, log q(T.n can be adapted to prevent samples from deviating from the generative manifolds when the measurements are dete- riorated, as illustrated in Fig. 2] In contrast to linear inverse modeling. we endeavor to leverage the underlying mechanism of generalist modeling by universally formulating the degradation as Iin = To+ 1o+1%% +.. Such a model exhibits versatility across various degradation scenarios. order to seek tractable approximation for q/inn [T,), we utilize surrogate functions to maximize the likelihood, yiclding approximations of universal posterior sampling (UPS). Specifically, given the formulation 444n 4)) Erp. 1f (lgl9(((in Lo, [l, we make approximations by replacing the outer expectation of q (o, IPes) over the posterior distribution with imner expectations:
q( (4) = q( To == ElTo To Peeal: Pes. == El [T)).
(17)
As the clean images continuously disrupted by Gaussian noise in the diffusion process, we can derive the closed form upper bound of the Jensen gap for the general degradation modeling.
Theorem For the measurement model Tin To + Ires + n in linear verse problem with (O I²I), we have (proof in Appendix C
q(Inn (1) qlin Trea):
(18)
the approximation error can be quantified with the Jensen gap, which is upper bounded by:
d
J(o
exp
M.
(19)
V2T0²
20
where M := f [[ + Ires) - (fo + Tres))
(rrrs)ll((Tn (9(())9((ooo) Ireal I
Note that M finite for most of the distribution in practice, which can be considered as the generalized absolute distance between the observed reference and estimated data. J(o, M) can approach 0 as
+ 0 or oc, suggesting that the approximation errors reduce with extreme measurement noise. Specifically, if the predictions of Iy and Tres are accurate, the upper bound J (o, M) shrinks due to the low variance and distortion. Oppositely, if the predictions lack accuracy, the upper bound of the J(o M) MM-Maar shrinks as well for large variance and limited distortion. According to Theorem I universal posterior sampling stabilize the multi-step step inverse sampling process and provide the tractable gradient approximations of log- lieelioel, as measurement distribution is given:
V,, log (m. It) 1I, log q(T fo. Phc.)) = -1/0"V,, J. - (fo + frex))1.
(20)
serves as a compensatory mechanism to furnish reliable gradient guidance for optimization the latent manifold, thereby mitigating the restoration burden and efficiency deterioration. Notably, our method is formed through the collaboration of diffusion generalist solvers and universal posterior sampling. Both components exclusively depend on 1 and eq predictors, which are intended for

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Table 1: Quantitative comparisons of five image restoration tasks. The FLOPS is calculated in the inference stage with 256x256 resolution. The best results of task-specific models and universal models are shown in blue and red, respectively.
Mathool
Year
Deraiiin
Eshaneement ISNR
Dessoningg SSIM
Dcharing PSNRT
Debiuring
Assrage SSEM
Complesit
PSNRT SSMM
SSEM
PSNR
SSIMI
PSNRT
SSIM
PSNRT
Parame/M
Tesk-specific Method
SwinlR
2021
23.81
0.645
17.49
0.715
25.00
0.913
0.856
0.810
0.67
58.71
Revoomer
2022
38.02
0.878
23.07
0.774
33.24
0.958
32.04
0.969
32.87
0.039
26.12
140.99
MAXIM
S7
7032
20.50
0.625
23.33
0,893
29.05
0.954
18.
0.911
32.86
0.919
22.14
158.16
10-SDE
0.782
18.76
0.625
27.91
137.13
379.33
RODM
2016
25.00
0.919
19:90
0.722
20.79
01.99
27.87
01961
27.80
0.818
15:47
05.87
Universal Method
Resions
26.54
0.647
21.75
0.742
28.53
0.919
0.790
27.61
0.569
26,09
140,99
AirNet
24.78
0.774
13.05
0.485
25.30
0.885
18.53
25.76
24.01
0.509
5.76
301.27
Prompe-ll
33
2023
25.97
0.856
20:97
0.333
20.52
0.838
35.80
0.939
26.25
0.707
27.80
0.576
$2.96
158,14
PooRes
13
2023
22.42
0.752
20.31
0.741
24.55
0.859
24.81
0.838
26.05
0.792
24.08
0,514
370.26
97.17
27.77
AstaDilk
2024
20.30
0.863
15.65
0.707
15:31
0.706
19.01
0.839
0.864
22.43
0.799
115.35
63.58
DA-CUP 31
2014
28.63
0,854
19.50
0,730
28.25
0.934
27.26
03541
26.47
0.818
27.54
0.891
32,96
158,14
DIBUIR
21
7034
90.67
0.887
21:31
0.789
10.70
0.943
19:
0.877
20.91
0.500
7.21
12.91
Ours
25.10
19:48
0.719
25.47
0.590
25.95
0.45
O
25.99
0.852
20.36
0.749
26.95
0.905
25.73
0.906
25.80
0.785
25.93
0.861
1.07
8.01
Ous
29.50
0.574
20.55
0.747
31.04
0.948
28.12
0.937
27.74
0.850
29.16
0.898
3.68
23.97
Ours-L
0.501
32.69
0.955
Pocz
NACA
EMTEN
Figure 4: Visualization comparison with state-of of-eeoff-hheart methods on different restoration tasks.
inverse inference. Once the neural networks are adequately optimized, restoration improvements can be achieved without re-training. For the different solvers in Eq. (14), employ universal posterior sampling to modify the noise E-(-) predicted by neural networks as follows (See Appendix D):
Co Co Jins 4n) = co(Te,, In, + B,,//0 102 I. I - (fo + (Te, Iin: t.)
(21)
Our computational overhead primarily originates from the total derivatives of 7 (.), ofd))) ()), and the approximation of V,, log qI It) To accelerate the inverse inference, we introduce a queue- based sampling strategy to improve efficiency, as illustrated in Fig. 3 Specifically, we precompute the items of intermediate time points during the initial sampling, and subsequent sampling operations will use these items of intermediate time points as their starting points. Furthermore, each sampling iteration solely necessitates the calculation of updated intermediate time points obtain the final results. In this context, for k-order solver, naive sampling from time to t requires interpolating k 1 intermediate points within the interval, resulting in k number of function evaluations (NFEs) per step and ((n)) computational complexity for n steps. In contrast, the basee sampling precomputes
time points and caches them for reuse in subsequent steps, reducing total complexity to On + k 1). In conclusion, this approach improves efficiency by effectively reducing the consumption of computational resources. For more details, please refer to the Appendix [
4
Experiment
4.1
Benchmarks and Implementations
Extensive experiments are conducted on five image restoration tasks, encompassing deraining
17
43
[66|
69
3,
24
[ 46 Datasets details are summarized in Appendix E. and we use PSNR [6 and
low-light enhancement [6] 30 36). desnowing [1, dehazing [21 8
and
deblurring
39
SSIM 60
to evaluate the performance on RGB space. For fairness, five task-specific methods
27
67
33
32
are trained separately on task-related datasets and all universal methods
72
33
75
except for [18 are re-implemented the mixed datasets.
7

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Table 2: Effects of universal posterior sampling and solver orders of our method.
Configurations
Deraining
Enhuncement
Desnowing
Dehazing
Deblurrigg SSIM:
Average
Order
UPS
PSNK1
SSIM1
PSNR1
SSIM1
PSNR1
SSIM
PSNRT
SSIM
PSNK"
PSNR1
SSIM
()
X
30.99
0.886
23.04
0.795
32.05
0.952
31.47
0.947
29.33
0.887
30.70
0.913
(i)
31.10
0.892
23.07
0.798
32.32
0.954
31.54
0.947
29.80
0.895
31,01
0.918
(ii))
0.948
30.07
(iv)
b =
31.46
0.896
23.84
0.801
32.69
0.955
31.68
0.946
30.15
0.899
31.36
0.920
(v)
x
31.35
0.894
23.13
0.799
32.52
0.955
31.71
0.948
30.09
0.898
31.23
0.919
[vi)
à =
31.45
0.896
23.84
0.801
32.69
0.955
31.67
0.946
30.15
0.899
31.36
0.920
US
US
US
US
US
US
US
US
Balp koag
[eb
Greeddieh
Figure 5: Visualization effects of varied solver and universal posterior sampling.
Our method is trained using 8 Nvidia A800 40GB GPUs with PyTorch 41 framework for 210h. Adam optimizer and L1 loss are employed for 500k iterations with learning rate of le-4. We set the batch size 20 and distribute it evenly to each task. We randomly crop patches with size of 256 x 256 from the original image as network input for training and use timesteps for full-resolution testing. Additionally, we offer an adaptation for the Ascend 910B 64GB NPUs, aligned with our GPU settings. We use U-Net [47] architecture and change the channel number of the hidden layers C to obtain different versions with varied parameter quantities: Ours-T(C = 32, 32, 32, 32), Ours-S(C= 32, 64, 64, 128), Ours-B(C = 64, 128, 128, 256), Ours-L(C== = 64, 128, 256, 512).
4.2 Comparative Experiments
Quantitative results are presented in Tab. In contrast to universal methods, we achieve substantial performance improvement across all tasks, indicating our highest accuracy. Compared with task- specific methods, still achieve competitive results with the lowest number of parameters and computational overhead. Evidently, the performance our method gets better as the network capacity increases, fully validating its scalability. Visual comparisons are shown in Fig. [ and for more results please refer to Appendix E.2
Our method outperforms others and is the most similar to ground-trute
4.3 Ablation Study
To thoroughly harness the effectiveness of each component, we conduct ablation studies that include three categories: (i) Effects of varied solver orders and universal posterior sampling, (ii) influence of the value of ôr, and (iii) impact of the sampling steps.
Effects of varied solver orders and universal posterior sampling. We present the qualitative comparisons and quantitative analysis in Fig. [5] and Tab. []] respectively. Obviously, metric assessments improve as order increases and the performance of our method reaches saturation in configuration (iv). Besides, the activation of UPS can steadily enhance the solution accuracy and its effect weakens as solver order increases. Therefore, we select h = 2 with UPS enabled as our default settings to strike a balance between performance and efficiency.
Table 3: Quantitative results of our method with different values of dy.
Deraining
Enhuncement
Desnowing PSNRT
Dehazing
Deblurring PSNR*
Average
PSNRT
SSIM
PSNR*
SSIM
SSIM
PSNR
SSIMt
SSIM
PSNR
SSIM
0.1
29.44
0.875
21.03
0.751
32.24
0.955
29.11
0.942
29.31
0.883
29.99
0.908
0.25
29.44
0.877
21.75
0.765
32.25
0.956
29.57
0.945
29.36
0.884
30.10
0.910
0.5
30.83
0.887
22.56
0.786
52.12
11,53
31.58
0.944
30.13
0.897
30.91
0.915
0.75
31.37
0.896
22.46
0.786
32.32
0.955
31,52
0.948
30.26
0.899
31.15
0.919
0.9
31.40
0.896
23.75
0.807
32,34
0.954
32.05
0.946
30.13
0.897
31.26
0.919
31.46
0.896
23.84
0.801
32.69
0.955
31.68
0.946
30.15
0.899
31,36
0.920

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Table 4: Restoration performance of different sampling steps.
Deraining PSNR
Enhancement
Desnowing PSNR1
Dehazing PSNR
Dehlurring
Average
Steps
SSIM
PSNR1
SSIM
SSIMT
SSIMT
PSNR1
SSIM
PSNR1
SSIM1
4
31.35
0.895
23,30
0.801
32,56
0.956
31.70
0,948
30.09
0.898
31.25
0.920
31.33
0.894
23.22
0.800
32.52
0.956
31.66
0,948
30.08
0.898
31.22
0.919
23.84
0.801
32.69
0.955
31.68
30.15
31.36
0,920
31.31
0.894
22.99
0.798
32.46
0.955
31.67
0.948
30.05
0.897
31.18
10
31.28
0.893
23.09
0.799
32,46
0.955
31.57
0.948
30.04
0.897
31.16
0.919
NOAA
NOMA
NOAA
Nighttime-haze haze
(1)
(2)
(3)
(4)
(5)
Figure 6: Results of low-light enhancement and dehazing for five iterations on NHRW 70
Star megs
Inderwer
Autte
Fouede
devdit
Figure Visualization of real-world scenes generalization.
Influence of the value of J.. 87 determines the commonality of degradation representation in positively correlated manner. We show the restoration performance with various 8y settings in Tab, 3 Clearly, the optimal commonality ratio varies across different tasks, and the performance under low commonality markedly inferior to that under high commonality. To fully verify that our method can handle complicated distribution conditions, we perform image restoration with d7 = 1. Impact of the sampling steps. Restoration performance and efficiency of the inverse inference depend on sampling steps, as shown Tab, and Tab. A4 respectively. As a result, increasing the number of sampling steps beyond certain point leads decline in PSNR/SSIM. This counter- intuitive phenomenon may stem from two main factors. (1) Our approach for image restoration is motivated by the principle of commonality, aiming handle diverse degradation types within a unified framework. The model tends to first remove the primary degradation before addressing secondary ones. Consequently, in cases where samples from the deraining deblurring datasets also suffer from additional degradations (e.g., low-light conditions), the restored output may diverge from the available reference, leading reduced evaluation metrics despite better perceptual quality, as presented in Appendix E.3 (2) The stochasticity in both the forward and reverse processes, combined with network estimated errors, can slight influence on performance. Considering the balance between efficiency and performance, we select 8 as the optimal number of sampling steps.
4.4 Compound Image Restoration
To demonstrate that our method can handle more complex restoration tasks where various degradations occur one image, we repeatedly perform our method on the degraded images, as displayed in Fig. 6 Evidently, our method eliminates the former degradation before the latter one within its capability range. incurs no additional information interference for well-restored data, ensuring high fidelity. For more visualization, please refer to Appendix E.3

---

Soae kooe
Frabes
2010
Dun
sires
Figure 8: Visualization typical remote sensing restoration tasks.
Table 5: Performance comparison of our method with other universal image restoration methods in remote sensing image restoration tasks. The best results are highlighted in bold.
Method
Year
Denoising PSNR1
Enhancement
Deblur
Docloud
Super-resolve
Averae
SSIMI
PSNR1
SSIM
PSNR1
SSIM1
PSNR1
SSIM
PSNR
SSIM
PSNR
SSIM1
Restomer 67
2022
29.21
0.773
32.61
0.976
28.14
0.781
20.13
0,520
28.58
0.809
28.97
0.810
AirNet 22
2022
0.881
0.761
Prompt--B IR 4
2023
30.05
0.796
33.03
0.982
25.72
0.6399
19.54
0.498
27.97
0.788
28.56
0.778
Proies 15
0.932
0.508
27.95
27.57
0.764
IDR 72
2023
27.56
0,695
30.32
0.967
25.32
0.625
18.98
0,476
27.81
0.784
27.07
0.740
DA-CLIP 33
2024
29.98
0.780
33.69
0.986
27, 70
0.747
19.06
0,463
27.11
0.762
29.01
0.793
DiFUIR 75
2024
29.59
0,740
35,28
0.988
30,01
0.837
20.34
0.554
28.63
0.818
30.17
0.821
Ours-1
25.37
0.671
28.34
0.963
24.78
0.610
17,40
0.425
27.63
0.786
25.79
0.725
Ours-S
27.38
0.772
Oun-B
29.22
0.748
32.00
0.980
29. 48
0.829
18.96
0.510
28.75
0.823
29.10
0.817
31.83
0.875
29.47
4.5 Zero-shot Generalization in Real-world Scenes
To evaluate the generalization ability, conduct the zero-shot restoration real-world scenes. We generalize all methods to the degradation scene that matches our task specification. Due to the absence of ground-truth, we merely provide the qualitative results, shown in Fig. 7 Obviously, outperform others and achieve the best visual effects. More comparisons are placed in Appendix E.4
4.6 Remote Sensing Image Restoration
Remote sensing imaging possesses different depth attributes and degradation types. To further validate the superiority of our method, we adapt all universal methods to remote sensing image restoration tasks, including denoising, illumination adjustment, deblurring, cloud removal, and super-resolution. Metric evaluations and visual comparisons are presented in Tab, Tab.5land Fig. S respectively. We achieve the best quantitative and qualitative results (See more details in Appendix E.5)
5 Conclusion
In this paper, we introduced DGSolver, a diffusion generalist solver with universal posterior sampling for image restoration. We derived exact ODE formulations for diffusion generalist models and developed customized high-order solvers with queue-based sampling strategy to reduce cumulative discretization errors and enhance efficiency. Additionally, our universal posterior sampling improves the approximation of manifold-constrained gradients, leading to more accurate noise and residual estimation. Extensive experiments across natural and remote sensing datasets demonstrate that DGSolver consistently outperforms ttate-of-the the-art art methods in accuracy, stability, and scalability.
Acknowledgments and Disclosure of Funding
This work was supported part by the National Natural Science Foundation of China (62276192, 622251 13, 624B2109, 623B2079), in part by the Zhongguancun Academy Project (20240308), and in part by the Fund of National Key Laboratory of Multispectral Information Intelligent Processing Technology (614211323022
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A Mathematical Theory of Diffusion Models
Diffusion models that are most pertinent our methods are theoretically discussed as follows, including diffusion prohabilistic models (DPMs), diffusion residual models (DRMs), and diffusion generalist models (DGMs). For each category. we analyze their discrete formulations perturbed generation process [44] and deterministic implicit sampling [50]. and continuous formulations of score-based stochastic differential equations (SDEs).
A.1 Forward Process of Diffusion Probabilistic Models
Diffusion probabilistic models consist of a non--onnan con-aaaaetrri noise addition forward process and a denoising reverse process. In the forward process, it gradually disrupts the original image content by adding the Gaussian noise. The sampling distribution in markovian manner between the adjacent time step, i.e. 1t and 4-11, can be defined as q 4-1) (( 8,T,, which is equivalent
1 = xx + Sic,
(22)
where az and 3. /I of are predefined coefficients over time to modulate image content and noise intensity, respectively. é is a random variable following the normal Gaussian distribution A I).
A.2 Perturbed Generation Process of Diffusion Probabilistic Models
By the property the markovian chain, Eq. can be converted to the below formula that DPMs can diffuse from the clean image To to any intermediate vanable J at time step t in just one step:
I, ToJo + B,c,
(23)
where τ = =o as and B, = VI az are denoted for simplicity. Based on Bayesian formula, we have:
(lc4-11) N(aule-1, (44--1140) Ni B. (44:44)) ((accoo,3.1) 3,1).
(24)
(111111,, ((9144-1)9(44-114 [o) exp (1
((+)oo))
To,
(1+ - aahh-1)
(1--1 11))
(I+ (1o)
x exp
2
23,-1
23
(25)
1 ai
1 21.-1 Ø
ocl 83
+ Cl1,1o)).
x exp
B
3,-1
Apparently, Eq.
25 can be supposed general Gaussian distribution, ( E.-11
ai
1
2T-1(Te-1 Io) =
(26)
B7
8-11
Pe-1 (o)
a
σe To
32
01
1.
01-17
82 Io- (27)
B
3
B.
3.
Consequently, the inverse process regarded as sampling from the following Gaussian distribution step by step:
42
3
q Tt, To) =N(
T
To
r)
(28)
B
a133
82
B
T 1-1
To
(29)
8
8
8
A.3 Deterministic Implicit Sampling of Diffusion Probabilistic Models
Discarding the markovian modeling qTT (T-1) in the forward process and assuming that the sampling distribu- tion still follows the Gaussian distribution in the reverse process, we can obtain:
g(4/44) = NN4, 3 + 1a = Tolo + B,61,
(30)
q(I-- (T)) -1 B,,,1) I-1 T. To B.-152.
(31)
9441//4, 4)) -1 dela, oela, + 1:-1 rele Xelo 0:63.
(32)
21

---

Substituting Eq.
30
into Eq.
32
and comparing the coefficients with Eq. 31 we have:
le-1- (Mer + Ae) + Ke3,e1 decs = (Nece + xe)
KEB off.
(33)
0 NitE + Aos Bi-1 =K, to:
(34)
these parameters he calculated
/ 1 of of
T B oz
Kt
(35)
B,
B.
Consequently, a common reverse process in the non-mante man--arooviin manner is:
(1e-1ll,, 1:) = =N( V B
Te
p o2
1. (a.11
Ho, o
(36)
B.
3.
#2
If we set the variance equal to can derive another representative sampling distribution by setting variance di equal to 0:
Eq. (36 holds the identical distributions with Eq.
28
Typically, we
37
c(Tc-1/7, To)o.=0 = NN - I, + (or-1 T. Vo, of T),
(37)
B,
in this way, the inverse process is deterministic for ignoring the random noise:
32-1 1. + (ū.1-
0r.
I
Lo.
(38)
3:
3.
A.4
Training Objective of Diffusion Probabilistic Models
The training objective for the noise estimators in DPMs is calculated as follows:
L = to.e ||| ca + +3,
(39)
A.5
Stochastic Differential Equations of Diffusion Probabilistic Models
The formulation of SDEs of DPMs' forward process be reformulated from continuous perspective with diffusion term fr(1t) and drift term gr as follows:
dli f.()) gedwz,
(40)
where wi the standard Wiener process. Its formulation of inverse SDEs can be proved as follows 20
dI, = (1,) g¹ V log (I.)]dt + gidb,
(41)
Similarly, we modify the formulas in Eq.
30
32 and consider the linear solution f1(4,) = Sile:
((4++1144) NN444 Tee a A.,,, Be+ar() + 41S Tesallo + Betac
(42)
(TT(To)) NTT, 3.1) a.To + B,ez.
(43)
((1+)))11)) NUtea (1 f:At)l,, LItA (1 f.11)4 Ates
(44)
We can obtain:
ō4t (1+ f.Lt)a.,
(45)
Betsi = +1+ ftAt) "3, 92
(46)
By setting At - 0, we can derive:
fi dexe dt
3
(ln..)
gi
25. B
(47)
dt
di
di
A.6 Forward Process of Diffusion Residual models
Diffusion residual models additionally incorporate a residual term Tre. to assist the forward process. The forward process between adjacent steps is defined as:
g (te Nilei atdres, 3.1),
(48)
22

---

= I--1 + culres + 3tc,
(49)
where Tres = Tin To- By the properties of Markov Chain and reparamekerizzation, we have:
a(TelTo, Irea) = NT To + ai aelr,,, 5,
(50)
I = 1--1 + ares + Bee [T-2 + a: Irea + 1cc- actres Beer
1 lar-1 + aijllre + v 1 c3.ec
(51)
lu Celres + D,c,
where Ci Eis1 and B, Vi=z 88.
A.7 Perturbed Generation Process of Diffusion Residual Models
Evidently, if a. 1, the forward process makes the representations of Io be the degraded version with additive noise I = Jnn + B,c. Similarly, we can obtain:
t/=://-1, Tres) NV44; To-1 rre, 3TT)
(52)
44+--1/4 Ares)
au- ( To Trea) WVs,; To T.lre,, B,
(53)
(54)
Suppose the form of g(1c-1 1 Lo, Aven) belongs to Gaussian distribution family as all the distributions are Gaussian, denote:
a(4-1 14,, Jo, (res) = 14 Jo, dres), 11 Lo, Ares)).
(55)
To derive the expressions of variahles (tallesss/tteeeeeeeeeeeeeeee To, Tres) and Ex-1 (Te, To, Tres), we take the logarithm directly and ignore the constant term:
q(T--ll,, To, Tres) ((1// Lo Imw )g((e- (roa)
(56)
q( To, Tres)
g(4-11 To, Tress===xppp
log To, Tree) + log gl Irea) log Logg(Te|To,,ll
(Ie - (4-1 cetrea)))
(I-1 (To T. 1ree))
(I. (To + [elree))
exp
(57)
25²
28 aeters
20
To + an-slree dore }.
exp
Mr 211-1
B3
B
B
Ben
Comparing the coefficients, we can derive the variables below:
325 7-1
Σ (J Jo, Incs)
(58)
83 + B
B.
Pt-1(Te, To, Tres) = (1e cetres
ai llres
B
B7
I 0r
Jres
(59)
8²
B,
A.8 Deterministic Implicit Sampling of Diffusion Residual Models
For generality, we discard the assumptions that the forward process follows a Markov Chain and denote:
q(Te-1/To Tre.) WI To Ti I 1 B(n1))
(60)
To Tres) = NTT: To ar (,,1111111 B,1,,
(61)
then the conditional distribution 4 14, 1t) can be defined as:
q(Ic-114, I, Irea) M Aelo kelt + mrrres I).
(62)
reformulate the probability formulas into algebraic expressions:
--1 Io + iIres + B,-16,
(63)
1. lo aetres B,c,
(64)
23

---

4-1 Aelo + Nole + mhrres + oet
(65)
lo+ (Kr + (xi,5 + +
By comparing coefficients of Eq. 65) and Eq. 63 we can get the following quadratic equation:
A + Ki = 1,
Kg00 + = 0:
(66)
27 K.P, tor = B
To complete the derivation, treat of as predefined parameters:
Ke B of 8 7 =
B B of
3
of
d
(67)
8
Consequently, the inverse process can be formally simplified as:
Pt-1 1 lres - (3, -B-11-1---7
(68)
Obviously, the mean g(( Ma, 1,, Ares) ) is manipulated by the variance intensity de If we make of equals that of Eq. (58) then the value of j in 48. equals that of Eq. (59.. Besides, if we make of equals to 0, then the deterministic inverse process be denoted as:
1-1=1 aclres - (3, - -B-11))
(69)
A.9 Training Objective of Diffusion Residual Models
The training objective of DRMs is calculated follows:
B E (T+ - are Thes
32
Tres
60
(70)
B
Leveraging Eq.
5T then Eq. 70 can be simplified into two equivalent formulas:
B. [Ares Tres (Te Iin, +)]]],
(71)
C = 15 B[Aelle co(le, lin, t)]l] where the weights, i.e., Ares, d. {0, 1), the balance coefficients.
(72)
A.10 Stochastic Differential Equations of Diffusion Residual Models
Like the derivation from Eq. 42 to Eq.
46 we have:
I Jo, Tress Im.) = N(I; Te Telr B,I),
(73)
glletae Lo, Ires, 4m) =
= ((4+aG + aetsetres Bi+
(74)
((o+ [e, Tres, Jmm) = -V(++arr (1 f(f) + h(t) o (t)
(75)
the marginal distribution
can computed as:
I+ (1 + (f) Trer B,61) + h(t) g(t) Atez.
(76)
Comparing the undetermined coefficients Eq. 74 with Eq.
1+ f(t 1,
(77)
h(t) (1 (tt) = Oc+A6
(78)
(1+ f())))))) + = B4A
(79)
then the above four functions can he solved follows:
82 Lt
- 8
f(t) = h(t)
Ct-At
(²())
(80)
At
At
Let At + 0, then:
1(11 = da:
d5
d3
1(1) =
(1))
23,
(81)
di
dt
dt
24

---

A.11 Forward Process of Diffusion Generalist Model
Based on the formulation of DGMs, diffusion generalist models additionally incorporate the degraded image manipulated by 6, in the forward process:
lin) = (1 (1:414111 1--1 + aulres 8,Lsn, 1)
(82)
I = 1-11 + aelres + fiet - Selmn
(83)
Leveraging the property of Markov Chain and reparameterization, we have:
1 I-1 + apTres B+4 - -n
[T-2 + at-1 + fc-1 - 1 + or Tree + Bere deTin
(oc-1 T
+ ai) Ires + 188 + 07 (ōc-1 + 56)11n lin
(84)
To - athres + 3,e--elly
= (a, 1)lres + 3ee + (1 D.)Lins
where ae = Eim 0s 3, = 2im 88
= Eim 8c of + 1, clean images are diffused as:
I. 4= (1 3,) 3,e.
(85)
Thus, we obtain an expression for degraded image representation, which & plays a role in controlling the commonality, 8, + 0, Eq. (85) degrades to the pattern of diffusion residual models; whereas if ó, + all representations are unified into pure Gaussian distribution.
A.12 Perturbed Generation Process of Diffusion Generalist Models
By repeating the aforementioned derivation process, akin to Eq.
52 EEq. 57
we have:
q(Tel Tres: Iin) NN,; T--1 art 8e T), Ares, lin) (l4e-1/4;
(86)
-N at-1lres - 1lin, B 1),
(87)
9(Ta|To. Ires, Ien) NVir; To Telres J.len, 3,1).
(88)
(Tle-1lee: Trees eexp log Tres, (mn) log q(tt-1( Inn) log q( Tress 1nn)
(Ie (Te (I-- artres 6,T;n))
(1-1- (at-1
1)I (1 8 dm
x exp
233
27,-1
+ (le - (Tr -1)lres (1 1-8))lma)
2
exp
1
1
1 -21-11
T ou Irea + Ten
( 1)Tmx (1 8, 31-1)
+C(ee Jo, Ire
B7
3
83
3,
(89)
Comparing the coefficients of cach term, we obtain:
B2B²2
2 (1e, Lin, Lrc))
(90)
87 B
Br
Me-1(lt, Lin, Ares) = 11-1a- aitres +ôlmn
1o + ai Ti- idme
BBB
37
B
B
1
(3 -0 Ires Lrei
+0,82111
3,10
B.
(91)
-
T0.-187 -as
=2
-
1it
Inea
In (Ui-arlrc--B,cttbb.
3,
B.
B²
3.
B2
l Tres Tn
3,
the inverse process can be considered as sampling from the following distribution:
$3
B1
g(Ic lo, Areny Ann) ~ N(1 le - andr + delsn
1).
(92)
B.
3
25

---

A.13 Deterministic Implicit Sampling of Diffusion Generalist Models
A common forward process can be determined follows:
aToe-1To, Tres: Tin) (1 Te- - 1 (imn:,P111 Be-1 q(Ie Jo, Arexs Iin) = (1;;; to + Tedrex d,Lin B.
(93)
(94)
For simplicity, we still define the formula of the conditional distribution q To, Is) as:
q(1-1/4, 1,, To fres, In) (Te- Kel + nelres + %lo + Gelens 02 I),
(95)
then we can reformulate the probability formulas into algebraic expressions:
4-11 To + 0t-1 8i-1 11-1511
(96)
4 1o + Telres - d,lin + B,ez,
(97)
Ie-1 Kili Milres + nlo Selin + 0cc3
= -Kr 1o acdme 8elin B,C2)
+ Milres + nlo + Sihin + tics
(98)
(Ne +o (Kot + n Mree (N.ō. Se)l (,i,5 of)
By comparing coefficients Eq. (96) and Eq. 98 we have:
Re + i = 1,
(99)
K + 1/s C4.
(100)
k:O; 6: 5:-1;
(101)
RB oi B
(102)
To complete the derivation, we treat of as known variable:
8 1-1
3
TI 000-1 di
1-1
(103)
3
8 2
3 o
S1
dig
0
(104)
3
B
The inverse process in Eq.
98
reformulated as:
(1
8:-
of
B o?
4-1 =
I, +
a
Irea
(105)
82
07
8² of
To
8zn
8 Im + ozc.
8
Furthermore, Eq. 105p can be simplified and decomposed into mean 14-1(1e, Io. Iree) and variance EZ_, To, Ires), which can be denoted
(t- Io, Ires) = 1t - (axe - at-1) + (ō. - dr-1) - (B.-VBr- Br_ -o? -0?)e
(106)
Similarly, we set the variance to zero for deterministic inverse inference:
I- (a. Ir- + (ō, d.-1) (3, -3
(107)
A.14 Training Objective of Diffusion Generalist Models
The training objective of DGMs is calculated as follows:
82
R oreres J.
(J+ Th
Eo)
3
3
(108)
82
R as (T
Trea) + (Eg
t)
8
26

---

Leveraging the Eq. 84 then Eq. 108 can be simplified into the following formula:
a.)
(Ires Frealee leng t))
(Ires Prea (le Leyy t))[
Be
(109)
Arel fee Lin, ()
where the coefficient Area is constant. The detailed training algorithm summarized Alg. Algorithm 1: Training of diffusion generalist model.
Input: Tens To A In Io. f 1 {B,, N f5,, J {ts h=o- repeat
Io q(To)
Uni "niiforn(!......"
c I)
1t = lo + at, Lres + Bec dc, Lin
Take the gradient descent step
VollEres Ihes (le,, Isn, 4)]l2
until:
Output: Ieg.
A.15
Stochastic Differential Equations of Diffusion Generalist Models
The forward process is not only constrained by time-related variables It but also limited by Irea and Iin. Therefore, the stochastic differential equations and the inverse equations can be denoted
di, = f(Te, t)dt + h(Tree, t)dt + I(Tmn t)dt + g())dur, To (o()),
(110)
d/. (f 1) (I 1) ((Iim. 1) 9 (L)V log q (4.) + g())to,, ~ gr()),
(111)
di
where [I,, 1) ((),,,((rra. () h(1)Tras. and ((Iin, 1) ((1) According definitions of Eq. 84). we can derive:
( To, Ires, Jmm NN: lo Telr - (eIm,B,1) al Fo. Fre. Is.) = (I tn + T44aebrs - detaetin F(4Aet), we assume that the general form of the onc-step forward process is:
(112)
(113)
qI=+ Le, Ires, )In) = [ h(t))drres + 1(1))llin 9()))))
(114)
similarly, we compute its marginal distribution / q (1, ao (4)4(44)))))
It+ = (1+ (()) + a,lre J. + 3,c,) + h(t) + ((t)aetT, + g(1)
(115)
comparing the undetermined cnefficients in Fq. 113 with Eq. [15] we have:
1 ftt) 1,
(116)
((t) + + f(t) = at+
(117)
1())1 - (1+ (() -J
(118)
(1+ (1) o (t) =B+AAr
(119)
then the above four functions can be solved as follows:
Pttar 3
((1) = 0,,((()
dets Tee .(1)
3, d.-At
.9 ,9((1
(120)
AL
Al
A!
let At + 0, then get:
= da ((1)
dog
d3
((1) = 0,1(t)
= dt -2((1)
(121)
dt
dt
B
Customized Solver for Diffusion Generalist Model
B.1 Reverse ODEs Formulation of Diffusion Generalist Model
For generality, follow the notations in Section The inherent parameters time continuous satisfying
a(t) = Cee 8(t) = 3. B(t) = 3, The inverse ordinary differential equations can be concretely formulated
151:
dlc do Ircs dt dt
db, 1m dt
d dt
v log gı (I.).
(122)
27

---

Given initial value I, time O, the solution I; al each time
< LL < T can be computed as:
[ da, dr
Ireat - J. db dr
1 d3 J.
4. 1- 1a
lindT
2 dr log q, (1, )dr.
(123)
According to Tweedie's Formula in Lemma parameters Na can be computed follows:
Given a Gaussian variable 2 ~ (2; E,), the estimated
= V log P(z).
(124)
forward process Eq. 94 there exists:
Celres Je D. log P(Ti) It B,c,
(125)
V log 9: (1)
(126)
3
Hence, substituting the relationships between score functions and noise into 123)
[13
have:
= 1. + [:
dār dr Ireadt- [: dō. dr
S: d3 edr,
lnnd
(127)
Furthermore, we have:
Tt 10 J: dir dr
(haaTee (ee Tin T)dT Inn S: Jōr d S: dizeo(Te Tiny T)dT, dr
d3 dr
(128)
where lres = (eea(le, Len,t) and c0 Lnn t) are time-relevant, In is input degraded image and can be considered as constant. Since the above inherent parameters are time continuous and monotonic for linear noise schedule, there must exist inverse functions satisfying (=(a(()) (T(((t)) tz(((t).. We denote Tn = (oT (J,, Tan, T) = Peem (Tr, Tsn, @) and I = I co Im,, T) = Eo Im. F), and then we have:
4= (6. Jn + [_ (Ta a)
//,
(.) co Im B)d3.
(129)
With the simplicity of the continuous formula of Eq. 129 we can propose high-order solvers that correspond to the ODEs. Specifically, given an initial point Ir at time 7 and total M + 1 time steps (t, ) decrcasing from lo = T to LM Let {1e == the sequence iteratively computed at time steps [4: k= using the presented solvers. Starting with the previous solution I at time 44-1, the exact solution It,-1-4, 1-1,, at target time I; is:
[ (Te (Ta, Iin, a)da + for B B.,-1
Bes
I1-1 = I - (b.. 1
Co(TA, Fin. B)d3.
(130)
We need to approximate the integral of both Poc. (Ja, Imn, σi) from ai, to aie and Co I 33 from B Br... to B. We utilize the Taylor expansion make & orders approximation:
R-1
at; a),-))" )" To (Tor, Iin .....) +0((a -...))))
Po (Ta (in
(131)
Tel
n=0
k-1
B3 B. )
Co(T Tin,
Fo(Tz, Jin ((
(132)
n=0
substituting above Taylor expansion into Eq.
130 we can obtain:
a Co. )a
1G 1-4; L1 145-1165 16,, -d.,
1 +
p (Tre. Lon 0c, 1)
da
ne!
==0
(133)
p.,
(3
B
)m d ~
Σ Colf Jen, B
n!
since the integral is lincar, its analytical expression can he easily written as:
(oi, Tr,-) ) n po(n) (T...) lsn,ti-))
I It,, (B., - B
(n (n+1) 1)!
(134)
3 1u41
(n + 1)!
n
28

---

Therefore. to approximate Is 1. Ie we only need to approximate the w-th order total derivatives
Iyes (T Inn 1) and 2 (N)
(T1, Iany 11 for - k 1. the case of 1, Eq. [34 becomes:
3,,, I, = -ريا (ō., - Man (Ti, Thy ) (1ا Lin (:1 (1))+ - ) (L Len, (e1). (135)
By dropping the high-orde aigh-order error term O(..-), k = 1, we call this Diffusion Generalist Solver-1, I, and the detailed algorithm
obtain the first-order approximation for 14--1 Iu. As
follows.
First-order Diffusion Generalist Solver Given initial value Ir and M 1 time steps (t: 1= decreasing from to T to lar 0. Starting with Itg Ir, the sequence [I,, / computed iteratively follows:
4 سدح - (ō., س Min + (Tr, - M (1 Lin, L + (3,, Beg
Lin, 1.)).
(136)
Apparently, it consistent with Eq. 107 which is a special case of first order solver within our methods by omitting the discrete error. The overall algorithm for first-order diffusion generalist solver is presented in Alg
Algorithm 2: First-order Diffusion Generalist Solver.
Input: I Inn, hass cp. for b. (B,,). ). (o.,) (ti)
To = IT ~ NN,,, T)
2 for i = to M do
Ires = - I 11-1)
? (I,--1 (at.-1 1)lres 011-0-11-000 (1-
To 11--1 Tue Ires Br, E+ de, lin
Lt. 14-1 (8e, Min + (Tte 0t. Trea + (3r, Bu__)e
Output: Ita.
For k = 2. we need an auxiliary time variable L, = rl4 + (1 r)); where is the controllable variable measuring the intermediate points the range of [4., 1). Then Eq. (133) becomes:
1 I (7., de._ Din + (o,, C.. Iee (Fe. Jin 4--1) + (B,; - B,, )ea (Te Iin (in)
2 (a., Ci a. 1200 (Ir Lmy 1-1 + (P.. P20' (Te (s
(137)
leveraging the finite-difference methods [2], we can approximate the derivative part:
1000 (Te,,, ti-1) x Leea (Ir,, Lin, a..) Tr The--
Le (1 Lim (i..)
1 Leee (Te,, Lmn to) - L APeaTTe lim)l-1))
Ts T1,-1
(138)
(Tar (ic colTz,; Ba) co B,-
Tiny Band colfen Lini tu) co (inlinsh) k-1)
3 3
B. - -Bem
(139)
Therefore, the simplified result of Eq.
137 is:
I It (o., Je Iin (ar, Ceea 1oca (T, Iin 4-- (B,, -B, )eo(Te fin
+ (Tn, -it, ) (Tr (Tens len, tw) Lien Lins tz-1))
(140)
2r
+ 2r BB, Meo(Trn Im, ta) cal co( Ien, bn-1))-
Second-order Diffusion Generalist Solver Given an initial value Ir and M + 1 time steps [4) decreasing from to = T to tar = O, starting with Tr=o = Jr, the sequence {Te, hiso computed iteratively as follows:
J1 I ( d. Imn (Ti, - Tee (Te Ten, h- (3., De, Im, t-)1
2r (Zi, - Ti Tr.-1 (1ree Lmn tw) - 1 (Tr.-19 Linn tz-1))
1
(141)
2r (3,, - B,,_ (eo Lin tw) (e ces lenyta-a)).
1
The overall algorithm for second-order diffusion generalist solver is presented in Alg. 3
29

---

Algorithm 3: Second-order Diffusion Generalist Solver.
Input: Iin Desr ( (B,,), (ouh (4)
It < IT (0, (00,3:1011
for i to M do
bu rt; + (1 r)ti-1
Tres 1 (1 Lsnn tj-1)
E = (- (1,-, - 1) 1) - (1
Ho -1--1 - a p Br, Er.m1 + -i,
Lt_ 111 (Ōr, de Uii (Tte 0t, Tres + (Be, De
Pe 1 (Te Iinitul tw)
Ta [t - (at - 1) -1 1(1--(111 -(1-
Th- lt, Ct. T. Br, Et + or, lin
L. = I (81, dc, lin + (Th, 0t, (rest + +Be, - -Be,,)E + 27 (Tn.
11
0t Prus Trea)) + 2r
(3 - ( - -((,))
Output: Lo.
For k- = 3, Eq.
133
becomes:
14 = 1,-1
(ō,; 1e,-))lin + (as, ate (ee ( Lin,iiii1 1i-1) + (B,g - Do ti-1)
2 (as, at.-1) (2((1) (Te.-19 In 1 +
2 (3,,
B.,,) )200 (Te.-19 tin:te-1)
(a., 0t.-1) (T (Te Iim 14-1) + (3, )8 (Fe fin
(142)
we need two additional points to obtain the second-order derivative. Suppose the Taylor expansion of function ((x at + by and l2 can be:
f(x f(x) (x) MP (*)
(143)
h3
f(x bo) hof
f" (x) o(1))
(144)
2
Then, we can derive the formulations by eliminating (xx)
(" h12 1n) [hi h2) ha (x + h1) (h2 hi)f(x))
(145)
Similarly, two time variables L, = rT4; + (1 11)4:-1 and L. = r24 + (1 12)4:-11 satisfying >>>>00 i 1. Therefore, we can compute that:
(22)) (Te-1s (mn 2(ā., )..-)) -2
[relies - r2lte. + (r2 - (Te (mn
(146)
M1P2(12- - r1)
c0:2) (Te, 19 (-m 2( ((,-1)--
rir2(12 - [rice, 13em. + (r2 ri)ca (.liiii-1)). (147)
where 1::. (in Jony ts), I' 1 (ens Jum and et., co Tong (ons eo(Ten Iins tu). Consequently, can derive the formula follows:
1: 1,11 (8.; dc, Iin - ) Aeees(ie Jeny (0 (mmti
(āv, - o (Te,, Imn ta) - leen (Ter ti-1)
+ 2r1
+ 1
(B - -B,,,) (anstt =o(Te, 11 (enntr-1))
2r1
(148)
+
(ar, - [..-1)
(T |n 11 (Te, lin, (.) rrz lin in bu) (r2 - r1) the (T, Iin 4-1]
3rirz(r2 - m1)
(d,, - -
|nico Lin, to) rrea (Te, Lin,ta) (r2 (ra-r1)co (1) lin, ti-1]]
3rirz(r2 r1)
The overall algorithm for second-orde diffusion generalist solver is presented in Alg.
30

---

Algorithm 4: Third-order Diffusion Generalist Solver.
Input: Iin Less EB, J. (B,,), (or (4h T1, 72.
It < IT [0, B10
for i to M do
3
bu = rit; (1 11) =22;
r2)4--1
Pres = =Ires (L Isn 1)-1), E = (1in- - (āt, 1)(res -(1 /
:o -= = 0tt, Tree - Be,-, Plos + dir-iiin
lt = 11 (6en - db, Min + (att - Ot (Tres + (Bt, - -B,,_1 )oc,
Pe =I (It, Ii tu); 2, = =I - -āa 1) - -(1 - (mm /
res
10" = =I. Ct I. B Ct. + 5t, Inn
T, = =,,,- - (ōt, B(. Iin + (Tt, ( : + (B,, - 3 pra- + 22 (T+,
7
at. ( - Tree') 1 + 12 B3,,- - -Be_,) (2t, - -(t,-1)
2r1
Pr. 1 (T,, Im 1e), E = (1,, (T), - Pies - -(1 -1- -1,, (nn) /
To' = =1,,, Tt. les Be, E, + +ō,
Diea = -uus Pre-1 Dí= = Cr. -,,--
Dree 11ra(12-1) m1) (rr T r2 + (r2 p1 )rees))))
11
D2 (122(22-11) (riê,, r22t, + (r2 - 11 (1)((..))
2
I. = 4h-1 (dt, - (r...)) Min + (ātz - -(,-1) 1os + +B, - -B,,, )r,, + 2r1 (ar,
0t Dres + 2r1 (B,, - B B Di + Ha -0t,-) Dr + (Br, - 1 B
Output: Itx.
B.2 Cumulative Error Bound of Diffusion Generalist Solvers
For k-order diffusion generalist solvers, we make the following assumptions:
Assumption 1.
The total derivatives (0(m) 196n) (Iz Lon, (
a(n)) Pre Jin and (m) (1z B) dal(n)
(( Me. exist and are continuous for 0 S ke 1.
43
Assumption 2. The function (Jx, Fan, at) and ca (Tz,) Jon, 33 are Lipschitz with respect to Te-
Assumption 3. The step sizes for a and B are limited by max(a,, oct.,) 1) O(1/Mz) and max(B,,
Be__,) O(1M7)), respectively. Besides, 1/Mx < 1/M and 1/M3 < 1/M.
Proof of Taking n 0, t. t,-1 Eq. 134.. obtain:
I = -,-, - (6e, 6 + (To. Ttt_1 Ie (l (mn + ( - -7)
(149)
F B, Dea(Te
( -3,) ?),
Based on the Assumptions and Eq. (366). we can derive:
F, = Pe.., - (ō,, δ,,_, Min + (Tt, 1 Iin (-1) (3,, - Be, )eo(TTy.r Iinshi
= =T,-- - (br, ),,,)) Ti + (ai, - -. + (B,, - B (ea 1-1) + - - -(4-1))
ti + O I,,-,))
I1, O(max(a )u-))² O(max(3, B,,_,)² )2) 1 lu-1).
(150)
By repeating the process recursively M times, get:
Tex It, + OM max(a., Te.,) O(M max(B, - Br, ,)²)
(151)
It, + Omax((a, - at; .)) O(max(B, ;),,)). Proof of ks = 2. We consider the following update for 0 < s <s<u<re M in Alg,
Ir.. le, (3... din (ar.. ar, (ee Lin, (P... (omeee
(152)
31

---

It It, - -(ô1, - Mnn + (āe, - -a., Tee Ln, ts) + (B,,- - De,,eo(le,
+ 1 ( - Tt, (R. i longtn) 18 ( (1e, Lin, t.))
(153)
2r
+ 2r (8,, - 3,,)(ea( Lon, bu) - -o (cn,
1
Taking n = 1, we can obtain:
Tt. It, (ō.. Jin + (at. - at, T. (Jo,) Iim t2) + BB,, (eTs
2 (ar. -Or,) 1ec (Tre, Imm a.,) + ( - ac,,)
(154)
(B., B,, (e0 (T,,, Inn: 0 -(..))))))
From Eq. 153 we have:
Lee (Heus Aini tw) = 1rea (He, Lins (a) + (Tht. - Ti, 1rc0 (LaTe, (in + O(TT.. - -.,)),,
(155)
ca(Teu, lun tu) = Eo (Te, Tin;tal 1s) + (Be, - -,, ) (Tp., -inn :.in:3,,) (13rr. -3,,) (1)
(156)
Te -1, (61.- - dm + (ote, Tu. I (le, Lin toa (3,,- (0,) (saits)
+ (Te, Tr, ( (in: fren (le Lins tu) + (Tis. -Te,) (IT,, Linn (i.)) + +O((a.. - =x,,)²))
+ BB. - (eo) bu) co( (sn, + (B... - 3Eg (Tz., Imn B,,) + O(3,,. - -B,,)
(157)
According to the Lipschitzness of Lies and ep and the conclusion from Eq. 150). we can compute:
IPraaTen, Jenytut
tu) (en (Tw, Im tw)ll = O(I -t. II) = Olaas,. - -a.,) +). - )), (158)
[[eo(Teus Lan, lu) en lins 11 O -.,. O((a,, - )).))) + +O,,,, )3. (159)
As (Tr... [t,) = ra,, ai,)) and (B,, - -,,)) = =r(3,, - -B,,) for linear generation schedule, we subtract
Ie in Eq. 157 from L,, in Eq. 154
It. T, (oe. oc,)( (kt, (i,) (a,, (.a)
(Ta Tsn, T.,,) O((Tr. -or,)")
2
2r
+B,,- B,,) (3,, - -B,)) (B
Be,d)
)ob (12,, (o) Im B,))
O(13, - -3,,)))
(160)
2
2r
O(a,, )o, + O((B,, - -3,,)²
Proof of k = 3. We consider the following update for i - <
Alg and we can obtain:
Iep I1, - (ō.r d., Man + (a,, - T., Lres (le,, Lins (.) + (B,, - co
+ 1 (a,, - (Treaeaaaaa Jin, vew) Io (Tae, Iins )),))
2r1
+ 1
2r: (3,, - B,, -B,,)ol(Ts,. Jin Ben) =o(TT,, Jen,
(o,, - a.,) (rr (ritres (To,, Iin, (t,)) eetreolTa,, Iin, (tt.)) + (12 -oa--rr)tooe 11) (i,))
3r1r2(12 -1)
(3,, - -B.)
3rir2(12 -r) (riea(Tn,,. 3..) -rae (Tx, Imm (12 ri)eo : I.m.B..))
+ ( - ))) + ( - ))
(161)
Similar to aforementioned proof, we have:
- It_ Ir + -(8,,--8,,) OO -a,))")))))))) ))),
(162)
Ie, = Ir,, + (1B,, - -B,,') + 0 ( - -),,
(163)
I, It, - -(8,. 6c, Min + (at, - Tu, Prea (le, ta) + (3,, - -B,, -B,,)co (,,)),((,,,,,n
+ (aur an.) )2 2/1 11(1)) (1 Linn at1)) + (arr ac, )3 (1002) (Iae, Lmn, at.)) + 0 - a.,)") ((,)' (164)
(B,,- 3,,))² (Zz, Jens B,,) (B,, - -B,,) (0 Lim B,,) ) + (1B, ((,BB,B,)))) -),))).
By applying Taylor expansion, can get the conclusion follows:
lt, - TT O(3,, B', O ( )7
(165)
32

---

C
Universal Diffusion Posterior Sampling
Lemma Tweedie's 's formuda. Let ((yln) be the exponential family distribution:
((uin) qo(u) exp(1) (v) p(1));
(166)
among them, is called the natural parameter, ((1) is called the cumulant generating function for normalizing
the density, and qoly) is the density up the factff when 1j O. Then, the posterior îj B(y should satisfy:
(V,T(w)) Vylogo(v) Vv log (u()).
(167)
Proof. Marginal distribution 7(y) could be expressed as:
q(v)= a(uin)a(n)dn = gu(v) exp (1" p(n)) g(n)dn,
(168)
Then, the derivative of the marginal distribution ( with respect becomes:
/ exp(n" (()))(()d + (V,I(y))) (go(v) exp(n"I(s)) ((n))((n)d)
Voe(u)= - Vugo(u) Vugu(u)
/
((xlm)a(n)dn (V,T(w))) (u(vln)(()dd go(y)
Vugu(u)
qNv) + (V,T(M)) nu(u.,)dd. (o(y)
(169)
Therefore:
Vgg(y) ((V)
Vyyo(y) (V,T(N))" / nalulu)dn.
(170)
(u(y)
which is equivalent to:
V. log (()) V log (o)) "V
(171)
This concludes the proof.
Proposition 2 Jensen gap upper bound (12). Define the absolute centered as mp == /B - 1
and the mean as j = E{X). Assume that for a O, there exists a positive number such that for any E IR, If(c) f(u)) < K| z n(" Then:
El(f(x) - ((B(X)) S (1(()) f(w))dg(())
(172)
s K f x---~1 -l do(X) S Mm
where an upper bound estimator constant that can be taken K or other constants related to the function and the distribution ((X).
Lemma 2 Let <b-) be univariate Gaussian density function with mean μ and variance o², there exists Lipschitz constant L such that:
(x(x) - +(x)) S L/x 7).
(173)
where I. exp
2
V2na²
Proof. As o' is continuous and bounded, we use the mean value theorem to get:
((, v) R² lø(x) p(v)) 1| ||~0 /zr vl.
(174)
Since the I Lipschitz constant, we have that S || 1| Taking limit y + x gives ||00 S L,, and thus
|| S L. Hence:
110' = x dxx))loo.
(175)
Since the derivative of o' is given
a (1 - 2x p1) )
(176)
Setting ø" = 0, we
21 -2 (x 2) 0, which gives + ar and we have:
11/02
= ||0l| V2n0²
(177)
33

---

Let Φ(.) be isotropic multivariate Gaussian density function with mean i and variance o² I. There exists constant such vx, y :
Lemma 3
(d(z) o(v)) S L/x v/,
(178)
where L V2n0? exp 2n
Proof.
[[0(x)) p(v)l max V=0(=)]] || yll
exp(
2 ) 1|a vll:
(179)
V2TO²
each element dimensions V=)(2) are bounded by V2
exp(
z0z):
Proposition 3 For the measurement model Iin = Alu + n. in linear verse problem with n ~ NN0 I), we
have:
g(Lin)de) q(Lin|To),
(180)
where the approximation error can be quantified with the Jensen gap, which is upper bounded by:
(- 20 20
S
exp
A(o))lmm
(181)
2
where (V == max, V A(T)) and mi := f I| Ta (alla(aa(4i)dl.
Proof The measurement model In ATo + n can be formulated as:
(Lin Alo)²
(( NNAAo, o2
exp (
V2TO²
20
1
ATo Lin
(ATo)²
12t
exp
eepp (
(182)
2
20²
20
12
Ale
(Alo))²
2
exp
2
exp
Tin
20²
Hence, as member of the exponential family distribution ((vln) qo(y) exp(11" (y) (0(())())...)))))))))())) the undetermined functions go (Lw))
12 1/" (4m) o²
Alo
Alo (ATo)²
exp
Im. x(10) exp 0²
1in
(183)
22
2x
20
In order to exploit the measurement model (/mmn/ we factorize q(lnn (1) as follows:
((IinlI,) 9 o(Tinllo))((ol44 (d) = [1n-q(()1
(184)
here, f(-) h((A()),, where A is the measurement operator and h(-) is the multivariate normal distribution with mean A(To) and the covariance I²I. Therefore, we have:
If. -
s(E(4)) [E]((L)) 1 - (fo)..
(185)
[E(H(A(TO)) -
(186)
/ (I(A(1))) ((A(((()))))))))) -((A(o)
(187)
/ [IA(To) A(fo)))Q((((A.).
(188)
V2T0"
d
T A[ / [14- (1)1))())1)))
$
(189)
V2T0z
S
A(o)((m)m
(190)
V2T02
where dQ((T]T:) qTTo
Proposition 4 For the measurement model I.v. In Irox in linear inverse problem with o2).
we have:
((Isalte) = q(In La, Ires),
(191)
the approximation error can quantified with the Jensen gap, which is upper bounded by:
J <
Toak
(192)
2
20
where my motttres ; I1lo + Ires) (To + Ilae)), free (eldoddre...
34

---

Proef. The measurement model In 1o + Lrex can be formulated as:
(Tn
(To + Tres))
9(4m No) NV((Lo + 0²
V2T0² exp
20²
(I0 + Loca)
(L1 + Lraa))?
1in
2 exp
Iin
exp
(193)
o²
20² (To Trea)
20z
1
((To Trea))
exp (
Lim
V2T0²
2o
o²
20
In order to exploit the measurement model To), we factorize q [T) as follows:
(Tinl1t)== q Io, Tee (T Tres I
(194)
here, f(-) == h(Io + Ir and h(-) is the multivariate normal distribution with mean lo + Lres and the
covariance 0² I. Therefore, have:
=
- s(B(T))) [B[](T)) f(fo)). = [Ej + [rna)] )] 11 + I:..)l: Ih(To - (res) - -hTTo Ires))dQ(Uu, Irea/fe)
(195)
(196)
(197)
d
F 242 /
$
111oo + Tres) - (Fo Trea))la(To, Tres
(198)
V2T0" d 20 m torerr
(199)
2
where glo,, Lrea l:) is the distribution of the clean image Io and the residual components Lres. mi considered a generalized absolute distance loss because measures the mean absolute error between the predicted values and ground-truth values.
toee is
04
04
14
(a) J
(b) T
17
Trea
(c) T
V2se
20 m1
Io+trer
Figure Al: Relationships among the upper bound error I and its variables and m toen
Notably, mot+rres is finite for most of the distribution in practice once the predicted networks are optimized well. It can be regarded as generalized absolute distance hetween the observed reference and estimated data. The Jensen gap J(f, mo Trea) approach to as o ++ or suggesting that the approximation eтror reduces with extreme measurement noise. Specifically, if the predictions of To and Trea are accurate, the upper bound Ire. ))__.. fo lot/ves shrinks due to low variance and distortion. Oppositely, the predictions J(a, mi," mnf
lack accuracy, the upper bound of the Jaa M)) o-xo0,m." frex fertrrrrsmmad smax((n to treay shrinks as well for large variance and limited distortion. We further illustrate the relationships among the upper bound error and its variables Fig. Evidently, the results support the aforementioned analysis. For universal measurement model Icn
To Ires + n, we can derive the probability formulation as:
(Jnn
(To + Trea))?
qlmm Lo, Irex) ~ (No.. + Lrea, o²1)
2 expl
(200)
20
Consequently, the formulation of universal posterior sampling is:
V1, log (e) ~ Vr, log q To, Tres) = -1/o"V 1m - (To Ihen (Te, Tony +))l
(201)
according the DPS (7,. the value of a varies with time and be calculated as ||4m (Lo 1 (efes(( Lin, t))
35

---

D Diffusion Generalist Solvers with Universal Posterior Sampling
The inverse process ordinary differential equations of diffusion generalist model are:
dI
dec dā
1 d3 V
Ire
In
log
(I)).
(202)
dt
dt
dt
dt
According to the Bayes' theorem, we can derive:
Vi, logg V, log ((d)g(Lm)) V1, ( Vi, log (lin [+),
(203)
((Iin)
where V4, log P(Ie) the representing the gradient direction of inverse process and the discarded item P(Jo)) irrelevant /t. By incorporating the universal posterior sampling as conditional guidanee, can get:
dl
dat
dō:
d (Vi, log (ogg)(1t) + V,, (ogn((()))
Irea
lin
dt
dt
dt
dt
dô:
1 dB;
co(le 1/02 11 (To +Iee (Tea ()))) (204)
Ire
Tin
di
3.
da
dō,
d3, (co(I, Im, B.--BB/o'V.ll (Jo + Pres (12, Tany f))
Irea
Inn dt
dt
dt
Eq.
204 the detailed description of our method. Based on the proposed algorithms in Alg. 2 Alg. and
Alg.
we can further present varied orders with universal posterior sampling, as depicted in Alg. S
Alg. 6 and
Alg.
From the ablation experiment results in the paper, method has the best balance between restoration
performance and efficiency when k 2 and universal posterior sampling activated. Therefore, customize an accelerated algorithm for this configuration.
Algorithm 5: First-order Diffusion Generalist Solvers with Universal Posterior Sampling.
Input: Lin, Pee. {ar, (B,,). (0c,). (4.)/0 Io IT (0,,, 3.0.)
for i = 1 to M do
Ire. = 1fe 19
= (J - (ā 1)Tres - 8,,,))
To -1-11 ate, B,, - 2+ deg
== 2+ (B,_,// | (Io + [rea))l)V V |1 (Io + Ires)]l
I I: (ō.; Jn Iin + (Ti, ai Tree (3e, Be,
Output: less-
Algorithm 6: Second-order Diffusion Generalist Solvers with Universal Posterior Sampling.
Input: Iin, 17 cg. jor.h 1B,,h ( (t
do +- 17 ~ NO0, 3
to
tu = rt: + (1 r)t;-1
Tees 1 = 19es (It 11 lim 1-11)
: =Jtt-1 (at;-1 1)Tr: -1 Dim
Ho = Ct; Imma B,,, Z2; 82; p/n
E. =,,,1 (3,, //1 Iin ( + Ite )l)
Iin P P:--1
I = lt, -(8r, - den (ate, oi. I (3r. B, Yee,
Pe T It,, tu; lin)
a. [It, (ai, 1)
lmn) /3,.
11
To It, at, T2.
B.
En + dr, Iin
12
Tte = ē., + Mm - (Io" T. + P2)))l) Se,, |1 - T. Pral
V J, = I - (b,, 3, - Tm + (ou, az_;)
) T + (3,, - B B,,, F + L
13
a; (ires P )) + B. Mee, Ct
Output: Its-
36

---

Algorithm 7: Third-order Diffusion Generalist Solver with Universal Posterior Sampling.
Input: Lmn, Lress Co. { J. 1,,, r..
ho < lr ~ (0,,0,
for: 1o
4. bnn44 + (1 1444) r24: + (1 12)41-1
Pres 1 = Ihes (T Tin 11-1), P = (J+,-1 -att1-1 1111 Teee - 80, (n /
To 14:11 ar; Irie B 2; δG, ptin
E. = =ū, (3 / lin - (To + 1ree V 1 To- 1 Phr-t1
1. lt, -(or, - de /in + (ate - ā..)) (3 - -B,,
Pres = Tee (Tu: tu), ; Er, = = -at. - 1) 1)rre - -1 3rw (in))
Ton I._ Tt. Pra. -B,, Et_ + b
C. E. (3, / [ (I V || Th P.m
La = I1- (ō,, 81 Lin (ās, 04 Troal + (Br, Be, )Et2 + 32 (at.
2r1
0e ( Prea + 12 (3,, Be- [)t, -((
Tr Tee Juns to), Ct = (Tz, -aas, - -1)Pies - (1-6e,) lin)/
10
Po =,,-- Ti, Po B, E, + 33, lin
G, = E, + +(3 (3,,/ /11m - (Ig [.aa)llVV || - -iu- -Paall
11
D Thn P Di ttn 6
Djes 11rg(22-1) 113 (ritres rilte + (r2 ri (Triea))
13
14
2 D2 112(1221) (rit,, r22t. + (r2 - rı (ec-)))
J, Ii-1 (6,, d lin + (Ti, -o,_,) 1 1 + (3
Min
F - 24 aa,,
a: Die + 1 (3, - B Di + (ar, at Dige + 13,, - 3 D2
2-1
Output:
Tex-
Algorithm 8: Queue- Blsee Second-order Solvers with Universal Posterior Sampling. Input: Iin. Pee. [.. (3,, J. (.4.)...))).. Io < IT NO0, I).
R = D.
(Je, Prt,, F,, P.1) = 1s order Solver(tz) in Alg. R = queuc_puuh((e,) Pri,, E,, );
while < tm do
-1-iit tu =t,, t4 t.41. r = (tu tp)/ tp) (Lep, P.e (lpp Cty Tr, Fo = queue (op(Ri,4p)
(Tem: Ph PrasFty, Pou) = - orddrr Solver(t in Alg.
by = Iep (0tg din + (are, axpp) Ties + (3 - 3, Yap + (a,, (Prcs T22.) +
(B,, Bes, Tep)
queue_pusuhee (I,,, Pra, Tiu To ). i ty
10
Output: Iey-
Queuc-Based Accelerated Strategy. The introduction variables pertaining to intermediate time point tu incurs additional computational overhead, making the process inefficient. To alleviate this problem, we construct queue-bace based accelerated strategy by collecting the previous solutions, as illustrated in Fig. Bin the paper. Specifically, for the initial sampling from f t, introduce the intermediate step to assist the approximation at time point Subsequently, is regarded as intermediate time point the sampling from tu to 1 iteratively. In this context, the sampling steps are controlled by the predefined parameters r without introducing extra calculation. Relevant implementation details are summarized in Alg.
37

---

E Experiment Supplementary
E.1
Summary about the Datasets
We evaluate the proposed method on five natural image restoration tasks, including deraining, low-light hancement, desnowing, dehazing, and deblurring, We select the most widely used datasets for each task, as summarized Tab. AT
Table A1: Summary of the natural image restoration datasets utilized in this paper.
Task
Dataset
Synthetic/Real
Train samples
Test samples
DID 69
Synthetic
1,200
DeRaindrop
4
Real
861
307
Rain13K
Synthetic
13,711
Rain_1001
666
Synthetic
100
Deraining
Rain_100L
66
Synthetic
100
GT-Rain 3
Real
26,125
2,100
RealRain-Ik-H
Real
896
224
RealRain-Ik-L
22
Real
896
224
LOL 61
Real
485
15
Low-light
MEF 36
Real
Enhancement
VE-LOL-L 30
Synthetic/Real
900/400
100/100
NPE [57
Real
CSD 4
Synthetic
8,000
2,000
Desnowing
Snow 100K-Real
Real
1,329
SOTS [2]]
Synthetic
500
ITS ITS_v2 v2
Synthetic
13,990
Dehazing
D-HAZY
Synthetic
1,178
294
NH-HAZE 2
Real
55
Sense-Haze
Real
55
NHRW (70
Real
150
GoPro 339
Synthetic
2,103
1.111
Deblur
RealBlur [46
Real
3,758
980
Image deraining. We use the merged datasets mentioned in 13 17 and DeRaindrop 4
as training parts, which wide range of different rain streaks and rain densities. We test model both deraining Furthermore, we perform zero-shot generalization on real-world datasets such as GT-Rain 3) RealRain- Ik-H (24), and RealRain- (24).
and deraindrop scenes using the mixed datasets [66
Low-light enhancement, We utilize the LOL 61 and VE-LOL-L 30 datasets with real and synthetic paired images as the benchmark, which consist of a large number of indoor and outdoor scenes with different levels of light and noise. Furthermore, generalize our method on the MEF 36 dataset with images of multiple exposures to conduct the compound restoration experiments.
Image desnowing. We use the CSD dataset the desnowing benchmark and employ the Snow 100K- Real (31] to perform world generalization.
g datasets as the dehazing benchmark, which are widely used synthetic fog datasets with different levels of fog and various scenes. Besides, the outdoor fog dataset SOTS (1 is selected for quantitative evaluations. The real-world fog datasets Dense-Haze (I]. NHRW [70]
Image dehazing, We use ITS 21 D-HAZY
and NH-HAZE 2 utilized for generalization validation real-world scenes.
Image deblurring. We the GoPro 39 dataset the deblurring benehmark. which contains 2,3 training pairs and 1,111 testing pairs. The dataset contains various levels blur obtained by averaging clear images captured in very short intervals. To further validate the generalization ability of our model, we perform zero-shot generalization on the RealBlur [46] dataset.
E.2 More Visual Comparisons on Comparable Experiments
We show the visualization results of other degradation categories in Fig. A2 to further demonstrate our superiority. Evidently, Our method generates more stable image samples with high fidelity than other universal image restoration methods. Benefiting from the accuracy of diffusion generalist solvers and stable guidance of universal posterior sampling, achieve the outstanding reconstruction of the missing details.
38

---

IM-10P
Figure A2: Visualization results of other degradation categories.
Nighttime-blar
(2)
GrouddTruth
Figure A3: Results of low-light enhancement and deblurring for five iterations on RealBlur 46
multi-exposure
(2)
33)
Figure A4: Results of multi-exposure restoration for five iterations on MEF 36
E.3
Compound Restoration on Real-world Scenes
To demonstrate that our method can handle the more complex restoration tasks that various degradation occurs in one image. We generalize our well-optimized model on RealBlur [46] datasets for low-light enhancement and deblurring, and NHRW [70] datasets for low-light enhancement and dehazing, and MEF (36] datasets for multi-exposure restoration. Specifically, we repeatedly perform our method on degraded images five times, and the visual comparisons are displayed in Fig. A3 Fig. and Fig. [4] respectively. As a result, as our method is trained on multiple degraded datasets, it is capable of excellently accomplishing relevant composite tasks. Furthermore, it possesses the ability to eliminate the former degradation and subsequently the latter
39

---

degradation within its capability range. For well-restored samples, our method incurs additional information interference, thus ensuring the high fidelity of the restored images.
E.4 Real-world Scene Generalization
Except for the desnowing on the real-world scene, other restoration results of different degradation types are illustrated in Fig. A5 We utilize the RealRain- 24 dataset for deraining. NPE [57] dataset for low-
light enhancement, Snow 100k real [3]] dataset for desnowing. Dense-Haze [1] dataset for dehazing, and RealBlur [46] dataset deblurring. The results show that our method can generalize real-world scenes and achieve competitive visual results.
Fabic Looe
Pymydh
Frohin
TXTTER
Buo Emag
Kertormet
Froeget
IM-CPP
setra
Figure A5: Visual effects of zero-shot generalization on sollwwrldd image restoration.
E.5
Generalization on Remote Sensing
Table A2: Summary of the utilized remote sensing image restoration datasets.
Task
Dataset
Synthetic/Real
Train samples
Test samples
Cloud Removal
Sen2_ LMC 15
Real
9,784
467
Denoise
AID 63
Synthetic
38,173
1,827
Deblur
NWPU-RESISC45 (5)
Synthetic
10,000
1,094
Super-resolve
Second 65
Synthetic
22,603
1,141
Light Enhancement
PatternNet 76
Synthetic
28,905
1.495
We further validate the scalability of method
sensing image restoration tasks, including cloud removal, denoising, deblurring, super-resolution, and low-light enhancement. Except for the cloud removal task, there are no existing real-world datasets for other restoration tasks [44). so we adopt synthetic data generation approach to simulated the datasets, presented in Tab. A2 Specifically: i) Cloud Removal: We use the Sen2 MTC dataset [15] the benchmark, which contains paired cloudy and cloud-free images. ii) Denoising: Due the lack of real-world noisy remote sensing images, we synthesize noisy dataset based the AID dataset (63] considering three mainstrcam noise types (Gaussian noise, stripe noise, and speckle noise) and their random combinations. iii) Deblurring: We primarily simulate Gaussian blur, motion blur, and defocus blur, and conduct experiments the
dataset (5]. iv) Super- Resolution: We generate low-resolution images using four-fold downsampling degradation model with Gaussian blur and construct paired images simulate low-light imaging conditions on PatternNet dataset (76 through linear transformation, gamma correction, and additive Gaussian noise. The visual comparisons of each method are presented in Fig. A6 and Fig. 8 and quantitative evaluations are reported in Tab. of this paper. All methods have been e--mmllmeeted except for AutoDIR, released for it. Evidently, visual effect surpasses that of others and achieves the best metric evaluations.
on the Second dataset 65 v) Low- Light Enhancement:
training code has been
Table A4: Time efficiency of different strategies.
Sign
Nasslkett
OURS
11120
11:14
11.633
01322
6
0.401
0:066
D.574
(L585
10
(1:6]
0.127
0.320
00309
0.476
0.533
0.55
0.725
Nave 0e:3)
02011
ILS78
12
13
(1977
19
1.100
21
13222
1,440
27
1.533
30
1.344
Uscan (ki)))
0.197
1L318
6111
0.47
0.38
D457
6779
40

---

Table A3: Efficiency comparisons among universal methods. '-' means out of memmory.
256x256
512x512
1024x 1024
Method
Mem.
Time(s)
FPS
Mem.(G)
Time(s)
Mem.(G)
Time(s)
FPS
Restomer
1.959
0.105
9.563
6.670
0.381
2.622
25.419
1.773
0.564
AirNet
1.039
0.194
5.159
3.480
0.738
1.355
11.244
20.499
0.049
PromptIR
2.544
0.111
8,981
7.255
0.399
2.508
26.005
1.845
0,542
ProRes
2.027
0.318
3.149
2.514
0.766
1.305
6.025
1.715
0,583
IDR
1.340
0.052
19.253
4.313
0.136
7.373
16.110
0.615
1.626
AutoDIR
7.023
6.266
0.160
11.021
11.986
0.083
DA-CLIP
2.119
2.585
0.387
6.775
7.937
0.126
58.548
60.893
0.016
DifTUI((1=3)
1.563
0.118
8,450
3.528
0.206
4.862
18.060
0.911
1.098
Ours- Ou=s-L(n=3)
1.561
0.112
8.908
3.528
0.199
5.014
18.059
0.907
1.103
Naive (n=8), A: 1, XUPS
Ours-T
0.777
0.277
3.605
2.291
0.385
2.594
15.306
1.705
0.587
Ours-S
0.787
0.290
3.450
2.300
0.401
2.494
15.316
1.755
0.570
Ours-B
0.942
0.291
3,431
2.907
0.492
2.033
17.438
2.280
0.439
Ours-L
1.562
0.293
3,407
3,527
0.529
1.889
18.058
2.402
0.416
Naive (n=8), is 1, VPPS
Ours-T
1.764
0.500
1.998
5.793
0.829
1.206
32.581
4.454
0.225
Ours-S
1.905
0.502
1.993
6,265
0.865
1.156
34,237
4.545
0.220
Ours-B
2,762
0.520
1.923
9.645
1.078
0,928
41,357
5.770
0.173
Ours-L
3.613
0.535
1.868
10.593
1.144
0.874
43,815
6,022
0.166
Naive (n=8), A: 2, x XUPS
Ours-T
0.784
0.535
1.869
2.297
0.717
1.395
15.313
3.194
0.313
Ours-S
0.794
0.548
1.824
2.308
0.747
1.339
15.323
3.296
0.303
Ours-B
0.948
0.555
1.802
2,913
0,924
1.082
17.444
4.279
0.234
Ours-L
1.563
0.571
1.751
3.527
0.989
1.011
18.059
4.522
0,221
Naive (n=8), h: 2, UPS
Ours-T
1.770
0.974
1.027
5,800
1.559
0.641
32.587
8.150
0.123
Ours-S
1,933
0.987
1,013
7.508
1.871
0.535
33.586
10.482
0.095
Ours-B
2,803
0.990
1.010
9.688
2.089
0.479
41,399
11.276
0.089
Ours-L
3,829
1.019
0.981
10.679
2.219
0.451
44.154
11.723
0.085
Queue (n=8), 1: = 2, x UPS
Ours-T
0.780
0.303
3,301
2.294
0.431
2.321
15.310
1.905
0.524
Ours-S
0.791
0.321
3.115
2.304
0.460
2.174
15.320
1.980
0.505
Ours-B
0.946
0.323
3.096
2,911
0.547
1.823
17.442
2.573
0.388
Ours-L
1.563
0.324
3,086
3.527
0.598
1.673
18.059
2.694
0.371
Queue (n=8), l: = 2, UPS
Ours-T
1.771
0.557
1.795
5,800
0.925
1.081
32.587
5,012
0.200
Ours-S
1,920
0.561
1.782
6.279
0.961
1.041
34,250
5.101
0.196
Ours-B
2.892
0.590
1.695
9.677
1.201
0.833
41,388
6.472
0.155
Ours-L
3.816
0.600
1.667
10.601
1.273
0.786
44.074
6.741
0,148
Naive (n=8), t: 3, x XPP
Ours-T
0.790
0.759
1.317
2.303
1.053
0,950
15,319
4.687
0.213
Ours-S
0.799
0.775
1.290
2.312
1.099
0.910
15.328
4.836
0.207
Ours-B
0.954
0.780
1,283
2.919
1.347
0.742
17.450
6.253
0.160
Ours-L
1.562
0.801
1.248
3.527
1.446
0.692
18.058
6.586
0.152
Naive (n=8), h: 3, UPS
Ours-T
1.782
1.414
0.707
5.812
2.338
0.428
32.599
12.384
0.081
Ours-S
1.958
1.428
0.700
6.318
2.474
0.404
34,288
13.238
0.076
Ours-B
2,909
1.433
0.698
9,694
3.100
0.323
41.405
16.755
0.060
Ours-L
4.048
1.496
0.668
11.050
3,302
0.303
48.273
17.461
0.057
Queue (n=8), 4: 3, XUPS
Ours-T
0.777
0.338
2,959
2,290
0.481
2.077
15,306
2.132
0.469
Ours-S
0.787
0.364
2.744
2,301
0.502
1,991
15.316
2.195
0.455
Ours-B
0,942
0.366
2.730
2.907
0.615
1.626
17.438
2.845
0.351
Ours-L
1,562
0.369
2,710
3.527
0.657
1.522
18.058
2.999
0.333
Queue (n=8), 4:
3, /PPS
Ours-T
1.765
0.624
1.602
5.794
1.060
0,943
32.581
5.664
0.177
Ours-S
1.907
0.646
1.547
6.267
1.090
0.917
34.238
5.789
0.173
Ours-B B
2.765
0.661
1.512
9,648
1.369
0.730
41.359
7.339
0.136
Ours-L
3.613
0.695
1.440
10.593
1.470
0.680
43.814
7.656
0.131
41

---

Lem-treliati
Inder
Praylla
Fretes
24-4P
santa
del
Figure A6: Visual results of remote sensing image restoration.
E.6 Efficiency Comparisons
For fairness, collect and mix the datasets used by comparison methods, with image resolution ranging from 256 1024 pixels. Accordingly. we evaluate model efficiency under three representative resolution settings, as summarized in Tab. A3 Let & denote the solver order and n be sampling steps. Obviously, our method (n = 3) and baseline DifFURR remain competitive in computational cost and efficiency. As increases, memory consumption remains stable while time cost grows proportionally. Activating UPS that requires gradient backpropagation introduces per-step computational overhead and additional memory usage. Hence, we adopt the queue-hased sampling strategy that significantly improves efficiency by reducing the computational complexity from (nn) to O(k + 1). Tah. A4 also proves that queue-based solver achieves approximately a 2x efficiency improvement over naive solvers when & 2, and around a 3x improvement when ht = 3.
F Discussions, Limitations, and Future Work
Limitations and broader impact. The main challenge lies in improving the precision and robustness of a unified restoration model when handling highly mixed representations. While we have theoretically proposed general formulation for k-order solver, experiments limited first-, second-, and third-order solvers GPU memory constraints. This because unified posterior sampling requires storing gradient map during inference compute V log (T; To. PPcx) and time points needed compute derivatives. In addition, is observed that the second order solver with universal posterior sampling reach the performance bottleneck. However, it remains unknown whether it will be saturated in more complex scenes or using larger models. Besides, our solver and strategy are training free and can potentially be extended to other methods after appropriate modifications. Future work could explore more efficient network architectures reduce memory usage, enabling larger batch sizes, higher-order solvers, and better task-specific training strategies, such as adaptive learning rate schedules. Despite current limitations, we helieve our unified model offers strong foundation for advancing image restoration.
Future Work. There are several promising directions to enhance our method: (1) With the rise of high-resolution imagery (e.g.. 4K, 8K) driven by advances in sensor technology, developing an interpretable multi multi-dimensional latent diffusion model is crucial to address the computational demands that current architectures struggle with. (2) Apply model distillation reduce memory overhead and parameter size. (3) Introduce additional single or compound degradation types, expand parameter ranges, and enhance model versatility. (4) Design adaptive learning rate schedules to reduce sampling steps and improve restoration quality.
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Reviews:
[{"review_id": "KGFMM3E8VS", "reviewer": "Reviewer_9bRU", "summary": "Overall, this work is a strong paper with both theoretical depth and clear empirical improvements over previous universal methods.\nThe authors reformulate the restoration process as a semi-linear ODE that can be solved analytically, and they derive a closed form integral solution for the ODE. This derivation uses an inverse transformation for the nonlinear parts, assuming those parts are monotonic and invertible functions. They expand these nonlinear integrals with Taylor series and retain terms up to a chosen order k, and obtain a family of k-th order update formulas. Authors show that earlier diffusion restoration methods can be considered as the first order case of their solver.\nOverall, the proofs around proposition 1 are solid and novel in the context of diffusion restoration.\n\nNext, authors provide a theoretical upper bound for the approximation error (Jensen's gap where the outer expectation over the posterior is replaced by inner expectations with respect to the data distribution). They show the error depends on the variance of the measurement noise and the accuracy of the model’s predictions, guaranteeing that the error is controlled (for linear inverse problems with Gaussian noise) in Theorem 1.\n\nAuthors' claims are well supported by the empirical results, showing strong SoTA performance among universal methods and on par performance compared to task specific methods.", "questions": "1. Equation (4) has a typo; $q_{0t}$ should be $q_{0}$?\n\n2. Equation (21), is this $l_{2}$ norm $∥⋅∥^2 $?\n\n3. Could you show a direct ablation against another method like DiffUIR using their own solver vs. DGSolver to isolate the solvers contribution?", "limitations": "Theorem 1 provides a rigorous bound for the posterior approximation error only for linear measurement models, although many degradations ops are inherently nonlinear and hence not covered.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The paper proposes the novel idea to combine exact ODE solution + universal posterior sampling, generalizing prior diffusion restoration methods. \n2. Theoretical derivations are clear.\n3. Empirically, they show strong SOTA results across tasks and datasets. \n\n\nWeaknesses:\n1. Authors don't provide FPS/runtime throughput data on full-resolution images. Also, they don't provide data on actual inference time or memory usage.", "was_revised": true, "final_justification": "Authors have addressed my concerns and I continue to support acceptance of this work.", "rebuttal": "### W1: Efficiency comparisons among universal methods\n\nWe appreciate the reviewer’s concern regarding inference efficiency. For fairness, we collect and mix the datasets used by comparison methods, with image resolution ranging from 256 to 1024 pixels. Accordingly, we evaluate model efficiency under three representative resolution settings, as summarized in Tab. r1. Let $k$ denote the solver order and $n$ be sampling steps. Obviously, our method (n = 3) and baseline DiffUIR remain competitive in computational cost and efficiency. When increasing $n$, memory consumption remains stable while time cost grows proportionally. Activating UPS that requires gradients backpropagation introduces per-step computational overhead and additional memory usage. To alleviate these issues, we employ a queue-based sampling strategy that significantly improves efficiency by reducing the computational complexity. Specifically, we use the number of neural function evaluations (NFEs) to evaluate the complexity. For a $k$-order solver, naive sampling from time $s$ to $t$ requires interpolating $k-1$ intermediate points within the interval, resulting in $k$ NFEs per step. Consequently, the overall computational complexity for $n$ steps is $O(nk)$. In contrast, the queue-based sampling strategy precomputes the values at $k−1$ time points and caches them for reuse in subsequent steps, offering high efficiency by reducing total complexity to $O(k+n−1)$. Tab. r1 also demonstrates that the queue-based solver achieves approximately a 2× efficiency improvement over naive solvers when $k=2$, and around a 3× improvement when $k=3$. \n\n\n|Size||256x256|||512x512|||1024x1024||\n|-|-|-|-|-|-|-|-|-|-|\n|Method|Mem.(G)|Time(s)|FPS|Mem.(G)|Time(s)|FPS|Mem.(G)|Time(s)|FPS|\n|Restomer|1.959|0.105|9.563|6.670|0.381|2.622|25.419|1.773|0.564|\n|AirNet|1.039|0.194|5.159|3.480|0.738|1.355|11.244|20.499|0.049|\n|PromptIR|2.544|0.111|8.981|7.255|0.399|2.508|26.005|1.845|0.542|\n|ProRes|2.027|0.318|3.149|2.514|0.766|1.305|6.025|1.715|0.583|\n|IDR|1.340|0.052|19.253|4.313|0.136|7.373|16.110|0.615|1.626|\n|AutoDIR|7.023|6.266|0.160|11.021|11.986|0.083|-|-|-|\n|DA-CLIP|2.119|2.585|0.387|6.775|7.937|0.126|58.548|60.893|0.016|\n|DiffUIR(n=3)|1.563|0.118|8.450|3.528|0.206|4.862|18.060|0.911|1.098|\n|Ours-L(n=3)|1.561|0.112|8.908|3.528|0.199|5.014|18.059|0.907|1.103|\n|Naive(n=8)|k=1|×UPS|\n|Ours-T|0.777|0.277|3.605|2.291|0.385|2.594|15.306|1.705|0.587|\n|Ours-S|0.787|0.290|3.450|2.300|0.401|2.494|15.316|1.755|0.570|\n|Ours-B|0.942|0.291|3.431|2.907|0.492|2.033|17.438|2.280|0.439|\n|Ours-L|1.562|0.293|3.407|3.527|0.529|1.889|18.058|2.402|0.416|\n|Naive(n=8)|k=1|√UPS|\n|Ours-T|1.764|0.500|1.998|5.793|0.829|1.206|32.581|4.454|0.225|\n|Ours-S|1.905|0.502|1.993|6.265|0.865|1.156|34.237|4.545|0.220|\n|Ours-B|2.762|0.520|1.923|9.645|1.078|0.928|41.357|5.770|0.173|\n|Ours-L|3.613|0.535|1.868|10.593|1.144|0.874|43.815|6.022|0.166|\n|Naive(n=8)|k=2|×UPS|\n|Ours-T|0.784|0.535|1.869|2.297|0.717|1.395|15.313|3.194|0.313|\n|Ours-S|0.794|0.548|1.824|2.308|0.747|1.339|15.323|3.296|0.303|\n|Ours-B|0.948|0.555|1.802|2.913|0.924|1.082|17.444|4.279|0.234|\n|Ours-L|1.563|0.571|1.751|3.527|0.989|1.011|18.059|4.522|0.221|\n|Naive(n=8)|k=2|√UPS|\n|Ours-T|1.770|0.974|1.027|5.800|1.559|0.641|32.587|8.150|0.123|\n|Ours-S|1.933|0.987|1.013|7.508|1.871|0.535|33.586|10.482|0.095|\n|Ours-B|2.803|0.990|1.010|9.688|2.089|0.479|41.399|11.276|0.089|\n|Ours-L|3.829|1.019|0.981|10.679|2.219|0.451|44.154|11.723|0.085|\n|Queue(n=8)|k=2|×UPS|\n|Ours-T|0.780|0.303|3.301|2.294|0.431|2.321|15.310|1.905|0.524|\n|Ours-S|0.791|0.321|3.115|2.304|0.460|2.174|15.320|1.980|0.505|\n|Ours-B|0.946|0.323|3.096|2.911|0.547|1.823|17.442|2.573|0.388|\n|Ours-L|1.563|0.324|3.086|3.527|0.598|1.673|18.059|2.694|0.371|\n|Queue(n=8)|k=2|√UPS|\n|Ours-T|1.771|0.557|1.795|5.800|0.925|1.081|32.587|5.012|0.200|\n|Ours-S|1.920|0.561|1.782|6.279|0.961|1.041|34.250|5.101|0.196|\n|Ours-B|2.892|0.590|1.695|9.677|1.201|0.833|41.388|6.472|0.155|\n|Ours-L|3.816|0.600|1.667|10.601|1.273|0.786|44.074|6.741|0.148|\n|Naive(n=8)|k=3|×UPS|\n|Ours-T|0.790|0.759|1.317|2.303|1.053|0.950|15.319|4.687|0.213|\n|Ours-S|0.799|0.775|1.290|2.312|1.099|0.910|15.328|4.836|0.207|\n|Ours-B|0.954|0.780|1.283|2.919|1.347|0.742|17.450|6.253|0.160|\n|Ours-L|1.562|0.801|1.248|3.527|1.446|0.692|18.058|6.586|0.152|\n|Naive(n=8)|k=3|√UPS|\n|Ours-T|1.782|1.414|0.707|5.812|2.338|0.428|32.599|12.384|0.081|\n|Ours-S|1.958|1.428|0.700|6.318|2.474|0.404|34.288|13.238|0.076|\n|Ours-B|2.909|1.433|0.698|9.694|3.100|0.323|41.405|16.755|0.060|\n|Ours-L|4.048|1.496|0.668|11.050|3.302|0.303|48.273|17.461|0.057|\n|Queue(n=8)|k=3|×UPS|\n|Ours-T|0.777|0.338|2.959|2.290|0.481|2.077|15.306|2.132|0.469|\n|Ours-S|0.787|0.364|2.744|2.301|0.502|1.991|15.316|2.195|0.455|\n|Ours-B|0.942|0.366|2.730|2.907|0.615|1.626|17.438|2.845|0.351|\n|Ours-L|1.562|0.369|2.710|3.527|0.657|1.522|18.058|2.999|0.333|\n|Queue(n=8)|k=3|√UPS|\n|Ours-T|1.765|0.624|1.602|5.794|1.060|0.943|32.581|5.664|0.177|\n|Ours-S|1.907|0.646|1.547|6.267|1.090|0.917|34.238|5.789|0.173|\n|Ours-B|2.765|0.661|1.512|9.648|1.369|0.730|41.359|7.339|0.136|\n|Ours-L|3.613|0.695|1.440|10.593|1.470|0.680|43.814|7.656|0.131|\n\n*Table r1: Efficiency comparisons among universal methods. '-' means out of memmory.*\n\n### Q1&Q2: Definitions about $q_{0t}$ and norm $\\|\\cdot\\|$\n\nWe consider $q_0$ as the data distribution of $I_0$. In Eq. (4), $q_{0t}:=q(I_t\\vert I_0,I_{res},I_{in})$ denotes a conditional probability distribution of $I_t$ conditioned on $I_{0},I_{res},I_{in}$ for simplicity. In Eq. (21), $\\|\\cdot\\|$ is $l_2$-norm. We appreciate your comments and promise to refine the notations to avoid ambiguity in the revision.\n\n### Q3: Isolated contribution of our Solver\n\nWe appreciate your insightful suggestion. Our solver is built upon the prediction of residuals and noise, and is thus applicable to any diffusion-based models that adopt similar formulations. To validate its generality, we adapt our solver to DiffUIR. As shown in Tab. r2, DiffUIR consistently benefits from our solver across various configurations, leading to notable performance improvements.\n\n|DiffUIR|n=3|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|Order|UPS|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|\n|k=1|x|30.67|0.887|21.21|0.769|30.70|0.943|30.29|0.944|29.00|0.877|29.93|0.907|\n|k=2|x|30.76|0.888|21.28|0.771|30.82|0.946|30.58|0.949|29.04|0.879|30.05|0.910|\n|k=3|x|30.81|0.890|21.30|0.772|30.85|0.948|30.61|0.951|29.05|0.880|30.08|0.911|\n|k=1|√|30.72|0.887|21.24|0.770|30.81|0.945|30.43|0.946|29.02|0.878|30.01|0.908|\n|k=2|√|31.17|0.899|21.52|0.786|31.01|0.951|30.76|0.954|29.34|0.885|30.32|0.917|\n|k=3|√|31.21|0.901|21.54|0.787|31.08|0.953|30.80|0.955|29.35|0.886|30.37|0.918|\n\n*Table r2: Adapt our solvers to DiffUIR*"}, {"review_id": "tx8M0nJok6", "reviewer": "Reviewer_1zc3", "summary": "The paper introduces DGSolver, a novel diffusion generalist solver with universal posterior sampling for image restoration. It addresses the limitations of existing diffusion models in universal image restoration, which suffer from cumulative errors in reverse inference and the trade-off between degradation representation commonality and restoration quality. The authors derive exact ODEs solution for diffusion generalist models and develop high-order solvers with a queue-based accelerated sampling strategy to enhance accuracy and efficiency. Additionally, they integrate universal posterior sampling to improve noise estimation and correct errors in inverse inference. Extensive experiments demonstrate that DGSolver outperforms state-of-the-art methods in restoration accuracy, stability, and scalability across various tasks and datasets.", "questions": "1. What challenges or opportunities arise from applying the generalist diffusion process reformulated with ODEs and customized high-order solvers, which are common in image generation, to the field of image restoration?\n2. Given that k=2 with UPS already achieves near-optimal performance, what is the practical benefit of higher-order solvers (k=3) in terms of restoration quality versus computational cost? Can you provide a trade-off analysis (e.g., FLOPs vs. PSNR) for different k?\n3. The paper mentions the use of a queue-based accelerated sampling strategy. Could this method remain effective when using fewer sampling steps? Typically, high-order solvers are designed to accelerate sampling with fewer steps.\n4. The optimal $\\bar{\\delta}_T=1$ is used for complex degradations, but does this generalize to all tasks? For example, does deraining benefit from a different $\\bar{\\delta}_T$ than deblurring?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### Strengths:\n1. The reformulation of the generalist diffusion process using ODEs and the development of customized high-order solvers with a queue-based accelerated sampling strategy are novel contributions to the field of image restoration.\n2. The integration of universal posterior sampling provides a versatile and effective way to enhance the accuracy of noise estimation and correct errors, leading to improved restoration quality.\n3. The proposed method is validated through experiments on multiple image restoration tasks, such as deraining, low-light enhancement, etc., and shows superior performance over existing methods.\n\n### Weaknesses:\n1. The reformulation of the generalist diffusion process using ODEs and the development of customized high-order solvers have been extensively explored in image generation, and their application to image restoration is a natural extension.\n2. The paper proposes a queue-based accelerated sampling strategy. However, its effectiveness might be limited in few-step sampling scenarios, and the paper lacks ablation studies to verify its performance in such cases.\n3. While $\\bar{\\delta}_T$ controls degradation commonality, the paper does not systematically explore its effect on specific degradation types.\n4. Although zero-shot results on real-world datasets are presented, the evaluation lacks quantitative metrics for unpaired real data, relying solely on visual comparisons", "was_revised": true, "final_justification": "The authors' response during the rebuttal phase has addressed my concerns.", "rebuttal": "### W1&Q1: Challenges and opportunities arise from image restoration ODEs solvers\n\nTo the best of our knowledge, in the field of image generation, high-order solvers are primarily employed to accelerate the sampling process from random noise to image samples, with the only requirement being that the generated samples conform to the target distribution. There is no strict constraint enforcing a one-to-one correspondence between samples from the prior and those from the data distribution. Consequently, various solvers can be employed to approximate the data distribution without explicit constraints, and the potential of high-order terms remains largely underexplored due to the weak coupling between the prior and data. In contrast, image restoration involves a strong dependency between the prior (i.e., the degraded image) and the target data (i.e., the high-quality image). Therefore, we argue that the key challenge and opportunity for designing solvers in image restoration lies in effectively exploiting the relationship between these two distributions and embedding this relationship throughout the diffusion process. To this end, we propose DGSolver. Our key insights are: (1) We fully incorporate the residuals derived from both the prior and data distributions into the high-order solver. (2) The potential of residuals in designed solvers is further exploited through a Universal Posterior Sampling (UPS), which reinforces the connection between the prior and data distributions, thereby enhancing restoration performance, inference stability, and sample fidelity.\n\n### W2&Q3: Disscusion about queue-based sampling\n\nHigh-order solvers can reduce the sampling steps, but they do not necessarily reduce the number of neural function evaluations (NFEs) in few-step sampling scenarios. NFEs directly measure the number of neural network calls with respect to total runtime. For a $k$-order solver, naive sampling from time $s$ to $t$ requires interpolating $k-1$ intermediate points within the interval, resulting in $k$ NFEs per step. Consequently, the overall computational complexity for $n$ steps is $O(nk)$. In contrast, the queue-based sampling strategy precomputes the values at $k−1$ time points and caches them for reuse in subsequent steps, offering high efficiency by reducing total complexity to $O(k−1+n)$. Therefore, only when $n=1$ do both naive solvers and queue-based solvers have the same time complexity $O(k)$. When $n > 1$, the naive solvers are increasingly less efficient than queue-based solvers for $nk \\ge k-1+n$. \n\nTo verify that, we report runtime comparisons in Tab. r3 with a fixed image size of 512 and UPS deactivated. Apparently, queue-based strategy is more efficient than naive sampling. Additionally, as shown in Tab. r1 in Reviewer (9bRU-W1), enabling UPS further increases the computational cost, making the efficiency gap between the two strategies even more pronounced. Moreover, Tab. r9 in Reviewer (iqQ1-Q2) presents a performance comparison between the queue-based and naive strategies. Queue-based solvers possess a 2–3× speed advantage with comparable restoration quality to naive solvers.\n\nIn conclusion, queue-based accelerated sampling still remains effective when using fewer sampling steps\n \n\n|Steps|1||2||3||4||5||6||7||8||9||10||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|NFE|Time(s)|\n|(k=1)|1|0.063|2|0.129|3|0.194|4|0.263|5|0.322|6|0.401|7|0.466|8|0.529|9|0.596|10|0.661|\n|Naive(k=2)|2|0.120|4|0.247|6|0.393|8|0.523|10|0.644|12|0.769|14|0.910|16|0.989|18|1.147|20|1.251|\n|Queue(k=2)|2|0.127|3|0.191|4|0.262|5|0.320|6|0.399|7|0.476|8|0.533|9|0.598|10|0.659|11|0.728|\n|Naive(k=3)|3|0.201|6|0.401|9|0.578|12|0.769|15|0.977|18|1.169|21|1.322|24|1.446|27|1.585|30|1.814|\n|Queue(k=3)|3|0.197|4|0.258|5|0.318|6|0.410|7|0.479|8|0.528|9|0.594|10|0.657|11|0.719|12|0.806|\n\n*Table r3: Time efficiency of differnent strategies*\n\n### W3&Q4: Discussion about $\\delta_T$\n\nWe appreciate your constructive suggestion. We respectfully clarify that performance comparisons under different values of $\\overline{\\delta}_T$ are presented in Tab. 3 (Page 8). As observed, the choice of $\\overline{\\delta}_T$ significantly influences restoration performance across different tasks, with higher values generally being more suitable. Specifically, the model achieves the best performance on deraining, low-light enhancement, and desnowing tasks with $\\overline{\\delta}_T = 1.0$; for dehazing, the optimal value is $\\overline{\\delta}_T = 0.9$; and for deblurring, $\\overline{\\delta}_T = 0.75$ yields the best results. Nevertheless, considering overall performance across tasks, $\\overline{\\delta}_T = 1.0$ emerges as the most favorable setting within our framework.\n\n### W4: Real world quantitative evaluation\n\nTo systematically evaluate the zero-shot capability of each method on real-world datasets (Appendix E, Tab. A1), we adopt PSNR for paired data and NIQE [1] for all data to assess perceptual quality, as reported in Tab. r4. Our method consistently achieves superior PSNR across various tasks. However, the non-reference metric (NIQE) exhibits patterns that diverge from the objective PSNR trend, where our performance is at a moderate level. This observation is consistent with prior studies [2,3], which highlight the inherent instability of NIQE. NIQE is highly sensitive to content variations and often fails to align with human perception, particularly under complex degradations. In conclusion, our method still remains highly competitive, as evidenced by both quantitative metrics and visual results (Appendix E, Fig. A6).\n\n\n|Method|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n||PSNR|NIQE|PSNR|NIQE|PSNR|NIQE|PSNR|NIQE|PSNR|NIQE|PSNR|NIQE|\n|Restomer|21.19|47.45|-|13.66|-|14.13|10.46|12.30|20.19|11.00|20.60|30.34|\n|AirNet|19.57|50.49|-|13.63|-|14.34|10.21|12.20|13.55|14.13|17.67|32.54|\n|Prompt-IR|21.38|51.34|-|12.98|-|14.29|10.46|13.06|24.83|12.56|21.98|32.66|\n|ProRes|16.26|45.57|-|13.12|-|15.99|10.54|13.08|21.38|12.19|17.47|30.12|\n|IDR|21.06|49.09|-|13.09|-|13.82|10.28|13.02|22.28|11.81|21.06|31.25|\n|AutoDIR|19.60|54.68|-|13.02|-|14.47|10.41|15.38|25.43|12.71|20.89|34.48|\n|DA-CLIP|20.93|51.90|-|12.76|-|14.45|10.40|14.11|22.25|12.52|20.96|32.99|\n|DiffUIR|21.38|51.85|-|12.57|-|14.34|8.57|12.79|24.77|12.21|21.91|32.85|\n|Ours|21.49|53.23|-|12.63|-|14.28|10.61|13.25|25.67|12.34|22.29|33.57|\n\n*Table r4: Evaluation on Real-world image restoration tasks*\n\n\n[1] Mittal A, Soundararajan R, Bovik A C. Making a “completely blind” image quality analyzer[J]. IEEE Signal processing letters, 2012.\n\n[2] Liu Y, Zhao H, Gu J, et al. Evaluating the generalization ability of super-resolution networks[J]. TPAMI, 2023.\n\n[3] Zhang L, Zhang L, Bovik A C. A feature-enriched completely blind image quality evaluator[J]. TIP, 2015.\n\n\n### Q2: Disscusion about solver order\n\nThe performance and computational cost of different solvers are summarized in Tab. r5. Notably, UPS increases computational cost by ~2× due to additional backward propagation[1]. In our configuration, the solver with $ k=2 $ and UPS yields the best performance and is thus used as the default setting in the paper. The case of $k=3$ demonstrates the solver’s scalability, although performance reaches a saturation point within our framework. Nevertheless, when generalized to related frameworks such as DiffUIR, both $k=3$ and UPS bring additional improvements, as shown in Tab. r2 in Reviewer (9bRU-W3). In terms of efficiency, due to the queued sampling strategy yielding neural function evaluations (NFEs) of $n+k−1$ (as proved in W2&Q3), the computational difference between solvers of different orders is determined solely by the order gap. Consequently, FLOPs increase approximately linearly with $k$. In conclusion, the solver complexity grows linearly with order, while performance varies depending on the target framework.\n\n|Order|UPS|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||NFEs|Forward FLops(G) per NFE|Backward FLOPs (G) per NFE|TotalFlops(G)|\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|||PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|||||\n|k=1|×|30.69|0.886|23.04|0.795|32.06|0.952|31.47|0.947|29.33|0.887|30.70|0.913|8|32.93|-|263.50||\n|k=2|×|31.31|0.894|23.08|0.798|32.51|0.955|31.62|0.948|30.07|0.897|31.20|0.919|9|32.93|-|296.44||\n|k=3|×|31.33|0.894|23.13|0.799|32.52|0.955|31.71|0.948|30.09|0.898|31.23|0.919|10|32.93|-|329.37||\n|k=1|√|31.10|0.892|23.07|0.798|32.32|0.954|31.54|0.947|29.80|0.895|31.01|0.918|8|32.93|≈65.86|≈790.32||\n|k=2|√|31.46|0.896|23.84|0.801|32.69|0.955|31.68|0.946|30.15|0.899|31.36|0.920|9|32.93|≈65.86|≈889.11||\n|k=3|√|31.46|0.896|23.84|0.801|32.69|0.955|31.68|0.946|30.15|0.899|31.36|0.920|10|32.93|≈65.86|≈987.91||\n\n*Table r5: FLOPs vs. PSNR for differnent $k$ with **queue-based sampling** strategy (Steps=8)*\n\n \n[1] Hobbhahn M, Sevilla J. What’s the backward-forward flop ratio for neural networks?[J]. Published online at epochai. org, 2021."}, {"review_id": "sQM3HwX9RQ", "reviewer": "Reviewer_Y9z3", "summary": "The paper proposes DGSolver, a training-free diffusion-based solver for image restoration that unifies a custom reverse-time ODE, high-order solvers, and universal posterior correction. It introduces a semi-linear forward process using a residual-based degradation model and derives tailored solvers to efficiently recover clean images.", "questions": "1.\tSection 3.3 appears heavily influenced by Diffusion Posterior Sampling (DPS), yet the original paper is not cited or discussed in this section. Can the authors clarify what is novel in their Universal Posterior Sampling (UPS) approach beyond the omission of an explicit measurement operator? Additionally, why is the original DPS work not acknowledged in the relevant discussion?\n2.\tThe results in Table 4 feel a bit unintuitive to me. It’s not clear why using exactly 8 steps leads to better performance than 9 or 10. I would normally expect performance to improve as the number of sampling steps increases. Is there a natural explanation for this behavior?\n3.\tCan the authors comment on sample variety?", "limitations": "Yes, the authors have addressed the limitations in the Appendix F.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n1. The use of high-order solvers tailored to the semi-linear structure of the reverse process introduces a novel and principled way to reduce the number of sampling steps in diffusion-based restoration.\n2. The solver design is clearly presented, with a queue-based implementation that enables efficient computation of higher-order derivatives without redundant evaluations.\n3. Evaluation is very thorough and shows improvement over comparison methods.\n4. Ablation studies are well-designed and support the impact of key components such as solver order and universal posterior correction (UPC).\n\n**Weaknesses:**\n1. The main ideas are valuable but somewhat difficult to follow due to dense presentation and unconventional notation. A more streamlined explanation with clearer separation of key components would improve accessibility.\n2. The forward process relies on the definition $I_{res}=I_{in}-I_0$, which implies additive degradations in aligned domains. While appropriate for denoising-like tasks, this formulation does not generalize to inverse problems such as super-resolution, MRI, or operator-unknown deblurring, where the degradation is either non-additive or defined in a different domain. Given that $I_{res}$ plays a central role in both the forward and reverse processes, this assumption limits the applicability of DGSolver to a narrow set of problems, despite its claim to universality.\n\nMinor comments:\n- The blue zoomed-in insets in Figure 5 appear inconsistent as the regions shown for the rainy input and ground truth differ from those in the restored results.\n- Line 94, “…computationally intractability...” -> computationally intractable\n- Line 111, “when” -> When", "was_revised": true, "final_justification": "The authors have satisfactorily addressed my concerns, and I find the paper to offer a meaningful and well-supported contribution that warrants acceptance.", "rebuttal": "### W1: Steamlined explanation of our key components \n\nWe thank the reviewer for pointing out the issue with presentation clarity. We will add a more streamlined method description for readability in Sec. 3. Simplified contents can be concluded as: _The overview of our DGSolver is illustrated in Fig. 2. In the forward process, diffusion generalist SDEs map degradations into a shared, degradation-agnostic latent space. In the reverse process, our DGSolver integrates diffusion generalist solvers and universal posterior sampling to jointly enhance inference accuracy and stability, with respect to red and blue trajectories in Fig. 2. The former component reduces cumulative error by solving a semi-linear ODE, thereby guiding the solution toward the ground truth; the latter component further optimizes the solutions through gradient guidance along the learned data manifold._\"\n\n### W2: Residual Formulation and Degradation \n\nFrom a degradation modeling perspective, residual-based and kernel-based approaches emphasize additive degradation restoration and inverse problem solving, respectively, with each carrying inherent limitations. However, we respectfully clarify that our use of residual modeling is not tied to a strict pixel-wise alignment assumption, but instead rooted in a signal decomposition perspective. From this broader view, residual decomposition offers a more general and widely applicable framework, extensively adopted across various image restoration tasks, such as super-resolution, deblurring, and denoising [1,2,3]. In contrast, kernel estimation often suffers from ill-posedness and increased complexity, making the decomposition less stable and kernel estimation harder to solve. In our DGSolver, residual modeling is seamlessly embedded into the diffusion generalist SDEs, enabling the unified design of both high-order solvers and the UPS mechanism. As a result, UPS becomes a plug-and-play component for diffusion-based methods that adopt residual modeling, whereas kernel-based approaches still depend on explicit kernel priors or estimation. That said, we acknowledge the limited capacity of residual modeling to generalize across all inverse problems. For instance, as shown in Tab. 2, UPS yields more substantial improvements in deraining (i.e., 0.15 dB (k=2), 0.12 dB (k=3)) than in deblurring (i.e., 0.07 dB (k=2), 0.06dB (k=3)). We believe that kernel-based modeling may perform well in specific inverse problems, but its applicability to broader restoration scenarios remains somewhat constrained, especially in the compound restoration tasks. \n\nIn a broad sense, our residual-based modeling exhibits the potential to be applied beyond the realm of image restoration. It can be seamlessly extended to other tasks wherein both prior and target data distributions are accessible, such as image translation.\n\n[1] Zhang Y, Li K, Li K, et al. Residual non-local attention networks for image restoration[J]. ICLR, 2019.\n\n[2] Liang J, Cao J, Sun G, et al. Swinir: Image restoration using swin transformer[C]. CVPR, 2021.\n\n[3] Cui Y, Ren W, Cao X, et al. Revitalizing convolutional network for image restoration[J]. TPAMI, 2024.\n\n\n### W3: Typos and figure errors\n\nWe appreciate your attention to detail regarding typos and figure inconsistencies. We will revise \"computationally intractability\" to “…computationally intractable” in line 94 and capitalize “when” to “When” in line 111. Besides, we will ensure that all zoomed-in regions are spatially aligned across input, ground truth, and restored images in Fig. 5. All these issues will be thoroughly checked and addressed in the revision to ensure clarity and accuracy.\n\n### Q1: Disccusion about UPS and DPS\n\nWe thank the reviewer for pointing this out. We sincerely apologize for overlooking the discussion and connection to DPS in Section 3.3. We fully acknowledge and appreciate that UPS is partially inspired by DPS, and in fact, we have dedicated Section 2.2 to introducing the theoretical foundations and related methods of DPS. We will explicitly include a discussion in Section 3.3. The simplified addition is as follows: “_DPS[1] circumvents the intractability of posterior sampling in diffusion models via a novel approximation, which is generally applicable to noisy inverse problems. Inspired by this, we propose universal posterior sampling from the perspective of residual modeling._”\n\nBoth UPS and DPS aim to incorporate gradient guidance into the diffusion sampling process. By omitting the measurement operator, our residual modeling is generalizable across compound degradations. Beyond omission of an explicit measurement operator, we believe UPS can be a plug-and-play component for diffusion models in field of image restoration. Given that the prior distribution (e.g. degraded images) and data distribution (e.g. high-quality images) are both available, we believe future variants of diffusion model will increasingly integrate residual prior into their formulation. In this context, UPS can be seamlessly coupled into their reverse process. Moreover, UPS can be seamlessly extended to other tasks wherein both prior and target data distributions are accessible, such as image translation. In contrast, kernel-based models are often limited by their formulation. In future work, we plan to explore the adaptability of UPS in other tasks.\n\n[1] Chung H, Kim J, Mccann M T, et al. Diffusion Posterior Sampling for General Noisy Inverse Problems[C]. ICLR, 2023.\n\n### Q2: Performance and sampling steps\n\nThanks for your insightful observation. We agree that the non-monotonic performance trend in Tab. 4 appears unintuitive at first glance. This counter-intuitive phenomenon may stem from two main factors. First, the stochasticity in both the forward and reverse processes, combined with network approximation errors, can cause slight performance fluctuations across neighboring sampling steps, where the variations are within an acceptable range. Second, since we project all degradations into a shared degradation-agnostic representation and train the model on mixed degradation types, the model implicitly learns to handle compound degradations, as discussed in Appendix E.2. Consequently, when the number of sampling steps increases, the model may begin to address secondary degradations present in the image. For example, some samples in the deraining or deblurring datasets, also exhibit low-light conditions. After removing the primary degradation, the model may perform low-light enhancements, where the available ground truth typically reflects only the removal of the primary degradation. This mismatch means that additional sampling steps do not necessarily bring the output closer to the reference, and may even result in a slight drop in evaluation metrics.\n\n### Q3: Sample variety \n\nThanks for raising the concerns about sample variety. In field of image restoration, the primary focus is on achieving high-fidelity and consistent restoration results, rather than promoting sample diversity. Our work is therefore motivated by fully leveraging diffusion models to recover a unique, high-quality target, as opposed to sampling from a distribution of plausible outputs. As a result, sample variety is expected to be low. However, we still consider that our sample variety may arise from two sources: (i) the randomness of initial states sampled from the Gaussian prior, and (ii) the model’s capacity to handle compound degradations. For (i), we conduct a thorough evaluation across 10 random seeds, reporting the average and variances of metrics in Tab. r6. The results indicate strong robustness, with highly consistent outcomes across seeds. Notably, the use of UPS further stabilizes the inference process and reduces variance. For (ii), we conduct generalization experiments on compound degradations in Appendix E.2. Since the initial states are degradation-agnostic representations, more sampling steps allow the model to progressively eliminate secondary degradations, resulting in diverse yet plausible restorations.\n\n|DGSolver(k=2)|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|UPS|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|\n|x|31.31 $\\pm$ 0.00118|0.894 $\\pm$ 0.000004|23.08 $\\pm$ 0.00180|0.798 $\\pm$ 0.00008|32.51 $\\pm$ 0.00068|0.955 $\\pm$ 0.00001|31.62 $\\pm$ 0.00655|0.948 $\\pm$ 0.000012|30.07 $\\pm$ 0.000291|0.897 $\\pm$ 0.000014|31.20 $\\pm$ 0.00118|0.919 $\\pm$ 0.000001|\n|√|31.46 $\\pm$ 0.00008|0.896 $\\pm$ 0.000002|23.84 $\\pm$ 0.00053|0.801 $\\pm$ 0.00002|32.69 $\\pm$ 0.00011|0.955 $\\pm$ 0.00001|31.68 $\\pm$ 0.00187|0.946 $\\pm$ 0.000002|30.15 $\\pm$ 0.000182|0.899 $\\pm$ 0.000002|31.36 $\\pm$ 0.000125|0.920 $\\pm$ 0.000001|\n\n*Table r6: Performance of our DGSolver with different seeds*"}, {"review_id": "nPKvGtORfj", "reviewer": "Reviewer_iqQ1", "summary": "This paper presents DGSolver, a novel framework designed to balance the commonality of degradation representations with restoration quality in diffusion models. DGSolver mitigates accumulated discretization errors through tailored high-order solvers with a universal posterior sampling strategy, and further enhances sampling efficiency by introducing a queue-based accelerated sampling strategy.", "questions": "1.The Universal Posterior Sampling (UPS) strategy demonstrates clear performance benefits in this work. Could UPS also be applied to other existing methods?\n\n2.Is the queue-based sampling strategy affecting model performance while improving efficiency?\n\n3.Does increasing the order of the high-order solvers influence the efficiency of inference?", "limitations": "The limitations have been discussed in the supplementary materials.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper provides a thorough and rigorous derivation of the proposed method, demonstrating a solid theoretical foundation.\n\n2. The proposed high-order solvers and universal posterior sampling strategy are intuitively illustrated in the figure, effectively clarifying their fundamental principle.\n\n3. Experimental results demonstrate the effectiveness of the proposed method from qualitative, quantitative perspectives.\n\nWeaknesses:\n\n1.The ablation study should quantitatively compare model parameters, sampling steps, and inference time with baseline DiffUIR[1] to better demonstrate the superiority of the proposed method.\n\n2.Lack of quantitative comparison with the latest state-of-the-art methods proposed in 2025. \n\n3.Table 4 shows that increasing the number of sampling steps beyond a certain point leads to a decline in PSNR/SSIM. Could the authors provide insight into this counterintuitive result?\n\n4.Why were the experimental settings (datasets and training parameters) from the baseline DiffUIR not adopted?\n\n5.The motivation for designing the forward process as a stochastic differential equation (SDE) is unclear. Its practical significance remains questionable, as no explicit analysis or justification is provided in the experiments.\n\n6.The expression “total M+1 time steps” in line 146 is inconsistent with the notation presented in Figure 2.\n\n7.In Figure 5, the position of blue boxes is inconsistent.\n\n8.In Tables 2, 3, and 4, the best results should be highlighted in bold.", "was_revised": true, "final_justification": "I will retain my original recommendation.", "rebuttal": "### W1: Ablation and comparison with DiffUIR\n\n\nWe sincerely appreciate this constructive suggestion. We perform a detailed efficiency comparison with baseline DiffUIR, as presented in Tab. r7. Both methods maintain identical model parameters (7.73M), confirming our enhancements require no additional network capacity. Our method outperforms the baseline in terms of performance, but the overall time consumption increases. For a fair comparison, we apply our solver to DiffUIR for inference process, and the results show a significant improvement in restoration quality, as presented in Tab. r2 in Reviewer (9bRU-W3). This verifies our solvers can be integrated into related methods to enhance restoration accuracy without retraining. For detailed efficiency comparison, please refer to Tab. r1 in Reviewer (9bRU-W1).\n\n\n|Metric|DiffUIR|Ours (k=1)|Ours (k=1+UPS)|Ours (k=2)|Ours (k=2+UPS)|Ours (k=3)|Ours (k=3+UPS)|\n|-|-|-|-|-|-|-|-|\n|Param.(M)|7.73|7.73|7.73|7.73|7.73|7.73|7.73|7.73|\n|Steps|3|8|8|8|8|8|8|8|8|\n|Time (s)|0.206|0.529|1.144| 0.598 |1.273|0.657|1.470|\n|Average PSNR (dB)|29.93|30.70|31.01|31.20|31.36|31.23|31.36|\n|Average SSIM |0.907|0.913|0.918|0.919|0.920|0.919|0.920|\n\n*Table r7: Ablation and comparison with DiffUIR*\n\n\n### W2: Comparisons with 2025 SOTAs\n\nFor a fair comparison, we reimplemented and retrained several representative universal image restoration models, namely AwRaCLe[1], MaIR[2], and DeepSNNet[3], on our collected dataset using their released code and official settings. The quantitative results are presented in Tab. r8. Evidently, our method outperforms others. Thanks for your constructive comments, and we will include these results in revision.\n\n|Method|Year|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||Complexity||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|||PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|Params(M)|FLOPs(G)|\n|DA-CLIP|2024|28.63|0.854|19.50|0.730|28.23|0.934|27.26|0.941|26.47|0.818|27.54|0.881|32.96|158.14|\n|DiffUIR|2024|30.67|0.887|21.21|0.769|30.70|0.943|30.29|0.944|29.00|0.877|29.93|0.907|7.73|32.93|\n|AwRaCLe[1]|2025|29.42|0.868|22.22|0.773|28.90|0.944|24.60|0.883|27.54|0.846|27.94|0.888|94.18|165.42|\n|MaIR[2]|2025|29.17|0.859|20.09|0.726|28.85|0.931|26.79|0.936|26.74|0.812|27.91|0.880|20.71|110.44|\n|DeepSNNet[3]|2025|28.97|0.847|17.84|0.659|30.17|0.927|28.00|0.939|25.80|0.768|28.20|0.865|17.31|71.79|\n|Ours|-|31.46|0.896|23.84|0.801|32.69|0.955|31.68|0.946|30.15|0.899|31.36|0.920|7.73|32.93|\n\n*Table r8: Quantitative comparisons with latest methods*\n\n\n[1]Rajagopalan S, Patel V M. AWRaCLe: All-weather image restoration using visual in-context learning[C]. AAAI, 2025.\n\n\n[2]Li B, Zhao H, Wang W, et al. Mair: A locality-and continuity-preserving mamba for image restoration[C]. CVPR, 2025.\n\n\n[3]Deng X, Zhang C, Jiang L, et al. DeepSN-Net: Deep semi-smooth Newton driven network for blind image restoration[J]. TPAMI, 2025.\n\n\n### W3: Counter-intuitive phenomenon between performance and sampling steps \n\nThanks for your insightful observation. We agree that the non-monotonic performance trend in Tab. 4 appears unintuitive at first glance. This counter-intuitive phenomenon may stem from two main factors. (1) Our approach to image restoration is motivated by the principle of commonality, aiming to handle diverse degradation types within a unified framework. In Appendix E.2, we discuss the model's behavior under compound degradation scenarios. It can be observed that across multiple sampling steps, the model tends to first remove the primary degradation before addressing secondary ones. Consequently, in cases where samples from the deraining or deblurring datasets also suffer from additional degradations (e.g., low-light conditions), the restored output may diverge from the available reference that typically reflects only the removal of the primary degradation, leading to reduced evaluation metrics despite better perceptual quality. (2) The stochasticity in both the forward and reverse processes, combined with network estimated errors, can cause slight influence on performance within an acceptable range. \n\n### W4: Datasets setting\n\nOur task setting is closely aligned with that of DiffUIR. However, since each method adopts different dataset configurations for image restoration, we attempt to collect and standardize all the datasets used by these methods for fairness, enabling direct and meaningful comparisons. Besides, we strictly follow their open-source code and retrain all of them except autodir.\n\n### W5: SDEs motivation\n\nThe motivation behind adopting forward SDEs is that SDEs offer a broader perspective for analyzing the transition of probability distributions in a continuous manner. As discussed in Appendices A and B, both the perturbed process and deterministic implicit sampling can be viewed as first-order instances of the reverse-time SDE and ODE, respectively. Therefore, defining the diffusion process from the perspective of probability distribution in Eq. (4) is fundamentally equivalent to describing it through continuous SDEs in Eq. (5). Their performance comparisons can be found in Tab. 2, where the results for $k=1$ without UPS correspond to deterministic implicit sampling, while our high-order solvers demonstrate improved performance. Besides, SDEs formulation offers higher scalability, enabling the integration of our high-order solver and UPS, which are tightly coupled with the dynamics of the SDEs. \n\n### W6,7,8: Figure and table errors\n\nThank you for your careful observations regarding inconsistencies in notation, figures, and formatting. We will correct the inconsistency in the description of Figure 2, as well as the misaligned highlighted regions in Figure 5. In addition, Tables 2, 3, and 4 will be revised to include clearly marked highlights for improved readability. We will thoroughly check and address all similar issues in the revision for clarity and accuracy.\n\n### Q1: Generalization on other methods\n\nThank you for the thoughtful question. Within the domain of image restoration, UPS can indeed be generalized to related frameworks such as DiffUIR. We apply UPS to DiffUIR, which yields performance gains, as shown in Tab. r2 in Reviewer (9bRU-W3). Broadly speaking, UPS depends on residual information from given distribution to provide gradient guidance for diffusion reverse process. It can be seamlessly extended to other tasks wherein both prior and target data distributions are accessible, such as image translation. Therefore, we believe UPS exhibits strong scalability and generalizability. We leave a more comprehensive exploration of UPS in other vision tasks as promising future work.\n\n### Q2: Ablating sampling strategies on performance\n\nIn Appendix B.2, we theoretically show that cumulative error is positively correlated with the sampling interval $\\Delta t$ . For a k-order solver, naive sampling from time $s$ to $t$ requires interpolating $k−1$ intermediate points, whereas queue-based sampling leverages precomputed points cached in a queue. As a result, the effective sampling interval in the queue-based strategy is larger than that of naive sampling, which may lead to higher theoretical error. To validate this, we conduct experiments, and the results are shown in Tab. r9. Performance of queue-based sampling is slightly lower than naive sampling. However, queue-based approach significantly reduces neural function evaluations (NFEs), achieving 2–3× higher efficiency than naive sampling (please refer to Tab. r1 in Reviewer (9bRU-W1) ). \n\n\n|Sampling|Order|UPS|Deraining||Enhancement||Desnowing||Dehazing||Deblurring||Average||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|-|-|-|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|PSNR|SSIM|\n|Naive|k=2|×|31.33|0.894|23.10|0.799|32.52|0.956|31.64|0.948|30.09|0.898|31.22|0.919|\n|Queue|k=2|×|31.31|0.894|23.08|0.798|32.51|0.955|31.62|0.948|30.07|0.897|31.20|0.919|\n|Naive|k=2|√|31.47|0.897|23.85|0.802|32.70|0.955|31.69|0.946|30.17|0.899|31.37|0.920|\n|Queue|k=2|√|31.46|0.896|23.84|0.801|32.69|0.955|31.68|0.946|30.15|0.899|31.36|0.920|\n|Naive|k=3|×|31.35|0.895|23.12|0.800|32.55|0.956|31.65|0.948|30.11|0.898|31.24|0.920|\n|Queue|k=3|×|31.33|0.894|23.13|0.799|32.52|0.955|31.71|0.948|30.09|0.898|31.23|0.919|\n|Naive|k=3|√|31.50|0.897|23.87|0.803|32.72|0.955|31.70|0.946|30.19|0.900|31.39|0.920|\n|Queue|k=3|√|31.45|0.896|23.84|0.801|32.69|0.955|31.67|0.946|30.15|0.899|31.36|0.920|\n\n*Table r9: Performance for differnent $k$ with different strategies*\n \n\n### Q3: Solver order and inference efficiency\n\nWe use neural function evaluations (NFEs) to measure inference efficiency. Let $k$ denote the solver order and $n$ be sampling steps. For a $k$-order solver, naive sampling from time $s$ to $t$ requires interpolating $k-1$ intermediate points within the interval, resulting in $k$ NFEs per step. Consequently, the overall computational complexity for $n$ steps is $O(nk)$. In contrast, the queue-based sampling strategy precomputes the values at $k−1$ time points and caches them for reuse in subsequent steps, offering high efficiency by reducing total complexity to $O(k−1+n)$. Therefore, Both naive solvers and queue-based solvers have the same time complexity $O(k)$ only when $n=1$. When $n > 1$, the naive solvers are increasingly less efficient than queue-based solvers because $nk \\ge k-1+n$. Under the queue-based sampling strategy, the difference in NFEs among different solvers equals the difference in their orders. As a result, their efficiency gap scales linearly. For quantitative results, please refer to Tab. r1 in Reviewer (9bRU-W1) and Tab. r3 in Reviewer (1zc3-W2&Q3)."}, {"review_id": "fMXdh3oPmr", "reviewer": "Reviewer_92Km", "summary": "The paper proposes a training-free solver for diffusion generalist models, termed as DGSolver. First, it reformulates the diffusion process as ODEs and exploits the semi-linear integral structure to better approximate the solutions. This avoids the computational overhead of training a series of networks for quality refinement. Second, it integrates the estimated residual term I_{res} into diffusion posterior sampling to construct the universal sampling mechanism, which guides the sampling process universally without the prior knowledge of degradation formulation. These two components enhance the restoration accuracy and efficiency across a variety of degradation tasks. Overall, the proposed method demonstrates notable theoretical novelty, practical applicability, and broad adaptability.", "questions": "1.How do you choose the sampling steps or variable $t$ in the reverse process? What are the relationships between time step $t$ and predefined parameters $\\alpha(t)$, $\\beta(t)$, and $\\gamma(t)$ in Eq.(4)? I think it is important for readers to understand the paper and implementation.\n\n2.Please provide the computational complexity analysis before and after applying the queue-based sampling strategy.\n\n3.When dealing with compound degradation, the model can restore degraded images in a specific order, which seems to be determined spontaneously by the unified model. Could the authors share some insights on achieving controllable restoration in the given order or handling all degradation types in one execution for compound degraded images based on the proposed method?", "limitations": "Yes, the authors have discussed the related limitations.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1.This paper presents a rigorously derived and technically sound ODE integrator with accelerated sampling strategy for accurate sampling from diffusion models to achieve quality refinement, which can be widely exploited and benefited by other related methods.\n\n2.The proposed UPS is novel and clever, and does not require the prior knowledge or estimation of degradation formulation. It can cope with various types of degradation without complicated designs with high potential.\n\n3.All the framework is well motivated and the explanations are clear. The authors effectively provide a theoretical derivation and error analysis of the proposed method.\n\n4.Comprehensive experiments are carried out on multiple datasets, various tasks, and different domains, fully demonstrating the proposed method’s superior performance over state-of-the-art methods both quantitatively and qualitatively.\n\n**Weaknesses**\n\n1.The expressions for $\\alpha(t)$, $\\beta(t)$, and $\\gamma(t)$ in Eq.(4) lack clarity.\n\n2.Some of the notations in the appendix lack definitions and some typos need to be corrected (e.g., $I_\\text{res}^{\\theta}$ is misspelled in Appendix F). It would be worthwhile to spend some effort to make that better.", "was_revised": true, "final_justification": "After further discussion with the author, I still maintain my acceptance decision.", "rebuttal": "### W1&Q1: Experssion and selection of noise schedule\n\nFollowing the linear noise schedule, the formulations of $\\bar{\\alpha}_t,\\bar{\\beta}_t,\\bar{\\delta}_t$ are:\n\n$\\beta_t = \\beta_{min} + (\\beta_{max} - \\beta_{min})\\frac{t}{T},\\beta_{min} = 0.0001,\\beta_{max} = 0.02$,\n\n$\\delta_t = \\delta_{min} + (\\delta_{max} - \\delta_{min})\\frac{t}{T},\\delta_{min} = 1\\times 10^{-6},\\delta_{max}=0.002$\n\n$\\bar{\\beta}_t = \\sum_0^t \\beta_i$, $\\bar{\\alpha}_t = 1 - \\sqrt{\\prod_0^t (1 - \\beta_i)}$, $\\bar{\\delta}_t = \\sum_0^t \\delta_i$\n\nNotably, the total timesteps $T=1000, t\\in\\mathcal{Z}, 0\\le t < T$. Since $\\beta_t$ and $\\delta_t$ are non-negative within the range of [0,1], three variables $\\bar{\\beta}_t,\\bar{\\alpha}_t,\\bar{\\delta}_t$ exhibit strict monotonic increase.\n\n### W2: Typos \n\nWe sincerely appreciate your meticulous review, and we will correct the misspelled $I_{res}^{\\theta}$ in revision. We promise to revise other typos for readability.\n\n### Q2: Complexity of naive and queue-based sampling strategies\n\nWe utilize neural function evaluations (NFEs) to evaluate the computational complexity. Let $k$ denote the solver order and $n$ be sampling steps. For a $k$-order solver, naive sampling from time $s$ to $t$ requires interpolating $k-1$ intermediate points within the interval, resulting in $k$ NFEs per sampling step. Consequently, the overall computational complexity for $n$ steps is $O(nk)$. In contrast, the queue-based sampling strategy precomputes the values at (k−1) intermediate time points and caches them for reuse in subsequent steps. This reduces the total complexity to $O(k−1+n)$, offering a clear computational advantage for multi-step sampling. For quantitative results, please refer to Tab. r1 in Reviewer (9bRU-W1) and Tab. r3 in Reviewer (1zc3-W2&Q3). \n\n\n### Q3: Insights about controllable restoration.\n\nOur core idea is to map different degradations into a shared representation space, and then leverage high-order solvers to enhance reconstruction accuracy, the queue-based sampling to improve efficiency, and the UPS mechanism to provide effective guidance for restoration. Hence, our model handles compound degradation through a unified framework, but the restoration order is indeed implicitly learned during training rather than explicitly controlled. Below we clarify some insights and potential extensions: (i) Add task-specific prompts as controllable factors to condition the solver. For example, we can mine degradation priors $c$ either from the image itself or from foundation models, and train model $I_{res}^{\\theta}(I_t,I_{in},t,c)$ using paired data with restoration order. With learnable parameters, image restoration becomes controllable. (ii) Project the gradient guidance $\\nabla \\log p(I_{in}|I_t)$ onto orthogonal subspaces for degradation-specific restoration. For instance, we assign orthogonal bases $ v_i, i=1,..,k $ for $k$ degradation types during training, and utilize the UPS term to project gradients $\\sum_i^k \\theta_i P_{v_i}\\nabla \\log p(I_{in}|I_t)$, where $\\theta_i$ is a switch for activating the $i$-th degradation subspace and $P_{v_i}$ is a projection operator. This facilitates controllable and interpretable restoration. We sincerely appreciate your insightful comments, and we will conduct further research in our future work."}]

Author rebuttal:

Meta-review: The paper improves diffusion-based image restoration by proposing a training-free solver for diffusion models, treating them as ODE. They demonstrate enhanced restoration quality and efficiency across a wide range of degradations and benchmarks and compared to several existing baselines. 

Five expert reviewers refereed the work and found the approach sound, well-motivated, and the empirical evidence generally convincing. On the other hand, the main concerns were regarding ablation studies, some similar prior work, experimental setup, clarity, and computational complexity. 

The authors provided a thorough rebuttal which addressed all the main concerns and led to all reviewers eventually suggesting acceptance. 

The AC does not see any major flaw with the paper and on the other hand finds the contribution novel with significant empirical evidence. Therefore, the AC suggests acceptance.
Decision: Accept (poster)