Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models
Authors: Jinhui HOU
Zhiyu Zhu
Junhui Hou
Venue: ICLR 2026 Poster (2026)
Keywords: image restoration
bridge diffusion
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Diffusion bridge models have shown great promise in image restoration by explicitly connecting clean and degraded image distributions. However, they often rely on complex and high-cost trajectories, which limit both sampling efficiency and final restoration quality. To address this, we propose an Energy-oriented diffusion Bridge (E-Bridge) framework to approximate a set of low-cost manifold geodesic trajectories to boost the performance of the proposed method. We achieve this by designing a novel bridge process that evolves over a shorter time horizon and makes the reverse process start from an entropy-regularized point that mixes the degraded image and Gaussian noise, which theoretically reduces the required trajectory energy. To solve this process efficiently, we draw inspiration from consistency models to learn a single-step mapping function, optimized via a continuous-time consistency objective tailored for our trajectory, so as to analytically map any state on the trajectory to the target image. Notably, the trajectory length in our framework becomes a tunable task-adaptive knob, allowing the model to adaptively balance information preservation against generative power for tasks of varying degradation, such as denoising versus super-resolution. Extensive experiments demonstrate that our E-Bridge achieves state-of-the-art performance across various image restoration tasks while enabling high-quality recovery with a single or fewer sampling steps. Our project page is https://jinnh.github.io/E-Bridge/.

Published as a conference paper at ICLR 2026
ENERGY-ORIENTED DIFFUSION BRIDGE FOR IM- AGE RESTORATION WITH FOUNDATIONAL DIFFUSION MODELS
Jinhui Hou', Zhiyu Zhu and Junhui Hou
"Department of Computer Science, City University of Hong Kong;
2 DDeaarmeent of Computer Science. City University of Hong Kong (Dongguan). jhhou3-cemp ci tyu.edu........... hk, zhiyu. zhuecityu-dg, edu. en, jh. www.houucitye www.eu...uuuueeuu.ooo edu.hk
ABSTRACT
Diffusion bridge models have shown great promise in image restoration by ex- plicitly connecting clean and degraded image distributions. However, they often rely complex and high-cost trajectories, which limit both sampling efficiency and final restoration quality. To address this, we propose an Energy-orientel diffusion Bridge (E-Bridge) framework to approximate set of low-cost mani- fold geodesic trajectories to boost the performance of the proposed method. We achieve this by designing novel bridge process that evolves over shorter time horizon and makes the reverse process start from an entropy--egullrized point that mixes the degraded image and Gaussian noise, which theoretically reduces the required trajectory energy. To solve this process efficiently, we draw inspi- ration from consistency models Iearn a single-step mapping function, opti- mized via continuous-time consistency objective tailored for our trajectory, so as to analytically map any state on the trajectory to the target image. Notably, the trajectory length in our framework becomes tunable task-adaptive knob, allowing the model to adaptively balance information preservation against gen- erative power for tasks of varying degradation, such denoising versus super- resolution. Extensive experiments demonstrate that our E-Bridge achieves state- of-the-art performance across various image restoration tasks while enabling high- quality recovery with a single or fewer sampling steps. Our project page is https: / //ti//jjnnn////////// gii.i.i.ttttii.... github. to/ lo/E-Bridge/.
INTRODUCTION
Diffusion models (Ho et al., 2020; Song et al., 2021) have established a new state-of-the-ee art in generative modeling, with a profound impact on image restoration (IR). Two dominant paradigms have emerged. The first involves guiding a standard generative process, which starts from pure Gaussian noise, using the degraded image as a condition (Dhariwal & Nichol, 2021: Saharia et al., 2022). Although versatile, this approach suffers from slow inference speeds, it must traverse high-energy trajectory across vast entropy gap from a noise distribution to the image manifold.
To create
direct transformation, bridge diffusion models (Li et al., 2023; Liu et al., 2023; Yue et al., 2024; Zhu et al., 2025a; Wang et al., 2025b) have gained traction by constructing a stochas- tic process from the degraded image distribution to the clean one. By starting closer to the target, these models significantly reduce the number of sampling steps. However, the diffusion trajecto- ries in these models are often predefined and sub-optimal, which is typically fixed polynomial interpolation. Such trajectories may not represent the most efficient path, or geodesic, on the com- plex data manifold. They often enforce a redundant "re-noising" phase before denoising, leading to unnecessarily high trajectory energy and limiting both restoration quality and computational effi ciency. Although recent work has focused on leveraging powerful pretrained generative priors, they often rely on complex "stitching" mechanisms (Wang et al., 2025a) to connect two distinct diffusion processes, which can introduce inconsistencies.
"Corresponding author.

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Published as a conference paper at ICLR 2026
Instead of patching existing models, we argue that the optimal restoration trajectory should be low- energy geodesic on the data manifold. To this end, we propose an Energy-or -rieened diffusion Bridge (E-Bridge) framework that fundamentally re-engineers the restoration process. We first construct a more efficient, low-energy trajectory by designing a novel bridge process that evolves over a shorter, more controllable time horizon. This process starts the reverse (restoration) trajectory from an regulyy-reuularized point, which is a mixture of the degraded image and Gaussian noise, thus theoretically reducing the required control energy and bypassing the inefficient re-noising phase.
To ensure this low-energy trajectory approximates a geodesic, we innovatively leverage a pretrained denoiser not as a separate component, but as an integrated dynamic geodesic guidance field, con- tinuously pulling the trajectory toward the natural image manifold. To traverse this trajectory with maximum efficiency, we draw inspiration from consistency models (Song et al., 2023; Lu & Song, 2025). We learn single-step mapping function, optimized via continuous-time consistency ob- jective tailored for our trajectory, which can analytically map any state on the trajectory directly to the final restored image. This eliminates the need for slow, iterative sampling. Notably, the trajec- tory length in our framework becomes a tunable, task-adaptive knob. This allows our E-Bridge to adaptively balance information preservation (for mild degradations like denoising) against genera- tive power (for severe degradations like super--esolution). Extensive experiments demonstrate that E--Bridge achieves state-of-the-art performance across a variety of image restoration tasks, while enabling high-quality recovery in a single or very few sampling steps.
In summary, the main contributions of this work are:
we propose a novel energy-oriented diffusion bridge framework with low-energy data manifold geodesic trajectories and adaptation to current large pretrained models, and give corresponding theoretical analysis compared with state-of-the-art methods;
we give closed-form consistency solver that circumvents the need for numerical iteration, abling direct and single-step inference along the geodesic trajectory.
RELATED WORK
Image Restoration via Bridge Models. Recent advances in diffusion-based restoration have moved beyond simple conditioning towards bridge models (Bortoli et al., 2021; Luo et al., 2023; Zhou al., 2024; Shi al.. 2023) that define a stochastic trajectory from degraded to clean images. Early works formulated this trajectory a standard Brownian Bridge (BBDM) (Li et al., 2023; Liu et al., 2023) or as an optimal transport problem solved via Schrödinger Bridges (SB) (Wang et al., 2025b). Frameworks like UniDB (Zhu et al., 2025a) further unify these concepts. However, common thread unites these models: they learn a score or drift function that requires a slow, iterative sampling process for inference. To overcome the inference bottleneck, UniDB++ (Pan et al., 2025) proposes a post-hoc acceleration, where a multi-step model is compressed into a single-step one. The Implicit 12SB (I3SB) (Wang et al., 2025b) introduces mor-Mvakovian sampling scheme for pre-trained I2SB models.
Image Restoration using Conditional Diffusion Models. Another major category of methods con- ditions a standard generative process that begins from pure noise. Early works (Saharia et al., 2022; Lugmayr al., 2022; Hou et al., 2023; Zhang al., 2023) train conditional model from scratch, generating a high-quality image guided by the degraded input (Saharia et al., 2022). A different strategy involves guiding pre-trained unconditional model during the inference stage. Notable ex- amples include DDRM (Kawar et al., 2022), which leverages spectral projections to solve various linear inverse problems, and DifiPIR (Zhu et al., 2023), which employs Plug-and-Play optimiza- tion to incorporate degradation-speciff priors. Recently, the focus has shifted towards efficiently adapting large-scale pre-trained models. This is typically accomplished by fine-tuning task see--specffic modules, such as the time-aware encoder in StableSR (Wang et al., 2024) or the multi-task, prompt- based adapter in UniRestore (Chen al., 2025).
PRELIMINARIES
The task of image restoration be conceptualized transporting degraded image Y, residing a low-quality manifold My, to its corresponding clean counterpart X, on a high-quality manifold Mr. Existing diffusion models approach this via two primary ways:
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Guided Generative Trajectories. These methods (Dhariwal & Nichol, 2021; Saharia et al., 2022; Zhu et al., 2025b) guide a standard reverse diffusion process, which starts from pure Gaussian noise (XT ~ NO,, towards the clean image X, using the degraded image Y as a condition. From a Stochastic Optimal Control (SOC) (Kappen, 2008; Berner et al., 2022; Park et al., 2024; Zhu al., 2025a) perspective, the efficiency of this trajectory is measured by its energy, a cost functional representing the total transportation between distributions:
T J(u =E 0 2
|(A())1 li((t) Elbrotal (t. Xt)ll dt,
(A)
(R)
where j(t) denotes the instantancous velocity vector of the mean curve, while [[btotai] represents the effective drift field provided by the network. The term (A) defines the transportation energy, quantifying the kinetic cost associated with the velocity profile along the trajectory of jit).. In con- trast, term (B) corresponds to the control energy, measuring the cost required to steer the dynamics along the path, which can also be parameterized y| YI?.. Consequently, the optimal scenario is to achieve transport via a low-energy geodesic trajectory that requires minimal control intervention.
Advanced methods like IRBridge (Wang et al., 2025a) attempt to mitigate this by leveraging power- ful pre-trained generative models. However, this introduces its own complexities, such as the state distribution mismatch between the pre-trained model's process and the desired restoration process. To bridge this gap, IRBridge proposes a transition equation that mathematically maps state X² from the restorative trajectory to a state X; on the generative trajectory, using an estimate of the
clean image Xo:
(2) However, this approach involves the complexity of coupling two distinct and separately defined processes, creating an indirect and potentially fragile connection between the degraded and clean image domains.
= X + 3 X, +
Direct Bridge trajectories. These approaches (Liu et al., 2023; Zhou et al., 2024; Yue et al., 2024) construct a more direct stochastic process from Y to Xo- The simplest form is the standard Brownian Bridge, defined by the stochastic differential equation (SDE) that pins Wiener process to start at Xo and end at Y. In practice, rather than directly solving this SDE, models like 12SB (Liu et al., 2023) are built upon analytical posterior distribution p(x, Y). For example, I2SB defines the state X, as:
X, N' (a,Xo (1- Yoot)
where the variance of is often symmetric, arch-shaped function (e.g., of x f(1 - f)). The sym- metric noise schedule generally forces the reverse process to undergo a "re-noising" phase before denoising. For an asymmetric task whose goal is unidirectional denoising and restoration, this initial noise addition process is redundant in terms of energy, increasing the random process energy of the path. In pursuit of a theoretically optimal path, other works explore Schrödinger Bridges, which reframe the problem finding a trajectory measure P that minimizes the Kullback-Leibler (KL) divergence from a reference Wiener process measure W, subject to the marginal constraints:
min KL(P|(W) s.t. Po = p(Xo). Pr = p(Y).
(4)
Although the formulation in Eq. (4) aims to find the lowest-energy path, its standard solver, the Iterative Proportional Fitting (IPF) algorithm, is computationally prohibitive and often unstable for high-dimensional data, rendering it impractical. A more fundamental limitation, common to all aforementioned paradigms, their reliance iterative, multi-step sampling to approximate the solution of the underlying differential equation. This process, Xt-At R X v(x,, t)At, has a computational complexity of O(N C), where N is the number of sampling steps. This inherent sequential dependency fundamentally restricts inference speed and prevents real-time applications.
4 PROPOSED METHOD
As previously analyzed, existing diffusion-baasee image restoration bridges are often encumbered by complex, computationally expensive, iterative sampling along high-energy restoration trajectories. To overcome these fundamental limitations, we introduce the Energy-orientes diffusion Bridge (E- Bridge), novel framework that re-engineers the restoration process for unparalleled efficiency and
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X
X
(a) Standard Diffusion
(b) Diffusion Bridge
(c) Ours
Figure 1: Illustration of diffusion processes for image restoration. (a) Standard Diffusion Models: These traverse long, high-energy trajectory starting from pure Gaussian noise the clean image manifold, conditioned on the degraded image. (b) Conventional Bridge Models: These construct a path from the degraded to the clean image but often follow a sub-optimal, high-energy trajectory that includes a redundant "re-noising" phase before denoising.
(c) Our E- E-Bridge: It starts the reverse process from an entropy-regularizee point, which a mixture the degraded image and noise, thus bypassing the inefficient re-noising phase and creating more direct and shorter path for restoration.
fidelity. Our approach is built on dual-pronged strategy. To be specific, first, we introduce the Energy-oriented Diffusion Bridge (E-Bridge), a novel, low-energy, data manifold geodesic trajec- tory. This trajectory is theoretically more efficient and better suited to the unidirectional nature of image restoration, avoiding the unnecessary re- re-noising phases of previous bridge models (Sec. 4.1). Second, we derive the E-Bridge-Solvent
principled, single-step mapping function obtained through the analytical inversion of our trajectory equation. Crucially, based on the E-Bridge-Solver we train E-Bridge by enforcing a continuous-time consistency optimization objective to further boost its performance. This ensures the solver produces stable, high-fidelity output from any point the trajectory, effectively learning the underlying geometry of the data manifold (Sec. 4.2).
4.1 ENERGY-ORIENTED DIFFUSION BRIDGE
We begin by framing image restoration through the lens of data geometry, positing that the task fundamentally a transport problem between distinct data manifolds. Within this framework, the ideal restoration trajectory between clean image Xo and its degraded version Y should be the geodesic (t) on the underlying data manifold, representing the most efficient transformation. To regularize the diffusion trajectory following the bridge trajectory, we have formulated the following Proposition 4.1.
Proposition 4.1 (Kinetic Energy Minimization of the Geodesic Expectation). Let X₃ be a stochastic diffusion process defined the controllable time horizon t [o) If the process follows the manifold geodesic (1), defined as the trajectory that minimizes the kinetic transport energy between the starting distribution Y and the target distribution X, then the deterministic component (expectation) of the resulting trajectory, p(t) E(),), must satisfy the following linear transport equation:
(14 To
Xo + (1))
To 1 Y.
E[X.]
(5)
Proof. Please refer to Appendix B.1.
Based on that, we formulate novel diffusion bridge defined for controllable time horizon t [0, To] with To E (0, 1]. which is formulated as:
x, (1-(1-(2)) X
L To
X, X, + B,XT.
where
Y
Diffusion Trajectory
Inherent Data Geodesic Transition
However, a2 and B, still under-defined. Given that leverage the prior of the pretrained rectified flow neural network (Andreas et al., 2024) to boost the reconstruction performance, we further utilize this regularization to explore the optimal a and 3, selection in Proposition 4.2.
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Proposition 4.2 (Minimal Adaptation Energy via Lipschitz Stability). Let co be pretrained Rectified Flow network with Lipschitz constant K, originally trained on the trajectory Z, (1 - - + tZ, When fine-tuning this network on a new bridge process X, = a,X; + B,XT, the Adaptation Energy, defined the expected squared deviation in the network's output caused by trajectory mismatch, is minimized if and only if
04 = 1 - 1, a.d B1 = L.
Proof. Please refer to Appendix B.2.
These selections a; and Br ensure the process matches the linear hignal-to-noiss evolution inherent to Rectified Flow models. In the following selection, we introduce the consistency-based tuning to further boost the effectiveness and efficiency of the reconstruction algorithm.
4.2 CONSISTENCY E-BRIDGE SOLVER
Although the E-Bridge defined in Sec. 4.1 provides a theoretically efficient trajectory blueprint, it presents two subsequent challenges: first, how to traverse this trajectory without resorting slow, iterative ODE solvers, and second, how to ensure this traversal remains faithful to the complex, underlying data manifold. which is not an iterative sampler but direct single-step mapping function. Its design is a synthesis of two core principles: a principled architecture derived from the analytical inversion of the E-Bridge trajectory equation and powerful training objective based on continuous-time geodesic consistency.
address these issues, we introduce a consistency E-BBrigge solver,
Solver Derivation via Hanifold-Guided Trajectory Inversion. A key theoretical insight of our work is that the endpoint X, of the E-Bridge trajectory can be solved analytically. This requires reframing the problem from simulating stochastic trajectory to solving a deterministic equation. The challenge lies in the fact that the noise term Xy is unknown. We resolve this by positing that for any point X, on the trajectory, a pretrained flow-matching network 60 can provide an instantaneous estimate of the vector pointing to the terminal noise:
EB (X,) Xx Xo.
(8)
For any state X, residing in the ambient space (potentially off the data manifold), the denoiser co(-) allows us to define an instantaneous projection onto the learned manifold of clean images. This projection, denoted X,(X), represents the denoiser's estimate of the clean image corresponding X,, Consequently, the vector X, (X.) X, forms correction field, providing the precise direc- tion needed to steer the state X, back towards the manifold. It is this ability to provide geodesic transition that makes ca(-) the ideal tool for our trajectory inversion.
By substituting the relation from Eq. (8) into our E-Bridge trajectory definition in Eq. (6), we lever- age this geometric correction to transform the trajectory into single implicit algebraic equation where X, is the sole unknown:
X, = (1-1) 1) (1-(()) X +
Y + 1(E0 (X.) + Xo).
To
The linearity of this equation with respect to X, permits closed-form solution via algebraic inver- sion. This yields the principled architecture of our E-Bridge solver, (.):: Fo
X. - A(t)Y - B(t)co(X,)
X = FX, Y,t)
(10)
C(t)
are derived directly from the algebraic manipulation of Eq. (9). This formulation reveals the true power of our E- Bridge model: it is not merely an interpolator, but a direct solver for the endpoint a geometrically motivated trajectory.
where the coefficients A(t) = (1- To -1)=
Training as Energy-oriente Geodesic Consistency Enforcement. The training objective is en- dowed with clear geometric meaning. We are not just forcing a network to be self-consistent; are enforcing mergy--rreente geodesic consistency. This principle asserts that for any point on a given geodesic trajectory, perfect solver must map it to the same, invariant endpoint Xo.
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Algorithm 1: Training Process of Energy-oriented Diffusion Bridge
Input Diffusion model parameter dataset distribution (Xo, Y); Hyperparameters:: trajectory
length To.
Output Updated parameter 0
Initialize parameters 8 with the pretrained diffusion model.
while not converged do
// Sample data pairs, timestep, and noice.
[Xo, YJ ~ P P(Xo Y), ~ 21 Tol. XT N I);
// Initialize state interpolation by (6)
X,
X
Y:
To
X - (1 x) + IXT:
// Rstimate clear image through Rg. (10)
Xo Fo,,, Y,I);
1/ Model upéate dF, (X..
consistency optimization Eq. (11)
- Vo di
Return: O.
In the continuous-time domain, this is equivalent requiring the solver's output have total derivative of zero along the path. We enforce this directly by adopting the continuous-time consis- tency objective (Song al., 2023; Lu & Song, 2025).
Instead of a discrete distance metric, our training objective is defined by its gradient with respect the model parameters 0. Following the formulation of continuous-time consistency models, this gradient is given by:
[Fo (
(X,, Y,,) )
V V E-ridgo
E,, Vo =0)=
Fo, sg
(11)
where dFa= is the total derivative of the target solver's output along the geodesic trajectory, and dt
sg(-) ) is the stop-gradient operator.
This objective enforces geodesic consistency in the continuous domain. The term sg( an inconsistency tangent vector; it measures the instantaneous "drift" of the solver's output at time The optimization process minimizes the inner product between the online solver's output Fg and this inconsistency vector. The only stable equilibrium for this process across the entire trajectory for the inconsistency vector itself to become zero. This elegantly forces the solver satisfy the condition dfa ensuring its output remains constant and thus adheres to the principle of oi geodesic consistency.
acts as
By optimizing this continuous objective, train the underlying network e9 accurately capture the local geometry of the data manifold. This, in turn, allows the solver Fe to function as an effective and efficient geodesic solver. This training enables both high-fidelity single-step restoration and the possibility of iterative refinement along the learned optimal path. The training process is illustrated Algorithm
4.3 TRAJECTORY ADAPTIVITY OF E-BRIDGE FOR IMAGE RESTORATION
significant, yet often unaddressed, limitation of many diffusion- based restoration methods their reliance on a one-size-fits- restoration path. Whether for denoising, super-resolution, or other tasks, these models typically employ fixed diffusion trajectory (e.g., a reverse process from / to 0). This approach ignores the heterogeneous nature of image degradations: the information gap between a noisy image and its clean version is vastly different from the gap between a low- quality image and its high-quality target. Our E-Bridge framework provides principled mechanism for addressing this through the dynamic trajectory length parameter, Ty. which is more than just a trajectory shortener, it is powerful task-adaptis aadativee control knob that adapts the restoration process to the specific nature the input degradation. The choice of Tp directly determines the composition of the restoration's 's starting point, X7, (1 To)Y + ToXx. This allows to precisely balance

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Published as a conference paper at ICLR 2026
Table 1: Quantitative comparisons of different diffusion-based methods across multiple image restoration tasks. The best results are highlighted in bold. i represents that we utilized the offi-
cial public code and recommended settings train the baseline on the same training dataset as ours.
Task
Mathod
PSNR
I.PIPS
FID
NIQE
MUSIQ
DiIBIR
77.767
4.333
64.767
AusoDIR (liang 2024)
0.419
64.995
5.599
UniDB (2hu 2025a)
20,778
0.477
87.135
3.882
62.004
IRDridge (Wang et al., 2025a)
100
23.615
0.415
59.539
5.215
65.535
Super resouatin
MaRS! (1. et 2025)
20
23.325
0.525
6.328
53,162
MaRS! (L)
23.786
0.539
101.978
7,251
49,489
UmDB+++ (Pan et 2025)
10
23.283
0.469
79.383
5.709
57.237
UniB++ (Pan el 2025)
24.393
0.494
70.664
5,904
63,087
E- Baddee (Ours)
10
21:282
0.346
57.837
3.839
70,968
E.Brdge (Ours)
22.477
0.356
55,236
4,375
65,342
UniD (Pan il., 3025)
83.718
7.596
33,222
E- Bridge (Ours)
24.094
0.152
72.001
6.251
45,605
DA CLIP (Loo et al., 2024)
100
27.027
0,301
39.526
5.091
58.098
AmtaDIR (Jiang 2024)
25
25.063
0.282
28.452
5.095
58.399
UnDD (2bu et 2025g)
100
27.956
0.212
28.256
3.019
63,782
IRBridge (Wing
2025a)
41,936
4.376
69.053
MaRS (L) et al. 2025)
20
28, 465
0.220
29.338
3.993
63.004
Denobsing
MaRS (Li et 2025)
28.252
0.293
33.204
5.185
57,677
UmDB++ (Pan all, 2025)
28.515
33,310
4,144
68,863
UniB+++ (Pan et 2025)
29.308
0.214
34,872
4,394
69.048
E-Badge (Ours)
26,009
01184
26.611
3.514
69.736
E-Bridge (Ours)
0.182
26.256
4,492
67.507
UniDB++
(Pan et all, 2025)
21.844
0.613
92.362
8.197
45.150
E-Badge (Ours)
25:241
0.258
36.967
4,535
58,954
DA-CLIP (Luo et al., 2024)
100
31.241
0.061
24.066
4,817
67.593
AutaDiR (liang 2024)
32.332
26.789
3.365
68.723
Unigg (2bu et al., 2025a)
100
30.344
0.071
25,570
4,736
67.599
IRBridge (Wing el 2025a)
100
28.771
0.125
51.814
3.959
65,012
Raindrop Removal
MaRS! (1.i 2025)
24-554
66.249
MaRS (Li et al., 2025)
10
30.359
0.167
41.186
7.749
59,468
UnDB++ (Pan et 2025)
32.151
0.068
26.184
3.577
69.761
UniD+++ (Pan 2025)
28.866
70.498
E-Bridge (Ours)
29:222
0.079
30.551
3.221
70.006
Briddge (Ours)
29.611
0.080
29.726
3,449
67.379
UnaDBe= T (Pan 2025)
25.420
0.174
61,693
3.683
64,523
E.Bridge (Ours)
67.301
DA-CLIP (1ae 2024)
23.091
39.428
4.359
71,345
AmtDDRR Olang 2024)
26.961
0.098
38.118
5.208
70.500
IRRridge (Wang 2025a)
23.913
0.116
41.007
4,664
64,926
Low light
DITUIR (Zhung al., 2024)
0.160
71.703
4,904
71,390
E: Bridge (Ours)
10
25.338
0,997
41.197
4.198
71,684
E.Brdge (Ours)
25.847
0.107
9969
4,211
70,868
AdaRR (Cui 2025)
22
0.168
58 106
4,713
70,901
Bridge (Ours)
$1.787
UmDB (Zbu et al., 2025g)
100
19.719
0,283
61,063
5.768
61.031
IRBridge?
2025a)
21.441
36.367
6.158
61.199
MaRS (Li at al., 2025)
20
19.965
0.239
40.067
5.697
60.894
Demoirding
MaRSf (1.i 2025)
0.346
41.156
7.679
55,420
(Pan T
20.109
57.558
5.416
61.466
U (Pan et 2025)
20.527
0.183
37.335
5.452
62.054
E-Badge (Ours)
20:263
0.313
42.973
5.223
63.932
F.Bridge (Qurs)
0.230
5.437
62.155
U (Pan et all, 2025)
20.181
0.266
49.057
5.603
$5.618
E-Badge (Ours)
20.715
0.254
34,414
5,342
56.788
the trade-off between preserving information from the degraded input and leveraging the generative power the diffusion model. We call this the information-entroon trade-off.
For mild degradations like denoising, the degraded image Y contains substantial amount of reli- able nigh-frequency structure and is already strong prior for Xo. Here, we desire high information preservation. By choosing small To, the starting point Xx, remains very close to Y. The restora- tion process becomes a short, targeted refinement path, correcting deviations without unnecessarily destroying the trustworthy information already present.
For severe degradations like super-resolution, Y is a poor prior for Xn- It lacks fundamental high- frequency details, and its structure may contain misleading artifacts. Relying heavily on it would

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Super resollution
ERSK
ERSK
ERSK
Denaising
Raindrop
NCS
NUS
NUS
NUS
NUS
NUS
NUS
Demoirding
Uns
Uns
&I+GDAX
L. EGUN
LI LINN
LQ
GT
UnIDB(NFE=100 100
M=RS(NFE=20)
MNRS(NFE=10)
Our1(N
Ours(NFE:5)
Low fight
LQ
GT
AdaIR(FFE11)
DiFUURINFE=33
AutoDIR(NFE=50) Ours(NFE=10)
Ours(NFE:5)
Figure 2: Visual comparison of different methods across various tasks.
cap the restoration quality. Here, need high generative power. By choosing large To (c.g., To > 1), the starting point Xr, becomes dominated by Gaussian noise, effectively crasing the unreliable details of Y while retaining it faint structural guide. This provides the model with a longer, higher-entropy path, giving it the necessary room to hallucinate complex, realistic details from scratch, conditioned the low-frequency cues from
EXPERIMENTS
5.1 EXPERIMENT SETTINGS
Datasets. We conducted a comprehensive evaluation our proposed method across five chal- lenging image restoration tasks, including super-resolution, denoising, raindrop removal, low-light image enhancement, and image demoiréing. Specifically, for super-resolution, we trained on 3,450 images from DIV2K (Agustsson & Timofte, 2017) and Flickr2K (Lim al., 2017). with degrada-
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Table Ablation study on the impact of the proposed geodesic trajectory. Specifically, we replaced the geodesic trajectory with two other types of trajectories, i.e., a) the standard diffusion trajectory, starting from pure Gaussian noise to the clean image manifold, and b) the conventional bridge trajectory, which constructs a trajectory from the degraded to the clean image, often follows a "re- noising" phase before denoising. Best results each row are in bold.
Tisk
NFF
(b)
Ours
PSNRT
LPIPS
MUSIQU
NIQUL
PSNR1
LPIPS]
MUSIQU
NIQU,
PSNR1
LPIPS.
MUSIQU
NIQEL
10
20.527
0.464
63.476
19.945
69.157
4.95%
21.282
0.346
70,66
Supervesolattin
31.016
0.95
56,789
5.657
21.443
0.502
63.068
3.902
22.477
0.356
66.343
4.373
19.667
0-486
31,919
7.166
22,099
0.481
35,552
4.267
24.094
0.452
44.615
6.251
10
24,241
0.288
64.724
4.952
24.391
0.261
69.976
5.575
26,060
0.184
3,514
Desoising
23,701
0.316
60.500
5.235
25.275
0.235
65.870
3.807
26.649
0.182
67.507
4,492
21,693
0-460
45.74 747
5,889
24.516
0.371
51. 381
4.552
25.241
0.258
$8,954
4,535
10
34,919
0.136
68.559
26.346
0.148
67.546
3
29.222
0.179
70,006
3.221
Raindrop
24.702
0.155
66.448
4.023
27.471
0.118
64.260
3,462
29.611
0.1000
67.379
3.449
23.5900
0.319
52.946
6.189
26.202
0.252
55.173
3.528
29.220
0.085
67.301
3.522
10
23,375
0.148
68.498
3,881
22.176
0.181
70.109
4.234
25.538
0.097
71,684
4,198
Luw light
5
22.706
0.145
65.551
3,746
22.316
0.157
89 19.690
4,300
25.847
0.107
70,868
4,211
21,634
0.333
48.405
4.365
23,661
0.358
52.948
4,359
24.656
0.133
66,005
4.331
10
19,200
0.344
60.277
5.628
19.956
0.253
60.216
5,697
20.263
0.313
63,933
5.223
Demoiréing
19,601
0.429
56.406
6.342
20,076
0.250
59.877
5.161
20.849
0.230
62.155
5.437
19,501
0.469
44.049
6.840
19,468
0.372
51.601
5.776
20.715
0.254
56.768
5,34
tions generated by Real-ESRGAN (Wang et al., 2021). We tested on 100 images from the DIV2K validation set. For the denoising task, we utilized the same high-quality images from the SR task, synthetically corrupted with Gaussian noise (a 50). For low-light enhancement, we employed the LOLvI (Wei et al., 2018). We used the UHDM dataset (Yu et al., 2022) for image demoiréing, with a split of 4,500 training and 500 testing images.
Implementation details. We constructed our E-Bridge framework based on the large-scale FLUX- dev model (Andreas et al., 2024). The model was then fine-tuned for 10,000 iterations using the Adam optimizer with a learning rate of 2 x 10-5, We used a training patch size of 1024 x 1024 and effective batch size of 128, achieved via gradient accumulation. The trajectory horizon To was sampled from the interval [0.2, 0.95] during training, which endows the model with strong adaptability. Consequently, inference time, To becomes tunable knob that can be set demand to precisely control the restoration process. This allows us to select a smaller To for tasks requiring high fidelity and larger To for tasks demanding stronger generative power. All experiments were conducted on NVIDIA A800 GPUs
Evaluation metries. For quantitative evaluation, employ a comprehensive suite of metries: full-reference scores such as Peak Signal-to-Noise Ratio (PSNR) within the YCbCr color space and LPIPS; no-reference perceptual scores including NIQE (Mittal et al., 2012), and MUSIQ (Ke et al., 2021); and the distributional metric FID. We also report the Number of Function Evaluations (NFE) to measure computational efficiency. To leverage the full generative capacity of the pre- trained FLUX model and account for the iol-po nature of image restoration, we avoid overly constraining the output to the ground-truth reference. Consequently, our evaluation relies heavily on non-reference and perceptual scores such as NIQE and MUSIQ.
5.2 COMPARISON WITH OFATE-OF-THE-AAE METHODS
As presented in Table 1, our proposed E-Bridge consistently achieves state-of-the-art or highly com- petitive performance on a comprehensive suite of perceptual and distributional metrics, including LPIPS, FID, NIQE, and MUSIQ. Crucially, it accomplishes this with a remarkable reduction in computational cost, requiring only 5 to 10 NFE. This highlights the effectiveness and efficiency of our geodesic trajectory and consistency- based solver. Although not always leading in PSNR, our approach consciously prioritizes perceptual realism over pixel-wise accuracy. Methods that narrowly optimize for PSNR are known to produce overly smoothed results that lack convincing high-frequency details. Our framework, by contrast, is capable of generating perceptually rich im- ages. As visually substantiated in Figure 2, our method can even produce results with sharper details and more pleasing textures than the original GT, which may contain slight softness or compression artifacts. This underscores E- Briddge's ability to generate an idealized, visually plausible outcome rather than simply mimicking potentially imperfect ground-truth.
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Table 3: Comprehensive efficiency comparison between our consistency-based solver (E-Bridge) and traditional iterative ODE solver (DDIM). (P: PSNR1, L: LPIPSI, M: MUSIQI, N: NIQE)). Best results in each column are bold.
Denoising
Super
Raindrop rem.
Low light Eih.
Demoireing
Method
NPR
r.]
M1
1.4
M1
PT
1-3
M1
PT
N2
NJ
69.01
3.55
0.330
69.37
(1990
0.112
4,08
5.15
67:44
22.27
0.362
67:17
29.72
0.866
3.57
35.57
30.43
62.02
5.50
24.55
0.270
57.07
4.30
17.92
0.591
46.07
8.97
27.48
0.135
02.42
0.39
20.96
0.177
63.78
4.27
17.44
0,407
64.21
5.33
Ours
0.182
67.51
4.49
22.48
0.355
29.61
0.880
62.38
3:45
25.85
0.107
70.87
4.21
28.85
0.230
62.16
5.44
25.24
0.258
58.95
4.54
24.09
0.452
48.01
6.25
29.22
0.085
67.30
3.32
24.56
0.133
64.81
4.33
20.72
0.254
56.79
5.34
Table 4: Analysis of the trajectory horizon parameter To. (P: PSNRT, L: LPIPSI, M: MUSIQI. N: NIQEL). Best results for each task are in bold.
0.3
To 0.9
Task
L4
Mf
Pt
L4
Mt
Super-raaoution
21.12
0:157
62.16
1.88
21.19
0.397
67.00
1.36
21.82
0.359
68.77
3.84
21.28
0.346
70.87
3.84
Denoising
24.43
0.276
66.13
4.48
25.35
0.206
69.24
3.74
26.06
0.184
69.73
3,51
26.53
0.185
69.13
4,04
Low-light
Raindrop removal
29.61
0.080
67.38
3.45
29.74
0.081
66.73
3.57
29.70
0.083
65.23
3.821
29.21
0.092
63.98
3.97
Demoirding
20.85
0.234
61.67
5.34
20.85
0.230
62.16
5.44
19.76
0,219
61.32
5.60
16.47
0.328
59.04
5.68
5.3 ABLATION STUDY
Effectiveness of the E-Bridge. We compare our geodesic trajectory with two baselines from Fig. 1. It be observed that our method consistently outperforms both alternatives in perceptual metrics (LPIPS, MUSIQ, NIQE) and fidelity (PSNR) across all five image restoration tasks. These results confirm that the proposed low-energy geodesic path significantly enhances restoration quality com- pared to sub-optimal traditional trajectories.
Effectiveness of the Consistency Solver. As shown in Table our consistency-based solver demonstrates overwhelming efficiency compared to the traditional iterative solver. For denoising, our E-Bridge achieves the best perceptual quality (MUSIQ/NIQE) with only 10 NFE, surpassing the iterative solver, which requires 50 NFE. For super-resolution, the advantage is even more pro- nounced: our E-Bridge not only secures the highest generative quality (MUSIQ) at NFE but also achieves the best fidelity (PSNR) with just 5 NFF. Overall, our E-Bridge delivers superior or competitive performance a fraction of the computational cost (5-10 NFE vs. 50 NFE), proving its fundamental efficiency across different tasks.
Analysis the Trajectory Horizon To- We investigate the role of the trajectory horizon To as a task-adaptive control knob, with results presented in Table 4. This analysis validates hypothesis that To effectively balances the trade-off between information preservation and generative power, outlined in Sec. 4.3. For the denoising task, a moderate To 0.7 provides the best perceptual quality. Conversely, for the super-resolution task, a larger To = 0.9 is required to unlock the model's generative potential, yielding the best perceptual scores (LPIPS and MUSIQ). For tasks such as raindrop removal and demoiréing, where the degraded images retain significant useful information, smaller values of To are preferable. This study confirms that To is a critical control parameter that allows the restoration process to be precisely adapted to the demands of different degradations, balancing information preservation against generative power.
CONCLUSION
We have presented the Energy-oriented diffusion Bridge (E- (E-Bridge), a new framework that a benchmark for efficient, high-fidelity image restoration. By constructing low-energy geodesic trajectory and deriving single-step consistency solver, our method overcomes the core limitations of previous diffusion bridge models. The proposed E- E-Bridge framework avoids redundant re-noising phases and circumvents the need for slow iterative sampling, enabling high-quality restoration with fewer sampling steps. Extensive experiments demonstrate that E-Bridge achieves state-of offt---f-te-ant perceptual quality across various tasks.
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ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China under Grant 62422118, 18, and in part by the Hong Kong Research Grants Council under Grant 11218121 and Grant N.CityU111425.
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CONTENTS
Appendix Overview
15
Energy-oriented Diffusion Bridge
15
B.1 Proof of Energy-derived Geometric Linearity
15
B.2 Proof of Minimal Adaptation Energy via Lipschitz Stability
16
B.3 Proof of Geometric Linearity (Geodesic Trajectory Expectation)
17
B.4 Proof of Energetic Optimality (Zero Control Energy)
18
C GCB V.S. Ornstein-Uhlenbeck Process: Optimality Under Constraint
19
D Analysis of Schrödinger Diffusion Bridge
20
Analysis of E-Bridge within the Stochastic Interpolant Framework
22
F Additional Experimental Results
24
F.1
Computational Efficiency Comparison
24
F22 Real-world Image Restoration
24
G Limitation and Future Work
24
G.1
Limitation
24
G.2 Future Work
24
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A APPENDIX OVERVIEW
This appendix is organized as follows. In Sec. B, we present rigorous proofs and analyses of the proposed method. Specifically, Secs. B.1 and B.2 demonstrate the parameterization process of the proposed SDEs with the energy-based meegyy-based criteria, which establishes both transportation and control en- ergy superiority of the proposed method. Then, we also validate that the designed scheme geodesic and energy-optimal in Secs. B.3 and B.4. Building these energy-based criteria, we then validate the advantages of the proposed energy-oriented geodesic bridge over the Ornstein-Uhlenber cess in Sec. C. Because our method has the same trajectory expectation the Cmmstin-UUhlenber Process, the control energy is the same. We only analyze the transportation energy. Finally, Sec. I) provides further analysis of the Schrödinger diffusion bridge.
Pro-
B ENERGY-ORIENTED DIFFUSION BRIDGE
B.1 PROOF OF ENERGY-DERIVED GEOMETRIC LINEARITY
Our proof formally derives the linear interpolation formula B[X,] (1 1/To) ( by demonstrating that it is the unique solution to the geodesic energy minimization problem.
The proof proceeds by establishing that the geodesic 7(t) on the data manifold corresponds the path of constant velocity, and subsequently solving for the specific trajectory given the boundary conditions Y and X.
Variational Definition of the Geodesic. Let M be the data manifold. The geodesic (t) connecting the starting distribution (at 0) and the target distribution X (at t To) is defined the curve that minimizes the kinetic energy functional:
Ehi = L. [1((t]]
(12)
Energy Minimization via Jensen's Inequality. To find the optimal trajectory (ut) = E[X,] that coincides with this geodesic, analyze the energy required to traverse the displacement. By ap- plying Jensen's Inequality to the convex function f(x) = ||=|| over the interval [o, To]:
To /
To
[a(t)) N
ū(t)dtt
(13)
To
The term inside the norm on the RHS is the average velocity, which determined solely by the endpoints:
To A(t)dt ((T)) - (0)
Y Y-X.
(14)
To
To
Thus, the lower bound for the energy is:
To 2
Y
X
Y -xol1
E44
(15)
To
2T'
Equality holds if and only if the velocity ju(t) is constant throughout t E [o,
Derivation of the Deterministic Trajectory. Let this constant velocity be v². Since the velocity is constant, we have 2² Y Y-X,, The trajectory is obtained by integrating the velocity from the starting point:
n(1) m(0) + [- -''d==Xx Xo +1.
(x-xx To
(16)
Rearranging the terms yields the linear interpolation formula stated in the proposition:
(a()= = (1 (1-
t
X
+
To Y.
(17)
To
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Regularization via Finite Horizon To- As discussed in Appendix E, the introduction of the explicit horizon To is critical for ensuring the stability of the transport. In standard Stochastic Interpolant frameworks (e.g., Brownian Bridge) defined over t [0, 1]. the velocity field typically scales as v(())) & - As t + 1, this leads to a singularity where || + x., resulting in an unbounded Lipschitz constant and unstable training.
By defining the geodesic transport over controllable horizon [0, Tol. we ensure that the velocity V" Y-Xo remains strictly finite and bounded (provided To > 0). Consequently, the kinetic
energy of the trajectory is finite:
1
To
Y
X To
1 Y Xxll dt
Emin
(18)
0
2 To
Thus, the linear interpolation derived above represents the unique, energetically optimal, and nu- merically stable geodesic connecting Xo and Y.
B.2 PROOF OF MINIMAL ADAPTATION ENERGY VIA LIPSCHITZ STABILITY
Definition of the Adaptation Energy Functional. We define the Adaptation Energy & as the ex- pected L cost incurred by the displacement between the target distribution of the pretrained model (Z,) and the input distribution of the new task (X,). This quantifies the "spectral gap" the fine-tuning process must bridge:
E(a, / E (X,,, (Ze, () dt
(19)
where: Z, -1 1Z is the canonical Rectified Flow trajectory. X, a,X,, + B,X, is the
proposed bridge trajectory. We assume asymptotic alignment of domains: X, Zo (clean/structure) and Xx ~ Z₁ (noise) in distribution.
By the Lipschitz continuity of the neural network eg with respect its input state X, we have the inequality:
[c)(2) - co < K|| 2/12-
(20)
Substituting this into the energy functional yields a tractable upper bound:
K2 / E [|X, Ze Zell3] dt.
E(a 9) s <2
(21)
Minimizing & requires minimizing this upper bound, denoted as [aa, = B[[| -z.11²).
Minimization of Trajectory Displacement. We analyze the instantaneous displacement A, X, Z: inside the expectation. Substituting the trajectory definitions:
A = (a + 3xX,) - ((1 - t)x, + tX+).
(22)
Rearranging terms by basis vectors X, and Xr:
A. = (ar (1 ()) + ((XT))
(23)
The squared norm is:
|| (ar (1-1)) + 1)3 2(xx (1 (1-1) ())( (X,, XT).
(24)
Taking the expectation as follows:
E E || [1 C1 0, (Finite energy of signal); X | = d, (Expected norm of d-dimensional Gaussian noise);
(25)
X X = O, (Signal and terminal noise are uncorrelated);
Thus, the objective function simplifies to sum of non-negative quadratic terms as
3(),,, Ci 1))² + d(B -
(26)
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Global Optimality. Since C1 to a₁ and 3,. The global minimum of zero achieved if and only if each quadratic term vanishes independently:
and d > 0, the quadratic form J. is strictly convex with respect
(or (1 1))² (3
oo= o-=1-
(27)
8 =t
We can finally draw the following conclusion. Any choice where d # -r-tor 3r # yields Te >> 0, strictly increasing the upper bound on the Adaptation Energy &. Therefore, the linear schedule is the unique minimizer that preserves the pretrained signal-to-noise alignment.
B.3 PROOF OF GEOMETRIC LINEARITY (GEODESIC TRAJECTORY EXPECTATION)
Proposition B.1 (Validation of Trajectory Minimal Energy). Let X, be the stochastic process de- fined by the Energy-oriented Geodesic Bridge SDE X, (1 t)X, + tXT from Eq. (6). The expectation of this process, put) = E[X,], coincides exactly with the manifold geodesic 7()) for all E [0, Tul.
Proof.
Variational Definition of the Geodesic. Let M the data manifold equipped with Riemannian metric y. The geodesic (() connecting the clean distribution and the degraded distribution m7% the curve that minimizes the energy functional:
2 / dtttt
1 To
Eh
(28)
The critical point of this functional satisfies the Fuler-Lagrange equation, known the geodesic equation Kriz & Pultr (2013): v - 7(t) ((7)(()))
(29)
Construction of the Stochastic Process. We construct stochastic process X, governing the bridge via the following Stochastic Differential Equation (SDE) Song et al. (2021):
dX, u (X,, l)dl adW,
(30)
where the total drift u (X,, t) is designed. The design of (Xx, t) should enable the diffusion process trace the manifold geodesic trajectory. Thus, introduce error feed-back control dynamics. starts with defining the error in the Riemannian manifold of
c(t) log (r(t)).
(31)
where logx, (1() is the Riemannian logarithm map, representing the tangent vector at X, pointing towards the geodesic (t). To make the drift traces the geodesic trajectory, such error term should gradually decay. We thus design its gradient as
é(t) ate
(32)
where denotes a scalar factor for error feedback. By substituting the Eq. (31) in to the aforementioned equations, we have
(it) (t) + a log ()()))
(33)
Note that we have the following expectation drifting process as d E[X B(u(X,; ()) for Eq. (33).
To ensure the requirement of Eq. (33), we can parameterize w/ 1) as
X, () =(() + a (og
(34)
Derivation of the Mean Evolution ODE. We examine the time evolution of the first moment (mean), p(t) [.). Taking the expectation of the SDE integral form (noting E[dW,] = 0),
we have
du(t) Elu t)] = E[)(X,) + aE logx, (7(t)))
(35)
dt
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Linearization via Taylor Expansion. Assuming the distribution of X, is concentrated around its mean ju(t), we linearize the Riemannian logarithm map around p(t):
Eloog, (n(()) x (ogu(o (T()).
(36)
In a local coordinate chart where the manifold is approximately Euclidean (due to the noise), or assuming the ambient space Euclidean (as is common in diffusion models), the logarithm map simplifies to the vector difference:
Log (n((t) = r(t) ((t).
(37)
Furthermore, the transport velocity of the mean must account for the target's motion. The effective drift driving the mean tracking error dynamics is relative to the geodesic velocity (().. Thus, the evolution equation becomes
du(t) -(1) + a (x(() m(()]-
(38)
di
Analysis of Error Dynamics. Define the deviation from the geodesic as 8(t) p(t) - (t).
Differentiating with respect to time:
8(t) (it) (t),
(39)
Substituting the mean evolution equation derived in Linearization via Taylor Expansion:
8(t) (f(t) + a(>(t) - =((t)] (t).
(40)
8(t) -a(u(t)) - 7(t),
(41)
8(1) = -a6(1).
(42)
This is a homogeneous first-order linear ordinary differential equation. The general solution
8(1) = 6(e)e -at
(43)
The Energy-oriented Geodesic bridge is initialized exactly on the geodesic for TE(T))
Y,
lim d(e) = 0. c-
(44)
Therefore, the initial deviation 8(0) = Then, we have
8()) 6 aat 0
11,
(45)
p(t) = (t).
(46)
Thus, the mean trajectory of the process is identical to the geodesic y()) by construction.
B.4 PROOF OF ENERGETIC OPTIMALITY (ZERO CONTROL ENERGY)
Proposition B.2. (Energetic Optimality of the Geodesic Bridge) Let PE-Bridge be the probabil- ity measure of the Energy-oriented diffusion Bridge (F--Brigge) process X, defined on the interval [0, To). The kinetic energy cost Fxx ju] required to steer the mean trajectory p(t) = E(X,] along the manifold geodesic 7(t) is identically zero.
Definition of Control Energy (Onsager-Machlup Functional). From the perspective of Stochastic Optimal Control (SOC), the energy cost of a diffusion process is quantified by the kinetic term the Onsager- Machleu action functional Dür & Bach (1978); Raja et al. (2025). This function measures the deviation of the process's mean velocity from its expected instantaneous drift.
For process with mean path (t),, the trajectory energy is defined as:
Ex [m] = 2 ///
20 l1i(t) - Ex.ope [Broaa (t. X:)] dt,
(47)
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where j((t) is the actual velocity of the mean trajectory; (t, X,) is the total drift vector field governing the Stochastic Differential Equation (SDE); and (ll(()) is the Riemannian metric norm the point (t).
Dynamics of the Geodesic Bridge. The geodesic bridge process is governed by the following SDE on the manifold:
dX. (b(X:) + bgeo X.)) di oXx))dW.
(48)
(f, x,)
The specific control drift Uge is constructed to enforce tracing geodesic trajectories:
(t, (t, X,) a logx, (r(()) + -x (f(t)).
(49)
The second term the parallel transport of the geodesic velocity, often implicit in the Euclidean formulation.
Evolution of the Expected Drift. We calculate the expected total drift at time Substituting the definition of booua:
E brou (t, x = E (X) + aE|logx, (n(t)) + - (f(t))
(50)
By proposition 4.1 (Geodesic Consistency). we have proven that the trajectory expectation coin- cides with the geodesic, i.e., (t) 7t). At the mean jut) (t), the logarithmic map (error term) vanishes: E[logx, (y()) = (1 (v(t)) = O. The transport term at the mean becomes the
geodesic velocity itself: ((). Assuming the base drift b(X,) is zero (standard Brownian motion) or symmetric around the geodesic, it does not contribute to the mean transport velocity.
Thus, the expected total drift is exactly the geodesic velocity:
TE Bioaal (t. Xx)] =()),
(51)
Evaluation of the Energy Functional. We substitute the actual velocity of the mean (A(t) = (t)) and the expected drift derived above into the energy functional equation:
1 To 2
EK [u]
(t))
E E'boonal (t, X,)
dt,
(52)
7(0)
7(t)
Ex [x] = 2 //20
Lo
[17(t) - =(1) dt,
(53)
To
0 dt = 0. Ex(Ø] = 2
(54)
Then, conclude that
Ex (alll--BBiggee = 0.
(55)
This formally proves that the proposed E-Bridge requires net control energy to steer the mean of the distribution along the restoration path.
In contrast, for standard bridge (e.g., Doob's h-transform) constrained to hit fixed endpoint Y, the drift is bada = V log x1) The expected drift E[boad] generally does not equal the velocity of the mean path ju(t) except in the trivial case of zero eurvature and no constraints. The "forcing"
required to satisfy the boundary condition X₃ = Y creates non-zero residual || E[b]l > 0, resulting in strictly positive energy EK > O. We prove in the next section.
C GCB V.S. ORNSTEIN- PROCESS: OPTIMALITY UNDER CONSTRAINT
Expectation Integral. We compare the two processes based on the objective: Move the distribution mean ju(t) from the clean image X, to the degraded image Y over time E [0, 1).
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Proposed E-Bridge (Geodesic): the mean trajectory is a Linear Interpolation (constant velocity). We can simplify the trajectory of the expectation of the state as
PriBridge (t) = (1 - X) +tY.
(56)
While the corresponding velocity can be calculated as
VE Bridge (1)
Xo.
(57)
Note that the velocity is constant. Moreover, for the general OU Process (Mean-Reverting) mean trajectory is an Exponential Decay (variable velocity) towards Y.
The
pou (4) Y (Xo Y) 0t
(58)
The velocity is
You (t) (X.- Y) 61
(59)
Then, transportation energy is the integral of squared velocity:
E
[[((t)])dtt
(60)
Let's calculate the energy required to cover a specific displacement = (lx(1)-w(0)) By Jensen's Inequality McShane (1937) (applied to the convex function f(x) x²),, for any path satisfying the displacement D:
Lo (o(t))) N (ollcoolaa) 2 |D|
(61)
Mean Squared Speed
Squared Mean Speed
Equality holds if and only if the velocity v(t) is constant.
Due to the E-Bridge Has constant velocity. Thus, achieves the theoretical minimum energy for the distance covers.
Eg- Bridge = Y X
(62)
However, General OU has variable velocity (starts fast, slows down). Therefore, for the same displacement, energy strictly higher.
Foo > Fg-Bridge
D ANALYSIS OF SCHRÖDINGER DIFFUSION BRIDGE
The further mathematical analysis of the Schrödinger Diffusion Bridge, framed continuation of the previous analysis of Doob's h-bridge, which proves its inherent drawbacks in terms of compu- tational complexity and energetic inefficiency.
Analysis of the Schrödinger Diffusion Bridge
The Schrödinger Bridge (SB) problem offers powerful and flexible framework for connecting two probability distributions, representing a significant conceptual advance over the standard Doob's h-transform. It is deeply connected the theory of entropy-regularizze optimal transport (EOT). The SB finds the stochastic process that transports an initial distribution To to a final distribution Ti while being "closest" (KL) divergence-to a reference process, such as Brownian motion. While this provides an elegant theoretical solution, we will prove that this entropic optimality comes at the cost of significant computational complexity and, like the Doob's 's bridge, results in a trajectory that is not energetically optimal in the sense of Onsager-
probabilistic Spece-specifically, by minimizing the Kullback-Leible
action.
Mathematical Formulation of the Schrödinger Bridge Problem
Let H be the trajectory measure of reference diffusion process on a manifold M (e.g., Brownian motion) over the time interval. The Schrödinger Bridge problem seeks to find an optimal trajectory measure IIe that solves the following variational problem:
II" arg min (KL(III[Irre))
s.t. II Tu.
Il = mr)
(64)
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Here, (L((I[[ree/) is the Kullback-Leibles divergence, and Hg and Hy are the marginal distributions of the trajectory measure II at times = 0 and = T, respectively. This formulation seeks the "most likely" random evolution connecting the two distributions, given the reference dynamics.
The solution to this problem, II", is a Markovian diffusion process described by a pair of forward and backward stochastic differential equations (SDEs). For reference process dX, adW, e. the optimal forward process (X,) and backward process (Y,) are given by coupled system of SDEs:
dX. 6f X, dt + odWr, Xo no,
(65)
dY, = (t. dt + odW,, Yr TT.
(66)
The drifts by and by are learned to ensure that the marginal distribution of X, evolves to कт at time T, and the marginal of Y, (evolving backward in time from 2)) matches To at time O.
Proof of Drawback 1: High Computational Cost via Iterative Proportional Fitting (IPF)
The primary method for numerically solving the Schrödinger Bridge problem is the Iterative Propor- tional Fitting (IPF) procedure. a continuous-space analogue of the Sinkhorn algorithm. This iterative nature is the source of the method's 's significant computational burden.
The IPF algorithm generates sequence of trajectory measures (II,) that alternate between satisfying the initial and terminal marginal constraints. Starting with the reference measure HIg fr the iterative steps are defined as:
Find 11 arg minn (KL(II)I[22)) s.t. Il = Tr). This step adjusts the process match the target distribution Ty.
Backward Step:
Forward Step: Find 112442 arg minn (KL(I[| s.t. 1lg mo}. This step adjusts the new process to match the initial distribution To.
In modern implementations like Diffusion Schrödinger Bridge (DSB), each of these steps requires training a neural network to approximate the drift of the corresponding time-reversed SDE This leads to several computational drawbacks:
Dual Training: Unlike standard diffusion models that only learn the backward (generative) process, the SB framework requires training both a forward and a backward process drift function at each iteration of the IPF algorithm. This immediately doubles the training load per iteration;
Iterative Refinement: The IPF procedure is iterative. The entire process of simulating tra- jectories and training both forward and backward drift networks must be repeated multiple times until the marginals converge. This multiplicative effect makes the total computational cost much higher than that of a single-past diffusion model; and
Error Accumulation: Numerical methods for approximating SBs, particularly those based on IPF, are known to be susceptible to the accumulation of errors across iterations, which can affect the stability and accuracy of the final solution.
Therefore, while the SB framework is more flexible than a simple noising process, this flexibility is achieved at substantially higher computational cost, which significant drawback for practical applications.
Proof of Drawback 2: Energetic Sub-optimality
The Schrödinger Bridge is optimal in an entropic sense (minimizing KL divergence), but this does not imply optimality an energetic sense (minimizing the trajectory taken by the mean of the SB process is an "entropic interpolation" between the two distri- butions, not a geometric one.
kinetic action). The
The objective of the SB is to find the most probable random evolution, not the most efficient de- terministic path. The resulting process is complex balance between the underlying diffusive ten- dency of the reference process and the control drifts required to meet the boundary conditions. The two variational problems, minimizing kinetic action (as in the Energy-oriente Geodesic Bridge) and minimizing KL divergence (as in the SB), are fundamentally different. Their relationship is characterized by a Fisher information functional, which quantifies the difference between the two objectives.
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Let (t)(t) be the trajectory expectation of the Schrödinger Bridge process. The total drift of the forward process is by (t, 2). The trajectory energy of the mean is given by:
1 / (IAsn(() - Elbres (X, (X,))s dtt EK - 2
(67)
where bref the drift of the reference process (e.g.. zero for Brownian motion in Euclidean space, or the geometric drift on manifold), Since iss(()) = E(br(t, X,)]e=t, the energy is:
EK 1." [[]bb(t, X.) bres (x)))=eelBBsnt
(68)
The control drift, by Syef, is learned through the iterative IPF procedure to satisfy the terminal condition II, øy. This drift is generally non-zero and is not designed to follow a geodesic. steers the mean along a trajectory dictated by entropic considerations. Consequently, the trajectory energy Eg manifestly greater than zero.
While the SB is more structured than the Doob's h-transform bridge (as it learns both forward and backward processes), does not achieve the zero-energy mean transport of the Energy- oriented Geodesic Bridge. The control energy is expended to steer the mean along a trajectory that is entropically optimal but geometrically and energetically suboptimal. This proves that the Schrödinger Bridge, despite its theoretical elegance, still represents high-cost solution compared to geometrically-deree derivve bridge.
E ANALYSIS OF E-BRIDGE WITHIN THE STOCHASTIC INTERPOLANT FRAMEWORK
General Stochastic Interpolant Framework
The general Stochastic Interpolant (SI) framework Albergo et al. (2023) defines a time-dependent process X, that interpolates between two distributions po and #1 using stochastic map. The inter- polant is generally defined as:
X, a + ((1)x, + (()Z, 1 LE 10,1)
(69)
where Xo ~ Po represents source distribution, X1 P1 is the target distribution. Z ~ (O.. is an auxiliary Gaussian variable. att), are smooth interpolation coefficients satisfying a0))
1, a11) = 0 and 3(0) 0, (() = 1. (t) controls the noise level, typically with 7()) = 7(1) 0 for bridge processes.
The objective of SI methods is to learn a velocity field (z(r (and potentially score field s, (z)) that generates the probability flow ODE:
dX,
(X,).
(70)
dt
The optimal velocity field of (x) is defined as the conditional expectation of the time derivative of the interpolant:
of (x)
E E[ = x].
(71)
Brownian Bridge Specific Instance of SI
The standard Brownian Bridge (BB) is specific instance of the SI framework. By selecting the following coefficients:
a(t) 1 B(t)= =t, 7(t) = o01(111
(72)
The interpolant becomes:
x BB t)Xo + 1X1 +oV((1 -)z
(73)
Taking the time derivative BB
x BB (X, Xx) + o(1 - 21)
Z.
(74)
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To find the velocity field u/ uf"((z) (x), express Z in terms of X, using the interpolant equation:
Z X, - (1 - x)x 1 o /v -)
(75)
Substituting Z into X and taking the expectation E r). we obtain the vector field for the standard Brownian Bridge:
X
E X
BB (x)
(76)
1
noise, this
Specifically, focusing the bridge structure connecting data data (or data noise), the dominant term governing the dynamics as l + is:
X
BB
(1)
(77)
1-
Theoretical Limitations of Standard SI (Brownian Bridge)
Although mathematically elegant, the standard BB formulation within suffers from two critical issues at the boundary = 1:
A. Singularity of the Vector Field Ast the denominator (1-t) + O. Unless the numerator (X1 z) converges to O faster than the denominator (which is not guaranteed during training, especially with neural network approximation errors), the velocity field diverges:
lim | (x)ll
(78)
This singularity leads to: 1) Unstable Training: The regression target for the neural network be- comes unbounded. 2) Numerical Stiffness: The ODE solver requires extremely small steps near t = 1, increasing inference cost.
B. Unbounded Lipschitz Constant The difficulty of learning the flow characterized by the Lipschitz constant L, of the velocity field us (xr) For the Brownian Bridge:
BB (z) V V
(79)
The Lipschitz constant L R I-t
is not integrable over [0, 1]: ('''
L. [(
1 dt = 0o.
(80)
1 1-t
This implies that the flow becomes infinitely sensitive to perturbations 1, making the learning problem ill-posed at the boundary.
E-Bridge: Regularization via Horizon Truncation (T₂)
Our proposed E-Bridge method solves the limitation of Brownian Bridge by introducing truncated horizon To The process is defined on & [0, To]-
A. Bounded Vector Field By restricting < To, the denominator in the velocity field is strictly bounded from below by 1 To > The maximum norm of the velocity field is bounded:
sup ||0 E- Bridge (a--Bridge(x)))
(x)ll < C
1 To < xo.
(81)
eE(O.Ta]
where C depends on the bounded domain of the data. This eliminates the singularity.
B. Finite Lipschitz Constant The Lipschitz constant for the E-Bridge flow is bounded:
LE Bridge
sup
V D
E- Bridge
(x)
T-T To
(82)
rE[o,Ta]
Consequently, the total accumulation of the Lipschitz constant (which relates the error bound of the ODE solver) is finite:
1 Tadtt = / To
1Th
-11 In
1
-
x0.
(83)
T-
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F ADDITIONAL EXPERIMENTAL RESULTS
F.I COMPUTATIONAL EFFICIENCY COMPARTSON
Table F-5 highlights our method's exceptional efficiency: with the largest model parameters, incurs the lowest total inference time by requiring significantly fewer NFEs, significantly outper- forming other state-of-the-ee approaches.
F.2 REAL-WORLD IMAGE RESTORATION
We provide qualitative results on real-world super-resolution. We evaluated our E-Bridge model real-world low-quality images from Ji et al. (2020). Note that the model has specific real-world degradation distributions during training. As illustrated in Fig. F-3, our E- Bridge demonstrates robust generalization capabilities. This experiment empirically validates that the dependence on paired data is not a hindrance to practicality. By leveraging robust synthetic data pipelines, the proposed E-Bridge can also achieve gesh-peeforancce real-world restoration without requiring test-time optimization or degradation estimation, offering superior trade-off between offline training cost and online inference quality.
these
Table F-5: Computational efficiency comparison the 1024 x 1024 image denoising.
Method
Training Parameters
Inference Time per Step
NFE
Total Inference Time
AutoDIR
115,9
0.823s
20.58s
UniDB
137.1
1.243s
100
124,30s
IRBridge
361.3 M
0,263s
100
26.30s
UniDB++
137.1 M
1.224s
10
12.24s
Ours
743.8 M
0.673s
10
6.73s
G LIMITATION AND FUTURE WORK
Although method demonstrates superior perceptual performance and inference efficiency, sev- eral limitations remain that guide our future research directions.
G.1 LIMITATION
Spatial Heterogeneity Conffict: As observed in tasks like raindrop removal, our method achieves SOTA perceptual scores but can be numerically inferior in reference-based metrics (e.g., PSNR). This is direct consequence of using global scalar parameter To- A uniform To enforces sin- gle trade-off across the entire image, making it difficult to simultancously optimize for generative power (required inpaint occluded regions like raindrops) and information preservation (required to maintain fidelity in clean background regions).
Despite significantly reducing the Number of Function Evaluations (NFE) to 1-10 steps, the reliance on the massive Flux-dev backbone necessitates substantial VRAM. This computational footprint currently hinders deployment on consumer-grand grade hardware or edge devices compared to other baselines.
Computational Overhead:
G.2 FUTURE WORK
Spatially Adaptive To: To resolve the heterogeneity conflict, we plan extend the E-Bridge frame- work from global scalar to Spatially Adaptive Map Th, w). This will allow the model to dynamically apply high generative strength locally to degraded regions (e.g., raindrops) while en- forcing high fidelity in clean areas, optimizing the trade-off pixel-by-pixel.
We plan to compress the generative capabilities of the large-scale Flux-dev backbone into lightweight network. This process could employ trajectory alignment to minimize perceptual discrepancies between the student's 's single-step predictions and the teacher's geodesic mapping, feature-level distillation to align intermediate semantic representations, and adversarial objectives to ensure the compact model retains the sharp, photorealistic textures of the original E- Bridge without over-smoothing.
Model Compression:
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leareets
Bxx Eociiuul Veterenns
staey
76185
2H 76185
Boutque Vetements
Boutique Veternents
51
STOP
STOP
LQ
Ours
LQ
Ours
Figure F-3: Additional visual results on image super-resolution the wild.
25


Reviews:
[{"review_id": "hQeO8j2o8T", "reviewer": "Reviewer_JqCr", "summary": "The paper proposes the Consistency Geodesic Bridge (CGB) framework for image restoration, aiming to improve efficiency and restoration quality by constructing low-cost manifold geodesic trajectories. The method evolves over a shorter time horizon and starts the reverse process from an entropy-regularized point that mixes the degraded image with Gaussian noise, reducing the required trajectory energy. A pretrained denoiser is used as a dynamic geodesic guidance field, and a single-step mapping function is learned via a continuous-time consistency objective to efficiently map any state on the trajectory to the target image. Experiments show that CGB achieves state-of-the-art performance across multiple image restoration tasks while allowing high-quality recovery with a single or very few sampling steps.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The method tackles an interesting problem which is the problem of image restoration.\n\n1. The method achieves competitive results compared to existing methods.", "weaknesses": "1. The method relies on paired data for training, which limits its practicality compared to zero-shot and blind methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JCx6w0J62U", "reviewer": "Reviewer_12df", "summary": "This paper proposes a Consistency Geodesic Bridge (CGB) framework based on pretrained diffusion models for efficient and high-quality image restoration. The framework constructs low-energy trajectories that are near-geodesic paths, effectively avoiding redundant re-noising. In addition, the proposed consistency solver further improves both the efficiency and quality of image restoration.", "questions": "1.The paper points out the limitations of IRBridge [1], but why is there no performance comparison with it in the experiments?\n\n2.Is the strong performance of the proposed model primarily attributed to the powerful Flux-dev [2] model? Would similar results be achieved if this model were replaced with another restoration model or generative model?\n\n3.Regarding the parameters κ and T_{0}, the paper does not specify their values for different tasks. What are the specific hyperparameter settings for each task? Could additional ablation studies be provided to perform a sensitivity analysis?\n\n4.On page 7, last line, it is unclear why “underwater image enhancement” appears here. Please clarify whether it is part of the proposed tasks or a mistake.\n\n5.On page 8, line 411, why is PSNR evaluated in the YCbCr color space? Please provide the PSNR results in the RGB space for comparison.\n\n[1] Hanting Wang, Tao Jin, Wang Lin, Shulei Wang, Hai Huang, Shengpeng Ji, and Zhou Zhao. Irbridge: Solving image restoration bridge with pre-trained generative diffusion models. In International conference on learning representations, 2025a.\n\n[2] Blattmann Andreas, Sauer Axel, Lorenz Dominik, Podell Dustin, Boesel Frederic, Saini Harry,Muller Jonas, Lacey Kyle, Esser Patrick, Rombach Robin, Kulal Sumith, Dockhorn Tim, Levi Yam, and English Zion. Flux, 2024. URL https://blackforestlabs.ai/.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.This paper develops a novel consistency geodesic bridge framework, with rigorous theoretical derivation, demonstrating a solid theoretical foundation.\n\n2.The proposed method demonstrates highly competitive performance across diverse restoration tasks while supporting high-quality few-step sampling.", "weaknesses": "1.While the proposed consistency solver is interesting, its level of novelty appears somewhat limited, as it seems to build upon and combine ideas from existing approaches [1,2] rather than introducing a fundamentally new concept.\n\n2.The experimental evaluation is insufficient. Although the paper claims the method can handle five different tasks, the ablation and qualitative experiments are only presented for super-resolution and denoising.\n\n3.The experimental details are incomplete. The paper does not specify how the baseline models were trained, making it difficult to ensure the fairness of the comparisons.\n\n4.The ablation study only analyzes the consistency solver, without examining the role or impact of the geodesic trajectory.\n\n5.In Table 1, for the \"Denoising\" and \"Demoiréing\" tasks, the results for LPIPS and FID show that CGB (NFE=5) performs better than CGB (NFE=10), whereas the opposite trend is observed for the MaRS model. No corresponding explanation for this observation is provided in the text.\n\n6.The paper claims to achieve single-step inference along the geodesic path，however, it does not provide qualitative and quantitative results for the true single-step case (NFE = 1). The best performances reported in Tables 1 and 2 is obtained with NFE = 5 or 10.\n\n7.The paper lacks a more comprehensive efficiency analysis, including metrics such as the number of parameters, computational complexity, and actual inference time.\n\n8.The proposed method performs poorly in the raindrop removal task, however, it excels in all other tasks. The paper did not analyze this contrast.\n\n[1] Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In International conference on machine learning, 2023.\n\n[2] Cheng Lu and Yang Song. Simplifying, stabilizing and scaling continuous-time consistency models. In International conference on learning representations, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "V8RHsagdMK", "reviewer": "Reviewer_pMyF", "summary": "This paper proposed a novel geodesic diffusion bridge framework through constructing a efficient and geodesic trajectory, which effectively avoids redundant re-noising phases in traditional diffusion bridges. To realize one-step mapping on the data manifold, a pretrained denoiser is proposed and the continuous-time consistency objective is adopted to analytically map any state to the target distribution. Five different image restoration experiments demonstrate state-of-the-art performance of CGB while ensuring a single or fewer sampling steps.", "questions": "1. There seems to be some typos:\n   + In line 364 $X_0$ should be bold, yes? \n   + Eqn 19 appears some error. \n   + Commas and periods should be added at the end of the equations in Appendix as in main paper. \n   + There are some errors of the best results highlighted in bold in Table 1. \n   + ''to approach the a manifold geodesic transition process'' contains an extra article \"a\" in Line 206.\n\n2. Some motivation statements lack clarity. There are many kinds of interpolant coefficients as mentioned in Stochastic Interpolants [1], why choosing $1 - (t/T_0)^\\kappa$ and $(t/T_0)^\\kappa$ in Eqn. 6? Is there any relationships between this kind of coefficients and geodesic trajectories?\n\n[1] Albergo et al. \"Stochastic Interpolants: A Unifying Framework for Flows and Diffusions.\", 2023.\n\nAfter these clarifications, I would be better able to evaluate the overall contributions and potentially raise my rating.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed concept \"geodesic bridge\" looks interesting, reasonable and works as effective tools to solve the inefficient, re-noising trajectories used in traditional diffusion bridge models.\n\n2. The main experiments are reasonable, covering five different image restoration tasks (super-resolution, denoising, raindrop removal, low-light image enhancement, underwater image enhancement, and image demoir´ eing) and six metrics (PSNR, LPIPS, FID, NIQE, MUSIQ, NFE). \n\n3. The paper demonstrates the superior performance across different image restoration tasks with better perceptual realism and achieves a trade-off between efficiency and quality in image restoration.", "weaknesses": "1. The novelty and contribution of CGB solver seem incremental. The solver (Eqn. 9) is pratically the relationship between the noise prediction $\\epsilon_\\theta$ and data prediction $x_\\theta$ (predict $x_0$) in many diffusion models [1] [2] and the distillation training objective (Eqn. 10) is not in new formulation since it is directly adopted from [3].\n\n[1] Lu et al. \"DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models.\", 2022.\n\n[2] Zhou, Linqi, et al. \"Denoising diffusion bridge models.\", 2023.\n\n[3] Lu et al. \"Simplifying, stabilizing and scaling continuous-time consistency models.\", 2025.\n\n2. The paper's main contribution of enabling \"**direct and single-step** inference\" is misleading. The proposed CGB solver is described as a single-step mapping and the training objective is for distillation. However, all experiments are conducted with **5-10 NFEs** inference, and no real single-step results are shown. The experiments are contradicted with the statements of contribution.\n\n3. The motivation for geodesic trajectory and the related Proposition 4.1 seem intriguing but I'm not convinced by Proposition 4.1. Proposition 4.1 claimed CGB trajectory defined on $[0, T_0]$ for $T_0<1$ achieves a lower total energy (Eqn. 4) than standard bridge models operating on [0,1] for two reasons: (1) reduced integration upper bound, (2) a smaller initial control at time $T_0$. Although the integration upper bound is reduced from 1 to $T_0$ which results in the reduced trajectory cost (first term in Eqn. 4), the terminal cost $\\gamma/2 ||X_T-Y||^2_2$, second term in Eqn. 4, could not be ignored: the terminal cost is near or even equal to zero in standard bridge models (e.g. DDBMs and UniDB), while, as for CDB, the starting point $X_{T_0}$ of its reverse trajectory is a noise mixture distinct from $Y$, which appears to result in a non-zero and potentially substantial terminal cost. Therefore, it's not obvious for the two total costs to demonstrate which is smaller and it's better for the authors to provide a rigorous mathematical proof of Proposition 4.1 instead of only the text explanation. Otherwise, the correctness of Proposition 4.1 remains to doubt.\n\n4. The comparison to baselines seems insufficient. Since the authors included DA-CLIP [4], MaRS [5] (the training-free accelerated-sampling version of DA-CLIP), and UniDB [6] in their main experiments, they should compare UniDB++ [7] with the same NFEs as CGB/MaRS, which is a specific training-free acceleration algorithm for UniDB and achieves better results in lower NFEs, as mentioned in Related Works.\n\n[4] Luo et al. \"Controlling Vision-Language Models for Multi-Task Image Restoration.\", 2023.\n\n[5] Li et al. \"MaRS: A Fast Sampler for Mean Reverting Diffusion based on ODE and SDE Solvers.\", 2025.\n\n[6] Zhu et al. \"UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Control.\", 2025.\n\n[7] Pan et al. \"UniDB++：Fast Sampling of Unified Diffusion Bridge.\", 2025.\n\n5. The ablation study seems also insufficient. As in Inherent Data Geodesic Transition (Eqn. 6), except for $T_0$, another hyperparameter is the curvature parameter $\\kappa$, which should also be tuned and tested in ablation study.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uoH93BbEux", "reviewer": "Reviewer_Em12", "summary": "This paper aims to address an important and challenging problem in the field of image restoration: how to achieve extremely high sampling efficiency while maintaining high-quality restoration. The authors propose a novel framework called the **Consistency Geodesic Bridge (CGB)**, whose core idea is to ingeniously integrate three cutting-edge research directions: (1) bridge diffusion models that directly connect the distributions of degraded and clean images; (2) a geometric perspective that views the “geodesic” path on the data manifold as the most efficient restoration trajectory; and (3) consistency models that enable single-step inference without the need for ordinary differential equation (ODE) solvers. The CGB framework claims to construct a low-energy geodesic trajectory and derives a closed-form “CGB solver” that can be trained through a consistency objective function. As a result, it achieves state-of-the-art performance in perceptual quality metrics with a very small number of sampling steps (NFE = 5–10).", "questions": "See weekness.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper proposes the CGB framework, which remodifies the generation process based on the idea of minimum energy, demonstrating a certain level of insight.\n2. It achieves industry-leading inference efficiency with very few function evaluations, significantly enhancing the practical value of generative tasks.\n3. By introducing a single tunable parameter $T_0$, it achieves task adaptivity and maintains a balance between performance and flexibility across different restoration tasks.", "weaknesses": "**Theory:**\n1. The core motivation of the paper is to construct a path of minimum energy. The authors state in the appendix that the energy of the geodesic trajectory is 0, so they attempt to construct a geodesic named CGB. However, the construction on the right-hand side of equation (6) is merely a linear interpolation. Although the authors mention that this construction starts from $T_0$, reducing time and energy—thus seemingly consistent with the motivation of minimizing energy—this does not provide a reasonable scheme for constructing a geodesic. Hence, there is a certain conflict between the motivation and the construction.\n\n2. In Proposition 4.1, the authors claim that the CGB trajectory achieves lower energy, providing two reasons: (i) a shorter integration interval; (ii) at $t = T_0$, the denoiser is given a more “in-distribution” input. This reasoning lacks rigor. The first point is trivial: integrating a non-negative function over a shorter interval naturally yields a smaller value, but this does not imply that the path itself is more efficient—a shorter but sharper path may have higher energy than a longer but smoother one. The second point is more of an intuitive conjecture. Theoretically, given the energy formula, the construction in equation (6) and a certain diffusion bridge equation could be quantitatively compared.\n\n**Experimental:**\n\n3. The comparisons are unfair. The paper uses Flux-dev as the base model, which already has strong generative capabilities, while some other methods, such as UNIDB, are based on SD. Since the paper mainly presents a theoretical improvement and claims superiority over diffusion bridges, to prove that the CGB path is optimal, comparisons should be made between models that are as similar as possible but differ in theory.\n\n**Overall:**\nI believe the authors essentially constructed a model:\n$\\mathbf{X}_t = (1-t)\\left[\\left(1-\\left(\\frac{t}{T_0}\\right)^{\\kappa}\\right)\\mathbf{X}_0 + \\left(\\frac{t}{T_0}\\right)^{\\kappa}\\mathbf{Y}\\right]+ t\\mathbf{X}_T,$\nwhich can essentially be viewed as learning a mapping from $X_0$ to the distribution $N((1-T_0)Y,T_0^2I)$, representing a conditional mapping from a high-quality image to a low-quality one plus a certain amount of noise. Moreover, both training and inference are conducted within the range $[0,T_0]$ (note: $X_T$ is independent of $T$ and just a purely noise). Therefore, although the integration length is nominally reduced, theoretically it is not fundamentally different from previous bridge models ending in Gaussian distributions, and in fact it is essentially time varying Ornstein–Uhlenbeck process—it is merely a scaled version. Considering points 1, 2, and 3, this does not demonstrate that such a construction is truly of minimum energy or superior. I recommend that the authors further verify the theoretical optimality of CGB from two aspects:\n\n1. **Theoretically**, directly provide an inequality proof showing that the energy of CGB is lower than that of other bridge models;\n2. **Experimentally**, validate the superiority of CGB through experiments using models that are as similar as possible but differ in theoretical design.\n\nFinally, I would like to discuss the fundamental motivation proposed by the authors—that minimizing energy leads to an optimal path. In the paper, the authors state that typical diffusion-bridge models add noise to the image during the reverse process and then denoise it, with the noise level peaking at intermediate steps. They argue that this noise-adding process during inference is unnecessary and thus propose to start generation directly from $T_0 < T$, restoring high-quality images directly from low-quality ones, which they claim yields minimal energy and an optimal path.\n\nHowever, I believe that the noise-adding process in the reverse procedure serves an important purpose: it helps to destroy certain information (such as degradation information) before reconstructing a high-quality image, thereby achieving better results. If, as the authors suggest, one wishes to directly transform a low-quality image into a high-quality one without the add-noise-then-denoise process, then models like Rectified Flow (RF) could be directly applied, since RF can progressively transform a low-quality image into a high-quality one as described by the authors. Moreover, RF is also an energy-minimizing model in the context of optimal transport. However, according to experimental results, using RF for image restoration does not perform as well as diffusion-bridge models that incorporate stochastic noise. \n\nIf my understanding is incorrect, I welcome the authors to provide clarification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)