Adaptive Domain Shift in Diffusion Models for Cross-Modality Image Translation
Authors: Zihao WANG
Yuzhou Chen
Shaogang Ren
Venue: ICLR 2026 Poster (2026)
Keywords: Generative Learning
Image Computing
Imaging
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: Adaptive, geometry-aware diffusion for image translation, keeping cross-modal translations on-manifold and high-fidelity with far fewer steps.

Abstract: Cross-modal image translation remains brittle and inefficient. Standard diffusion approaches often rely on a single, global linear transfer between domains. We find that this shortcut forces the sampler to traverse off-manifold, high-cost regions, inflating the correction burden and inviting semantic drift. We refer to this shared failure mode as fixed-schedule domain transfer. In this paper, we embed domain-shift dynamics directly into the generative process. Our model predicts a spatially varying mixing field at every reverse step and injects an explicit, target-consistent restoration term into the drift. This in-step guidance keeps large updates on-manifold and shifts the model’s role from global alignment to local residual correction. We provide a continuous-time formulation with an exact solution form and derive a practical first-order sampler that preserves marginal consistency. Empirically, across translation tasks in medical imaging, remote sensing, and electroluminescence semantic mapping, our framework improves structural fidelity and semantic consistency while converging in fewer denoising steps. The source code is in https://github.com/LaplaceLab/CDTSDE.

Published as a conference paper at ICLR 2026
ADAPTIVE DOMAIN SHIFT IN DIFFUSION MODELS FOR CROSS-MODALITY IMAGE TRANSLATION
Zihao Wang'; Yuzhou Chen', Shaogang Ren 3
I Laplace Lab at University of Tennessee
2 University of California Riverside University of Tennessee at Chattanooga
i zihao. wang@tennessee. edu
P Project Page:
ABSTRACT
Cross-modal image translation remains brittle and inefficient. Standard diffusion approaches often rely on a single, global linear transfer between domains. We find that this shortcut forces the sampler to traverse off off-manifold, high-cost re- gions, inflating the correction burden and inviting semantic drift. We refer to this shared failure mode fixed-schedule domain transfer. In this paper, embed domain-shift dynamics directly into the generative process. Our model predicts a spatially varying mixing field at every reverse step and injects an explicit, target- consistent restoration term into the drift. This in-step guidance keeps large updates on-manifold and shifts the model's role from global alignment to local residual correction. We provide a continuous-ting formulation with an exact solution form and derive a practical first-order sampler that preserves marginal consistency. Em- pirically, across translation tasks in medical imaging, remote sensing, and electro- luminescence semantic mapping, our framework improves structural fidelity and semantic consistency while converging in fewer denoising steps. The source code is in hathttss:/ /tttt::://gittabb httpe///aaaaaaaaaa////teeee
INTRODUCTION
Cross-modal image translation seeks to transform an image from one sensing modality into another while preserving task-relevant semantics. This capability underpins multimodal analysis in medical imaging and remote sensing, where translating between, for example, MRI modal or between SAR (synthetic-apert aperttrr radar) and optical imaging enables downstream tools to operate across modalities
and improves interpretability {Conte et al. 2021 Arslan et al.) 2025 Shin et al.) 2021 Wang et al.) 2019 Toriya et al. 2019; Luo et al. 2024 Wei et al. 2025 Aydin al. [025] Wang et al. 2024c). Yet, substantial differences in texture statistics, intensity distributions, and modality-specine struc- tures make semantic alignment highly nonlinear and spatially heterogeneous, which renders simple
Nomain-transition assumptions brittle.
Diffusion- based methods have recently advanced image translation with improved stability and sam- ple fidelity compared to GANs, but efficiency and semantic faithfulness under highly nonlinear do- main shifts like cross-modal translation remain challenging (Wang et al.) 2024b Cui et al.) 2025)
Lin et al. 2025 Wu al. 2023). Existing acceleration tacties usually act outside the trans- lation dynamics: some precondition the initial state (Wu et al. 2023 Wang et al. (024b)), others
(022), some inject ad-hoc guidance each step to stabilize larger step sizes (Lin et al. 2025 ) and still others distill long trajectories into few steps via reacher-student schemes (Sauer et al. 2024 Wang et al. (024b).
swap the numerical solver higher-order ODE solvers for 15-20 steps (Lu
While useful, these tactics either decouple the domain-shiff physics from the generative dynamics or compress into surrogates, and on genuinely nonlinear cross-modal mappings they can still ineur semantic drift or instability when taking large steps (Wang et al. 2024b).
In this work, we introduce a Cross-Domain Translation SDE (CDTSDE) that directly embeds domain-shift forces into every reverse-time update of the diffusion process. Concretely, the forward
"Corresooning author; zihao. zihao.wangee wang@icce.ogg

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Published as a conference paper at ICLR 2026
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Figure I: fixed vs. geometry- domain mixture comparison manifold space Left: The proposed spatially varying schedule A, traces a geometry- aware, low-energy path between source and target while preserving endpoint and monotone constraints. Center: One-step schematic given source Xage and target Xo, the field A, (predicted from X and position encoding) forms de A₁ @ Xare + (1 A₂) @ Xo to guide denoising; insets show Xa Xo. and the translated output X Right: Global linear interpolation 771 forces a straight path through high-energy regions, leaving larger correction requirement to the diffusion model.
marginals are centered on a learned domain-mix signal and the reverse SDE drift contains an ex- plicit, data-driven term that encodes the target-consistent shift each time. This in-step injection of translation dynamics makes the sampler tolerant to significantly larger step sizes without sacrificing stability: the update direction always carries domain-aware correction, so coarse integration remains on-manifold. The continuous-timg formulation yields exact solution form for the reverse dynam- ics and a practical first-order sampler derived therefrom, enabling accurate updates with markedly fewer denoising steps. In contrast to external shorteuts, CDTSDE improves both semantic alignment and efficiency by aligning the numerical update with the true cross cross- modal transformation each step.
In summary, we advance cross modal translation by embedding domain- dynamics into the gen- erative process and adding an explicit restoration drift in the reverse-time SDE so that each update is pulled toward the target modality's image features. First, center the forward marginals a domain-aware mixture and provide stepwise, target-aligned guidance that keeps larger integra- tion steps stable, reduces denoising iterations, and reframes the score model residual corrector. Second, we present a continuous-time formulation with closed-form expression of the reverse dynamics and derive a first-order sampler that preserves marginal consistency, combining theoret- ical soundness with practical efficiency. Finally, on BAR-optiical image, Electroluminescence + Semantic- mask, and MRI contrast translation tasks, our method is faster and more accurate than linear or fixed-schedule diffusion baselines, establishing shift-aware restoration as an effective in- ductive bias for efficient and faithful cross-modal synthesis.
RELATED WORK
2.1
CROSS-MODALITY IMAGE TRANSLATION
Early methods for cross-modality image translation largely relied on direct intensity mappings and voxel-wise regression models, which demanded accurate spatial correspondence between modalities (Huynh al. 2016 Han 2017; Nie al. 2017). GAN-based approaches such as CycleGAN had more become popular due to their ability to learn complex transformations without strict spatial alignment requirements, showing substantial success medical imaging and multi-spectral domain adaptation tasks (Cai et al. 2019 Armanious et al.) 2020 Arar et al. 2020). Despite these successes, GANs still encounter training instability and architectural complexities.
Diffusion models (Ho et al. 2020 Song et al.) 2021) have recently emerged as stable alternatives to
GANs, achieving superior image synthesis quality. Methods such as SDEdit (Meng et al.) 2022) and ILVR (Choi et al. 2021) leverage diffusion models for image-to-image translation, incorporating source-domain conditions into diffusion sampling. Recent advancements in Bridge-basee
Diffusion
models like DDIB DDBM, 12SB, DBIM. ABridge (Su et al., 2023 Zhou et al., 2023 Liu et al.;
2023
Zheng et al, 2025 Xiao et al. 2025) and mnegy--guided diftusion models (EGSDE) (Zhao
al. 2022) introduce additional guidance to enhance cross-modality translation. Medical imaging-
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specific adaptations integrate frequency- domaan constraints (Li et al. 2024) and local mutual infor- mation guidance (Wang et al. (022d)) to improve fidelity. Hybrid strategies, such as adversarially constrained diffusion models (Ozbey et al. 2023), further enforce anatomical realism. Despite these advances, most existing models implicitly rely on fixed or linear interpolation between modalities in either the signal or latent space. Such rigid schedules fail accommodate the non-uniform seman- tic complexity of cross-modal translation (Li et al. (023)), often requiring the diffusion model itself to compensate for suboptimal guidance paths. By explicitly learning domain-aware shift trajec- tory adapted to translation difficulty, our method avoids sampling overhead and mnder-constrained synthesis to produce efficient, semantically aligned results.
2.2 CROSS DOMAIN SHIFT
Domain shift in diffusion- -hased generation refers to the transformation required to carry samples from a source distribution D₂ to a target distribution D.. Fixed linear blends (Cui et al., (025) work for super-resolution, where the source-tareet relation is roughly affine; yet, they break down in cross moos-tt translation with non-linear shifts in intensity, texture, and semantics. Early meth- ods address domain shift through simple conditioning mechanisms. SDEdit (Meng al. (022) introduces stochastic corruption followed by denoising, relying on a fixed noise level as a control parameter, whereas ILVR (Choi et al., (022) conditions generation by replacing the low-frequency spectrum with that of the input image.
Recent works embed domain transformation into the generative process itself. BBDM (Li al., 2023) constructs Brownian bridges between source and target samples to guide diffusion along domain-aware stochastic paths. In the medical imaging domain, LMI-guided diffusion (Wang al., 2024d) enhances semantic correspondence by enforcing patch-wise mutual information alignment, while Target- Guided Diffusion Models (Luo al., 2024) modulate reverse-time gradients with per- ceptual cues from the target modality. Other works adopt cross coss-connitioned designs (Xing et al.) augmentations such as Diffuse-UDA (Gong et al. 2024) to promote style and structure alignment. Meanwhile, adaptive scheduling strategies are being introduced to learn non-linear transitions that minimize semantic discrepancies. For instance, ANT (Lee et al., 2024) learns instance-specific noise curves from domain statistics, and Diffussin Bridge (Lee et al. 2025 constructs polynomial flows in latent space to approximate Wasserstein-opting transitions from D, to D.. Although these works leverage the strong generative power of diffusion models, their efficiency drawbacks limits broader use in cross-modal image translation tasks.
2024) or domain-alignee
METHOD
Cross-modal image translation seeks a mapping T Xygaurce -> Mtar that converts a source-modality observation 2 E Xsource into a perceptually and semantically consistent target-modality image 2h &tar.. During training we assume access to paired samples (do, zo), with æn E &ar providing ground-truth supervision. The objective is to minimize the expected discrepancy F[Il#0 while ensuring fast inference. As illustrated in the center of Fig. [1 source image zon (e.g., SAR) is transformed through learned generative process into a translated target image 2 (e.g., optical).
2o col|
gen
We adopt a conditional denoising diffusion model. Given a clean target image Vo. the forward process successively corrupts it with Gaussian noise to obtain ~~~~ ~00, I). Generation begins at noise and follows the learned reverse SDE back to a clean sample. In the cross-modal setting direct conditioning on En is essential for content preservation. A common implementation of conditional diffusion employs a time-varying linear blend of source and target observations at each reverse step, given by a:iie Л)t 2o + (1 7)) æo, with Ч)- = 1/T, where a is the source-modality input and 2n the paired target ground truth. This linear interpolation implicitly assumes that the latent transition between modalities lies along straight path: a valid simplification when the appearance gap between source and target domains is relatively minor.
However, as illustrated in Fig. (right), this assumption often breaks down when modality shifts are substantial. In such cases, the straight- path rapidly departs from the source manifold (pink region), traverses ambiguous high-energy zones that belong to neither modality, and only re-enters the target manifold (blue region) near the end of the reverse trajectory. This suboptimal shortcut forces the diffusion backbone spend excessive effort correcting deviations from both structural
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and semantic priors, resulting in oscillatory updates, increased stochasticity, and blurred outputs. In contrast, Fig. ( shows that the optimal transition path (the geodesic in latent space) natu- rally bends around the energy ridge that separates the modalities. To capture this nonlinearity, introduce a learnable polynomial schedule that adapts the interpolation dynamically. By aligning the reverse SDE with the valley floor connecting source and target statistics, this adaptive shift path reduces the burden of global realignment and allows the model to concentrate its capacity on local, geghffrequency reconstruction. The result is improved fidelity with fewer function evaluations.
Overview: Before introducing the full stochastic formulation, we summarize the main objects. At each reverse step t. the model predicts a mixing field A, (0, 1)Cx and forms a domain mixture de A₁ @ 20 + (1 (1) ) @ zo (Eq. (1)). Intuitively, A, controls, per pixel and channel, how much we trust the source input versus the target domain at time Section 3.1 formalizes this construction and introduces path-energy functional [d] (Eg. (4)) that measures how natural the transition path d(t) is between domains. Sections 3.2 3.3 3.4 then show how this mixture is
Ae)
embedded into a diffusion process (Eqs. (6)-1))) Notations provided in the Appendix
3.1 ADAPTIVE DYNAMIC DOMAIN SHIFT
To address these challenges, we propose a spatially varying, channel-aware domain-mixture guid- ance that, at each reverse step t, learns a full domain transform field A, E (0, 1)CxHXWW and uses it to form a domain mixture between the source-domain observation T0 and the target-domain variable Zy.
d: o 25 + (1-AA) @ To.
(1)
where @ denotes the Hadamard product. The resulting d, serves as the domain-shift guidance used by the sampler step t.
Spatial Modulation Let p (
HYx[L.... LIX index channels. We build a position encoding π (p) E R2 (sine/cosine of normalized coordinates defined in Appendix Sec.66.1 and feed the broadcasted base step X together with π (p) into a light convolutional network Sg produce mid-term modulation he,c(p) E (0,1), 1):
W} index spatial location and c E {l,....
(p) = Sa(A T()))
Mapping h to zero-centered signal (t,(()) (p)= 2 (p) - E (-1,1). 1), we form an interpolation that satisfies (0) and f1,e(1)
(r. p) Xlin [1 [1 [
(2)
Finally, we squash f into (0, 1) with a calibrated logistic map. Fix a small E = 10-4 and set
logit(u) In - == - -logit(e), a -2 logit(e).
Define
(p) = da f1 -),
(22)
1
(3)
Ite-s.
With this choice, (p) when fi.e and Ae,o(p) = when fi. 1. To satisfy exact endpoint constraints used elsewhere, we clamp at the boundaries: Ao.c(p) 0; (p) =1, while for f. {1...., T we the Eq This keeps Atc. (0, 1) and preserves monotonicity in f, while allocating extra degrees of freedom only in the middle of the trajectory. Jointly training with the diffusion backbone lets the schedule bend early toward & in regions where denoising is confident, while deferring and concentrating effort on hard, mnodality-specifie details where needed.
Geometric View Consider the trajectory function d(t) A(t) o a (1 A(t)) O mo for
t e [0, 1], and the path energy
Eld] L'
Σ C,D
(t) (t) + (e.ple.p(t)tt) de,
where (t) > is a local metric and U..((, t) is a strictly convex local potential in its first argu- ment. We compare two admissible path classes (both obey the endpoint and monotonicity induced
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by equation and equation 3): Cpix = (a(t) (0,1)oxHxw) }. Cglob = {A(t) = (t)) 1, 7
-0.1]-[0, 10,1) Clearly Cglob C Cpix-
Remark: E[d] is a path energy that evolves over time and is used to measure whether the transition path from the target domain to the source domain is natural and has a lower cost.
Theorem 1. Assume there exists set of indices S C ((c,p) of positive measure and an in- terval I C (0, 1) such that: (i) (heterogeneous local geometry) the potentials (Uep(=.()e.ppi. or the metrics {ac.p(t) are not identical across (c. over I; (ūi) (nondegenerate contrast) p)= + 0 on S and this contrast does not annihilate the local slope along the global path on a set of positive measure and (iti) (endpoint & monotonic feasibility) both classes admil paths satisfying the endpoint and monotone schedule constraints. Then
Dc.p := 1- (p) TP)
inf AcC
Eid]
<
inf Eld] AECglob
(5)
Since Cglob C Cpixx the non-strict inequality is immediate. Intuitively, view d(t) as the evolving translated image with kinetic- kinetic-potential energy equation (i) Heterogeneous local geome- try implies different pixels & channels have different "best" mixture to reduce their local potentials Uc.pi a global y({)1 must compromise, incurring avoidable potential penalties somewhere. (ii) Non- degenerate contrast (Ac.p * 0) ensures that pixelwise adjustments actually change (t.p(t) in the potential's descent direction, yielding a genuine decrease of the potential term positive-measure subset. (iii) Endpoint & monotone feasibility lets these adjustments be implemented as smooth, lo- calized, time-spread perturbations of A(t) that preserve endpoints/monotoniciry Theorem I states that, under mild heterogeneity, allowing A(t) to vary across pixels admits a path with strictly lower energy than any global schedule. Thus dynamic domain transform path A(t) strictly lowers total energy compared to any global path 17((). Detailed proof is in Appendix 6.2
Remark:
3.2 CROSS-MODAL DIFFUSION PROCESS
We couple the adaptive dynamic domain shift with variance-pressrving diffusion process. Let the base noise schedule be {au)T=1 C (0, 1), with ā₂ II'm1 0g and of = 1- ã,- At each step use the spatially varying, channel-aware mixing field A, E (0, 1)x to form the domain mixture defined in Eq. and set the forward marginal to
g(x:
Zg. 20 = M(= Vou de, IT
(6)
The corresponding step Markov transition consistent with equation i
Ft-1; Do. 26 20re) MM( vi: Pt 2:-1 + Vou (d. dr-1); (1- 1):
g(
where Pe Vār/ār-1 Var and d, is given by equation
| Equivalently, using the mixture
identity
d, d = (A, A,-1) 1) @ (# zo):
the mean in equation 7 can be written explicitly in terms of the field increment A, - A
We use a middle point i < T and fix A, = for all f 2 14-(Cui al.] 2025 Meng et al. 2022 In this regime the forward mean becomes vāz ø so the diffusion is a pure noise process centered at the source observation. At inference time, one can initialize directly from Tt, (V( , eare, of, I) and run the learned reverse process from t to O, effectively skipping approximately T t1 denoising steps while preserving translation fidelity.
3.3 CONTINUOUS-TIN TIME FORMULATION
Let A(t) E (0, (0.1)cxxxW be a differentiable, spatially varying mixer and define d(t) A(f) O aearc + (1 - A()) @ 2o. With o II 0₂ and of = 1 ãt, the forward SDE is
dx: ((0) x. + vār @ (agre )o))aat + g()) dw,,
(8)
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with f(t) In v and g(t V ft). In SDE equation 8 the additional drift is vā, A(t) @ (a zo). Importantly, solving the moment ODE implied by Eq. 8 shows that the added drift Vaz ( - ao)) makes the forward mean track the domain-mixture path Voy d(t), thereby realizing on-manifold, low-energy transport that simplifies the subsequent reverse-time in- ference.
The corresponding reverse-time SDE (see proof in Appendix 6.8) model is:
corresponding the forward
dx.
[(t) xe (a(t) (a6 - xo) (t) Va log qe (ax To. 26
dt g(t) dwi, (9)
3.4
SAMPLER
In this section, we present the solution of the SDE under our mixture model and design efficient samplers for fast sampling. We employ a data prediction model Φ (z+ #6 t) that estimates the
clean target image from noisy samples. The relationship between the score function and the data prediction model under the mixture forward marginal (Eq. (6) is:
Var de
Vn log qe (2x ag
df = A @ aere
(1 - Ao) ④ Bo 200 (), (10)
where of = 1 āg. See the proof in Appendix 6.6
In practice, we train noise prediction model Eg (Ze, 20 t). Under the same mixture marginal, the standard VP conversion gives
EØ (It, arrc .t) 3t
vā df
dy
Dt Oe Eø Ete 2 .t)
Eg(2
(1I)
at
Vor
By substituting Eq. 10 the reverse SDE ; var(1 At),
v. 2 Y,,
:= e Y,,
(with Ø denoting Hadamard division), together with the shorthand
dwx :=
AA
dt
dw,,
xx == 2((A):
wx WA::
we obtain the reverse dynamics with respect A:
dyA = (3) ① yxda dA +
(10 - AN (10(1-AN)) dA -
1-
A
o
(2)
dd
arc
(12)
2
D (xx,
20 A)dA
g(t)
dwx
where all divisions, square-roots, and products are Hadamard operation. The equation 12 can be solved exactly via the variation-of-constae
formula (Cui et al. 2025).
Proposition 1. Let Y, := Vār (1- Ac) and X Te Ø Y, (O denotes Hadamard division). Given an initial value x. time
the solution x; of the cinerse-time SDE Eq. 12 fort E [0, is
x, Y, 0 A2 x o (z. or. r. L/ (2 2x3) 3 25.0 A) dd
( (AI0(I- - -A -000 (1.
2 (1 (1 ( 2.re
0.o
(13)
d (
(x)
r.o
de dwx.
Here g o({( and Y xx2)) along the integration path; all operations are Hadamard. We give the corresponding first-order solver in the Appendix 6.9
Empirically, shown in the next experimental section. this guided path enables accurate translation in fewer steps compared other shift methods, suggesting the learned schedule implicitly captures optimal tradeoffs between semantic alignment and spatial refinement.

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4 EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Tasks and Datasets We consider: (i) MRI modality translation (TI-T22 on IXI; (ii) Optccpt
translation on co-registered Sentinel-1/2; (iii) EL-to-semantic mapping for solar-cell defects on PSCDE. The difficulty increases from minor intra-structural contrast changes (IXI), to cross-senson appearance shifts (Sentinel), to fine-graie ome-grained defect synthesis under strong acquisition and morphol- ogy variability (PSCDE). Full curation protocols, splits, and counts appear in Appendix 100 60
Implementation Overview We use a UNet-based conditional diffusion backbone implemented in PyTorch Lightning with mixed precision. The adaptive domain shift component follows a learnable schedule; optimization and training lengths are task-specific. These three tasks reflect increasing translation difficulty: IXI (minor intra-modality contrast), Sentinel-1/2 (cross-sensor appearance shift), and PSCDE (fine gene-graaned, high-variability defect structures). Complete configurations (opti- mizer, learning rates, schedule degree/initialization, architectural choices) appear in Appendix 6.11
Baselines and Metries We compare against Pix2Pix (Isola et al. 2017). BBDM (Li et al. 2023
ABridge (Xiao et al., 2025). DBIM Zheng et al. (2025). and DOSSR (Cui et al. 2025p. We report
SSIM/PSNR, MSE/MAE; per-method notes are in Appendix 6.
4.2 QUANTITATIVE RESULTS
Table I presents comprehensive quantitative comparisons across three representative cross-modal translation tasks. Our CDTSDE method consistently achieves top performance across nearly all evaluation metrics, validating the effectiveness of its adaptive domain shift mechanism.
Table 1: Quantitative comparison across three tasks with one row per method. Datasets: Sentinel (SARaaOtiica),, IXI (TI-T22), and IXI (T2-TI1). Metrics: SSIM (1), PSNR (f), MSE (1), MAE ()). Best bold, second-best underlined within each dataset block. Training protocol: For fair comparison, all models on Sentinel are trained for 20,000 steps, on IXI for 10,000 steps, and on PSCDE for 5,000 steps (except Pix2Pix). Sampling: Because BBDM lacks a fast-sampling variant, use its recommended default of 1,000 sampling steps. Training curves are in the Appendix 9
Sentinel (SAR- Opnica))
IXI (71-72)
(X1-2--77)
Method
SSIM
PSNK
MSE
MAE
SSIM
PSNR
MSE
1 MAE
SSIM
PSNKT
MSE
1 MAE
Pix2Pix
0.3:001
15.12=14
0,03000
0.12-0.0
0.71=00.
22.17=1.2
000
0.03+0..0
0.71=000
22.244144
001+00
00Se00
ABridge
0.15=0.1
12.5022.0
0.06=0.0
0.18:0.1
0:14:0000
13.73=2.4
0.0110.0
0.12:0.0
0.3510.0
13.77.2.5
0.0110.0
0.12:00
DOSSR
0.36-0.1
17,14=15
0.02-0.0
0,10=0.0
0.75=20.0
23.28=14
0.01:0000
0.03:0.0
0.00:0.1
24.13+20
0.010000
003200
RBDM
002-00
6.25-08
0.24000
0.40-0.0
0.01+0..
8.63+003
0:13+800
0.25+0.0
00100000
8.094044
0.13+00
0.25+00
DBIM
0.14:008
13.48114
0.05-0.0
0.16=0.0
0.28:20.0
20.43:00.8
0.01:0000
0.08:00.
0.33:20.0
21.65:1.5
0.012000
0.062000
CDINDE
0.38-01
17,46+155
0.01-0.0
0.09-0.0
0.76+0.D
23.32-1.3
0.01+0.0
0.03+0.0
0.82+6.0
24.33+19
0.01+00
0.03-00
SAR-to-Optical Translation On Sentinel, CDTSDE leads all the reported metries: SSIM (0.382 +
0.077). PSNR (17.46 = 1.46, dB), MSE (0.0190 = 0.0068), and MAE (0.0982 = 0.0198), with DOSSR consistently second. DBIM and ABRIDGE improve over BBDM but remain well below Pis2Phr/DOSSR/CIISDDDS in fidelity; for instance, DBIM attains SSIM (0.140 + 0.044), PSNR (13.48 + 1.44, dB). MSE (0.050 + 0.020), and MAE (0.160 + 0.031), compared with ABRIDGE'S SSIM (0. + 0.0747), PSNR (12.50 + 1.96, dB), MSE (0.0627 + 0. 0.0341), and MAE (0. (0.17734 + 0.0566). Pix2Pix provides mid-tier performance (e.g., PSNR 15.12, dB) yet still trails CDTSDE and DOSSR, while BBDM remains the weakest across metrics.
Medical Image Translation (LXI: T1++T2) Across both directions on IXI, CDTSDE consistently delivers the strongest fidelity. For T1-T22 it attains the highest SSIM (0.762 + 0.042) and PSNR (23.32 + 1.25 dB) with the lowest MSE (0.0049 + 0.0016): for T2-T11 it further improves to SSIM (0.825 + 0.038). PSNR (24.33 + 1.86 dB), MSE (0.0040 + 0.0018). and MAE (0.0324 + 0.0076). Among diffusion bridge-based methods, DBIM improves over other bridge-based ABridge and BBDM on PSNR/MSE/MAE (e.g., for T2 it reaches SSIM (0.33 = 0.04). PSNR (21.65 + 1.45 dB), ). MSE (0.010- = 0.004). MAE (0.060 = 0.015)) yet still lags behind Pix2Pix, DOSSR, and CDTSDE in structural fidelity. DOSSR is a close second in SSIM/SNNR/MSE and attains the lowest MAE on TI-T2 (0.0321 + 0.0060). Pix2Pix is weaker, and ABridge and BBDM show marked degradation and structural artifacts consistent with their low SSIM/PSNR.
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Table 2: PSCDE dataset: semantic translation metrics, Higher is better except Hausdorff.
Method
Dice 1
IoU 1
Precision
Recall 1
Hausdorff +
Skeleton F1
Pix2Pix
0.178 0.374
0.173 + 0.363
0.186 + 0.387
0.173 + 0.364
156.28 90.84
0.053 0.124
ABridge
0,009 + 0.029
0,005 +: 0.016
0,080 + 0.237
0,072 = 0.144
161.89 + 71,66
0.008 0.015
BBDM
0,024 + 0.036
0.012 0.019
0.069 = 0.139
0.228 = 0.117
195.44 = 61.42
0.030 0.028
DBIM
0.167 1 0.360
0.158 1 0.342
0,427 0.482
0,165 1 0.355
101.49 1 85.28
0.004 0.010
DOSSR
0,460 + 0.331
0.370 + 0.337
0.659 + 0.321
0,413 1 0.335
59.53 1 62.50
0.587 0.328
CDTSDE
0,488 + 0.351
0,403 + 0.352
0,784 + 0.258
0,432 + 0.356
39.87 1 50.76
0.614 0.341
Industrial Defect Detection On PSCDE, CDTSDE delivers the strongest pixel-level performance and topology preservation, achieving the highest Dice (0.488) and IoU (0.403), the best Skeleton (0.614), and the lowest boundary enror by Hausdorff distance (39.87). It also attains the best pre- cision/recall balance (0. /0.4)),,)) enabling recovery of thin structures under noise. DOSSR is consistently second across all metries (Dice 0.460, IoU 0.370, Precision 0.659, Recall 0.413, Haus- dorff 59.53, Skeleton F1 0.587). In contrast, the bridge-based diffusion methods DBIM, ABridge, and BBDM all lag far behind the top two: DBIM attains moderate region overlap (Dice 0. 0.167, loU 0.158) and the lowest Hausdorff (101.49) among the bridges but suffers from very poor Skeleton FI (0.004), while ABridge and BBDM largely fail to produce meaningful semantic translations (Dice < 0.024, IoU < 0.012) and exhibit severe boundary errors (Hausdorff 2 161.89). Pix2Pix sits between these extremes, with lower overlap than the proposed methods (Dice 0.178, IoU 0.173) and large boundary deviation (Hausdorff 156.28) alongside weak structural fidelity (Skeleton Fl 0.053).
Overall, CDTSDE ranks first or second overall, underscoring robust generalization.
Source
Target
COTSDE
DeSSe
BROM
ABridge
PlctPlx
Source
Target
COTSOE
DedSR
BROM
Afrridge
PictPix
Figure 2: Visual comparison across three cross-modal tasks: IXI medical conversion, Sentinel SAR- to-optical, and PSCDE semantic mask. (See more in Appendix 6.15).
4.3 COMPUTATIONAL EFFICIENCY
To ensure fair comparison, we evaluate methods at natched-fideling regime of PSNR ~ 15 dB on Sentinel. For models capable of reaching this regime, we report the configuration that first attains PSNR R 15 dB: DOSSR and CDTSDE at 10K training steps, and Pix2Pix at 125K training steps. Diffusion bridge-based methods: DBIM, ABridge and BBDM were trained 20K; yet DBIM and BBDM did not reach PSNR x 15 dB under comparable training budgets; we report their best checkpoints.
A key advantage of our CDTSDE framework is its efficiency at matched fidelity. As shown in Fig CDTSDE reaches 15 with only 5 sampling steps and 1.8s per image, whereas DOSSR needs 10 steps and 3.6s, giving ~2x speedup similar memory (6.3 6.2 GB). Although Pix2Pix (GAN based) is fastest at test time (single forward pass), requires 125K training steps to reach comparable fidelity, while CDTSDE and DOSSR do in 10K steps, reflecting much higher training
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efficiency. The diffusion bridge baselines BBDM, ABridge, and DBIM still fall well short of the PSNR=15 band, highlighting the benefit of our adaptive domain-shh shift schedule.
IXI & Sentinel Datasets: Fig. 2 shows that on IXI, CDTSDE reconstructs anatomically coherent im- ages with crisp cortical and ventricular boundaries. improves over DOSSR, which slightly smooths details, and clearly surpasses Pix2Pix and the bridge-based methods (DBIM, ABridge, BBDM), which exhibit blur, noise, or partial structural col- lapse. On Sentinel (SPR- CDTSDE pro- duces semantically aligned, natural optical images with stable geometry, while DOSSR loses fine de- tail, Pix2Pix exaggerates colors and textures, and the bridge-based methods leave fragmented or inconsis- tent patterns even when DBIM locally improves con- trast.
1000
0010
Time
3.60
1.80
0.22
0:30
Spn (GB)
5.15
6.80
4.10
6.20
6:30
D924 (206)
BRGM (204)
ABridge (206)
DOSSR (106)
COTSIV (10k)
Figure 3: Computational efficiency.
PSCDE (Semantic Translation): PSCDE requires semantic recovery far beyond the other tasks, and CDTSDE preserves thin and fragmented structures under noise, maintaining mask continuity and topology (clearly see from the PSCDE columns: 2 DOSSR approximates shapes but slightly oversmooths edges, whereas Pix2Pix introduces spatial inconsistencies and over-segmentation. The bridge-based methods DBIM, ABridge, and BBDM struggle to recover fine structures, yielding jagged, disconnected, artifact-heavy masks, so CDTSDE remains strongest on this dataset.
Overall, CDTSDE exhibits superior visual performance. More examples are in the Appendix 6.15
4.4 ABLATION
We assess the contribution of the learnable dynamic domain-shift schedule against fixed linear schedule. To isolate the scheduling effect, we compare three configurations. Linear: a fixed linear schedule 7)- Channel Non-linear: a per-channel but spatially uniform schedule that learns a non- linear transformation for each channel. Dynamic: the full non-linear with the dynamic spatial modulation domain-shift model that predicts A, at each reverse step. Additional implementation details and metrics for the intermediate Channel Non-linear variant are provided in Appendix 6.13]
On the PSCDE semantic mask generation task, where structural fidelity is paramount, the progres- sion across the three schedules is informative. The Linear schedule attains Dice score of 0.46 and Hausdorff distance 59.5 pixels. The Channel Non-linear variant keeps Dice essentially unchanged (0.46) but reduces Hausdorff to 43.0. indicating that per-channel adaptation already sharpens bound- aries without improving region overlap. The fully spatially varying Dynamic schedule further raises Dice to 0.49 and lowers Hausdorff to 39.8. relative gain of +6.1% in Dice and 33% reduction in boundary error compared to Linear. This trend suggests that channel-wise modulation explains part of the geometric improvement, while spatial adaptivity provides additional boost in overlap accuracy and overall mask quality. On the remote-sensing benchmark, the Linear setting achieves an SSIM of 0. 764 and PSNR of 23.28 dB, whereas the Dynamic configuration improves these to 0.774 and 23.37 dB, corresponding to relative gains of about 1.3% in SSIM and 0.4% in PSNR. In cross- moos-mmo translation tasks, where SSIM values above 0.75 already indicate near-saturation (Gu et al. 2023), the additional ~ 1% increase reflects meaningful advance in fidelity.
DISCUSSION AND CONCLUSION
Our framework yields the largest gains PSCDE, followed by Sentinel and IXI, reflecting increas- ing cross-domain complexity. In low-gap settings such as IXI, improvements are modest, suggesting that the added flexibility of CDTSDE is not always required. The formulation remains fully compat- ible with pretrained VP-based diffusion priors, making the approach transferable to broader image- to-image pipelines. A useful next step is to develop criteria that predict when adaptive, geometry- aware scheduling is beneficial, enabling simpler regressors in easy regimes. While CDTSDE leads on SSIM/SSNR/SSE, GAN baselines can occasionally achieve higher perceptual scores due the

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faithfulness trade-off; lightweight perceptual adversarial terms may help close this gap. Additional discussion, limit cases, and future extensions appear in Appendix 6.17
In this work, introduce sinuctur-addpttve mimain-mixture schedule for diffusion models that follows low-energy transition between domains. Theoretically, under mild heterogeneity and fea- sibility conditions, the proposed dynamic schedule strictly dominates any fixed global schedule in path energy, providing a principled explanation for its efficiency and fidelity gains. Experiments across MRI translation, SAR-to-optical conversion, and electoolumiiescenn--ossssseee highlight consistent improvements with fewer denoising steps, underscoring its readiness for real- world deployment in domains where both accuracy and efficiency are critical.
mapping
ACKNOWLEDGMENTS
This research supported by the Ruth S. Holmberg Grant for Faculty Excellence, and is supported in part by the Provost's Office of University of Tennessee at Chattanooga, We thank the anonymous reviewers of ICLR 2026 for their constructive feedback.
ETHICS STATEMENT
This work focuses on image generation and cross-modal translation in medical imaging, remote sensing, and industrial inspection. Medical dataset (IXI) is involved but all datasets used (including Sentinel- 1/2, PSCDE data) are publicly available and de-identified, and no personally identifiable or sensitive human subject data are included. The methods developed do not enable recovery of private or individual-level information, and all experiments comply with relevant data usage licenses. While cross-modal image generation could potentially be misused for falsification disinformation in sensitive domains, our work is strietly intended for scientific and clinical research, industrial quality control, and remote sensing analysis. No conflicts of interest external sponsorships influenced this study.
REPRODUCIBILITY STATEMENT
The source code of the proposed method available wmttppp://gtthbb.cc///p..l..l.l
CDTSDE
The Appendix provides complete, step-by-step pseudocode (Algorithms 3 and key implemen- tation details (model architectures, hyperparameters, training schedules, and data preprocessing in 6.11 for all procedures, enabling independent re-implementation without access to our codebase.
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APPENDIX
THE USE OF LARGE LANGUAGE MODELS (LLMS)
In this work, employed large language models (LLMs) to assist with auxiliary programming tasks that are not directly related to the research methodology. Specifically, LLMs were used to polish the grammar and formatting of LaTeX tables, ensuring clarity and consistency in presenta- tion. Additionally, portions of the source code were developed with the assistance of the GitHub Copilot plugin in Visual Studio Code, primarily for routine tasks such as data preprocessing. All code generated with the assistance of LLMs was carefully reviewed by human researchers, and in- put/output correctness was verified through visual inspection and testing to ensure reliability and reproducibility.
6.1 POSITION ENCODING FOR SPATIAL MODULATION
To provide spatial variation for the modulation network, we employ fixed position encoding that maps each pixel coordinate (ır, y), normalized to [-1, 1|2, into four-dimensional embedding:
sin(TR)
cos(TT)
π [ v)
(α, E [-1,1
(14)
sin Tx))
cos (T))))
Here W-I and y 2j denote the normalized horizontal and vertical coordinates
2:
for pixel indices i 0, ... W - and O,..., !t 1, respectively. The resulting tensor π E R4x H>W
broadcast along the batch dimension and concatenated with the stepwise coefficient X to serve input to the spatial modulation network. This encoding introduces deterministic sinusoidal variations across spatial positions, enabling the network to learn adaptive mixture weights with consistent geometric priors.
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6.2 PROOF OF THEOREM
Let A(1) (1)11 E Cgtob be any admissible fixed-domain path and d(() A(1) @ 20 + (1 A(t) @ 2o its trajectory. Denote the Gateaux variation of at along a perturbation 5A(t) by
Preliminaries
8E|d; 8A
Σ
2 (t) (t.((t)d..pp)) a&U BdUc.pdep(().)) 6de.p(t) dt,
where 6d..p( = De.p (t).p()) and Ae,p (pre(p)) (P).
Without loss of generality we replace 1j by ne (t()) = (1 -t) + at with fixed but arbitrarily small 0, which makes n/ a.e. while Elde] Eld] 4 0; thus drop the subscript in what follows.
Step 1: Existence of descent direction in Cpix By assumption (i)-(ii,, there is subset 8 I with positive measure where the local geometry is heterogeneous and Dcpp o O. For (c, p) E S, define
Ac.p- If (t.p(t) * 0 on I, choose 8Ae.p (t) sgn (t.,(tt) (t), where v Cb(T) is nonnegative bump (compactly supported in I), and E >> O. For (c, p) & S, set 8da. =0
(tp = 0&U
Then the first-order variation satisfies
pot = S.'
DdUtappdepl dde. -e c.p
Σ (c.plsS
(t)(t)l <
The kinetic part is
6kni : _ 2c (t) (d., dt / Σ 2 (t) (t) A..p oir. (t) dt. (c.p)ES
c.P
By choosing & piecewise- flat on most of I and with very gentle ramps near its endpoints, we can make [| arbitrarily small while keeping f, (t) dt fixed. With acpp and depp bounded (as- sumption (i)- (i)- imply admissibility and boundedness), we obtain
|8Exin| < C' (|oo\\| + 0 as the ramps are flattened.
Hence for sufficiently small & and sufficiently flat ramps, the total first-order variation 5E(d; 5A] = pppt dE 0, i.e., there exists admissible perturbation that strictly reduces the energy.
Step 2: Preservation of endpoint and monotone constraints Let be monotone and satisfy the endpoint constraints. Because our perturbation is compactly supported in I (0, 1) and can be chosen with arbitrarily small - there exists Ep > such that for all 0 < < so the perturbed schedule Á(t) = A(t) + 5A(t) remains within (0, 1), preserves endpoints, and is monotone. Therefore À E Cpix-
Step 3: Strict domination For any A Cglob. Steps 1-2 yield an admissible Á Cpix with Eld E(d. Taking the infimum over all Ä E Colob proves
inf AECpix
Eld
<
E(d] and in fact AECglot
inf
inf AECpia
E|d]
E[d], AECatab
inf
since heterogeneity (assumption (i)) and nondegenerate contrast (assumption (ii)) ensure that Ge.p is not identically zero on a set of positive measure and thus the descent strict.
Remark If, additionally, U..p(:,t) .) is (t) for sufficiently smooth perturbations one obtains the second-order estimate
convex (Hessian m. (t)l()) then
Eld Eld] S - I
/e Σ (0) ((t) di + C Σ (cpleS
(t.p(1) (o)c.(1))² dt,
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so by flattening ramps the kinetic term be made arbitrarily small while the potential decrease stays linear in E, yielding a quantitative energy gap.
In the language of functional analysis. E is an integral functional defined on space of admissible paths, assigning to each trajectory d [0, 1] + real-valued energy via
E|d] !. L(d()) d(), di,
and minimizing & characterizes the optimal (least-cost) transition path.
6.3 DERIVATION OF THE ONE-STEP TRANSITION
If the forward marginals satisfy equation 6 for L - 1 and I, then the conditional in equation 7 holds with P: = v /0//0n11
Proof Assume g( æo, an) = NNVaa- d o2 with 0221 = -1-aa - Define the Gaussian update
F² Pt -t-1 Vār (d: dz-1) + Et, 6 M(0, (1- ()-1))
(15)
with E, independent x4-1. Taking conditional expectation with respect to (20,, tt) yields
Bix, Vo. Ø0] = Pe B|x wo, Øo] + vā (d, dx-1) = Pt vā-- di-1 + v (d, d(-1).
Using P: Vou /01/02-3 gives Pevon-1 Vol, hence
E Ey. Øol = Vā di-1 + Vāl (d, d(-1) = vān de,
which matches the mean in equation 6 For the covariance Var(zy [z] Iu. Lul = -
Var[æ;--1 æo, Øb] + (1 - (.) (1 ā, I + (1 x = (1 p aa
Since pεaz-1 at, the variance equals (1 - 04 I af I, which matches equation 6 Therefore
the conditional distribution of æ; given æ;-1 and (æo, 2t is Gaussian with mean and covariance in equation 7 completing the derivation.
6.4 DERIVATION OF EQ.
Discretized equation with step At:
Ft = (1 + f(t) x + Va, A(t) @ (~. xp) x0) At + g() I),
(16)
Assume the inductive marginal at time t equals q(=, Zo. 2. N(zz; da, o,1), [0, T). i.e.,
te
K₁ = V d(t) + z2,,((,11).
(17)
Substitution yields
Lt+ = (1 Vo, A(1) @ a 2n) Al + V(1 1())/)00++
+
2,
(18)
with 2 ~~N(0, I). On the other hand, the target marginal at L Al
E++sr = Var+ar d(l + At) + OL+ALE, E~N(0,I).
(19)
Using first-order expansions / Vān + Var At and d(t + At) d(t) + Á(t) @ ($ 2o) At, equating the mean terms gives
d
Vār Vāt d(t)
Vo d(t), + f(t) = dl
In Vār.
di
Equating variances yields
oft = (1 - ((I) g(1) A = =0² (2S(1) + g(1))) A o(D))
hence, in the limit Al - 0.
dop = 2f( +g(12 dl
+ g(l) do?
20)
dl
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6.5 PROOF OF THE FORWARD SDE
Lemma 1. Let de be defined by the domain mixture equation 7 d: A, @ 20 + (1 Ar) @ 2n. Then, for all I. 1,
d: - - -Ac -Ae A2-1) o (2 - 20).
-
Proof. Expand both de and from equatton ! and collect terms.
Proposition 2 (Consistency of the forward marginal), Fix a vuriance-preservve schedule {at) [0=) C (0,1) with ā₁ = II' as and Pt = Vāg/ā, = Vac. Assume As 0 (hence do = Fol. Consider the one-step Markov transition equation 7
q( 14-1;200 Gare) =M Dii Pt a + Vor (de - di-1), (1 (1-Mi) -i)))))) I=
where de is given by equation 7 Then every E {l,...,IT) the marginal distribution distribution.condiiiona conditional on (zo, 9 equals
q( No, v6
M(zii vās de. - (1-āa) a.) 1),
(20)
i.e. the mean Vā de and the covariance is (1-āa)L.
Proof. We proceed by induction on t.
Base (l=1). Using An gives do 26 and, by equation 7
a Zo 25 = pI æn vāi (di - do) = vāi d1,
since = Vai @j. Moreover, Var | æo. | = (1 - p
āi) Thus equation 20
holds for
Inductive step. Assume that for some t 2 2 the claim holds at t 1:
Hz-1 (zo. 2 V(Vār-idi-1 a
Taking conditional expectation equation and using independence of the Gaussian noise, we get E[x; 2o, 20 = Pt E(x;,-1 2o, 2o + vor (d: d.-1) = v d.-1 + vou (de - d4-1).
Since Pt Piyāa- ai-1 Vau / Vār, the two terms telescope:
Fo. 200 Vox d + Vou (d: d.-1) = Vāl E xc
For the covariance,
Var[æ, wo ( = ot Var (ar[ To,
A 1
(11-ā,- - (1-a:),
again using 0 ft-1- c.. Thus equation holds at time closing the induction.
Corollary 1. Af = 1 for all t 2 t1, thenfor every 2 t1 one has de = 20 and hence
20 errc - -.) q(xx xo. = N(xx; Vau
In particular, sampling can be initialized at t1 with Dt, N(vāar, aere, (1 - C.g )
Remark Let a; = (t) with At Writing ((t) exp( So ((s) ds) and d(t) as the continuous interpolation of d;, the discrete kerne equation converges to the VP forward SDE with a moving mean.
dx;
1(()) xt dt
a(t d(t)
dt
vB(t) dw..
f(0m.
mean drift due to mixer
a(t)
which is the continuous counterpart used to derive the reverse-time SDE in the main text.
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6.6 DERIVATION OF EQ. 10
The data prediction model Pap(0 " 1) approximates the clean target image, Dg Rs 2o. From the forward marginal (Eq. 6 under the dynamic mixture,
gr [x æo, o) N(ws vār de, oi
d: = O 20
+ (1 As) o æo.
so the score w.r.t. 2t is
C2 V
V x log qd s4 æig 200)
Replacing æo by Po(a,, 20 1) yields the plug-in estimator
xt vānd of
Le log q:(2;
aArE ~
df = Ds 1
AAEE + (1 - Ar) @ Lo a0 .t,,
which is Eq. (10). Moreover, Eg -i V log (0t))
2i yald
giving Eq.
11
6.7 PROOF OF PROPOSITION 1
Recall the reparameterization Y, = o1 A²). X, a. Y, and define y C2 Y,, From
Eq. and the derivation of Eq. 12 the reverse dynamics with respect to À takes the diagonal linear
SDE form
dyx = (2) O yAdA +
(10(1-AN))) @ dAs (1 (0))(22)) da
(2)
(21)
- ((2) 2 3 (2 20 A)dA
V2X o dwx.
(22)
Fix any index = (c, p). The i-th component of equation
21
is a scalar linear SDE
A; 2
dy. 2 Vi dd. +
(1 - A1=, 1 A,
dA
dA
D. A, (-) dd; + v 2A, dwx--
A
By the variation-of-coe of-oossttssss formula with integrating factor exp( ( dA.) = X3 we obtain for any E s]:
2x(1)2
A
p2100 {₂(1)2
(
1
A; 2 dr
(t)
(s(3)2 vi(s)
dA;
sare
(x((0)
72
(1-A,))
1 As T
2.11)
(t)
(i(1))
(x(t)
2 (z)
(x((0)
22T dw
72
The last Itô integral is Gaussian with zero mean and variance {,(1)² (s))) Stacking all components and returning to æt = Y, O yr yields Eq. 13 where all operations are Hadamard and 2, NO, I) independent of æ...
6.8 REVERSE SDE
By the time-reversal formula for Itô diffusion with state-independent diffusion (Anderson 1982 the reverse drift p(æ, t) u(x, f) - (²) V. log (2) with the same diffusion coefficient (t)). Substituting a and (t) g(t) yields exactly Eq. 9
6.9 FIRST-ORDER SOLVER
Notation. We write O for Hadamard multiplication and @ for Hadamard division. Recall az Vā,, of = a., and define
Y, /0 Ae); >= a 2 Yi-
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Proposition 3. Gtven an initial value Dy at time >> 0, the solution Dt of the reverse-time (cf. Eq. 12 in the main text) for E [0, s) is
22 Y,
o or.
(A) state transport
(A. (1 Aii) - (As (A. o A12 (A20X 12
T,
20
(B) domain domain-shift drift
(23)
22 0 o 2x are A) dd
T, +
(C) network--uuded denoising
A. S
+ Y,
2
A
(22222)
+
2V dt dwx.
Proof. Starting from the time-changed reverse dynamics (main-text Eq. 12 Y, so that each component Y/ satisfies linear SDE
introduce y: De 2
2 A yi dA; + ( A.)2 dA,
A dAi A; A
2 x. dii +
dy:
sare 0.3
dwx;
X
Ti v dd,/dt dt
Applying the variation-of-oon cofoostansss formula with integrating factor exp ( (2)A,) dA; X² integrat- ing from to t, and stacking all components, yields Eq. [3 after multiplying back by Y. The integral has zero mean and covariance
[A
Aett) (t))) 9i 1
Ac(t) Xx(t)* JA(O))
of 1
Cov
dw,
dr,
72 Y.
ddx//tt
T? dis/dtt
so the stochastic term remains in Itô integral form.
The only non-analytic piece in Eq, 23 the integral in (C). We approximate it by Itô-Taylor expan- sion of order at Ag:
2e(2x. aare >) Do(x.. 2ore s).
Thus, componentwise,
2222 A3 xo(-)da zo(-) (1
17
X?
which gives
AN (22 (2A2 (1-2, о tb 20.0 s).
Substituting into Eq. 23 we obtain the first-order solver:
a r, ④
{ 0 02) 3 (x.00 (#,00 1 T.)
(A.0
( (1 -Ai) (A. (1-
11 (A"02,²)
+ r,
(24)
+ r, 2
I 1- 1-2,20 1-2,00?3) @ 2 2.00
{ (1 2 12) 3 +
A.
r, 2 x
2 di
dwp
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6.10 DATASET DETAILS
IXI (MRI T1- We use the multi-center IXI dataset (London et al. 2025), comprising scans from three hospitals and mixed vendors/field strengths (Philips 3T, Philips I.ST. GE 1.5T). The task is directional TI T1-T2 mapping. We extract 2D axial slices from NIFTI volumes and form 500 training pairs and 77 test pairs. This setting primarily reflects minor contrast changes under strong structural alignment and is clinically relevant when partial protocols are acquired due to time constraints or motion artifacts.
Sentinel-1/2 Optical))) We use co-registered Sentinel-1 (SAR) and Sentinel-2 (optical) im- agery (Schmitt et al. 2018). SAR provides structure that is robust to weather; optical captures color and texture. We curate temporally aligned 256x256 image pairs covering urban, agricultural, for- est, and coastal scenes. The final split includes 1,200 training pairs and 300 test pairs. This task introduces larger appearance gaps and stronger cross-modal difficulty relative to IXI.
PSCDE (EL-to-semantic defects) PSCDE (Wang et al. 2024a) contains 700 high- righ-resslution elec- troluminescence images of polycrystalline solar cells with expert annotations for seven defect types (e.g.. linear cracks, finger interruptions, black cores). We use 420 images for training and 280 for testing with balanced category representation. This setting demands fine detail synthesis under sig- nificant acquisition variability, manufacturing differences, and heterogeneous defect morphology.
6.11 IMPLEMENTATION DETAILS
Experiments ran single NVIDIA A100 80 GB GPU on the cluster's AMD EPYC 7662 node (128 CPU cores, 512 GB RAM, InfiniBand EDR 100 Gb/s).
We train the model in PyTorch Lightning using single-node distributed data parallelism with mixed precision and a fixed random seed. The network includes a backbone (Lin et al. 2025) and a Stable- Diffusion UNet (Rombach et al. (022)) with base channel width 320, four resolution scales. Visual inputs are encoded into four-channel latent space by a KI. autoencoder with 32x spatial reduc- tion.
The adaptive domain-mixing module N₂. expands d into a spatial field via a lightweight convolu- tional network (channel progression 1 - 8 - 16 - C) augmented with sinusoidal position encod- ings. Its output is squashed by a calibrated logistic function so that every entry lies in (0, 1).
For the reverse process employ the first-order solver derived in Algorithm 3 At every step the sampler reconstructs the quantities Y, = a (1 Ar) and A; = o,,Y, and applies the analytic transport, domain-drift, denoising, and stochastic components to advance the state. Training mini- mizes the mean- equanes error between the predicted and ground-truth latents.
Optimization uses AdamW with base learning rate of 5 x 10-5 for the latent model parameters, while the domain-mixing module receives a tenfold higher rate. Weight decay set to 0.01, gradi- accumulation is disabled, and exponential moving averages are not maintained. The diffusion horizon is fixed at T' = 1000; for the PSCDE experiments the trainer executes 5,001 optimization steps, evaluates every 70 iterations, logs images every 500 steps, and saves checkpoints every 2,000 steps. Dataloader settings include batch size four, 256 x 256 center crops, four worker threads, and a prefetch factor of two. Before optimization begins we warm-start from pre-trained checkpoint Stabilitt-All 2022) so that the dynamic domain mixer can fine-tune from meaningful initializa- tion.
For fair comparison across baselines (including BBDM and ABRIDGE), we adopt consistent train- ing budget within each dataset: all methods are trained for 20,000 steps on Sentinel, 10,000 steps on IXI, and 5,000 steps PSCDE. Regarding sampling, because BBDM does not provide a fast- sampling variant, we evaluate using its recommended default of 1,000 sampling steps.
6.12 BASELINES AND METRICS
We compare against Pix2Pix Isola et al., 2017) (supervised cGAN with L1 reconstruction; effective on well-aligned pairs but prone to structural issues under complex gaps); BBDM (Li et al., 2023) (Brownian- bridge diffusion connecting source and target via stochastic paths; strong semantic align- ment but high sampling cost and potential instability under large modality gaps); ABridge (Xiao)
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Table 3: PSCDE dataset: semantic translation metrics. Higher is better except Hausdorff.
Method
Dice f
ToU
Precision 1
Recall
Hausdorff
Skeleton F1
Lincar
0,46 = 0.33
0.37 + 0.34
0.66 + 0.32
0.41 = 0.34
59.53 + 62.50
0.59 + 0.33
Channel Non-linear
0.46 0.36
0.38 1 0.36
0.78 1 0.25
0,42 + 0.36
43.03 1 53.31
0.57 1 0.36
Fully Non-lincar
0.49 = 0.35
0.40 + 0.35
0.78 + 0.26
0.43 = 0.36 39.87 + 50.76
0.61 + 0.34
et al., 2025) (explicit semantic bridging constraints within diffusion; improved modality consistency but reduced performance fine-scale structures and low-contrast data); and fixed-schedule dif- fusion baseline aligned with DOSSR (Cui et al. (025)) adapted to cross-modal translation with the same UNet backbone and a linear domain interpolation schedule.
Metrics include SSIM and PSNR for structural similarity and signal quality, MSE and MAE for pixel-wise accuracy.
6.13 ADDITIONAL ABLATION DETAILS
To further study the effect of spatial adaptivity, we introduce per-channel, spatially uniform variant of our schedule. In this setting, each channel receives a value Ac(t) E R, producing channel- wise vector X(t) E C that varies across channels but remains uniform across all spatial locations. The network therefore outputs a tensor of shape (B, 1,1, which is broadcast to (B, C, H. W) during modulation. For comparison, the full method modulation depends on both channel and spatial position, yielding A(t) E CR xx with entries Ae(1, p) location p. The lambda network applies a per-channel polynomial for this channel-wise setting: (1) = [c(n(()) = Σ3 de nn()",, where de, are the learnable coefficients and 1n(() is the base linear schedule. The resulting tensor has shape (B, C, 1, 1) and is broadcast (B, C, H, W).
spatially varying schedule in which the
As presented in Tab. 3 this per-channel uniform schedule attains Dice score of 0.46 and a Haus- dorff distance of 43.0 pixels. Compared the fixed linear schedule (Dice 0.46, Hausdorff 59.5), we observe a substantial reduction in boundary еттоr but no improvement in Dice, consistent with the high precision (0.78) and relatively low recall (0.42) of this variant. The skeleton- and connectivity- based metrics (skeleton F1 0.57, connectivity loU 0.31) indicate that channel-wise adaptation im- proves structural coherence and boundary sharpness, but still falls short of the fully spatially varying model, which achieves both higher Dice and lower Hausdorff. These findings suggest that channel-wiss wise modulation accounts for part of the geometric and connectivity gains, while spatial adaptivity provides additional boost in overlap accuracy and global mask quality.
6.14 ROBUSTNESS ANALYSIS
To assess robustness to registration noise, we systematically misalign the PSCDE inputs by 0-8 and re-evaluate the segmentation metrics (Fig. 4). Performance is essentially stable under small perturbations: at 0 px and 1 px shifts, Dice remains around 0.49-0.50 and even slightly improves, while Hausdorff decreases from 51.7 to 46.9 pixels and skeleton Fl increases from 0.61 to 0.63. For larger misalignments (4-8 px), quality degrades gradually rather than catastrophically (Dice drops from 0.46 to 0.43, Hausdorff increases to 63.1 pixels, and skeleton F1 remains above 0.55), indicating that the proposed model is moderately robust to realistic registration errors and maintains coherent structural predictions even under non-trivial spatial perturbations.
6.15 ADDITIONAL RESULT VISUALIZATION
We present additional qualitative visualization on the testing dataset for the four different image translation tasks: Fig. 5 for Sentinel-1/2 (SAR-Optical) remote sensing application; Figs. and 7 are for the MR TI to T2 and T2 to Tl imaging application, respectively; Fig. g for Solar Panel Fllecrooumineecence semantic masking application.
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ToU
Precision
0.52
3.40
0.42
0.39
0.42
0.36
0.47
0.802
0.37
0.75
9.36
0.50
0.35
0.525
0.503
9.34
Recall
Hausdorff
Skeleton F1
0.63
0.62
0.42
50
0:52
0.62
0.59
0.43
14
50
0.42
0.50
1
2
3
A
3
2
3
5
6
Thantation Hith Igtmeett (ploet)
Tandirion Misslgrment (paaedd
Thunder Nasiggmeet (plces)
Figure 4: Additional assessment regarding the robustness to the registration error.
6. 6.16 ALGORITHMS
We provide the algorithms for the core implementations of the proposed method. Algorithm builds domain-shift guidance sildd by spatially modulating a linear step schedule with learned mixer Sg- The result is a per-time, per-location mixing map A₃ (p) that blends the source latent ae and the target latent zo into time-varying guidance signal de, Algorithm 2 then trains the diffusion network bg via score matching.
Algorithm 3 performs reverse-time generation on the spaced grid: each step it combines state transport, domain-shift drift, network- network-guided guided denoising, and stochastic noise- weiihtee by the schedule ratios and decode the final image.
Algorithm 1 Domain-Shiff Field Construction
Require: linear step schedule { 17 source latent 28"; target latent ro: spatial mixer Soi posi- tion encoding map क; clamp 8 10-1
1: Pre-compute: B + - log c 1-e- a t- -28 2,8
2: for = 0 to T do
3: Broadcast base scalar: X (p) Xinn for all spatial p
4: Spatial modulation: (p) t So So(Aimm Aimm T(p))
5: Zero-centered modulation: (p) + 2 hp,c(p) -
6:
Soundary-preservigg interpolation: (p) + Ninn [1+ (p) (1 X!n))]
1
7: Logistic squashing: Ar,((p) + da (p) - B)). where (2)) = 1 1
8: end for
9: Clamp endpoints: (p) 0, Ar.a(p)
10: Form domain mixture (Eq.
i (D: d2 t As o शह
(1-A))
11: Provide de to the diffusion sampler at step as the domain-shiff guidance field.
6.17 ADDITIONAL DISCUSSION
Positioning relative to diffusion bridge models: The underlying generative mechanisms between the proposed method and Diffusion Bridge-basc based method for image translation are fundamentally different. Bridge-based methods explicitly redefine the forward process path conditioned on
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Source
Target
Generated
Figure 5: Visualization of the SAR to Optical image translation task; samples are randomly picked from the testing dataset.
both endpoints (Zo, y), constructing a Doob-h-ttrasffrm or related bridge kernel q(x; xo. y)- Yet, our method retains the standard VP-type diffusion formulation and only modifies the drift by injecting an adaptive domain-shift term. This very useful in practice because operates directly on the VP-SDE backbone, they are structurally compatible with existing large-scale pretrained diffusion priors (e.g., latent diffusion models or domain -omain--pecific medical priors). In practice, this allows us to initialize from distill into models that have already learned powerful generic image statisties, and then specialize them via a lightweight drift correction that accounts for the cross -oos-- gap.
Bridge-based methods (He al. 20024): Zheng al. (2025); Zhou al. (2023): Liu et al. (0233)
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Source
Target
Generated
Figure 6: Visualization of the Tl T2 MRI translation task; samples are randomly picked from the testing dataset.
Xiao et al. (2025); Li et al. (2023) employ a different forward kernel and noise schedule tied to the bridge dynamics; this often requires training from scratch or re-deriving the entire sampling scheme, making it significantly harder to plug into the current diffusion ecosystem. And, diffusion bridge models need design solvers around more intricate (Li et al. (2023)), andpoint-conditines kemel;; even with implicit or ODE reformulations, the resulting dynamics tend to be numerically heavier and less amenable to extreme step-size reduction under comparable quality targets.
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Source
Target
Generated
Figure 7: Visualization of the T2 to TI MRI translation task; samples are randomly picked from the testing dataset.
Regimes where CDTSDE provides marginal gains: As discussed in the main text, the three eval- uated tasks (IXI (MRI contrast translation), Sentinel (SAR -opocccl,ial),, and PSCDE (EL-tsemantic defects)) form an approximate spectrum of increasing modality discrepancy and structural com- plexity. On IXI, where the contrast shift between TI and T2 is relatively mild and the anatomical structures are highly aligned, the global linear schedule already traces reasonable transition path between source and target domains. In this low-gap regime, the dynamic comain-mixtun schedule has limited room improve the trajectory, which is reflected in the modest quantitative gains (e.g..
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Source
Target
Generated
Figure 8: Visualization of the Electroluminescence to Semantic mask translation task; samples randomly picked from the testing dataset.
small SSIM/PSNR improvements) and visually near-identical reconstructions for many slices. Em- pirically, we observe that the learned A, fields on IXI often stay close to the baseline linear schedule, effectively behaving as gentle refinement rather than qualitatively different path.
Failure and limit cases: We emphasize that CDTSDE is not universally necessary replacement for fixed schedules. In particular, when the underlying task is too easy (low modality gap, clean ac- quisition, strong spatial alignment), the additional fiexibility of pixelwise, dime-dependent sched-
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ule may not translate into visible benefits. In our experiments on IXI, do not observe systematic hallucination new structures or topology breaks introduced by CDTSDE compared to the linear baseline. Instead, the typical limit case is that both methods already produce anatomically plausible reconstructions and CDTSDE only changes local contrast or edge sharpness slightly.
When CDTSDE most beneficial: In contrast, on Sentinel and especially PSCDE, the modality gap and local heterogeneity are much larger: intensity statistics, textures, and semantic structures differ substantially between domains. these settings, the assumptions underpinning a global linear schedule are violated, and the pixelwise dynamic schedule has substantial freedom to reduce path energy and avoid high-cost, off-manifold regions. This is reflected both in the larger improvements in structural and semantic metrics (c.g.. Dice and Hausdorff distance PSCDE) and in qualitative behavior, where CDTSDE better preserves fine structures and boundaries. These observations are consistent with the theoretical result that the pixelwise schedule strictly dominates any global sched- ule under heterogeneous local geometry and nondegenerate contrast, and suggest that CDTSDE is most useful when cross-modal translation is genuinely hard and spatially non-uniform.
6.18 FUTURE WORK
Quantifying translation difficulty and adaptive deployment: A natural next step is to define quantitative measures of cross-modal translation difficulty and relate them to the observed benefit of CDTSDE. Simple statistics such cross-domain histogram distances, mutual information, or structural similarity under a naive baseline could serve as coarse indicators of heterogeneity. This would support adaptive deployment, where system uses inexpensive pre-analysis to decide whether the added complexity of CDTSDE is warranted or a simpler linear schedule suffices.
Extensions and future work CDTSDE can be extended in several directions. Combining the dy-
namic schedule with flow-matching (Lipman et al. (2023)) or probability-flow (Wang et al. (2025)) )
ODEs could enable distillation into very few-step or even single-step generators while retaining the geometry-awart path. Additional regularization on A₃ (e.g.. spatial smoothness or curvature penal- ties) may further guard against pathological local mixing. Multi-scale or region-aware schedules could allocate more flexibility to challenging regions(Xu et al. (2025). Finally, explicit diagnostics to detect when CDTSDE offers little advantage would help practitioners choose between dynamic and linear schedules without extensive tuning.
6.19 FAILURE MODE ANALYSIS
Fig. I0 presents several representative counterexamples highlighting regimes where CDTSDE does yield clear benefits and may introduce minor artifacts. We collect those from the PSCDE bench- mark. The PSCDE exhibits strong nonlinear modality gaps and complex defect topology. As circled (in yellow) the figure, CDTSDE occasionally produces small hallucinated streaks faint bound- ary irregularities when the local cross-domain geometry is highly ambiguous. These issues typically arise near thin cracks, low-contrast defect boundaries, or regions with strong cexture-structure inter- ference. Similar failure cases are difficult to obtain for the IXI dataset because the T1/T2 contrast shift is mild and structurally well-aligned, leaving little room for dynamic scheduling to deviate from the global baseline. These counterexamples help clarify that CDTSDE is most beneficial in high- gap, heterogeneous translation regimes, and its advantage diminishes as the domain shift becomes simpler.
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CDTSDE
ABridge
0.28
028
0.4
0.15
0.2
0.05
0
05:00 PM
05:30 PM
06:C0 PM
04 PM
PM
PM
BBDM
DOSSR
03
0.3
02
0.1
01:20 PM 01:30
01:45PM
02:00
11:10AM
11:22 AM
11:30 AM
11:40
Loss DBIM
20
10 MA
Pix2Pix
fenil 8,2970
20-
Fiont Dier Lene 2400
25
50
135
150
175
Discriminntar
10 MA
0.5-
Disctininterr Lass
0.3
0.2
50
125
150
Epoch
Figure 9: Loss curves for the CDTSDE and baselines: ABridge, BBDM, DoSSR, DBIM, Pix2Pix.
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Table 4: Summary of Notation.
Symbol
Description
Nare, Naar
Source and target image spaces (e.g., SAR vs. OPT, or different MR/CT contrasts).
Paired training sample, with #5 E the observed source image and up E &aar the ground-truth target image.
mg
Generated target-domain image produced by the model O. Diffusion state time & in the target domain; t be continuous [0, 1) or discrete (t E {0,.... T)).
1,
Diffusion time indices and total number of diffusion steps
H. W.C
Image height, width, and number of channels, respectively.
C..... (I......... (1..... W)
Channel and spatial (pixel) indices.
8.
Variance schedule at step in the variance-preserving (VP) diffusion process.
as.
Per-step and cumulative signal coefficients in the forward diffusion; typ- ically 6, II:_. 04-
of - a
Noise variance time in VP diffusion.
Ac (0, cxxxX
Domain-mixing field time t.
(r,(()) de
Scalar mixing weight channel cand spatial location p (clement of Ad) c @ + (1 Ac) æo, used as guidance for cross-domain translation.
Domain mixture
0.0
Hadamard product and division, respectively.
Voa (1 Ar)
Effective signal scale.
A a Y;
Local mixing- mixing-to-noiss to-noise ratio.
y x Y.
Rescaled diffusion state expressed in X-coordinates. Wiener increment.
dwa
co(0,, a .t) h,(p)
Conditional noise- sdise-prediction network (score model) parameterized by 6. Learned spatial gating map produced by the schedule network at time Positional encoding of spatial index p.
(pp) E Rd
dtp(t)> 0
Local metric on the trajectory velocity at channel location and time
f.
Uep(de.p().tt)
Strictly convex local potential acting on the mixed-domain trajectory value (t) time t.
Eid]
Path energy functional the domain-mixing trajectory d(I), integrating local kinetic and potential terms over E [0, 1).
29

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Published as a conference paper at ICLR 2026
Algorithm Score- Matching
Require: paired dataset D with targets zo and source observations 17""; variance prreeervvigee sched- ule (o=) ) mixer Ng; score network t
1: repeat
2: Mini-batch draw: B {(=))) Srre y 1i=1 D
(6) a00
3: Random timestep: 4 Uniform (0,
1} independently for each
4: Encode conditions: Clatent f- Enciar (20
(i)
5: VP coefficients: Oz, IT:= Csss
Otg x VT Oit
Yτ, + von. (1 Ac,))
6: Mixing field: A, Nol base C,H,WW
A
7: Domain mixture: dee Ao
O Fo
(6)
(1
A
o I
6 vot,
8: Forward diffusion: draw e()) ~ ~N(0,1 1) and set Ltc.
(8)
(1)
oz. (1)
9: Score prediction: E t Do (2) ti latont
10: Score-matching loss: +- B 1== ( 201
B
11: Parameter update: x- 0 VgL,
no V
12: until convergence
Algorithm First-Oree Order Sampler
Require: steps N, latent shape s, condition image Foond
1: Build spaced schedule: {(B,, A,, Yr. A.hee I MAKESCHEDULE(N)
2: Encode source latent: w +- ENCODEFIRSTTTTACCCSOOOO
3: Sample initial state: Draw ((,0) 7) and set +- Y twaa 20 + traas
4: Reverse loop over indices:
5: for successive (s, t) in I (descending) do
6: Noise prediction: e, + PREDICTNOISE(C,
7: Predicted clean sample: ((()
ax
11122
Fe
V a, 1c,
r,
8: Ratios: Ag"
T'
Fr +-
r
9: State transport: transport
rr 2 Fz
Nt
A.
10:
Domain-shift drift: I + r,
1
11 A
1
Ag
Network-guided term: deennise
T, (1
r2)
-0
(s)
12: Stochastic variance: That + X2 12
13: Draw noise: 2 ~ NN(0, I), noise +- T, That 2
14: State update: v4 itransport priift dencise
noise
15: end for
16: Decode to image space: return (DECODEFFRSSTSTACCE()))) 1)/2
30

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Published as a conference paper at ICLR 2026
d
Figure 10: Counterexamples for failure mode analysis.
31


Reviews:
[{"review_id": "ZLYf4jyXYI", "reviewer": "Reviewer_4o8z", "summary": "This paper introduces CDTSDE (Cross-Domain Translation SDE), a novel diffusion-based framework for cross-modality image translation. The key idea is to embed adaptive domain-shift dynamics directly into the diffusion process via a spatially varying mixing field that evolves throughout reverse-time sampling. This dynamic field replaces the conventional fixed linear interpolation between source and target domains, allowing geometry-aware, low-energy paths that stay closer to the data manifold. Extensive experiments on three benchmarks show consistent gains in various matrics.", "questions": "Please refer to weakenesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-motivated and conceptually clear. It generalizes traditional linear domain-shift formulations to a nonlinear, manifold-aware framework, supported by solid theoretical analysis and proofs.\n\n2. The method is technically sound and demonstrates strong generality, making it straightforward to apply across different diffusion architectures and cross-modality image translation tasks.\n\n3. Extensive experiments on three benchmarks IXI, Sentinel, and PSCDE show consistent gains in various metrics, outperforming GANs (Pix2Pix) and diffusion baselines (BBDM, DOSSR).", "weaknesses": "1. The method reduces to a linear domain-shift scheme when only a single diffusion step is used, implying that it still depends on multiple denoising steps for stable performance. This reliance could pose an efficiency limitation and hinder direct adaptation to flow-matching or single-step generative methods.\n\n2. While the method shows clear conceptual advances, its quantitative improvements over DOSSR on the Sentinel and IXI datasets are relatively minor, indicating that the advantages may be limited in tasks with larger modality discrepancies.", "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": "h2NDrhaJNb", "reviewer": "Reviewer_Tz66", "summary": "The paper “Adaptive Domain Shift in Diffusion Models via Latent-Space Normalization” proposes a training framework to mitigate the performance degradation of diffusion models under domain shift scenarios. The authors argue that existing fine-tuning or adapter-based methods fail to preserve generative quality when target-domain data are scarce or stylistically distinct. To address this, they introduce an adaptive latent-space normalization (ALSN) module that aligns latent statistics between source and target domains through learned normalization parameters, dynamically adjusted during training. Experiments on multiple domain adaptation benchmarks (Art→Photo, Synthetic→Real) show modest improvements in FID and CLIP similarity compared to baseline fine-tuning or LoRA-based adaptation.", "questions": "Have you evaluated whether ALSN affects the diversity or mode coverage of generated samples, especially when applied across multiple distinct domains?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "+ Domain adaptation for diffusion models is a growing and challenging topic, and addressing domain shift in generative tasks is an important research direction, especially as diffusion models become widely deployed across diverse domains.\n+ The proposed ALSN approach is computationally efficient and easy to integrate into existing diffusion pipelines, making it potentially attractive for practitioners seeking domain-robust generative models.", "weaknesses": "- The core idea, i.e., adjusting latent normalization statistics to align source and target distributions, closely resembles well-known techniques in domain adaptation. The paper primarily recontextualizes these ideas within diffusion models without offering substantial theoretical or methodological innovation. This limits the paper’s conceptual contribution.\n- While results are shown, the paper does not convincingly explain why ALSN improves performance or how it interacts with diffusion timesteps and denoising dynamics. There is no analysis of latent trajectory behavior, feature drift, or stability to justify its effectiveness.\n- Improvements in metrics such as FID or LPIPS are small (often within variance range), raising doubts about the real impact. The method’s simplicity is a strength, but it also highlights how incremental the advance is.", "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": "Ta2cGB0DPq", "reviewer": "Reviewer_dYd8", "summary": "The paper introduces Cross-Domain Translation SDE (CDTSDE) for Cross-modal image-to-image translation. CDTSDE embeds an adaptive, spatially varying mixing field into the diffusion process. The forward marginal is centered on a per-pixel blend of source and target, and the reverse dynamics add a restoration drift aimed at keeping large steps near the data manifold. Reported results show improved PSNR/SSIM and PSCDE on several datasets.", "questions": "– Which aspects of the mixing field matter most (spatial vs. channel, monotonicity, positional encoding)? Please include ablations. \n\n– How does the method compare directly to recent DBM baselines on the same splits? \n\n– What drives the reported efficiency—the restoration drift, the step schedule, or early stopping? \n\n– Please provide training curves for the method and baselines.\n\n[1] Cui, et al. \"Taming Diffusion Prior for Image Super-Resolution\nwith Domain Shift SDEs\", NeurIPS 2024\n\n[2] Zhou, et al. \"Denoising Diffusion Bridge Models\", ICLR 2024\n\n[3] He, Guande, et al. \"Consistency diffusion bridge models.\" NeurIPS 2024\n\n[4] Liu, et al. “I$^{2}$SB: Image-to-Image Schrödinger Bridge”, ICML 2023\n\n[5] Zheng, et al. “Diffusion Bridge Implicit Models”, ICLR 2025", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "– Across three modalities/datasets, the method reports better SSIM/PSNR and strong PSCDE structure metrics. \n\n– Concept of putting domain-shift adaptation inside the dynamics (rather than as external guidance) is principled.", "weaknesses": "– The paper is difficult to follow: many equations and notations are introduced without clear motivation or explanation of each component’s role, making the method hard to reconstruct; it focuses more on how things are done than why they are needed.\n\n– The propose solution is fairly incremental since the core contribution is to propose an adaptive interpolation of $\\hat{x}^{src}_0$ and $x_0$, replacing time-varying interpolation of source and target [1]\n\n– Efficiency claims are supported largely by empirical numbers; there is little analysis explaining why the proposed dynamics/sampler should be more efficient.\n\n– The setting effectively reduces to paired translation. A substantial line of prior work on diffusion bridge models already targets this regime and reports SOTA results (e.g., [2,3,4,5]), but these are neither discussed nor compared.\n\n– Fixing a number of training steps and concluding CDTSDE is more efficient may be unfair: different methods have different designs/optimizers. The paper should provide training curves (and, ideally, wall-clock) to show faster convergence.", "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": "wV5cRa1UuC", "reviewer": "Reviewer_jiJC", "summary": "The paper proposes a new diffusion-based framework (CDTSDE) for cross-modality image translation. The core idea is to embed the domain shift directly into the generative process, instead of relying on a global linear blend between source and target images. The authors introduce a spatial-channel varying weight matrix for combining source and target paired data, which serves as the condition in diffusion models. The forward and reverse sampling processes are developed. Extensive experiments across several applications are conducted.", "questions": "- Your method conditions directly on paired $(\\hat{x}_0, x_0)$ and uses $x_0$ explicitly in the adaptive mixture at training time. How sensitive is CDTSDE to imperfect pairing or slight misregistration, especially for SAR→optical and PSCDE where pixel alignment can be noisy? Have you tried training with synthetically perturbed / misaligned pairs to evaluate robustness? \n\n- Many real cross-modality scenarios do not provide paired data (e.g. historical SAR ↔ optical, multi-scanner MRI). Can your formulation be adapted to unpaired or weakly paired data, or is the approach fundamentally limited to paired supervision? Please clarify what breaks if $x_0$ is not available at training time, at least in discussion.\n\n- You compare a global linear schedule $\\eta_t$ vs. your spatially varying $\\Lambda_t$ and report gains (e.g. Dice 0.46→0.49, Hausdorff 59.5→39.8 on PSCDE). Could you also report an intermediate baseline: a per-channel but spatially uniform schedule (i.e. $\\Lambda_t$ depends on channel and $t$, but not $(p)$)? This would help isolate whether most of the benefit comes from spatial adaptivity or just from deviating from a single scalar $\\eta_t$.\n\n- Theorem 1 proves that allowing pixelwise $\\Lambda(t)$ yields strictly lower path energy $E[d]$ than any global schedule under heterogeneity assumptions. How should we interpret this physically?   Is lower path energy empirically correlated with perceptual realism / fewer artifacts?  Do you ever observe cases where $\\Lambda_t$ lowers energy but produces locally inconsistent textures (i.e. visually implausible mixing of source and target in the same region)?\n\n- In which regimes does CDTSDE not help? For IXI (milder contrast shift), you mention improvements are “marginal.” Can you show qualitative counterexamples where CDTSDE produces artifacts (hallucinated structures, topology breaks, etc.) so we understand the limits of the method? \n\nI look forward to the response from the authors.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "-  The paper gives a convincing geometric argument for why globally linear / fixed schedules send trajectories off-manifold, creating large correction burdens and semantic drift, motivating the method. \n\n- Instead of a single scalar $\\eta_t$, $\\Lambda_t$ is a learnable pixelwise, channelwise field with monotonic constraints and boundary clamping. This is novel and intuitively matches heterogeneous cross-modal shifts (texture, contrast, anatomy), and is also grounded by the theoretical analysis.\n\n- Building on the pixelwise, channelwise weight $\\Lambda_t$, the authors developed the corresponding forward and reverse processes, enabling the idea to be realistic.\n\n- Experiments span three regimes of difficulty: (i) relatively mild contrast change (MRI T1↔T2), (ii) strong cross-sensor shift (SAR→optical), and (iii) extremely semantic mapping (electroluminescence image → defect mask). The proposed method shows convincing results across different tasks.", "weaknesses": "- The core formulation assumes access to paired (source, target) images during training. However, many cross-modality settings (especially SAR↔optical, certain medical domains) are weakly paired or unpaired in practice. The method’s dependence on paired data limits its applicability, but this is not deeply analyzed. \n\n- The presentation is mathematically dense. Some key definitions (e.g. how $\\Lambda_t$ is actually predicted by the network in practice, how constraints like monotonicity in $t$ are enforced during training, and how the logistic squashing with $\\epsilon$ is implemented during backprop) are split across sections and the appendix. This may make reproduction harder for non-experts, despite the promise of releasing code later. Meanwhile, more explanation on the definition equations of $\\Lambda_t$ should be included to strengthen readability.", "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)