The Noise Plane: A Diffusion Bridge Across Exponent and Dispersion

Xuran Hu; Yuru Jia; Tengxi Wang; Miloš Daković; Ljubiša Stanković; Wufan Zhao

cs.CVOct 27, 2026

The Noise Plane: A Diffusion Bridge Across Exponent and Dispersion

Authors: Xuran Hu, Yuru Jia, Tengxi Wang, Miloš Daković, Ljubiša Stanković, Wufan Zhao

Organizations: The Hong Kong University of Science and Technology (Guangzhou), Guangzhou 511453, China · KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden · University of Montenegro, 81000 Podgorica, Montenegro

Abstract

Diffusion-based denoising commonly uses Gaussian perturbations, whereas sensor noise follows diverse signal-dependent statistical laws. In this paper, we propose $p$ -Bridge, a unified measurement-indexed diffusion framework for Tweedie noise. The noise exponent $p$ describes the variance--intensity relation, while accumulated measurement $ν$ controls the noise level. Together, they define a noise plane that includes Gaussian, Poisson, and Gamma models. The stochastic bridge connects measurement distributions through independent increments, giving denoising a physical interpretation as progressive measurement accumulation. Both the starting and stopping indices can be specified, allowing restoration from an observed measurement level to a prescribed higher level. Measurement superposition yields a shared reverse-mean update, paired with a deviance-based objective for learning the required conditional mean. A latent compound Poisson--Gamma construction enables sampling at intermediate exponents. A single exponent-conditioned network adapts to different Tweedie noise models without separate training for each exponent. Experiments show that exponent conditioning improves cross-exponent robustness over fixed-exponent models. Using estimated exponents, the shared network achieves robustness comparable to an oracle-selected set of five specialist models. Experiments on synthetic images, FMD fluorescence microscopy, and multi-sensor SAR images further demonstrate its effectiveness across different signal-dependent noise regimes.

Figures & tables

Fig. 1: Noise regimes along the Tweedie exponent axis. Top: illustrative noise at representative exponents. Bottom: real observations with estimated exponents $\hat{p}$ .

Fig. 2: Measurement-indexed denoising trajectories on the noise plane. $p$ controls signal dependence, while increasing $\nu$ reduces noise.

Fig. 3: Network architecture and training losses of $p$ -Bridge, with a deterministic restoration trajectory on an FMD crop.

$p$	$x_{\nu}$	index	$c^{*}(z;\nu)$	$q(z_{t}\mid z)$
$0$	$x_{\star}+\sigma\xi/\sqrt{\nu}$	frames	$\propto e^{-z^{2}/(2\nu\sigma^{2})}$	$\mathcal{N}(\alpha_{t}z,\sigma^{2}\alpha_{t}\Delta_{t})$
$1$	$\mathrm{Pois}(\nu x_{\star})/\nu$	exposure	$\nu^{z}/z!$	$\mathrm{Binomial}(z,\alpha_{t})$
$(1,2)$	$Z_{\nu}/\nu$	exposure	series ∗	Binom. $\rightarrow$ Beta ∗
$2$	$\Gamma(\nu,\nu/x_{\star})$	looks	$z^{\nu-1}/\Gamma(\nu)$	$z_{t}/z\sim\mathrm{Beta}(\nu(t),\Delta_{t})$

TABLE I: Bridge states and thinning kernels across the supported exponent axis.

severity	$p{=}1$	$1.25$	$1.5$	$1.75$	$2$	told $p_{\mathrm{data}}$	blind $\hat{p}$	$\Delta$	envelope
	fixed specialist, worst case over $p_{\mathrm{data}}$					conditional			envelope
severe	18.997	20.150	21.339	20.036	18.007	22.260	22.260	$+0.921$	22.254
moderate	20.921	21.899	23.112	22.903	21.747	24.635	24.637	$+1.525$	24.631
mild	24.142	25.052	26.022	26.336	25.551	27.436	27.435	$+1.099$	27.431

TABLE II: Minimum PSNR (dB) across tested noise exponents, averaged over three corruption seeds at

\mathrm{NFE}=5

. “Told” and “blind” use true and estimated exponents, respectively. “Envelope” uses oracle-selected specialists.

\Delta

is the blind model’s gain over the best fixed-exponent model. Bold highlights the best fixed-exponent result and the conditional result.

Classical, transform-based, and Gaussian-bridge baselines
method	params (M)	$\mathrm{NFE}$	severe, $\eta{=}1$			moderate, $\eta{=}0.5$			mild, $\eta{=}0.25$
method	params (M)	$\mathrm{NFE}$	$p_{\mathrm{data}}{=}1$	$1.5$	$2$	$1$	$1.5$	$2$	$1$	$1.5$	$2$
BM3D [ 1 ]	—	1	18.63/.493	17.59/.479	16.87/.467	23.19/.632	22.70/.622	22.08/.609	26.58/.763	26.21/.747	25.63/.724
VST + BM3D [ 17 , 43 ]	—	1	21.21/.485	20.12/.434	19.86/.462	23.87/.634	23.77/.638	23.62/.641	26.89/.778	26.76/.778	26.59/.776
Noise2VST [ 20 ]	32.64	1	21.99/.543	21.53/.521	21.05/.499	24.71/.691	24.56/.689	24.40/.685	27.66 / .812	27.55/.812	27.38 /.811
$p$ -Bridge, $p{=}0$	15.19	5	18.67/.407	17.51/.374	16.97/.357	21.39/.541	20.04/.503	19.13/.480	24.85/.701	23.52/.666	22.42/.642

TABLE III: Synthetic-image denoising results. PSNR (dB) / SSIM are averaged over three corruption seeds. Bold and underline mark the best and second-best results per metric. “—” denotes no trained parameters.

method	CF-FISH	CF-MICE	TP-MICE	mean
BM3D [ 1 ] ∗	27.01/.826	33.18/.942	31.63/.886	30.61/.885
VST + BM3D [ 17 , 43 ] ∗	25.73/.809	33.62/.946	31.55/.889	30.30/.881
Noise2VST [ 20 ] ∗	28.58/.879	34.02/.949	32.14/.895	31.58/.908
$p$ -Bridge, $p_{\mathrm{model}}{=}0$ ∗	24.68/.811	31.18/.932	30.45/.887	28.77/.877
NAFNet [ 7 ]	28.23/.878	34.06/ .951	32.61 / .898	31.63/.909
Restormer [ 8 ]	28.00/.873	34.19 / .951	32.39/ .898	31.52/.908

TABLE IV: FMD denoising results. PSNR (dB) / SSIM are averaged over three seeds on the held-out field. Bold and underline mark the best and second-best results per metric. ∗ Without fine-tuning.

method	$\mathcal{M}$ / $\mathrm{RIS}$ , lower better
method	miniSAR	FARAD-X	TerraSAR-X	mean
BM3D [ 1 ]	32.5/55.1	41.0/78.1	49.2/109.5	40.9/80.9
VST + BM3D [ 17 , 43 ]	34.7/46.4	35.8/64.2	45.4/77.9	38.6/62.8
Noise2VST [ 20 ]	25.5/37.9	30.5/50.3	39.6/77.5	31.9/55.2
$p$ -Bridge, $p_{\mathrm{model}}{=}0$	36.3/63.8	39.9/73.6	46.7/100.9	40.9/79.4
NAFNet [ 7 ]	12.0 / 17.4	21.8 / 34.7	32.2 /60.8	22.0 / 37.7

TABLE V: Real SAR despeckling results.

\mathcal{M}/\mathrm{RIS}

are averaged over sensors. Lower is better. Bold and underline mark the best and second-best results per metric.

Fig. 4: Effect of the noise exponent at matched relative noise. Top: clean image $x_{0}$ and noisy observations; percentages give the fraction of pixels at the intensity floor. Bottom: relative noise $\mathrm{SD}/x_{0}$ , normalized at $\mu_{\mathrm{ref}}$ ; values give the maximum-to-minimum ratio.

Fig. 5: PSNR (dB) across model exponents (rows) and data exponents (columns), averaged over three corruption seeds at $\mathrm{NFE}=5$ . Bold marks matched exponents, and shading shows the PSNR loss relative to the best result in each column.

Fig. 6: Synthetic-image restorations at moderate noise ( $\eta=0.5$ ), with one row per noise exponent. Corner values show PSNR (dB).

Panel of Figure 12 — Fig. 7: FMD restorations, one crop per configuration. Corner values show PSNR (dB).

component	full model	ablation	min. PSNR	$\Delta$
Full conditional model			$22.260$	–
Reconstruction weight	$\mathbb{E}_{t}[\nu(t)/\nu_{\mathrm{obs}}]$	$\nu(t)/\nu_{\mathrm{obs}}$	$21.916$	$-0.345$
Exponent-dependent weight	$\mu_{\mathrm{ref}}^{\,2-p}$	$1$	$22.248$	$-0.012$
Moment regularization	$\lambda_{\mathrm{mom}}=1$	$\lambda_{\mathrm{mom}}=0$	$22.251$	$-0.010$
Consistency regularization	$\lambda_{\mathrm{cons}}=5$	$\lambda_{\mathrm{cons}}=0$	$22.051$	$-0.210$
Index schedule	Geometric	Linear	$21.992$	$-0.268$

TABLE VI: Component ablations at severe noise and

\mathrm{NFE}=5

, using the true exponent. Each row changes one setting. Minimum PSNR (dB) is taken over the tested data exponents.

\Delta

is the difference from the full model.

Fig. 9: Qualitative component ablation at severe noise, $p_{\mathrm{data}}=1.5$ . Corner values report PSNR (dB).

Fig. 10: Excess relative error over the matched-exponent model, by intensity decile and signed exponent error. The box marks $|\hat{p}-p|\leq 0.25$ , and the gray strip shows the range of the estimator errors. Rows average exponent pairs with the same error.

vs. fixed $p{=}1.5$	$\delta{=}0$	$\delta{=}0.25$	$\delta{=}0.5$	$\delta{=}\infty$
severe	$+0.922$	$+0.830$	$+0.107$	$-1.707$
moderate	$+1.523$	$+1.311$	$+0.184$	$-1.863$
mild	$+1.414$	$+1.233$	$+0.177$	$-1.796$

TABLE VII: PSNR gains (dB) over the fixed

p=1.5

model under worst-case exponent errors. The conditioning exponent satisfies

|\hat{p}-p_{\mathrm{data}}|\leq\delta

within the tested exponent range.

\delta=0

uses the true exponent, while

\delta=\infty

removes the error constraint. Positive values favor the conditional model.

Fig. 11: Variance–mean relations by detector class. Each configuration is normalized by its fitted variance at the median signal. Dashed lines mark

p=0,1,2

; shaded bands span the fitted exponents, and corner values give the class mean.

Fig. 12: Fitted exponents by imaging configuration and detector class. Dots show medians, thick bars interquartile ranges, and thin lines full ranges; carets indicate values outside the displayed range.

Fig. 13: Real and simulated FMD frames using the fitted exponent and index. Rows show the clean signal, real and simulated observations, and middle-band noise histograms (real filled; simulated outlined). Numbers are simulated-to-real noise-standard-deviation ratios averaged over the dark and bright bands; values closer to $1$ indicate better agreement.

Fig. 14: Noise statistics versus candidate exponent for the fields in Figure 13 . Left: simulated-to-real noise-standard-deviation ratios; right: simulated skewness, for dark and bright bands. Curves closer to the horizontal match lines agree better with real data; vertical lines mark the fitted exponents.

	Gaussian	Poisson	Poisson–Gamma	Gamma
$p$	$0$	$1$	$(1,2)$	$2$
$\kappa(\theta)$	$\sigma^{2}\theta^{2}/2$	$e^{\theta}$	$b^{(2-p)/(1-p)}\!/(2-p)$	$-\log(-\theta)$
$\tau(\theta)$	$\sigma^{2}\theta$	$e^{\theta}$	$b^{1/(1-p)}$	$-1/\theta$
$V(\mu)$	$\sigma^{2}$	$\mu$	$\mu^{p}$	$\mu^{2}$
$\tau^{-1}(\mu)$	$\mu/\sigma^{2}$	$\log\mu$	$\mu^{1-p}\!/(1-p)$	$-1/\mu$

TABLE S1: Distributional functions of the supported Tweedie models.

severe	$\mathrm{NFE}{=}1$	$\mathrm{NFE}{=}5$	$\mathrm{NFE}{=}25$
diagonal mean PSNR	22.326	22.330	22.225
$p{=}1.5$ worst over $p_{\mathrm{data}}$	20.739	21.339	21.594
mean off-diagonal penalty	$+1.601$	$+1.256$	$\mathbf{+0.924}$
diagonal dark log-RMSE	0.2841	0.2831	0.2854
diagonal res. var.	0.926	0.949	0.968

TABLE S2: Effect of the number of function evaluations at severe noise. “Diagonal” denotes matched model and data exponents. “Worst” is the minimum over the tested data exponents. Bold marks the preferred value in each row.

Fig. S1: Synthetic-image restorations at severe noise ( $\eta=1$ ), with one row per noise exponent. Corner values show PSNR (dB).

Fig. S2: Synthetic-image restorations at mild noise ( $\eta=0.25$ ), with one row per noise exponent. Corner values show PSNR (dB).

sensor	nominal	$\hat{p}$ at ENL band					implied
sensor	$L$	all	5–95	10–95	25–95	40–95	$L$
miniSAR	1.00	2.150	2.042	2.014	2.005	2.002	1.06
FARAD-X	1.00	2.103	2.071	2.058	2.030	2.013	0.92
TerraSAR-X	1.16	2.106	2.067	2.044	2.014	2.005	1.37

TABLE S3: SAR noise-exponent estimates across local-ENL window selections (medians over images). Nominal and dispersion-implied looks,

L

, are included for reference.

Diffusion-Based Denoising

See all 49 papers

Jul 21, 2026

$γ$ -Bridge: A Look-Parametric Diffusion Bridge

Multiplicative Gamma noise is a signal-dependent degradation in coherent imaging; synthetic aperture radar (SAR) despeckling is its most prominent real-world instance. Existing diffusion denoisers parameterize their forward process by abstract signal-to-noise schedules rather than by the physical look number

L

, so different deployment scenarios typically require separately trained models, and transfer from synthetic Gamma training to real SAR remains challenging without clean ground truth. We introduce

γ

-Bridge, a look-parametric bridge whose schedule

L(t)

connects the noisy observation at

L_{obs}

to the clean limit through exact multiplicative Gamma marginals. Its closed-form Gamma--Lévy reverse posterior admits both stochastic and deterministic processes, while observation conditioning and a two-step consistency loss stabilize multi-step inference in the low-SNR single-look regime. Because bridge time directly represents

L

, one conditioned network can smart-start from any admissible input look and stop at a target look number. These two orthogonal controls enable zero-shot restoration over the full admissible grid after training only at

L_{obs} = 1

on natural images with synthetic Gamma corruption. Combined with a homogeneous-patch look estimator,

γ

-Bridge processes data from six spaceborne and airborne SAR sensors without sensor-specific fine-tuning, achieving leading results on standard synthetic benchmarks while providing physically interpretable input and output controls absent from prior denoisers. Codes are released here.

Xuran Hu, Yujie Zhu, Tengxi Wang +2

Urban Governance and Design Thrust, Society Hub, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · Faculty of Science and Engineering, Macquarie University, NSW 2109, Sydney, Australia · School of Geography and Planning, Ningxia University, Yinchuan, China

Diffusion Models for Image Restoration Physics-Guided Image Restoration Synthetic Aperture Radar Diffusion-Based Inverse Problems Image Denoising

Ask PDF

Aug 9, 2026

SDDBMs: Soft Denoising Diffusion Bridge Models

Diffusion bridge models leverage Doob's

h

-transform to construct stochastic transports between arbitrary endpoint distributions, and have shown strong potential in image-to-image translation and restoration. However, most existing bridge models rely on hard endpoint conditioning, which forces the terminal state to match a prescribed target exactly. This hard constraint induces terminal-boundary singularities: the terminal law collapses to a Dirac measure, and the resulting drift coefficients become ill-conditioned near the endpoint. In this paper, we propose Soft Denoising Diffusion Bridge Models (SDDBMs), a generalized framework that regularizes diffusion bridges directly at the level of their terminal constraints. Instead of imposing an exact endpoint, SDDBMs prescribe a non-degenerate Gaussian terminal marginal under the transformed path measure, with a flexible terminal center and variance. Starting from this prescribed marginal, we develop a complete closed-form construction of the soft bridge, including the Gaussian terminal reweighting and soft

h

-function, the induced Gaussian forward marginals and

\mathbf{x}_0

-free dynamics. Theoretically, SDDBMs provide a unified probabilistic perspective that encompasses existing diffusion bridge models, including DDBMs, GOUB, and UniDB, as special cases under specific parameter choices. Extensive experiments on image restoration tasks demonstrate that SDDBMs achieve improved numerical stability and superior generation quality over existing bridge-based methods.

Shiyi Qi, Kun He, Mingmou Liu

Nanjing University · Renmin University of China

Diffusion Models for Image Restoration Doob H-Transform Diffusion Models

Ask PDF

May 27, 2026

Resolving Endpoint Underfitting in Diffusion Bridges via Noise Alignment

Diffusion bridge models offer a powerful framework for connecting two data distributions, such as in image restoration and translation. Many existing methods learn this bridge by mimicking the score-matching formulation of standard diffusion models. In this work, we find that this way leads to an anomalous underfitting phenomenon near the target endpoint, as the process approaches the target distribution (

t \to 0

). This underfitting, characterized by significant drift in the predicted variance and direction, results from an excessively large discrepancy in noise levels between the network's input and its regression target.To resolve this issue, we propose the Noise-Aligned Diffusion Bridge (NADB).Our approach reformulates the diffusion bridge by first employing a mean network to provide a cleaner conditional target, and then introducing a novel, noise-aligned mapping relationship. This new formulation resolves the noise mismatch and corrects the underfitting near the target endpoint. Experimental validation across multiple image restoration and image translation tasks demonstrates the effectiveness of our approach. Code is available at https://github.com/gyr02/NADB.

Yurong Gao, Zicheng Zhang, Congying Han +2

University of Chinese Academy of Sciences

Diffusion Models for Image Restoration Image-to-Image Translation Diffusion Models

Ask PDF

Abstract

Diffusion-Based Denoising

Explore similar work

γγγ-Bridge: A Look-Parametric Diffusion Bridge

SDDBMs: Soft Denoising Diffusion Bridge Models

Resolving Endpoint Underfitting in Diffusion Bridges via Noise Alignment

$γ$ -Bridge: A Look-Parametric Diffusion Bridge