Hybrid++: The Bridge between PDE Models and Deep Learning for Gamma Noise Removal

Mahipal Jetta; Sujato Dutta

eess.IVOct 6, 2026

Hybrid++: The Bridge between PDE Models and Deep Learning for Gamma Noise Removal

Authors: Mahipal Jetta, Sujato Dutta

Organizations: Mahindra University

Abstract

Multiplicative gamma noise is one of the dominant noise factors in Synthetic Aperture Radar (SAR) and medical ultrasound images. They are dependent on pixel level noises due to which they are highly varying across the image and harder to handle as compared to additive noise. The denoising methods to address this noise currently include classical partial differential equation (PDE) methods which are good for interpretation but lack the restoration ability as compared to the state-of-the-art models while deep convolutional networks like DnCNN achieve high performance but at the cost of transparency due to which practitioners are skeptical to use them in high-risk critical fields like medicine. This paper presents Hybrid++, a novel trainable nonlinear reaction-diffusion architecture that addresses both the concerns - staying interpretable while offering performance close to huge black-box models. It combines a fully learnable PDE initialization with a 3-stage reaction-diffusion network having 64-channel multiscale filter banks, 4-layer Squeeze-and-Excitation attention-based influence functions and a 64-dimensional noise level embedding. It uses a two-phase training strategy, stage-wise optimization followed by joint end-to-end refinement which enables co-adaptation of all learnable parameters. On the FoE benchmark, Hybrid++ substantially improves over classical PDE, BM3D and the original TNRD baselines. In the severe-noise setting L=1, it comes within 0.23 dB PSNR of a separately trained DnCNN while using only about 8% of its parameters. We therefore position Hybrid++ not as a universal state-of-the-art image restoration backbone, but as a compact, physically structured reaction-diffusion model for multiplicative gamma noise.

Figures & tables

Panel of Figure 1 — Figure 1 : Hybrid++ architecture diagram. The noisy image $f$ passes through a differentiable PDE initialization with learnable $\alpha,\beta$ and three enhanced reaction-diffusion stages before the final output. Dashed arrows denote noisy-image skip connections.

Component	Original TNRD	Hybrid++
Initialization	Identity (noisy)	Learnable PDE
Filter input	1-channel	2-channels
Filter bank	8 single-scale ( $3{\times}3$ )	64 multiscale ( $3{\times}3$ + $5{\times}5$ )
Influence fn.	1-layer per-ch $\tanh$	4-layer with SE attention
Noise adapt.	None (separate models)	64-dim MLP embedding
Stages	7	3

Table 1 : TNRD

\to

Hybrid++: Component-wise Differences

Panel of Figure 2 — Figure 2 : Effective 1-D response of the TNRD influence function (blue, solid) vs. the Hybrid++ SE-attention $\Phi$ (red, dashed) along a single filter axis.

		$L=1$ (Severe)		$L=10$ (Mild)
Method	Params	PSNR	SSIM	PSNR	SSIM
BM3D	0	12.67	0.384	24.64	0.755
PDE Baseline	0	17.66	0.422	23.07	0.651
TNRD	21k	18.55	0.324	23.17	0.628
Hybrid++	44k	20.86	0.463	25.02	0.692
DnCNN	556k	21.09	0.484	25.94	0.761

Table 2 : Quantitative Comparison on FoE Test Set (40 images)

Figure 3 : Qualitative comparison on FoE test images under severe gamma noise ( $L{=}1$ ).

Figure 4 : Qualitative comparison on FoE test images under mild gamma noise ( $L{=}10$ ).

Figure 5 : Validation loss during stage-wise training. Stage 1 achieves the dominant loss reduction, with diminishing returns in subsequent stages, validating the choice of 3 stages.

Figure 6 : Qualitative comparison on real SAR images (Sentinel 1&2 dataset). PDE over-smooths, TNRD shows artifacts, while Hybrid++ achieves visually competitive results to DnCNN with preserved structural detail.

	$L=1$		$L=10$
Variant	PSNR	SSIM	PSNR	SSIM
Full Base TNRD	18.56	0.326	23.18	0.629
$-$ Nonlinear influence	10.46	0.144	17.63	0.456
$-$ Reduced filters ( $N_{k}{=}4$ )	16.91	0.245	21.99	0.559
$-$ Reduced stages ( $T{=}3$ )	17.06	0.289	22.24	0.578
PDE Baseline	17.65	0.421	23.07	0.651

Table 3 : Ablation Study: Impact of Different Components

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

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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.

Abstract

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