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

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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

Figure 1 : Hybrid++ architecture diagram. The noisy image ff passes through a differentiable PDE initialization with learnable α,β\alpha,\beta and three enhanced reaction-diffusion stages before the final output. Dashed arrows denote noisy-image skip connections.
ComponentOriginal TNRDHybrid++
InitializationIdentity (noisy)Learnable PDE
Filter input1-channel2-channels
Filter bank8 single-scale ( 3×33{\times}3 )64 multiscale ( 3×33{\times}3 + 5×55{\times}5 )
Influence fn.1-layer per-ch tanh⁡\tanh4-layer with SE attention
Noise adapt.None (separate models)64-dim MLP embedding
Stages73
Table 1 : TNRD →\to Hybrid++: Component-wise Differences
Figure 2 : Effective 1-D response of the TNRD influence function (blue, solid) vs. the Hybrid++ SE-attention Φ\Phi (red, dashed) along a single filter axis.
  L=1L=1 (Severe)L=10L=10 (Mild)
MethodParamsPSNRSSIMPSNRSSIM
BM3D012.670.38424.640.755
PDE Baseline017.660.42223.070.651
TNRD21k18.550.32423.170.628
Hybrid++44k20.860.46325.020.692
DnCNN556k21.090.48425.940.761
Table 2 : Quantitative Comparison on FoE Test Set (40 images)
Figure 3 : Qualitative comparison on FoE test images under severe gamma noise ( L=1L{=}1 ).
Figure 4 : Qualitative comparison on FoE test images under mild gamma noise ( L=10L{=}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=1L=1L=10L=10
VariantPSNRSSIMPSNRSSIM
Full Base TNRD18.560.32623.180.629
−- Nonlinear influence10.460.14417.630.456
−- Reduced filters ( Nk=4N_{k}{=}4 )16.910.24521.990.559
−- Reduced stages ( T=3T{=}3 )17.060.28922.240.578
PDE Baseline17.650.42123.070.651
Table 3 : Ablation Study: Impact of Different Components

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