Paper deep dive
Hybrid++: The Bridge between PDE Models and Deep Learning for Gamma Noise Removal
Mahipal Jetta, Sujato Dutta
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 10/8/2026, 6:25:45 AM
Summary
The paper introduces Hybrid++, a novel trainable nonlinear reaction-diffusion architecture designed for removing multiplicative gamma noise in Synthetic Aperture Radar (SAR) and medical ultrasound images. It bridges the gap between interpretable classical Partial Differential Equation (PDE) methods and high-performance but opaque deep learning models like DnCNN. Hybrid++ features a fully learnable PDE initialization, a 3-stage reaction-diffusion network with multiscale filter banks, Squeeze-and-Excitation attention, and noise-level embeddings. It employs a two-phase training strategy (stage-wise optimization followed by joint end-to-end refinement). On the FoE benchmark, Hybrid++ significantly outperforms classical PDE, BM3D, and TNRD baselines, achieving performance close to DnCNN (within 0.23 dB PSNR at severe noise levels) while using only ~8% of the parameters.
Entities (10)
Relation Signals (8)
Hybrid++ → addresses → Gamma Noise
confidence 95% · This paper presents Hybrid++, a novel trainable nonlinear reaction-diffusion architecture that addresses both the concerns... for multiplicative gamma noise.
Hybrid++ → approachesperformanceof → DnCNN
confidence 95% · In the severe-noise setting L=1, it comes within 0.23 dB PSNR of a separately trained DnCNN
DnCNN → isa → Black-box Model
confidence 95% · deep convolutional networks like DnCNN achieve high performance but at the cost of transparency
Hybrid++ → isbasedon → Reaction-Diffusion
confidence 95% · Hybrid++ is built directly upon the Trainable Nonlinear Reaction Diffusion (TNRD) framework
BM3D → iseffectivefor → Additive Gaussian Noise
confidence 95% · Traditional non-local methods like BM3D [3], are effective for additive Gaussian noise
Hybrid++ → outperforms → TNRD
confidence 95% · On the FoE benchmark, Hybrid++ substantially improves over classical PDE, BM3D and the original TNRD baselines.
Hybrid++ → outperforms → BM3D
confidence 95% · On the FoE benchmark, Hybrid++ substantially improves over classical PDE, BM3D and the original TNRD baselines.
Hybrid++ → uses → Squeeze-and-Excitation
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
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.
Tags
Links
- Source: https://arxiv.org/abs/2610.08892v1
- Canonical: https://arxiv.org/abs/2610.08892v1
Trouble viewing inline? Open PDF directly →
Full Text
31,847 characters extracted from source content.
Expand or collapse full text
Hybrid++: The Bridge between PDE Models and Deep Learning for Gamma Noise Removal Mahipal Jetta Sujato Dutta Affiliation: 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. Keywords: Gamma noise Image Denoising Partial Differential Equations (PDE). 1 Introduction Multiplicative gamma noise is a significant challenge to deal with in imaging systems such as Synthetic Aperture Radar (SAR) and medical ultrasound where the noise is signal-dependent and cannot be handled by standard additive-noise denoising models. Classical methods like partial differential equations (PDEs) [11] try to address this by adapting smoothing strength to local image gradients and intensity. However, they rely on fixed, handcrafted parameters and struggle to recover complex features from the noisy images. On the other hand, deep convolutional neural networks (CNNs) such as DnCNN [17] achieve remarkable performance in denoising but operate as black box models which can’t be easily interpreted and lack transparency. They also require training hundreds of thousands of parameters. Traditional non-local methods like BM3D [3], are effective for additive Gaussian noise but require ad-hoc adaptations (e.g. homomorphic transforms) for multiplicative noise and often fail to produce optimal results. Trainable Nonlinear Reaction Diffusion (TNRD) [2] was one of the breakthroughs offering a middle ground by unfolding the denoising process as a sequence of interpretable diffusion stages with learnable parameters. However, the original TNRD was meant mainly for additive noise and its compact parameterization limits its capacity for handling the more challenging multiplicative noise. More recently, image restoration has been strongly influenced by transformer and diffusion-based architectures. Models such as SwinIR [8] and Restormer [16] demonstrate the effectiveness of windowed or efficient self-attention for general image restoration and denoising, while recent diffusion-transformer restoration methods [14] further improve degradation modeling through generative priors and coarse-to-fine refinement. In the SAR despeckling literature[10], CNN, contrastive-learning [4], transformer, and diffusion-based methods have also been explored for removing speckle from real or simulated SAR images. These methods are important modern baselines for broad image restoration. However, our focus is different, we design our main architecture Hybrid++ as a compact, interpretable extension of the PDE/reaction-diffusion family for multiplicative gamma noise, rather than as a large generic restoration backbone. Our main contributions are: • A fully differentiable PDE initialization module whose parameters (α, β) are optimized via backpropagation in place of traditional grid search. • Parallel multiscale filter banks (32 filters of 3×33×3 and 32 filters of 5×55×5) producing a 64-channel feature representation that captures both fine textures and gradients while also handling broader structural context. • A 4-layer attention influence function with an expanded squeeze-excitation [7] bottleneck (64→ 16→ 64) (here the numbers denote the number of neurons in the respective layers of the network), enabling deep nonlinear feature weighting without normalization layers. • A 3-layer MLP [12] noise embedding (1→ 32→ 64→ 64) that conditions every stage on the noise severity allowing a single unified model to handle varying noise levels. • A two-phase training strategy combining stage-wise optimization with joint end-to-end refinement for co-adaptation of all parameters including the PDE ones. • Comprehensive evaluation showing that Hybrid++ improves strongly over PDE, BM3D, and TNRD baselines, while approaching DnCNN [17] specifically in the severe-noise setting L=1. Unlike a fully black-box CNN, Hybrid++ preserves structured interpretability through explicit reaction-diffusion stages, identifiable diffusion filters, bounded influence functions, learnable fidelity weights, noise conditioning, and a gamma-noise-motivated reaction term. 2 Proposed Method: Hybrid++ 2.1 Multiplicative Gamma Noise Model Let u denote the clean grayscale image and f the noisy observation. The multiplicative gamma noise model is: f=u⋅n,n∼Γ(L,1/L),f=u· n, n (L,1/L), (1) where Γ is the gamma distribution, L is the number of looks, yielding [n]=1E[n]=1 and Var(n)=1/LVar(n)=1/L. Low L (e.g., L=1L=1) corresponds to severe noise, while higher L (e.g., L=10L=10) represents milder noise levels. 2.2 Architecture Overview Hybrid++ is built directly upon the Trainable Nonlinear Reaction Diffusion (TNRD) framework [2], preserving its core mathematical ideation - a sequence of reaction-diffusion stages, each comprising learnable convolution filters, a nonlinear influence function and a data-fidelity reaction term while systematically upgrading them for suiting multiplicative gamma noise and giving better performance. Hybrid++ is not a purely hand-crafted PDE with all parameters fixed analytically. Rather, the denoising trajectory remains organized as explicit reaction-diffusion updates. The PDE initialization, multiscale diffusion filters, bounded influence function, stage-wise fidelity weights λt _t, and noise-conditioned reaction term each have identifiable roles in the restoration dynamics. This differs from a standard feed-forward CNN such as DnCNN, where intermediate feature maps do not directly correspond to a known iterative restoration process. Thus, Hybrid++ demonstrates structured interpretability. Table 1 shows the component-wise enhancements and Fig 1 shows the network architecture. Noisy Input ffLearnablePDE Init(α,βα,β learnable)Stage 1Stage 2Stage 3Restored Image u uu0u_0u1u_1u2u_2u3u_3skip ffNoise Embedding ∈ℝ64e ^64MLP: →641\!→\!32\!→\!64\!→\!64 Figure 1: Hybrid++ architecture diagram. The noisy image f passes through a differentiable PDE initialization with learnable α,βα,β and three enhanced reaction-diffusion stages before the final output. Dashed arrows denote noisy-image skip connections. Table 1: TNRD → Hybrid++: Component-wise Differences Component Original TNRD Hybrid++ Initialization Identity (noisy) Learnable PDE Filter input 1-channel 2-channels Filter bank 8 single-scale (3×33×3) 64 multiscale (3×33×3 + 5×55×5) Influence fn. 1-layer per-ch tanh 4-layer with SE attention Noise adapt. None (separate models) 64-dim MLP embedding Stages 7 3 Parameters ∼ 21k ∼ 44k The architecture consists of two principal components: (1) a learnable PDE initialization that replaces TNRD’s noisy initialization with a PDE-based prior, and (2) a sequence of 3 enhanced reaction-diffusion stages that iteratively refine this estimate using the upgraded components described below. 2.3 Differentiable PDE Initialization The process starts with a nonlinear smooth diffusion PDE [11]: ∂u∂t=div((uσM)α11+|∇uσ|β∇u), ∂ u∂ t=div\! ( ( u_σM )^\!α 11+|∇ u_σ|^β\,∇ u ), (2) where uσu_σ denotes the Gaussian-smoothed estimate, M is the local maximum intensity, and α, β control the intensity and gradient dependent diffusion behaviors respectively. Key enhancement: Unlike classical PDE [11] where α and β are fixed scalars decided by grid search, Hybrid++ makes these learnable, allowing the model to jointly optimize the PDE alongside the convolutional stages using backpropagation. This enables the PDE initialization to adapt its diffusion behavior to the specific noise it is dealing with during training. 2.4 Multiscale Filter Banks with Skip Connections At each stage t, the network receives a 2-channel input formed by concatenating the current estimate ut−1u_t-1 with the original noisy observation f. This skip-connection [6] provides access to high-frequency information that otherwise would have been lost through successive diffusion steps. This input is processed by two parallel and independent learnable filter banks: • 32 filters of size 3×3×23×3×2: A 3×33×3 kernel operating on the 2 channel input (the concatenation of the current estimate ut−1u_t-1 and the original noisy image f) to capture fine-grained local textures and sharp edge transitions. • 32 filters of size 5×5×25×5×2: A 5×55×5 kernel to capture broader spatial context and large structural patterns. The filter responses are concatenated along the channel dimension, producing a 64-channel feature map. This design is essential for multiplicative noise where noise patterns overlap a lot with structural high-frequency information. The dual-scale representation allows the network to separate noise from texture at multiple spatial frequencies simultaneously. All filters are initialized with Kaiming-style scaling [5] and are fully learnable throughout training. 2.5 Dynamic Noise Level Embedding To enable blind denoising across varying noise levels, a 3-layer Multi-Layer Perceptron (MLP) [12] maps the scalar noise strength parameter to a 64-dimensional conditioning vector: =W3⋅ReLU(W2⋅ReLU(W1⋅σL+b1)+b2)+b3,e=W_3·ReLU(W_2·ReLU(W_1· _L+b_1)+b_2)+b_3, (3) where σL=1/L _L=1/ L is the noise strength derived from the gamma distribution’s coefficient of variation, bib_i’s are biases and W1∈ℝ32×1W_1 ^32× 1, W2∈ℝ64×32W_2 ^64× 32, W3∈ℝ64×64W_3 ^64× 64. This 64-dimensional embedding e is broadcast and added to the 64-channel filter responses at every stage. This makes the feature representation biased based on noise level. This design allows Hybrid++ to use a single unified noise-conditioned model for both L=1L=1 and L=10L=10. In contrast, the TNRD and DnCNN baselines in our experiments are trained separately for each noise level. Thus, Hybrid++ is evaluated in the more flexible setting: it must learn one conditional mapping across noise severities rather than a separate optimum for each fixed L. 2.6 4-Layer SE-Attention Influence Function A very important component of the reaction-diffusion update is the nonlinear influence function Φ(⋅) (·), which decides the diffusion behavior. Hybrid++ introduces a deep, 4-layer architecture with integrated channel attention: 1. Layer 1: 1×11×1 convolution (64→6464→ 64) followed by tanh activation. 2. Layer 2: 1×11×1 convolution (64→6464→ 64) followed by tanh activation. 3. Layer 3: Squeeze-and-Excitation (SE) [7] block with bottleneck ratio 4: SE()=⋅σ′(Wup⋅ReLU(Wdown⋅GAP())),SE(x)=x·σ \! (W_up·ReLU(W_down·GAP(x)) ), (4) where Wdown∈ℝ16×64W_down ^16× 64 squeezes and Wup∈ℝ64×16W_up ^64× 16 excites, with GAP denoting global average pooling and σ′σ the sigmoid function. This mechanism dynamically re-weights the 64 channels based on their global importance, acting as a learned attention mechanism. 4. Layer 4: 1×11×1 convolution (64→164→ 1) followed by tanh , collapsing the attended features into a single spatial diffusion map. Design choice: The original TNRD [2] uses a simple 1-layer per-channel influence function. By expanding this to 4 layers with an intermediate SE-attention [7] block, Hybrid++ can learn highly complex nonlinear response mappings. The tanh activations ensure bounded outputs to stay consistent with the physical diffusion formulation. Three prominent differences emerge relative to TNRD [2]: • Cross-channel interaction. The SE block [7] computes a global channel descriptor via average pooling and uses it to re-weight each of the 64 channels based on their global importance. Filters aligned with dominant image edges are amplified and the ones aligned with noise patterns are suppressed. A per-channel TNRD cannot perform this selection. • Input-dependent asymmetric shaping. Because the SE channel weights depend on the global average of the current feature map, the effective response curve of Φ changes from region to region: texturally complex patches activate a different channel mix than flat smooth regions. • Sharper edge-stopping via depth. Hybrid++ has four successive tanh compositions that raise the effective degree of the approximated penalty function as compared to a single tanh in original TNRD [2]. Fig. 2 illustrates the effective 1-D response along a single filter axis. Hybrid++’s Φ shows a sharper transition and a lower saturation plateau demonstrating stronger selective suppression of small noise-like responses while better preserving large-magnitude edge responses. −3-3−2-2−1-100112233−1-10011Transition regionFilter response rΦ(r) (r)TNRDHybrid++ Figure 2: Effective 1-D response of the TNRD influence function (blue, solid) vs. the Hybrid++ SE-attention Φ (red, dashed) along a single filter axis. 2.7 Stage Update Rule Each stage combines the influenced diffusion term with a data-fidelity reaction term: ut=ut−1−w⋅Φ((ut−1,f)+)−λt⋅w⋅ut−1−fut−12+ε,u_t=u_t-1-w· \! (R(u_t-1,f)+e )- _t· w· u_t-1-fu_t-1^2+ , (5) where (ut−1,f)R(u_t-1,f) denotes the 64-channel multiscale filter response, e is the noise embedding, w is the intensity-dependent weight derived from the Gaussian-smoothed estimate and λt _t is a learnable fidelity parameter. The reaction term derives from the MAP formulation for multiplicative gamma noise [1]. 2.8 Two-Phase Training Strategy We use a two-phase optimization strategy: Phase 1: Stage-wise Initialization. Each of the 3 stages is trained sequentially for 100 epochs. When training stage t, all prior stages are frozen and only stage t’s parameters alongwith the PDE [11] and embedding parameters are changed. This ensures each stage provides a meaningful refinement step. Training uses Adam with lr=10−3lr=10^-3 (learning rate), early stopping with patience 10 and gradient clipping at max norm 1.0. Phase 2: Joint End-to-End Refinement. After stage-wise initialization, all parameters across the 3 stages, the PDE module [11] and the noise embedding are unfrozen and jointly optimized for 50 epochs. This phase uses Adam with lr=10−4lr=10^-4 and CosineAnnealingLR [9] (Tmax=50T_ =50, ηmin=10−6 _ =10^-6), allowing the entire network to co-adapt and fine-tune the inter-stage interactions. We use Mean Squared Error (MSE) as the loss function for both phases. The purpose of the two-phase strategy is not to keep the stages independent after training, but to provide a structured initialization before global co-adaptation. The stage-wise phase encourages each reaction-diffusion block to learn a meaningful refinement role. The later joint phase then fine-tunes the already structured initialization across the PDE module, noise embedding, and all diffusion stages. Therefore, joint training does not cancel the stage-wise phase; it refines an initialization in which the stages have already learned progressively useful denoising roles. 3 Experimental Setup 3.1 Dataset We use the FoE dataset [13] containing 400 grayscale images that are split into 80% training, 10% validation, and 10% testing (40 test images) with a fixed random seed of 42. For qualitative evaluation on real-world data, we additionally test on real SAR images from the Sentinel-1&2 dataset. 3.2 Baselines We compare Hybrid++ against four models including classical baselines, learning-based architectures and deep learning approaches: • BM3D [3]: Non-local patch-based method applied using homomorphic (log-domain) transform with noise variance σ2=ψ1(L)σ^2= _1(L) (trigamma function). • PDE Baseline[11]: Classical nonlinear smooth diffusion PDE (Eq. 2) with hand-tuned parameters (α=1.0α=1.0, β=2.0β=2.0). • TNRD [2]: Original 8-stage trainable reaction-diffusion. We trained separate models for each noise level. • DnCNN [17]: 17-layer deep CNN with batch normalization (approx. 556K parameters) for the black-box deep neural network architecture. We intentionally use DnCNN as the representative black-box deep CNN baseline rather than benchmarking against every recent transformer or diffusion restoration model. Recent restoration models are often designed as large general-purpose backbones for multiple degradation types, while our goal is to evaluate a compact reaction-diffusion extension specialized for multiplicative gamma noise. The selected baselines therefore isolate the main comparison axis of this work: classical PDE priors, non-local denoising, trainable reaction-diffusion, and deep CNN restoration. All learning-based models are trained with identical MSE loss, Adam optimizer and dataset split as our Hybrid++ architecture for fair comparison. Remark on log-domain vs. direct-domain processing. BM3D [3] is an additive-noise denoiser. To handle multiplicative gamma noise we apply it via a homomorphic transform: the log of the noisy image is computed, BM3D is applied with standard deviation σ=ψ1(L)σ= _1(L) (where ψ1 _1 is the trigamma function, the log-domain noise variance of a Γ(L,1/L) (L,1/L) variate), and the result is exponentiated. All other models - PDE [11], TNRD[2], DnCNN[17] and Hybrid++ are applied directly in the original image domain without any log transformation. BM3D is evaluated in its strongest available configuration for gamma noise (log-domain) so it is fair comparison with the learning-based methods which either use a direct gamma prior (PDE, Hybrid++) or a purely data-driven mapping (TNRD, DnCNN). 4 Results and Discussion 4.1 Quantitative Results Table 2 reports the average PSNR and SSIM [15] across all 40 test images for comparison of the five models. Table 2: Quantitative Comparison on FoE Test Set (40 images) L=1L=1 (Severe) L=10L=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 Under the severe noise regime (L=1L=1), Hybrid++ achieves 20.86 dB, surpassing the PDE baseline [11] by +3.20 dB, TNRD [2] by +2.31 dB and BM3D [3] by +8.19 dB. At the milder setting (L=10L=10), Hybrid++ leads with 25.02 dB dominating all the interpretable baselines by a considerable margin. For SSIM, it stays clearly better for the (L=1L=1) severe noise setting while trailing behind BM3D [3] by just 0.063 at (L=10L=10). When compared to the black-box model DnCNN [17], it comes close at severe noise level (L=1L=1) trailing by just 0.23 dB PSNR and 0.021 SSIM. For (L=10L=10), the gap is wider at 0.92 dB PSNR and 0.069 SSIM. However, Hybrid++ achieves this using one unified noise-conditioned model, while DnCNN is trained separately for each L. It also uses only 44k parameters as compared to DnCNN’s 556k while maintaining structured interpretability vs DnCNN’s black-box architecture. 4.2 Qualitative Analysis 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). Figs. 3 and 4 show us the qualitative comparisons under L=1L=1 and L=10L=10 noise levels respectively. Under the severe L=1L=1 regime, BM3D [3] fails due to the inadequacy of its homomorphic log-domain adaptation for extreme multiplicative noise. PDE [11] performs over-smoothing, thus losing structural information. TNRD [2] does comparatively better but it shows blurring of the edges and haziness over the image. Hybrid++ recovers significantly sharper edges and more coherent textures approaching the performance of DnCNN [17]. Under the milder L=10L=10 setting, the differences between methods narrow, but Hybrid++ consistently preserves finer granular detail that PDE and TNRD diffuse away. 4.3 Bridging the Gap to Deep Learning DnCNN, with approximately 556K parameters, achieves 21.09 dB at L=1L=1. Hybrid++, with 44,408 parameters, reaches 20.86 dB, leaving a gap of only 0.23 dB while using roughly 12.5× fewer parameters. This serves as an evidence that a compact physically structured model can approach a representative black-box CNN under severe multiplicative noise. At L=10L=10, DnCNN remains ahead, especially in SSIM. Hybrid++ trades some peak restoration performance for parameter compactness, unified noise conditioning and structured reaction-diffusion interpretability. 4.4 Training Convergence Analysis 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. Fig. 5 illustrates the validation loss trajectory over the stage-wise training phase. Stage 1 shows a sharp decline in loss as the model learns the primary denoising transformation. Stage 2 provides a modest further reduction, refining the residual details. Stage 3 shows near-flat convergence, confirming that 3 stages capture the essential representational capacity of our model and additional stages provide negligible improvement while increasing parameter count and training complexity. This convergence behavior also supports the role of the two-phase training strategy. The stage-wise phase gives the stages distinct denoising roles before the final joint refinement: Stage 1 performs the dominant restoration, Stage 2 provides residual correction, and Stage 3 performs only a small final adjustment. The later end-to-end phase therefore fine-tunes an already structured sequence rather than learning the full model from an unstructured initialization. 4.5 Qualitative Evaluation on Real SAR Images 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. To show performance on real world images and use-cases, Fig. 6 presents denoised results on SAR images from the Sentinel 1&2 dataset. Since real SAR images have no ground-truth clean reference, this comparison is purely qualitative. The PDE [11] produces over-smoothed results that lose structural information. TNRD [2], while improving over PDE, retains residual speckle artifacts. Hybrid++ produces cleaner results with well-preserved edges and structural coherence demonstrating high denoising quality coming close to that of DnCNN [17]. 5 Ablation Study To justify our architectural design choices in Hybrid++, we conducted an ablation study by progressively removing components from the base TNRD [2] framework and checking their impact on the performance. Table 3: Ablation Study: Impact of Different Components L=1L=1 L=10L=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 (Nk=4N_k=4) 16.91 0.245 21.99 0.559 −- Reduced stages (T=3T=3) 17.06 0.289 22.24 0.578 PDE Baseline 17.65 0.421 23.07 0.651 PDE initialization + Base TNRD 20.09 0.411 23.99 0.680 Hybrid++ (Full) 20.86 0.463 25.02 0.692 Table 3 reveals the importance and impact of the various design choices we made for our final architecture: Nonlinear influence functions - ϕφ is replaced by the identity function (ϕ(r)=rφ(r)=r), reducing the influence function to a plain linear filter response. This shows a catastrophic ∼8 8 dB drop at L=1L=1, reducing performance on severe noise levels. This confirms that the non-linear influence function is the foundation of the architecture and is irreplaceable. PDE initialization - The PDE initialization provides a strong starting point before the learned diffusion stages are applied. As shown in Table 3, the PDE baseline already achieves competitive structural preservation with SSIM [15] values of 0.421 and 0.651 on the L=1 and L=10 noise settings respectively. When combined with base TNRD [2], it improves performance from 18.56/23.18 dB PSNR to 20.09/23.99 dB while also boosting SSIM. These results indicate that the PDE initialization effectively suppresses multiplicative noise and stabilizes the subsequent learning stages. Filter capacity - Reducing filters from 8 to 4 drops PSNR by 1.65 dB at L=1L=1. This motivates our Hybrid++’s expansion to 64 channels (32×2 multiscale), which enables the dramatic +2.31 dB improvement over the base TNRD [2]. Stage depth - Reducing from 7 to 3 stages drops PSNR by 1.50 dB in the base TNRD [2] (with 8 filters). However, Hybrid++ demonstrates that with sufficient filter capacity (64 channels) and deep influence functions, just 3 stages suffice to provide its potential upper limit performance as demonstrated in Fig. 5. 6 Conclusion This paper presented Hybrid++, a physics-informed architecture for multiplicative gamma noise removal that bridges the divide between classical hand-crafted PDE-based methods and modern deep learning. By expanding the TNRD framework with learnable PDE initialization, 64-channel multiscale filter banks, 4-layer SE-attention influence functions, 64-dimensional noise embeddings and a two-phase training strategy, Hybrid++ achieved 20.86 dB at L=1L=1 and 25.02 dB at L=10L=10 on the FoE benchmark while only requiring to train a single unified model. Most significantly, a 44,408-parameter model closed the gap to the 556K-parameter DnCNN to within just 0.23 dB at severe noise levels. These results suggest that principled physical priors when augmented with targeted learnable components, can close a substantial part of the gap to black-box CNNs in the severe multiplicative-noise regime while preserving a structured reaction-diffusion interpretation. Future work should extend the comparison to larger transformer and diffusion restoration models, broader SAR datasets, and real-noise training protocols. References [1] G. Aubert and J. Aujol (2008) A variational approach to removing multiplicative noise. SIAM Journal on Applied Mathematics 68 (4), p. 925–946. External Links: Document, Link, https://doi.org/10.1137/060671814 Cited by: §2.7. [2] Y. Chen and T. Pock (2017) Trainable nonlinear reaction diffusion: a flexible framework for fast and effective image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence 39 (6), p. 1256–1272. External Links: ISSN 2160-9292, Link, Document Cited by: §1, 3rd item, §2.2, §2.6, 3rd item, §3.2, §4.1, §4.2, §4.5, §5, §5, §5, §5. [3] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian (2007) Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Transactions on Image Processing 16 (8), p. 2080–2095. External Links: Document Cited by: §1, 1st item, §3.2, §4.1, §4.2. [4] Y. Fang, R. Liu, Y. Peng, J. Guan, D. Li, and X. Tian (2024) Contrastive learning for real sar image despeckling. ISPRS Journal of Photogrammetry and Remote Sensing 218, p. 376–391. External Links: ISSN 0924-2716, Document, Link Cited by: §1. [5] K. He, X. Zhang, S. Ren, and J. Sun (2015) Delving deep into rectifiers: surpassing human-level performance on imagenet classification. External Links: 1502.01852, Link Cited by: §2.4. [6] K. He, X. Zhang, S. Ren, and J. Sun (2016) Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Vol. , p. 770–778. External Links: Document Cited by: §2.4. [7] J. Hu, L. Shen, S. Albanie, G. Sun, and E. Wu (2019) Squeeze-and-excitation networks. External Links: 1709.01507, Link Cited by: 3rd item, item 3, 1st item, §2.6. [8] J. Liang, J. Cao, G. Sun, K. Zhang, L. V. Gool, and R. Timofte (2021) SwinIR: image restoration using swin transformer. External Links: 2108.10257, Link Cited by: §1. [9] I. Loshchilov and F. Hutter (2017) SGDR: stochastic gradient descent with warm restarts. External Links: 1608.03983, Link Cited by: §2.8. [10] Y. Pan, L. Zhong, J. Chen, H. Li, X. Zhang, and B. Pan (2024) SAR image despeckling based on denoising diffusion probabilistic model and swin transformer. Remote Sensing 16 (17). External Links: Link, ISSN 2072-4292, Document Cited by: §1. [11] P. Perona and J. Malik (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12 (7), p. 629–639. External Links: Document Cited by: §1, §2.3, §2.3, §2.8, §2.8, 2nd item, §3.2, §4.1, §4.2, §4.5. [12] M. Popescu, V. Balas, L. Perescu-Popescu, and N. Mastorakis (2009) Multilayer perceptron and neural networks. WSEAS Transactions on Circuits and Systems 8, p. . Cited by: 4th item, §2.5. [13] S. Roth and M.J. Black (2005) Fields of experts: a framework for learning image priors. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), Vol. 2, p. 860–867 vol. 2. External Links: Document Cited by: §3.1. [14] L. Wang, Q. Yang, C. Wang, W. Wang, J. Pan, and Z. Su (2023) Learning a coarse-to-fine diffusion transformer for image restoration. External Links: 2308.08730, Link Cited by: §1. [15] Z. Wang, A.C. Bovik, H.R. Sheikh, and E.P. Simoncelli (2004) Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing 13 (4), p. 600–612. External Links: Document Cited by: §4.1, §5. [16] S. W. Zamir, A. Arora, S. Khan, M. Hayat, F. S. Khan, and M. Yang (2022) Restormer: efficient transformer for high-resolution image restoration. External Links: 2111.09881, Link Cited by: §1. [17] K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang (2017) Beyond a gaussian denoiser: residual learning of deep cnn for image denoising. IEEE Transactions on Image Processing 26 (7), p. 3142–3155. External Links: ISSN 1941-0042, Link, Document Cited by: 6th item, §1, 4th item, §3.2, §4.1, §4.2, §4.5.