Linear Hybrid Denoiser

• Linear Hybrid Denoisers are systems that integrate precise analytic linear operations with targeted neural corrections for robust denoising.
• They achieve high parameter efficiency and interpretability by separating linear signal processing from data-driven enhancements.
• These architectures excel in image and video restoration, offering real-time performance improvements and adaptability across inverse problems.

A Linear Hybrid Architecture Denoiser refers to any image or video restoration architecture that combines explicitly linear processing modules with nonlinear neural networks, typically occupying distinct functional roles within a larger denoising pipeline. In recent advancements, such hybrids leverage the interpretability and efficiency of analytic linear operators (e.g., wavelet transforms, local regressions, Wiener filtering, asynchronous linear filtering) and strategically integrate neural networks for adaptation, sparsity modeling, semantic enhancement, or data-driven priors. This hybridization yields denoising systems with substantially improved parameter efficiency, robustness, and adaptability across image, video, and signal restoration tasks.

1. Core Principles of Linear Hybrid Denoising

Linear hybrid denoisers are characterized by an architecture that separates strictly linear signal processing components from nonlinear, learnable mappings. The primary linear modules include undecimated wavelet transforms, local regression systems, Wiener filters, or continuous-time spatial-temporal filters. These provide well-understood analytic foundations (invertibility, sparsity, MMSE optimality, or explicit moment matching) and facilitate efficient computation and memory usage.

The nonlinear components are typically instantiated as small neural networks (CNNs, U-Nets, LISTA modules), which perform targeted tasks unaddressed by the linear backbone—such as coefficient shrinkage (soft-thresholding), guide generation, gain correction, or noise profile estimation. In some architectures, the nonlinear functions are restricted to act only on transform coefficients or guide channels; in others, they provide data-adaptive priors or correct systematic errors made by the analytic module.

This architectural dichotomy ensures that high-order semantic reconstruction is achieved via data-driven learning, while the bulk of denoising is governed by interpretable, robust linear processing.

2. Representative Architectures and Mathematical Formulations

2.1 LINN: Lifting-Inspired Invertible Hybrid Network

LINN (Huang et al., 2021) employs a learnable lifting scheme sandwiched between linear wavelet splitting and perfect-reconstruction inverse. The forward analysis pass performs

$$\begin{cases} z_d^{(i)} = z_d^{(i-1)} - P_i(z_c^{(i-1)}) \ z_c^{(i)} = z_c^{(i-1)} + U_i(z_d^{(i)}) \end{cases} \quad i=1,\ldots,I$$

yielding an over-complete code $$c = [z_c^I; z_d^I] \in \mathbb{R}^{4 \times m \times n}$$. Nonlinear shrinkage is applied either via channel-wise soft-thresholding,

$$\tilde{c} = S(c;\lambda), \quad S(z;\lambda) = \operatorname{sign}(z)\,\max(|z|-\lambda,0)$$

or by a compact LISTA network. The backward synthesis pass reconstructs $x$ via invertible linear maps using the denoised coefficients.

2.2 Fast Local Neural Regression (FLNR)

FLNR (Salmi et al., 15 Oct 2024) integrates a local linear regression model with a lightweight U-Net for guide enhancement. For neighborhood $T_k$ at pixel $k$,

$$J(A_k) = \| X_k A_k - Y_k \|_F^2 + \lambda \|A_k\|_F^2$$

$A^*_k = (X_k^T X_k + \lambda I)^{-1} X_k^T Y_k$

$I_k = x_k^T A^*_k$

Here, the U-Net supplies enhanced guides $X'_\text{model}$, $X'_\text{map}$, optimizing regression quality over raw physical channels.

2.3 Hybrid WienerNet

WienerNet (Bled et al., 7 Aug 2024) preserves a classical Wiener filter as its backbone,

$$H(\omega) = \frac{P_{xx}(\omega)}{P_{yy}(\omega)}, \quad \hat{P}_{xx}(\omega) = \max(P_{yy}(\omega) - P_{nn}, 0)$$

and refines coring estimates, window functions, and noise profiles using dedicated CNN modules. The key advantage is dramatic parameter reduction with near-state-of-the-art PSNR/SSIM.

2.4 Partially Linear Denoisers

DPLD (Ke et al., 2020) imposes a structural decomposition on the denoiser output,

$R(y) = g(x) + L n + e,$

where $g(x)$ is nonlinear in the clean image, $L$ is linear in the noise, and $e$ is a low-variance residual. Training utilizes noisy-only samples and exploits auxiliary noise vectors for unbiased surrogate MSE minimization.

2.5 Linear Combination Diffusion Denoiser (LCDD)

LCDD (Dornbusch et al., 18 Mar 2025) post-processes a noisy input by blending two outputs from a pretrained diffusion model: $I_D$ (distortion-focused, minimal steps) and $I_P$ (perception-focused, multi-step), yielding

$$\operatorname{LCDD}_\lambda(y) = \lambda I_D + (1 - \lambda) I_P,$$

with $\lambda$ trading off fidelity and perceptual quality on the denoising Pareto front.

2.6 Asynchronous Linear Filter for Event-Frame Fusion

The AKF architecture (Wang et al., 2023) maintains a per-pixel continuous-time state $\hat{L}_p(t)$, fusing event impulses and asynchronous frame updates:

$$\frac{d}{dt} \hat{L}_p(t) = e_p(t) - \alpha [ \hat{L}_p(t) - L_p^A(t) ]$$

Spatial denoising is achieved by convolving input events and frames with Gaussian kernels prior to fusion, maintaining full asynchronicity and real-time readout.

3. Algorithmic Procedures and Data Flow

Most linear hybrid denoisers exhibit separable module organization, with explicit data flow from linear transforms to neural enhancement, followed by synthesis or regression:

• Input signal $x$ undergoes linear transformation ($T_f$), e.g., wavelet split, local regression, or spectral filtering.
• Nonlinear subnetwork applies shrinkage, guide modeling, or correction ($S(c;\lambda)$, U-Net, coring CNN).
• Inverse linear map or synthesis ($T_b$) restores the full-resolution denoised estimate.
• Some architectures solve local closed-form equations, while others invoke neural correction only after principal linear denoising.

Implementation optimizations include blockwise processing, intelligent downsampling, GPU parallelization, and submillisecond execution.

4. Quantitative Performance and Efficiency

Linear hybrid designs consistently demonstrate competitive or superior denoising efficacy for both image and video domains, with drastically reduced parameter count and runtime:

Method	Params (k)	PSNR@25 (BSD68)	Time (ms, 1080p)
DnCNN	556	29.19	37.4
LINN (DnINN-ST)	134.7	29.08	–
FLR w/ AO	–	35.86*	0.64
FLNR w/ AO	–	37.24*	11.54
WienerNetBlind	290	33.1*	150

*PSNRs for video, not directly comparable to BSD68

The LINN architecture achieves nearly the same PSNR as DnCNN while using only ~1/4 of the parameters (Huang et al., 2021). WienerNetBlind remains within 0.2 dB of transformer-based VRT while being >10× faster and >100× smaller in parameter count (Bled et al., 7 Aug 2024). The FLNR method remains faster-than-real-time on commodity hardware (Salmi et al., 15 Oct 2024).

5. Adaptability, Interpretability, and Domain Generalization

The strict functional separation in linear hybrid designs enhances interpretability and facilitates domain adaptation:

• Linear modules are analytic, invertible, and memory-efficient.
• Nonlinear shrinkage or enhancement is easily modifiable for new noise levels, degradation types, or signal classes.
• The same hybrid skeleton can be extended to inpainting, super-resolution, deblurring by exchanging the core linear operator and adapting nonlinear network parameters (Huang et al., 2021).

For example, changing the invertible operator in LINN (from wavelet to other transforms) generalizes the hybrid system to multiple inverse problems. In AKF, any linear spatial kernel (Gaussian, Sobel, Laplacian) can be employed with identical asynchronous logic (Wang et al., 2023).

6. Theoretical Justification and Design Rationale

The rationale behind linear hybrid architectures is grounded in noise-separation, energy compaction, and structural modeling:

• Linear transforms or regression create overcomplete representations, which distribute noise sparsely, maximizing the effectiveness of thresholding or shrinkage.
• Neural modules compensate and adapt where analytic models fail—misestimated gain, missing semantic edges, nonstationary noise, or non-Gaussian corruption.
• MMSE optimality of classic linear filters is retained where applicable; learnable correction does not disturb this baseline but adds systematic improvements.
• In unsupervised and self-supervised contexts, partial linearity constraints enable surrogate MSE minimization using only noisy data, sidestepping the need for clean targets (Ke et al., 2020).

A plausible implication is that with further miniaturization and architectural refinement, linear hybrid denoisers can close the remaining performance gap with extremely large end-to-end CNN/transformer models in both real-time and high-fidelity applications, while remaining computationally accessible and interpretable.

7. Applications and Future Directions

Linear hybrid denoisers are utilized across image denoising, video restoration, rendering artifact removal (global illumination, path-tracing), event–frame fusion, and unsupervised inverse problems. Their modularity and efficiency suggest potential for edge devices, embedded vision systems, and high-throughput scientific imaging. Generalization to nonlinear inverse problems, multi-modal fusion, and reinforcement of physical priors are active research areas.

Future work may include the exploration of hybrid architectures with more granular control over linear–nonlinear module interaction, increased parameter sharing, and the integration of explicit uncertainty quantification within the analytic backbone, to provide further robustness and adaptability in challenging real-world scenarios.