Edge-to-Image Restoration Module

• Edge-to-image restoration modules are computational algorithms that use edge-aware priors to differentiate between smooth regions and critical edges for effective image recovery.
• They integrate variational formulations, wavelet-based models, and neural diffusion architectures to tailor regularization based on local edge information.
• Practical implementations employ modular designs and hardware acceleration to achieve robust, real-time restoration in embedded and cloud-based systems.

An edge-to-image restoration module is a computational component or algorithmic block designed to exploit edge information during image restoration. Its core function is to preserve or enhance edges, typically through edge-aware priors, regularization, or explicit neural architectural design, thereby recovering image structures that are otherwise degraded by noise, blur, or sampling artifacts. Edge-to-image restoration modules are central to modern inverse problems, appearing in variational frameworks, wavelet-based models, edge-adaptive regularization, implicit neural representations, and diffusion-based architectures.

1. Mathematical Principles and Model Classes

Edge-to-image restoration frameworks rest on the premise that recovery methods should treat smooth regions and edges differently to avoid blurring structural details. This is achieved by introducing explicit edge-driven or edge-adaptive mechanisms in the energy functionals or network architecture.

Edge-Driven Variational/Wavelet Models

A prototypical formulation is the edge-driven wavelet frame model, which seeks to recover an image $u:\Omega \rightarrow \mathbb{R}$ and an edge-set indicator $v:\Omega \rightarrow [0,1]$:

$$E(u,v) = \lambda \int_\Omega (1-v)\left( \sum_{\alpha\in\mathbb{I}} |\partial^\alpha u|^2 \right)^{1/2} dx + \gamma \int_\Omega v\left( \sum_{\alpha'\in\mathbb{I}'} |\partial^{\alpha'} u|^2 \right)^{1/2} dx + \rho \int_\Omega \left( \sum_{\alpha''\in\mathbb{I}''} |\partial^{\alpha''} v|^2 \right)^{1/2} dx + \frac12\|A u - f\|_{L^2(\Omega)}^2$$

where the smoothness and edge terms are spatially modulated by $v$, and $A$ models image degradation (e.g., blur, mask) (Choi et al., 2017).

Edge-adaptive hybrid regularization instead employs spatially varying weights in data-fidelity-driven Tikhonov and TV terms, adjusting regularization strength at each pixel according to a dynamic edge-inference field $E(i,j)$ (Zhang et al., 2020).

Neural and Diffusion-Based Architectures

The Edge-oriented Representation Network (EoREN) is an implicit neural representation partitioned into an edge-oriented module $g(x)$ (sine-activated MLP for edge fitting) and a channel-tuning module $h(z)$ (per-channel affine adjustment): $$\Phi(x) = h(g(x)),\quad g: \mathbb{R}^2 \to \mathbb{R}^3,\quad h(z) = \alpha\odot z + \beta$$ EoREN deploys a gradient magnitude adjustment (GMA) process on the target image gradient, training $g$ to fit these adjusted gradients via an edge-oriented loss and channel-tuning through a separate pixel-level loss, with strict back-propagation separation for stable optimization (Chang et al., 2022).

In Manifold-Preserving Guided Diffusion (MPGD), a diffusion process is guided at each denoising timestep by multiple inner gradient-descent projections onto a data-consistent, edge-preserving solution manifold. The multi-step guidance approach enables plug-and-play restoration even with zero retraining on out-of-distribution data (Chakravarty, 8 Jun 2025).

2. Edge Detection, Indicator Construction, and Adaptive Weighting

Edge-to-image modules are characterized by explicit or implicit identification of image singularities:

• Variational/wavelet models: Introduce an implicit edge map ($v$) as an auxiliary variable, initialized by thresholding wavelet coefficients and refined via alternating minimization. The edge map acts as a spatial gate that modulates regularization strength (Choi et al., 2017).
• EAHR: Dynamically computes a Gaussian-smoothed gradient field $M(i,j)$, and an edge-information map $E(i,j)=1/(1+|G\otimes\nabla u(i,j)|^2)$, binarizing at threshold $\tau$ to update local regularization weights $\alpha_1(i,j), \alpha_2(i,j)$ (Zhang et al., 2020).
• EoREN: Computes image gradients using normalized Sobel filters, correcting their magnitude based on coordinate ranges, and uses these as regression targets for the edge subnetwork (Chang et al., 2022).
• Diffusion frameworks: Do not rely on a standalone edge map but enforce edge-fidelity by iterative measurements consistency within a learned generative prior, with the latent space implicitly regularizing singularities (Chakravarty, 8 Jun 2025).

3. Optimization Algorithms and Architectures

The restoration process typically involves alternating or staged optimization, exploiting the edge map or its proxy:

• Wavelet Frame Models: Alternate between $u$-minimization (proximal shrinkage/Split-Bregman) and $v$-minimization, with all updates convex and efficiently solved via frame transforms and soft-thresholding. The process guarantees convergence to the minimizer of the discrete energy functional (Choi et al., 2017).
• EAHR: Applies a semi-proximal ADMM (sPADMM) scheme, splitting the TV and Tikhonov terms via an auxiliary variable $k$ and efficiently solving the resulting subproblems (pixel-wise shrinkage, FFT-based quadratic updates) (Zhang et al., 2020).
• EoREN: Employs two-stage neural optimization: Stage one trains the edge-oriented module to minimize gradient loss; stage two, channel-tuning is separated and optimized on pixel loss, with gradients to $g$ stopped for $L_\text{pixel}$ (Chang et al., 2022).
• MPGD (EIRM): Classical DDIM sampler for diffusion models is augmented with $K$ inner gradient-descent updates at each step, improving measurement consistency (super-resolution, deblurring) and robustness. Implementation on hardware like Jetson Orin Nano leverages FP16 quantization, operator fusion, and asynchronous compute (Chakravarty, 8 Jun 2025).

4. Quantitative Performance and Experimental Validation

Edge-to-image restoration modules consistently deliver superior quantitative and qualitative results in preservation of singularities and global fidelity.

Performance Highlights

Model/Method	Task	PSNR	SSIM	Notable Qualitative Observation	Latency/ Throughput
Wavelet-Frame (Choi et al., 2017)	Inpainting	33.7–36	–	Suppresses speckles, avoids staircase at smooth/edge	$O(N^2\log N)$/iter
EAHR (Zhang et al., 2020)	Deblurring	25.7–28	0.79–0.86	Avoids oversmooth/ ringing; edges sharp, noise removed	seconds/sub-second (FFT)
EoREN (Chang et al., 2022)	Image fitting	64–94	0.41–0.97	Exceeds pixel-only on edge-rich and MNIST; edges crisp	–
TomoGAN (Abeykoon et al., 2019)	Denoising (X-ray)	–	0.79	Comparable SSIM to GPU baseline, <1s per 1024² image	0.55–0.80s (<5W power)
EIRM/MPGD (Chakravarty, 8 Jun 2025)	SR/Deblur	20.9	0.88	LPIPS 0.32–0.35; robust on OOD UAV/Aerial scenes	50–90ms @ edge device

EAHR achieves the highest PSNR/SSIM across varied noise and blur patterns, outperforming TV, BM3D, TRL2, SOCF, and DCA baselines (Zhang et al., 2020). EoREN outperforms classical implicit approaches on edge-rich and handwritten digit datasets (Chang et al., 2022). MPGD-based EIRM attains SOTA restoration on natural and aerial imagery in real-time edge deployments (Chakravarty, 8 Jun 2025). GAN-based models, via quantization and tiling, permit real-time restoration on low-power edge hardware (Abeykoon et al., 2019).

5. Edge-to-Image Module Integration and Practical Deployment

Practical edge-to-image solutions are modular and compatible with embedded, cloud, and data-adjacent deployments.

Modularization and Embedding

• Block Design: Typical module block-structure includes edge-detection, adaptive weighting, and solver submodules, all differentiable and compatible with end-to-end learning in unrolled architectures (Zhang et al., 2020).
• Hardware Deployment: Quantization (8-bit, FP16), operator fusion (TensorRT/TFLite), tiling/stitching for large images, and fine-tune post-processors enable SSIM-preserving restoration on sub-10W platforms (Edge TPU, Jetson TX2/Orin Nano) (Abeykoon et al., 2019, Chakravarty, 8 Jun 2025).
• API Schemes and ROS: Restoration modules are exposed as callable Python/C++ APIs, suitable for robotic vision stacks, supporting variable fidelity/latency tradeoffs via dynamic adjustment of inner update numbers (e.g., $K=7$ for fast, $K=20$ for high fidelity) (Chakravarty, 8 Jun 2025).

Best Practices

Edge-to-image module deployment recommends calibration set monitoring, local data source colocation, profiling of end-to-end latency, and parameter scheduling for robust, sustained real-time operation (Abeykoon et al., 2019, Chakravarty, 8 Jun 2025).

6. Theoretical Guarantees and Future Extensions

• Convergence and Consistency: Discrete wavelet-frame algorithms rigorously $\Gamma$-converge to their continuous variational targets, ensuring solution consistency as grid size increases (Choi et al., 2017).
• Convexity and Convergence Rate: EAHR yields convex subproblems with guaranteed global linear-rate convergence by sPADMM under standard assumptions (Zhang et al., 2020).
• Robustness and Adaptivity: MPGD-based multi-step restoration is robust to distribution shift and does not require repeated offline retraining; the inner update schedule provides a natural degree of control over the quality-latency tradeoff for embedded AI (Chakravarty, 8 Jun 2025).

Future work may exploit further integration with end-to-end learned edge detectors, multi-scale fusions, blind operator estimation, and dynamic attention-based regularization as suggested by extension notes in (Zhang et al., 2020), with increasing algorithmic and hardware efficiency for application in broader inverse imaging domains.