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A Learned ADMM Framework with Fractional-Order Convolutional Regularization for Image Denoising
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DOI：10.1007/s00371-026-04650-4.png)
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Abstract
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Image denoising under complex noise models including additive Gaussian, Poisson, and mixed Gaussian-Poisson corruptions remains a fundamental challenge in computational imaging, where the underlying inverse problem is inherently ill-posed and the associated variational objective is both nonsmooth and nonconvex. Existing deep denoising networks either lack interpretability and convergence guarantees, or rely on hand-crafted regularizers that fail to capture the intricate geometric structure of natural image features. Model-based unrolling approaches have partially bridged this gap, but they typically impose fixed integer-order differential operators that cannot adapt to spatially varying image content, and they rarely provide rigorous convergence analysis for the resulting nonconvex learning problems. We propose FOCNet-LADMM, a model-based deep denoising framework that integrates three tightly coupled components. First, a Fractional-Order Convolutional Network (FOCNet) learns a spatially adaptive feature representation governed by a fractional-order differential operator of order $$\alpha \in (0,2)$$ , enabling the network to interpolate continuously between first- and second-order feature extraction according to local image geometry. Second, an $$\ell _{2,1}$$ group-sparsity regularizer is imposed directly in the learned FOCNet feature space, promoting structured sparsity across feature channels while remaining robust to outliers. Third, the resulting nonsmooth nonconvex composite minimization problem is solved by a Learnable Alternating Direction Method of Multipliers (L-ADMM) algorithm in which every proximal subproblem admits a closed-form group-shrinkage solution, and all algorithmic parameters? including step sizes, penalty coefficients, and fractional orders; are learned end-to-end from training data via back-propagation through the unrolled iterations. Under mild assumptions on the network architecture and the loss landscape; specifically, a Kurdyka–Łojasiewicz (KL) inequality on the augmented Lagrangian and bounded iterates; we prove that every limit point of the L-ADMM sequence is a stationary point of the nonconvex objective, and that the iterate sequence itself converges globally. We further establish a parameter efficiency bound showing that FOCNet-LADMM achieves a given approximation accuracy with $$\mathcal {O}(C / \epsilon )$$ fewer learnable parameters than a comparable integer-order baseline of the same depth, where C is a constant depending only on the fractional order $$\alpha $$ . Extensive experiments on standard benchmarks dataset demonstrate that FOCNet-LADMM achieves consistent improvements over state-of-the-art methods.
Keywords:
Image denoising
Gaussian–Poisson noise
Model-based deep learning
Learned regularization
Fractional-order convolution
Nonsmooth optimization
ADMM
Computational imaging
Medical image restoration.
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