MG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image Restoration

Jul 16, 2026

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MG-SpaIR presents a training-data-free restoration framework that reconstructs a clean image from a single corrupted observation containing blur, downsampling, noise, and missing pixels. Using implicit neural representations (INRs), it builds a multi-grade coarse-to-fine residual hierarchy to progressively refine reconstructions across resolution levels, enhancing representational fidelity and alleviating spectral limitations. An explicit sparse proximal regularization (ℓ0-type) is applied in the high-resolution image domain to stabilize optimization and suppress INR artifacts. A multi-grade proximal alternating solver is used with convergence guarantees, and experiments on mixed-degradation benchmarks show consistent gains over strong training-data-free baselines such as Deep Image Prior.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages40

Reading Time1 h 41 min

Introduction

Problem: single-observation mixed degradation and ill-posedness

Classical priors and their limitations

Implicit neural representations and their challenges

Proposed method: MG-SpaIR contributions

arXiv :2607 .00138v1 [ cs .CV] 30 Jun 2026 MG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image Restoration Jianmin Liao 1*†, Lei Huang2†, Ronglong Fang3 , Ashley Prater-Bennette4 , Lixin Shen 1 , Yuesheng Xu2 1* Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, 13210, NY, USA. 2 Department of Mathematics & Statistics, Old Dominion University, 2300 Engineering & Computational Sciences Building, Norfolk, 23529, VA, USA. 3 Department of Medical Physical, Memorial Sloan Kettering Cancer Center, 1250 First Avenue, New York, 10065, NY, USA. 4 Air Force Research Laboratory, 525 Brooks Road, Rome, 13441, NY, USA. *Corresponding author(s). E-mail(s): [jliao21@syr.edu](jliao21@syr.edu) ; Contributing authors: [lhuan001@odu.edu](lhuan001@odu.edu) ; [fangr1@mskcc.org](fangr1@mskcc.org) ; [Ashley.Prater-Bennette@us.af.mil](Ashley.Prater-Bennette@us.af.mil) ; [lshen03@syr.edu](lshen03@syr.edu) ; [y1xu@odu.edu](y1xu@odu.edu) ; †These authors contributed equally to this work. Abstract MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade coarse-to-fine residual hierarchy that progressively refines the reconstruction across resolution grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g. , ℓ0-type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a multi-grade proximal alternating scheme, and we establish convergence guarantees for the 1 associated updates under standard regularity conditions. Experiments on mixeddegradation benchmarks demonstrate that MG-SpaIR consistently outperforms strong training-data-free baselines such as Deep Image Prior, providing a stable, interpretable, and data-efficient alternative to conventional learning-based restoration methods. Keywords: image restoration, training-data-free, implicit neural representation, sparse regularization, multi-grade deep learning A revised version of this manuscript has been accepted for publication in the Journal of Mathematical Imaging and Vision, Collection: Special Issue on Mathematics of Imaging and Machine Learning. 1 Introduction We study the restoration of a clean image from a single observation corrupted by a mixture of degradations—blur, downsampling, noise, and missing pixels—reflecting common failure modes in real imaging systems. Because many clean images can explain the same degraded observation, image restoration is inherently ill-posed, making effective priors essential for accurate recovery. Classical restoration methods rely on explicit priors, most notably sparsity-based regularization such as total variation [1] and non-local self-similarity as in BM3D [2] . While effective, these priors are typically defined on discrete pixel grids and can struggle to represent complex long-range dependencies and fine continuous structures (e.g. , hair, fur, and thin lines) . Neural networks have been explored as implicit image priors for their ability to capture natural image statistics [3] . Implicit Neural Representations (INRs) [4–10] model images as continuous coordinate-to-color mappings, naturally recovering missing pixels and fine details, making them effective for training-data-free tasks like super-resolution and inpainting. Despite their appeal, standard INRs for restoration face two fundamental challenges. First, the inherent spectral bias [10, 11] impedes the accurate representation of high-frequency structures, leading to blurry textures and poor de

FAQs

What restoration setting does MG-SpaIR address?

MG-SpaIR restores a clean image from a single observation that is corrupted by a mixture of blur, downsampling, noise, and missing pixels, without relying on external training data.

How does MG-SpaIR improve reconstruction quality compared with standard INRs?

It introduces a multi-grade coarse-to-fine residual hierarchy to progressively refine details across resolution grades, mitigating spectral limitations that commonly cause blurry high-frequency content.

How does MG-SpaIR reduce artifacts during training-data-free optimization?

It adds an explicit ℓ0-type sparse proximal regularization directly in the high-resolution image domain, discouraging spurious high-frequency patterns while preserving sharp structures.

Jul 16, 2026

Upload by Jiven