[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83457-en":3,"doc-seo-83457-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83457,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","MG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image Restoration","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.","arXiv :2607 .00138v1 [ cs .CV] 30 Jun 2026  \nMG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image  \nRestoration  \nJianmin Liao 1*†, Lei Huang2†, Ronglong Fang3 , Ashley Prater-Bennette4 , Lixin Shen 1 , Yuesheng Xu2  \n1* Department of Mathematics, Syracuse University, 215 Carnegie  \nBuilding, Syracuse, 13210, NY, USA.  \n2 Department of Mathematics & Statistics, Old Dominion University, 2300 Engineering & Computational Sciences Building, Norfolk, 23529,  \nVA, USA.  \n3 Department of Medical Physical, Memorial Sloan Kettering Cancer Center, 1250 First Avenue, New York, 10065, NY, USA.  \n4 Air Force Research Laboratory, 525 Brooks Road, Rome, 13441, NY, USA.  \n*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) ;  \n[Ashley.Prater-Bennette@us.af.mil](Ashley.Prater-Bennette@us.af.mil) ; [lshen03@syr.edu](lshen03@syr.edu) ; [y1xu@odu.edu](y1xu@odu.edu) ;  \n†These authors contributed equally to this work.  \nAbstract  \nMG-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  \n1  \nassociated 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.  \nKeywords: image restoration, training-data-free, implicit neural representation,  \nsparse regularization, multi-grade deep learning  \nA 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  \nMachine Learning.  \n1 Introduction  \nWe 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.  \nClassical 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) .  \nNeural 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.  \nDespite 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","cbCaiaKYf5F8n1eB","https://ap.wps.com/l/cbCaiaKYf5F8n1eB","pdf",33123865,5,1,40,"English","en",105,"# Introduction\n## Problem: single-observation mixed degradation and ill-posedness\n## Classical priors and their limitations\n## Implicit neural representations and their challenges\n## Proposed method: MG-SpaIR contributions","[{\"question\":\"What restoration setting does MG-SpaIR address?\",\"answer\":\"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.\"},{\"question\":\"How does MG-SpaIR improve reconstruction quality compared with standard INRs?\",\"answer\":\"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.\"},{\"question\":\"How does MG-SpaIR reduce artifacts during training-data-free optimization?\",\"answer\":\"It adds an explicit ℓ0-type sparse proximal regularization directly in the high-resolution image domain, discouraging spurious high-frequency patterns while preserving sharp 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restoration setting does MG-SpaIR address?","Question",{"text":76,"@type":77},"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.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does MG-SpaIR improve reconstruction quality compared with standard INRs?",{"text":81,"@type":77},"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.",{"name":83,"@type":74,"acceptedAnswer":84},"How does MG-SpaIR reduce artifacts during training-data-free optimization?",{"text":85,"@type":77},"It adds an explicit ℓ0-type sparse proximal regularization directly in the high-resolution image domain, discouraging spurious high-frequency patterns while preserving sharp 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