Document Type

Article

Publication Date

2026

DOI

10.1007/s10851-026-01329-2

Publication Title

Journal of Mathematical Imaging and Vision

Volume

68

Issue

4

Pages

48 (27 pp.)

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 residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across 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 associated updates under standard regularity conditions. Experiments on mixed-degradation 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.

Rights

© 2026 The Authors.

 This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 

Original Publication Citation

Liao, J., Huang, L., Fang, R., Prater-Bennette, A., Shen, L., & Xu, Y. (2026). MG-SpaIR: Multi-grade sparse-guided implicit representation for training-data-free image restoration. Journal of Mathematical Imaging and Vision, 68(4), Article 48. https://doi.org/10.1007/s10851-026-01329-2

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