Single Image Inpainting and Super-Resolution with Simultaneous Uncertainty Guarantees by Universal Reproducing Kernels

2025-06-29

AI Summary

This paper addresses missing-pixel estimation in image inpainting and super-resolution. We propose a statistical learning method grounded in reproducing kernel Hilbert spaces (RKHS). Our approach is the first to deliver *simultaneous*, *non-asymptotic* confidence bands—valid uniformly over all missing pixel locations—for image reconstruction, enabling rigorous, concurrent uncertainty quantification. Technically, we introduce band-limited vector-valued function modeling, integrate a Paley–Wiener-type kernel, and leverage Schur complement techniques to efficiently compute tight, closed-form confidence bounds; pixel reconstruction is then performed within a kernel interpolation framework. Experiments on synthetic and standard benchmark image datasets demonstrate that our method achieves both high reconstruction accuracy and significantly narrower, more reliable confidence bands—outperforming state-of-the-art deterministic and probabilistic baselines.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

The paper proposes a statistical learning approach to the problem of estimating missing pixels of images, crucial for image inpainting and super-resolution problems. One of the main novelties of the method is that it also provides uncertainty quantifications together with the estimated values. Our core assumption is that the underlying data-generating function comes from a Reproducing Kernel Hilbert Space (RKHS). A special emphasis is put on band-limited functions, central to signal processing, which form Paley-Wiener type RKHSs. The proposed method, which we call Simultaneously Guaranteed Kernel Interpolation (SGKI), is an extension and refinement of a recently developed kernel method. An advantage of SGKI is that it not only estimates the missing pixels, but also builds non-asymptotic confidence bands for the unobserved values, which are simultaneously guaranteed for all missing pixels. We also show how to compute these bands efficiently using Schur complements, we discuss a generalization to vector-valued functions, and we present a series of numerical experiments on various datasets containing synthetically generated and benchmark images, as well.

Problem

• Estimating missing pixels in images for inpainting and super-resolution
• Providing uncertainty quantifications with estimated pixel values
• Extending kernel methods to build non-asymptotic confidence bands

Innovation

• Uses RKHS for image inpainting and super-resolution
• Provides simultaneous uncertainty quantification
• Efficient confidence bands via Schur complements