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

TL;DR

This paper introduces SGKI, a kernel-based method for image inpainting and super-resolution that not only estimates missing pixels but also provides simultaneous uncertainty quantification with guaranteed confidence bands.

Contribution

It extends kernel methods to include uncertainty quantification in image reconstruction, especially for band-limited functions in RKHSs, with efficient computation and broad applicability.

Findings

SGKI provides accurate pixel estimates with confidence bands.

The method guarantees uncertainty bounds simultaneously for all missing pixels.

Numerical experiments demonstrate effectiveness on synthetic and benchmark images.

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.