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.
Abstract (translated)
该论文提出了一种统计学习方法，用于解决图像中缺失像素的估计问题，这对于图像修复（inpainting）和超分辨率问题至关重要。这种方法的主要创新之一是它不仅提供了估计值，还给出了不确定性量化。我们假设底层的数据生成函数来自一个再生核希尔伯特空间（RKHS）。特别强调的是限带信号处理中的限带函数，它们构成了帕莱-维纳类型RKHS。 所提出的方法，即同时保证的核插值法（Simultaneously Guaranteed Kernel Interpolation, SGKI），是对最近开发出的一种核方法的扩展和改进。SGKI的一个优点是它不仅能估计缺失像素，还能为未观测到的所有像素构建非渐近置信区间，并且这些置信区间对所有缺失像素同时有效。我们还展示了如何使用舒尔补来高效地计算这些区间，并讨论了向向量值函数的一般化方法。此外，我们通过一系列在包含合成数据和基准图像的多种数据集上的数值实验来验证该方法的有效性。
URL
https://arxiv.org/abs/2506.23221
