Sampling Kantorovich operators for speckle noise reduction using a Down-Up scaling approach and gap filling in remote sensing images

TL;DR

The paper tackles gap filling and speckle noise in remote sensing imagery by developing multivariate sampling Kantorovich (SK) operators. It provides rigorous theoretical backing, including $C(\,\mathbb{R}^n\,)$ error bounds via Euler–Maclaurin and $L^1\cap L^2$ convergence with respect to the continuous SSIM ($cSSIM$), plus a linear prediction result for an LP-SK gap-filling algorithm. For despeckling, it introduces a Down-Up scaling workflow that couples SK-based upscaling with traditional filters, showing improvements in without-reference metrics on synthetic and real SAR data. Collectively, the methods offer robust, single-source gap filling and despeckling suitable for streaming or progressively acquired remote sensing data, with theoretical guarantees guiding practical performance.

Abstract

In the literature, several approaches have been proposed for restoring and enhancing remote sensing images, including methods based on interpolation, filtering, and deep learning. In this paper, we investigate the application of multivariate sampling Kantorovich (SK) operators for image reconstruction, with a particular focus on gap filling and speckle noise reduction. To understand the accuracy performances of the proposed algorithms, we first derive a quantitative estimate in $C(\R^n)$ for the error of approximation using the Euler-Maclaurin summation formula, under weak regularity conditions. We also establish a convergence result and a quantitative estimate with respect to the dissimilarity index measured by the continuous SSIM for functions in Lebesgue spaces. Additionally, we prove a multidimensional linear prediction result, which is used to design a new SK-based reconstruction algorithm to handle missing data, that we call LP-SK algorithm. To address speckle noise, we integrate SK operators into a newly proposed Down-Up scaling approach. Numerical tests are presented on synthetic and real SAR images to validate the proposed methods. Performance is assessed using similarity metrics such as SSIM and PSNR, along with speckle-specific indexes. Comparative analysis with state-of-the-art techniques highlights the effectiveness of the proposed approaches.

Figures (13)

• Figure 1: The dark gray cell indicates the missing pixel located at coordinates $(\nu,\mu)$. The light gray region represents the set of "past pixels" employed in the image reconstruction.
• Figure 2: The elementary ball in $4$-connectivity (a), and $8$-connectivity (b).
• Figure 3: Images with simulated loss: (a) Cameraman with $2.43$% missing pixels; (b) Bird with $4.47$% missing pixels; (c) Baboon with $3.79$% missing pixels.
• Figure 4: Reconstructed images using LP-SK Algorithm.
• Figure 5: Absolute error for each pixel reconstructed by LP-SK: (a) Cameraman; (b) Bird; (c) Baboon.

Theorems & Definitions (14)

• Lemma 1: Lemma 3.2 of CVbumi
• Definition 2
• Theorem 3: Theorem 4.3 and Theorem 4.4 of CNV2025
• Theorem 4: Formula 2.1.2 of EMS
• Theorem 5
• proof
• Theorem 6: Theorem 4.1 of CCNV2022
• Theorem 7
• Corollary 8
• proof