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Implementation of neural network operators with applications to remote sensing data

Danilo Costarelli, Michele Piconi

TL;DR

This paper extends multidimensional Kantorovich neural network operators to image-based remote sensing data, introducing two algorithms: one for modeling RS data and another for rescaling/enhancement, both using a hyperbolic tangent sigmoidal activation. It provides the underlying convergence theory for these operators and demonstrates, on RETINA RS images, that the NN-based methods can achieve high structural similarity (SSIM) and competitive PSNR compared to classical interpolation. The results show particularly strong SSIM gains, validating the approach for RS data processing, albeit with substantial computational cost. The work also outlines future directions to optimize computation and integrate Bayesian inversion techniques for improved robustness.

Abstract

In this paper, we provide two algorithms based on the theory of multidimensional neural network (NN) operators activated by hyperbolic tangent sigmoidal functions. Theoretical results are recalled to justify the performance of the here implemented algorithms. Specifically, the first algorithm models multidimensional signals (such as digital images), while the second one addresses the problem of rescaling and enhancement of the considered data. We discuss several applications of the NN-based algorithms for modeling and rescaling/enhancement remote sensing data (represented as images), with numerical experiments conducted on a selection of remote sensing (RS) images from the (open access) RETINA dataset. A comparison with classical interpolation methods, such as bilinear and bicubic interpolation, shows that the proposed algorithms outperform the others, particularly in terms of the Structural Similarity Index (SSIM).

Implementation of neural network operators with applications to remote sensing data

TL;DR

This paper extends multidimensional Kantorovich neural network operators to image-based remote sensing data, introducing two algorithms: one for modeling RS data and another for rescaling/enhancement, both using a hyperbolic tangent sigmoidal activation. It provides the underlying convergence theory for these operators and demonstrates, on RETINA RS images, that the NN-based methods can achieve high structural similarity (SSIM) and competitive PSNR compared to classical interpolation. The results show particularly strong SSIM gains, validating the approach for RS data processing, albeit with substantial computational cost. The work also outlines future directions to optimize computation and integrate Bayesian inversion techniques for improved robustness.

Abstract

In this paper, we provide two algorithms based on the theory of multidimensional neural network (NN) operators activated by hyperbolic tangent sigmoidal functions. Theoretical results are recalled to justify the performance of the here implemented algorithms. Specifically, the first algorithm models multidimensional signals (such as digital images), while the second one addresses the problem of rescaling and enhancement of the considered data. We discuss several applications of the NN-based algorithms for modeling and rescaling/enhancement remote sensing data (represented as images), with numerical experiments conducted on a selection of remote sensing (RS) images from the (open access) RETINA dataset. A comparison with classical interpolation methods, such as bilinear and bicubic interpolation, shows that the proposed algorithms outperform the others, particularly in terms of the Structural Similarity Index (SSIM).

Paper Structure

This paper contains 9 sections, 5 theorems, 33 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Approximation results by NN operators
  3. Implementation of the operators, pseudo-code of the proposed algorithms, and considered sigmoidal functions
  4. Analysis of the computational complexity of the proposed algorithms
  5. Methods: other algorithms for numerical comparison and similarity indexes for the evaluation of the results
  6. Numerical tests for the application to remote sensing data modeling and rescaling/enhancement
  7. Applications of Algorithm 1: modeling of RS data
  8. Applications of Algorithm 2: rescaling/enhancement of RS data
  9. Future works and conclusions

Key Result

Lemma 2.2

(a) For any $\underline{x} \in \mathbb{R}^d$, we have $\sum_{\underline{k} \in \mathbb{Z}^d} \Psi_{\sigma}(\underline{x} - \underline{k}) = 1$. (b) The following series $\sum_{\underline{k} \in \mathbb{Z}^d} \Psi_{\sigma}(\underline{x} - \underline{k})$ it turns out uniformly convergent on every com uniformly with $\underline{x} \in \mathbb{R}^d$. In particular, for every $\gamma > 0$ and for ever

Figures (5)

  • Figure 1: Screenshot of the RETINA website.
  • Figure 2: Screenshot of the dataset from which the images can be freely downloaded.
  • Figure 3: Satellite images from the RETINA dataset. (a) Rome.tif, (b) Berlin.tif, (c) Lisbon.tif, (d) Granada.tif.
  • Figure 4: Reference images: (a) Rome; (b) Berlin; (c) Lisbon; (d) Granada.
  • Figure 5: Comparison of rescaling methods in terms of PSNR and SSIM. (a) Rome, (b) Berlin, (c) Lisbon, (d) Granada.

Theorems & Definitions (6)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6