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Image resizing by neural network operators and their convergence rate with respect to the $L^p$-norm and the dissimilarity index defined through the continuous SSIM

Danilo Costarelli, Mariarosaria Natale, Michele Piconi

TL;DR

This work develops a theoretical and numerical framework for multivariate neural-network (NN) operators applied to image resizing, establishing $L^p$ convergence rates via averaging modulus techniques and linking convergence to a continuous dissimilarity index (cSSIM). It provides sharp bounds for $C^1$ and piecewise $C^1$ functions, extends to full $L^p$ spaces through the $\tau$-modulus and Riesz–Thorin interpolation, and introduces a concrete NN-based resizing algorithm with practical performance assessments. The study connects theoretical guarantees to image similarity metrics (SSIM/cSSIM, S-index, PSNR) and demonstrates competitive results against bilinear, bicubic, and u-VPI methods, especially on high-variance images, while maintaining a deterministic, training-free approach. Overall, the paper offers a rigorous foundation and practical NN-operator method for image resizing with provable convergence and favorable empirical behavior.

Abstract

In literature, several algorithms for imaging based on interpolation or approximation methods are available. The implementation of theoretical processes highlighted the necessity of providing theoretical frameworks for the convergence and error estimate analysis to support the experimental setups. In this paper, we establish new techniques for deriving quantitative estimates for the order of approximation for multivariate linear operators of the pointwise-type, with respect to the $L^p$-norm and to the so-called dissimilarity index defined through the continuous SSIM. In particular, we consider a family of approximation operators known as neural network (NN) operators, that have been widely studied in the last years in view of their connection with the theory of artificial neural networks. For these operators, we first establish sharp estimates in case of $C^1$ and piecewise (everywhere defined) $C^1$-functions. Then, the case of functions modeling digital images is considered, and specific quantitative estimates are achieved, including those with respect to the mentioned dissimilarity index. Moreover, the above analysis has also been extended to $L^p$-spaces, using a new constructive technique, in which the multivariate averaged modulus of smoothness has been employed. Finally, numerical experiments of image resizing have been given to support the theoretical results. The accuracy of the proposed algorithm has been evaluated through similarity indexes such as SSIM, likelihood index (S-index) and PSNR, and compared with other rescaling methods, including bilinear, bicubic, and upscaling-de la Vallée-Poussin interpolation (u-VPI). Numerical simulations show the effectiveness of the proposed method for image processing tasks, particularly in terms of the aforementioned SSIM, and are consistent with the provided theoretical analysis.

Image resizing by neural network operators and their convergence rate with respect to the $L^p$-norm and the dissimilarity index defined through the continuous SSIM

TL;DR

This work develops a theoretical and numerical framework for multivariate neural-network (NN) operators applied to image resizing, establishing convergence rates via averaging modulus techniques and linking convergence to a continuous dissimilarity index (cSSIM). It provides sharp bounds for and piecewise functions, extends to full spaces through the -modulus and Riesz–Thorin interpolation, and introduces a concrete NN-based resizing algorithm with practical performance assessments. The study connects theoretical guarantees to image similarity metrics (SSIM/cSSIM, S-index, PSNR) and demonstrates competitive results against bilinear, bicubic, and u-VPI methods, especially on high-variance images, while maintaining a deterministic, training-free approach. Overall, the paper offers a rigorous foundation and practical NN-operator method for image resizing with provable convergence and favorable empirical behavior.

Abstract

In literature, several algorithms for imaging based on interpolation or approximation methods are available. The implementation of theoretical processes highlighted the necessity of providing theoretical frameworks for the convergence and error estimate analysis to support the experimental setups. In this paper, we establish new techniques for deriving quantitative estimates for the order of approximation for multivariate linear operators of the pointwise-type, with respect to the -norm and to the so-called dissimilarity index defined through the continuous SSIM. In particular, we consider a family of approximation operators known as neural network (NN) operators, that have been widely studied in the last years in view of their connection with the theory of artificial neural networks. For these operators, we first establish sharp estimates in case of and piecewise (everywhere defined) -functions. Then, the case of functions modeling digital images is considered, and specific quantitative estimates are achieved, including those with respect to the mentioned dissimilarity index. Moreover, the above analysis has also been extended to -spaces, using a new constructive technique, in which the multivariate averaged modulus of smoothness has been employed. Finally, numerical experiments of image resizing have been given to support the theoretical results. The accuracy of the proposed algorithm has been evaluated through similarity indexes such as SSIM, likelihood index (S-index) and PSNR, and compared with other rescaling methods, including bilinear, bicubic, and upscaling-de la Vallée-Poussin interpolation (u-VPI). Numerical simulations show the effectiveness of the proposed method for image processing tasks, particularly in terms of the aforementioned SSIM, and are consistent with the provided theoretical analysis.
Paper Structure (10 sections, 17 theorems, 111 equations, 4 figures, 7 tables, 1 algorithm)

This paper contains 10 sections, 17 theorems, 111 equations, 4 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Some recall on the multivariate neural network operators
  3. Asymptotic estimates for multivariate operators: new approaches
  4. Error analysis for static greyscale images and implementation of the resizing algorithm
  5. Rate of convergence in $L^p$-spaces via a constructive density method by the averaged modulus of smoothness
  6. Convergence rate in terms of SSIM and cSSIM indexes
  7. Other indexes: S-index and PSNR
  8. Other methods of image rescaling
  9. Numerical simulations and experiments
  10. Final remarks, conclusions and future developments

Key Result

Lemma 1

Let ${\tt{x}}\in I$ and $n\in\mathbb{N}^+$, then

Figures (4)

  • Figure 1: Discrete dissimilarity index for the NN algorithm with ramp and logistic functions, plotted with $n = 5, 10, 15, 20, 25, 30$ on the horizontal axis, bilinear and bicubic interpolation, and u-VPI method for the images (a) montage, (b) france.
  • Figure 2: Discrete dissimilarity index for the NN algorithm with ramp and logistic functions, plotted with $n = 5, 10, 15, 20, 25, 30$ on the horizontal axis, bilinear and bicubic interpolation, and u-VPI method for the images (c) mountain, (d) library.
  • Figure 3: Histogram comparison of the PSNR, S-Index, and SSIM values for different test grayscale (left) and RGB (right) images (listed along the vertical axis), with the parameter fixed at $n = 30$. Each bar represents the corresponding quality index for a given image, highlighting the performance of the proposed method across multiple metrics.
  • Figure 4: Numerical dissimilarity index $1 - \text{cSSIM}$ obtained by NN algorithm with ramp and logistic functions is shown in (a) for montage and in (c) for france. In (b) and (d), the same numerical dissimilarity is compared with the theoretical dissimilarity $\frac{cf \cdot \log n}{n}$ as a function of $n = 5, 10, 15, 20, 25, 30$. In both (b) and (d), the numerical dissimilarity remains significantly below the theoretical curve.

Theorems & Definitions (35)

  • Lemma 1: COSP3
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 25 more