Image Resizing by Neural Network Operators and their Convergence Rate with Respect to the \(\boldsymbol{L}\)\(^{p}\)-norm and the Dissimilarity Index Defined through the Continuous SSIM

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 the 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 [Formula: see text]-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 operators, that have been widely studied in recent years in view of their connection with the theory of artificial neural networks. For these operators, we first establish sharp estimates in the case of [Formula: see text] and piecewise (everywhere defined) [Formula: see text] 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 [Formula: see text]-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, the likelihood index (S-index), and PSNR, and compared with other rescaling methods, including bilinear, bicubic, and upscaling de la Vallée–Poussin interpolation. 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.