Danilo Costarelli

Danilo Costarelli, Mariarosaria Natale, Michele Piconi

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.

Danilo Costarelli, Donato Lavella

In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and two- sided error estimates under non-restrictive assumptions for pointwise linear operators. % The proposed results have been proved exploiting the regularization and approximation properties of certain Steklov integrals introduced by Sendov and Popov in 1983. % By the definition of semi-discrete moduli of smoothness here proposed, we derive sharper estimates than those that can be achieved by the classical averaged moduli of smoothness ($τ$-moduli). % Furthermore, a Rathore-type theorem is established, and a new notion of K-functional is also introduced showing its equivalence with the semi-discrete modulus of smoothness and its realization. One-sided estimates of approximation can be established for classical operators on bounded domains, such as the Bernstein polynomials. In the case of approximation operators on the whole real line, one-sided estimates can be achieved, e.g., for the Shannon sampling (cardinal) series, as well as for the so-called generalized sampling operators.

Danilo Costarelli, Michele Piconi

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).

Francesco Asdrubali, Giorgio Baldinelli, Francesco Bianchi et al.

In this paper, we develop a procedure for the detection of the contours of thermal bridges from thermographic images, in order to study the energetic performance of buildings. Two main steps of the above method are: the enhancement of the thermographic images by an optimized version of the mathematical algorithm for digital image processing based on the theory of sampling Kantorovich operators, and the application of a suitable thresholding based on the analysis of the histogram of the enhanced thermographic images. Finally, an accuracy improvement of the parameter that defines the thermal bridge is obtained.

Danilo Costarelli, Federico Cluni, Anna Maria Minotti et al.

In this paper, we present some applications of the multivariate sampling Kantorovich operators $S_w$ to seismic engineering. The mathematical theory of these operators, both in the space of continuous functions and in Orlicz spaces, show how it is possible to approximate/reconstruct multivariate signals, such as images. In particular, to obtain applications for thermographic images a mathematical algorithm is developed using MATLAB and matrix calculus. The setting of Orlicz spaces is important since allow us to reconstruct not necessarily continuous signals by means of $S_w$. The reconstruction of thermographic images of buildings by our sampling Kantorovich algorithm allow us to obtain models for the simulation of the behavior of structures under seismic action. We analyze a real world case study in term of structural analysis and we compare the behavior of the building under seismic action using various models.