Danilo Costarelli

University of Perugia | UNIPG · Department of Mathematics and Computer Science

Human actions have accelerated changes in global temperature, precipitation patterns, and other critical Earth systems. Key markers of these changes can be linked to the dynamic of Essential Climate Variables (ECVs) and related quantities, such as Soil Moisture (SM), Above Ground Biomass (AGB), and Freeze-Thaw (FT) Dynamics. These variables are cru...

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)...

In this paper, we introduce the nonlinear exponential Kantorovich sampling series. We establish pointwise and uniform convergence properties and a nonlinear asymptotic formula of the Voronovskaja-type given in terms of the limsup. Furthermore, we extend these convergence results to Mellin-Orlicz spaces with respect to the logarithmic (Haar) measure...

The main aim of the present paper is to provide a full asymptotic analysis of a family of neural network (NN) operators based on suitable density functions within the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl...

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of...

In this paper we introduce a new class of sampling-type operators, named Steklov sampling operators. The idea is to consider a sampling series based on a kernel function that is a discrete approximate identity, and which constitutes a reconstruction process of a given signal f , based on a family of sample values which are Steklov integrals of orde...

This book covers contemporary topics in mathematical analysis and its applications and relevance in other areas of research. It provides a better understanding of methods, problems, and applications in mathematical analysis. It also covers applications and uses of operator theory, approximation theory, optimization, variable exponent analysis, ineq...

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes...

In this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the recon...

In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube $[0, 1]^d$ and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its square. In particular, we prove uniform convergence, by means of two different approaches, as well as a quantitative...

The aim of this paper is to compare the fuzzy-type algorithm for image rescaling introduced by Jurio et al., 2011, quoted in the list of references, with some other existing algorithms such as the classical bicubic algorithm and the sampling Kantorovich (SK) one. Note that the SK algorithm is a recent tool for image rescaling and enhancement that h...

PROJECT PRIN 2022 PNRR, Acronym: "RETINA", funded by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, under the MUR (Project Code: P20229SH29, CUP: J53D23015950001) PI: Prof. Danilo Costarelli, University of Perugia (Italy) What is the problem? The Earth is experiencing tangible climate chang...

The present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in “Sobolev-like settings”. More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type c...

In the present paper we study the perturbed sampling Kantorovich operators in the general context of the modular spaces. After proving a convergence result for continuous functions with compact support, by using both a modular inequality and a density approach, we establish the main result of modular convergence for these operators. Further, we sho...

In the present paper, convergence in modular spaces is investigated for a class of nonlinear discrete operators, namely the nonlinear multivariate sampling Kantorovich operators. The convergence results in the Musielak-Orlicz spaces, in the weighted Orlicz spaces, and in the Orlicz spaces follow as particular cases. Even more, spaces of functions e...

In this paper, we study the order of approximation for max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. F...

In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions defined on both bounded intervals and on the whole real axis has been considered. Here, under suitable assumption...

In this paper, we take advantage of the reconstruction properties of the sampling Kantorovich (SK) algorithm to estimate the volume of the human brain for the quantification of Alzheimer's biomarkers. At first, the goodness of the reconstructions is evaluated, comparing it to different interpolation methods by means of the Peak Signal to Noise Rati...

This paper deals with the study of the convergence of the family of multivariate Durrmeyer-sampling type operators in the general setting of Orlicz spaces. The above result implies also the convergence in remarkable subcases, such as in Lebesgue, Zygmund and exponential spaces. Convergence results have been established also in case of continuous fu...

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modul...

In this paper, we establish a quantitative estimate for Durrmeyer-sampling type operators in the general framework of Orlicz spaces, using a suitable modulus of smoothness defined by the involved modular functional. As a consequence of the above result, we can deduce quantitative estimates in several instances of Orlicz spaces, such as $$L^p$$ L p...

In the present paper, a new family of multi-layers (deep) neural network (NN) operators is introduced. Density results have been established in the space of continuous functions on [−1,1], with respect to the uniform norm. First, the case of the operators with two-layers is considered in detail, then the definition and the corresponding density res...

In this paper a new family of sampling type series is introduced. From the mathematical point of view, the present definition generalizes the notion of the well-known sampling Kantorovich operators, in fact providing a weighted version of the original family of operators by functions gk,w, \(k \in \mathbb {Z}\), w > 0, called noise functions. From...

In this paper, for the neural network (NN) operators activated by sigmoidal functions, we will obtain quantitative estimates in direct connection with the asymptotic behavior of their activation functions. We will cover all cases of discrete and Kantorovich type versions of NN operators in both univariate and multivariate settings since the proofs...

Image resizing is frequently used as a preprocessing step in many computer vision tasks, especially in medical applications. While tuning of the resizing method is usually omitted in the studies, there are many problems in which the exact influence of resampling on image textures and gradients is significant. The paper presents an in-depth analysis...

In real world applications, signals can be suitably reconstructed by nonlinear procedures; this justifies the study of nonlinear approximation operators. In this paper, we prove some quantitative estimates for the nonlinear sampling Kantorovich operators in the multivariate setting using the modulus of smoothness of L p (n). The above results have...

The aim of this article is to provide an improvement in the reconstruction and visualization of retinal superficial capillary plexus and choriocapillaris images from healthy subjects. The implemented method uses multiple Optical Coherence Tomography (OCT) scanned sequences, performs a registration, an average and a filtering on them using for this...

The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronov...

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in $$L^p$$ L p -setting, $$1 \le p \le +\infty $$ 1 ≤ p ≤ +...

In this paper, we consider the max-product neural network operators of the Kantorovich type based on certain linear combinations of sigmoidal and ReLU activation functions. In general, it is well-known that max-product type operators have applications in problems related to probability and fuzzy theory, involving both real and interval/set valued f...

In this paper the behaviour of the first derivative of the so-called sampling Kantorovich operators has been studied, when both differentiable and not differentiable signals are taken into account. In particular, we proved that the family of the first derivatives of the above operators converges pointwise at the point of differentiability of f, and...

In this paper, we establish a procedure for the enhancement of cone-beam computed tomography (CBCT) dental-maxillofacial images; this can be useful in order to face the problem of rapid prototyping, i.e., to generate a 3D printable file of a dental prosthesis. In the proposed procedure, a crucial role is played by the so-called sampling Kantorovich...

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and Lp convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampl...

In this paper, a procedure for the detection of the sources of industrial noise and the evaluation of their distances is introduced. The above method is based on the analysis of acoustic and optical data recorded by an acoustic camera. In order to improve the resolution of the data, interpolation and quasi interpolation algorithms for digital data...

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances...

An innovative technique based on beamforming is implemented, at the aim of detecting the distances from the observer and the relative positions among the noise sources themselves in multisource noise scenarios. By means of preliminary activities to assess the optical camera focal length and stereoscopic measurements followed by image processing, th...

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the dens...

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of continuity in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $L^{p}$-spaces, $1\leq p<\infty $, and in other well-known instances of Orlicz spaces,...

In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of smoothness in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of Lp-spac...

In this note we present smooth approximation to |x(1 −x) . . . (n−1−x)| using x(1−x) . . . (n−1−x)erf ((x(1−x)... (n−1−x))/µ) as µ → 0+. The above approximation results are of interest in some applications involving the so-called ”polynomial variable transfer”, such as modelling and simulation of filters characteristics and antenna charts. Numerica...

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of...

In the present paper we study the pointwise and uniform convergence properties of a family of multidimensional sampling Kantorovich type operators. Moreover, besides convergence, quantitative estimates and a Voronovskaja type theorem have been established.

In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich sampling operators with respect to the concept of multidimensional variation introduced by Tonelli. A sharper estimate can be achieved when step functions with compact support (digital images) are considered. Several examples of kernels have been prese...

In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such a...

In the present paper, a new family of sampling type operators is introduced and studied. By the composition of the well-known generalized sampling operators of P.L. Butzer with the usual differential and anti-differential operators of order m, we obtain the so-called m-th order Kantorovich type sampling series. This family of approximation operator...

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modul...

In the present paper, asymptotic expansion and Voronovskaja type theorem for the neural network operators have been proved. The above results are based on the computation of the algebraic truncated moments of the density functions generated by suitable sigmoidal functions, such as the logistic functions, sigmoidal functions generated by splines and...

In this paper the estimation of masonry characteristics by means of thermographic images, enhanced by sampling Kantorovich algorithm, is taken into account. In particular, the convergence of the Statistical Volume Element (SVE) to the Representative Volume Element (RVE) is analyzed. It is found that the enhancement, obtained by the proposed procedu...

The study of inverse results of approximation for the family of sampling Kantorovich operators in case of α-Hölder function, 0 < α < 1, has been solved in a recent paper of some of the authors. However, the limit case of Lipschitz functions, i.e., when α = 1, in which standard methods fail, remained unsolved. In this paper, a solution of the above...

In this paper we study the performance of the sampling Kantorovich (S–K) algorithm for image processing with other well-known interpolation and quasi-interpolation methods. The S-K algorithm has been implemented with three different families of kernels: central B-splines, Jackson type and Bochner–Riesz. The above method is compared, in term of PSNR...

In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces $L^{\varphi}$. The results here proved, extend those given by Costarelli and Vinti in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies...

In the present paper, we study the saturation order in the space $L^1(\R)$ for the sampling Kantorovich series based upon bandlimited kernels. The above study is based on the so-called Fourier transform method, introduced in 1960 by P.L. Butzer. As a first result, the saturation order is derived in a Bernstein class; here, it is crucial to derive t...

In this paper, convergence results in a multivariate setting have been proved for a family of neural network operators of the max-product type. In particular, the coefficients expressed by Kantorovich type means allow to treat the theory in the general frame of the Orlicz spaces, which includes as particular case the $L^p$-spaces. Examples of sigmo...

In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, w...

In this paper, we study the order of approximation with respect to the Jordan variation for the generalized and the Kantorovich sampling series, based upon averaged type kernels. In particular, we establish some quantitative estimates for the above operators. For the latter purpose, we introduce a suitable modulus of smoothness in the space of abso...

In the present paper we develop a numerical collocation scheme for integral equations based on the interpolation method which uses the so-called neural network operators activated by ramp functions. The proposed collocation scheme allows to solve both linear and nonlinear Volterra integral equations of the second kind; the convergence of the numeri...

In this paper, we develop an algorithm for the segmentation of the pervious lumen of the aorta artery in computed tomography (CT) images without contrast medium, a challenging task due to the closeness gray levels of the different zones to segment. The novel approach of the proposed procedure mainly resides in enhancing the resolution of the image...

A large class of multivariate quasi-projection operators is studied. These operators are sampling-type expansions ∑k∈Zdck(f)ψ(Aj·−k), where A is a matrix and the coefficients ck(f) are associated with a tempered distribution ψ˜. Error estimates in Lp-norm, 2 ≤ p < ∞, are obtained under the so-called weak compatibility conditions for ψ˜ and ψ, and t...

Introduction: Contrast medium (CM) use in computed tomography (CT) is limited by nephrotoxicity and possible allergic reactions. The purpose of this study is to introduce a tool for the diagnosis of abdominal aortic aneurysms (AAAs) by avoiding the use of CM. Methods: With and without CM CTs of patients with AAA were evaluated. A mathematical al...

In this paper we study the problem of the convergence in variation for the generalized sampling series based upon averaged-type kernels in the multidimensional setting. As a crucial tool, we introduce a family of operators of sampling-Kantorovich type for which we prove convergence in L^p on a subspace of L^p(R^N): therefore we obtain the convergen...

In this paper, we extend the saturation results for the sampling Kantorovich operators proved in a previous paper, to more general settings. In particular, exploiting certain Voronovskaja-formulas for the well-known generalized sampling series, we are able to extend a previous result from the space of $C^2$-functions to the space of $C^1$-functions...

In this paper we study the theory of the so-called multivariate sampling Kantorovich operators in the general frame of the Musielak-Orlicz spaces. The main result in this context is a modular convergence theorem, that can be proved by density arguments. Several concrete cases of Musielak-Orlicz spaces and of kernel functions are presented and discu...

In the present paper, we study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. In order to achieve the above result, we first need to establish a relation between the sampling Kantorovich operators and the classical generalized sampling series of P.L. Butzer. Further, for the latt...

In the present paper we establish a quantitative estimate for the sampling Kantorovich operators with respect to the modulus of continuity in Orlicz spaces defined in terms of the modular functional. At the end of the paper, concrete examples are discussed, both for what concerns the kernels of the above operators, as well as for some concrete inst...

We study the convergence in variation for the sampling Kantorovich operators in both the cases of averaged-type kernels and classical band-limited kernels. In the first case, a characterization of the space of the absolutely continuous functions in terms of the convergence in variation is obtained.

In this paper we study the max-product version of the generalized sampling operators based upon a general kernel function. In particular, we prove pointwise and uniform convergence for the above operators, together with a certain quantitative Jackson-type estimate based on the first order modulus of continuity of the function being approximated. Th...

In the present paper, an inverse result of approximation, i.e., a saturation theorem for the sampling Kantorovich operators is derived, in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, here we prove that the best possible order of approximation that can be achieved by the abo...

An asymptotic-numerical method to solve the initial-boundary value problems for systems of balance laws in one space dimension, on the half space is developed. Expansions in powers of \(t^{-1/2}\) are used, in view of the precise asymptotic behavior recently established on theoretical bases. This approach increases considerably the efficiency of a...

In this paper, we study the theory of a Kantorovich version of the multivariate neural network operators. Such operators, are activated by suitable kernels generated by sigmoidal functions. In particular, the main result here proved is a modular convergence theorem in Orlicz spaces. As special cases, convergence theorem in \(L^p\)-spaces, interpola...

In this paper, we study the convergence in variation for the generalized sampling operators based upon averaged-type kernels and we obtain a characterization of absolutely continuous functions. This result is proved exploiting a relation between the first derivative of the above operator acting on $f$ and the sampling Kantorovich series of f'. By s...

In this note, some approximation problems are discussed with applications to reconstruction and to digital image processing. We will also show some applications to concrete problems in the medical and engineering fields. Regarding the first, a procedure will be presented, based on approaches of approximation theory and on algorithms of digital imag...

In the present paper, quantitative estimates for the neural network (NN) operators of the Kantorovich type have been proved. Firstly, the modulus of continuity of the function being approximated has been used in order to estimate the approximation error in the uniform norm. Finally, a Peetre K-functional has been employed to obtain a quantitative u...

A numerical method for solving systems of nonlinear one-dimensional balance laws, based on multivariate sigmoidal functions approximation, is developed. Constructive approximation theorems are first established in both, uniform and $L^p$ norms. {\em A priori} as well as {\em a posteriori} error estimates are derived for the numerical solutions,...