Gianluca Vinti

Università degli Studi di Perugia | UNIPG · Department of Mathematics and Computer Science

This study evaluates two approaches applied to computed tomography (CT) images of patients with abdominal aortic aneurysm: one deterministic, based on tools of Approximation Theory, and one based on Artificial Intelligence. Both aim to segment the basal CT images to extract the patent area of the aortic vessel, in order to propose an alternative to...

In the multidimensional Euclidean space, except the classical real Hardy space, there are numerous product ones. We associate with each of them a class of functions related to the known variations and new ones. Such a characterization is fulfilled by means of the integrability of the Fourier transform

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 multidimensional Euclidean space, except the classical real Hardy space, there are numerous product ones. We associate with each of them a class of functions related to the known variations and new ones. Such a characterization is fulfilled by means of the inte-grability of the Fourier transform.

In this paper, we provide a study on eye fundus images of healthy and diabetic patients. Taking benefits from its reconstruction and enhancing properties, the sampling Kantorovich algorithm is used to process the considered images, after registration and averaging processes. Moreover, a hybrid segmentation procedure applied on superficial capillary...

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, the connections between the Sampling Kantorovich model and the sampling process are highlighted and exploited. Based on the theoretical framework of the Sampling Kantorovich operators, a sampling paradigm, here named Sampling Kantorovich by Difference (SKD), is introduced. In line of principle, SKD allows for overcoming the technical...

The main purpose of this article is to prove a result of convergence in variation for a family of multidimensional sampling-Kantorovich operators in the case of averaged-type kernels. The setting in which we work is that one of BV-spaces in the sense of Tonelli.

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...

In this paper, we carry out a study developed on 13,677 images from 15 patients affected by moderate/severe atheromatous disease of the abdominal aortic tract. A procedure to extract the pervious lumen of the aorta artery from basal CT images is exploited and tested on a large scale. In particular, the above method takes advantage of the reconstruc...

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...

We continue the work started in a previous article and introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group. We analyze the behavior of such nets and detect the fairest assumption, which are needed for the nets to converge with respect to th...

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...

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...

This volume is dedicated to the memory of our colleague and friend Domenico Candeloro (Udine, October 18, 1951-Rome, May 3, 2019). It collects contributions from several of his students, coauthors and from other prominent mathematicians who were close to him. Many of the papers are strictly related to the topics that fascinated Mimmo mostly and on...

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...

A new class of functions is introduced closely related to that of functions with bounded Tonelli variation and to the real Hardy space. For this class, conditions for integrability of the Fourier transform are established.

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...

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...

In the present paper, mechanical characteristics of masonry wall covered with plaster are estimated by means of thermography. Masonry wall samples have been purposely built with different textures: periodic, quasi-periodic and random. All the samples are covered with plaster. By means of a thermographic camera, images have been taken and the textur...

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 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 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 the present paper, the elastic mechanical characteristics of masonry samples, whose texture is not visible due to plaster, are estimated by means of homogenization technique applied through thermographic images. In particular, three masonry samples with different textures have been purposely built. The chosen textures were periodic, quasi-period...

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...

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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...

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 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...

A Learning Management System (LMS) is nowadays a pivotal element in the education environment of a modern university. However, though it generally has a beneficial and positive impact on the education, a part of the teachers is sometimes reluctant to adopt a LMS because of the perceived usage difficulty. Therefore, it is clear that a key step in or...

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...

In this paper we prove quantitative estimates for the Kantorovich version of the neural network operators of the max-product type, in case of continuous and p-integrable functions. In the first case, the estimate is expressed in terms of the modulus of continuity of the functions being approximated, while in the second case, we exploit the Peetre's...

The intervention on the existing building envelope thermal insulation is the main and effective solution in order to achieve a significant reduction of the building stock energy needs. The infrared technique is the methodology of the energy diagnosis aimed to identify qualitatively the principal causes of energy losses: the presence of thermal brid...

In this paper, we develop a procedure for the detection of the contours of thermal bridges from thermographic images, in order to study the energy 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 t...

There axe established some conditions for existence of solutions of a nonlinear integral equation Tf = f + g, where Τ is a convolution-type integral operator.

The paper deals with approximation results with respect to the φ-variation by means of a family of discrete operators for φ-absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in...

In this paper we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operato...

The theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform and $L^p$ norms, for continuous and $L^p$ functions, respectively. The above family of operators, are based upon kernels generated by sigmoida...

In the present paper, we obtain a saturation result for the neural network (NN) operators of the max-product type. In particular, we show that any non-constant, continuous function on the interval [0,1] cannot be approximated by the above operators $\F_n$, $n \in \N^+$, by a rate of convergence higher than $1/n$. Moreover, since we know that any Li...

In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous functions (signals) are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Finally, several examples of (du...

In this article, the theory of multivariate max-product neural network (NN) and quasi-interpolation operators has been introduced. Pointwise and uniform approximation results have been proved, together with estimates concerning the rate of convergence. At the end, several examples of sigmoidal activation functions have been provided.

In the present survey, we recall the main convergence results concerning the theory of neural network (NN) operators. Pointwise and uniform approximation results have been proved for the classical (linear) NN operators, as well as, for their corresponding max-product (nonlinear) version, when continuous functions defined on bounded domains are...

In the present paper, we develop the theory of max-product neural network operators in a Kantorovich-type version, which is suitable in order to study the case of L p -approximation for not necessarily continuous data. Moreover, also the case of the pointwise and uniform approximation of continuous functions is considered. Finally, several examples...

In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators...

The max-product neural network (NN) and quasi-interpolation (QI) operators are here introduced and studied. The density functions considered as kernels for the above operators are generated by certain finite linear combination of sigmoidal functions, and from them inherit very useful approximation properties. The convergence and the rate of approxi...

We introduce and study a family of integral operators in the Kantorovich sense for functions acting on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our...

We study the notion of φ-absolute continuity, providing several equivalent definitions, and we prove a characterization of the space of φ-absolutely continuous functions in terms of convergence in variation for a family of Mellin integral operators in the multidimensional setting.

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with res...

Computed Tomography images (C.T.) are currently part of the routine procedure in medical diagnostic techniques and can be used for the evaluation of occlusion rate of arterial vessels in presence of many diseases. The correct individuation of the morphology of these arterial anomalies allows specialists to diagnose the risk rate for the health of p...

We study approximation results for a family of Mellin integral operators of the form 􏰄 dtN (Tw f)(s) = N Kw(t, f(st))⟨t⟩, s∈ R+, w > 0, R+ where {Kw}w>0 is a family of kernels, ⟨t⟩ := 􏰁Ni=1 ti, t= (t1,...,tN) ∈ RN+, and f is a function of bounded variation on RN+. The starting point of this study is motivated by the important applications that app...

In this paper we show some new applications of the approximation theory, by means of the multivariate sampling Kantorovich operators, to thermographic images in seismic engineering.

In this note we consider the theory of sampling Kantorovich operators, and we show how it is possible to deduce an algorithm for Digital Image Processing starting from the general theory.

Sampling series play a crucial role in Signal and Image Processing. In this note, we illustrate a general method to construct Sampling series and to determine their convergence in Orlicz spaces. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)