Giuseppe Rodriguez’s research while affiliated with University of Cagliari and other places

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Publications (73)


Percentage of failures, on the left, and severe failures, on the right, for the various approach considered in RESC applied to the group of test problems
Percentage of experiments (s=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=10$$\end{document}) for which Q(tℓextr(s))≤q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(\textbf{t}^{(s)}_{\ell _\text {extr}})\le q$$\end{document}, with q∈[0.5,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in [0.5,1]$$\end{document}. This means that the extrapolated solution produces an error smaller than the optimal T(G)SVD solution by a factor of q
Some examples with Q(tℓextr(s))<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(\textbf{t}^{(s)}_{\ell _\text {extr}})<1$$\end{document}. On the left, the Phillips example with the given solution; in this case Q(t7)=0.74\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(\textbf{t}_7)=0.74$$\end{document}. On the right, the Foxgood example with the lin+sin2pi solution; here Q(t4)=0.41\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(\textbf{t}_4)=0.41$$\end{document}. In both cases, η=0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =0.1$$\end{document}, n=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=100$$\end{document} and s=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=10$$\end{document}
Parameter Choice Rules for Discrete Ill-Posed Problems Based on Extrapolation Methods
  • Article
  • Full-text available

February 2025

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1 Read

Journal of Scientific Computing

Andrea Azzarelli

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Claude Brezinski

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[...]

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Giuseppe Rodriguez

Linear discrete inverse problems are common in many applicative fields. Regularization consists of substituting to the original ill-conditioned problem an approximated formulation depending on a parameter, which has to be chosen so that the new problem is well-conditioned and its solution is close enough to the ideal solution. When the parameter is discrete, like in the truncated singular value decomposition (TSVD) and in the generalized TSVD (TGSVD), one has to choose a vector out of a sequence. In this paper we explore the possibility to employ a sequence of extrapolated solutions to estimate the best parameter, as well as substituting to the regularized solution an extrapolated one. We investigate the use of three classical vector extrapolation methods, MPE (minimal polynomial extrapolation), RRE (reduced rank extrapolation), and VEA (vector epsilon algorithm). For the VEA method we also develop a new computational scheme which reduces memory storage and computing time. Numerical experiments compare the performance of the newly introduced approaches with other well-known methods.

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Figure 1. Engraving found in the Domus de Janas of Corongiu, in Pimentel (Sardinia, Italy).
Figure 2. Lobby of the Neolithic tomb located in the Domus de Janas of Corongiu (Sardinia, Italy). The engraving is over the entrance to the internal chamber.
Figure 4. Relative errors in the ∞-norm, obtained by processing either the whole synthetic dataset or the reduced datasets suggested by the algorithms under scrutiny; δA is the distance of the light source from the object.
Figure 5. The Shell3 dataset is composed of 20 pictures of a seashell illuminated by sunlight; the light direction rotates counterclockwise; see [20] for details.
Figure 7. Color picture of the seashell from which the Shell3 dataset was created.
Ascertaining the Ideality of Photometric Stereo Datasets under Unknown Lighting

August 2023

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45 Reads

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2 Citations

Algorithms

The standard photometric stereo model makes several assumptions that are rarely verified in experimental datasets. In particular, the observed object should behave as a Lambertian reflector, and the light sources should be positioned at an infinite distance from it, along a known direction. Even when Lambert’s law is approximately fulfilled, an accurate assessment of the relative position between the light source and the target is often unavailable in real situations. The Hayakawa procedure is a computational method for estimating such information directly from data images. It occasionally breaks down when some of the available images excessively deviate from ideality. This is generally due to observing a non-Lambertian surface, or illuminating it from a close distance, or both. Indeed, in narrow shooting scenarios, typical, e.g., of archaeological excavation sites, it is impossible to position a flashlight at a sufficient distance from the observed surface. It is then necessary to understand if a given dataset is reliable and which images should be selected to better reconstruct the target. In this paper, we propose some algorithms to perform this task and explore their effectiveness.


Quadrature error estimates for Gm obtained by the averaged rules for the integral I 1 .
Averaged Nystr\"om interpolants for the solution of Fredholm integral equations of the second kind

July 2023

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32 Reads

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nystr\"om methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose. New stability properties of the quadrature rules used are shown.


Figure 1.1. Engraving found in the Domus de Janas of Corongiu, in Pimentel (Sardinia, Italy).
Figure 5.3. The shell dataset is composed of 20 pictures of a seashell illuminated by sunlight; the light direction rotates counterclockwise; see [25] for details.
Ascertaining the ideality of photometric stereo datasets under unknown lighting

July 2023

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32 Reads

The standard photometric stereo model makes several assumptions that are rarely verified in experimental datasets. In particular, the observed object should behave as a Lambertian reflector and the light sources should be positioned at an infinite distance from it, along a known direction. Even when Lambert's law is approximately fulfilled, an accurate assessment of the relative position between the light source and the target is often unavailable in real situations. The Hayakawa procedure is a computational method for estimating such information directly from the data images. It occasionally breaks down when some of the available images deviate too much from ideality. Indeed, in narrow shooting scenarios, typical, e.g., of archaeological excavation sites, it may be impossible to position a flashlight at a sufficient distance from the observed surface. It is then necessary to understand if a given dataset is reliable and which images should be selected to better reconstruct the target. In this paper, we propose some algorithms to perform this task and explore their effectiveness.



Photometric Stereo with Non-Lambertian Preprocessing and Hayakawa Lighting Estimation for Highly Detailed Shape Reconstruction

April 2023

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14 Reads

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2 Citations

In many realistic scenarios, the use of highly detailed photometric 3D reconstruction techniques is hindered by several challenges in given imagery. Especially, the light sources are often unknown and need to be estimated, and the light reflectance is often non-Lambertian. In addition, when approaching the problem to apply photometric techniques at real-world imagery, several parameters appear that need to be fixed in order to obtain high-quality reconstructions. In this chapter, we attempt to tackle these issues by combining photometric stereo with non-Lambertian preprocessing and Hayakawa lighting estimation. At hand of a dedicated study, we discuss the applicability of these techniques for their use in automated 3D geometry recovery for 3D printing.KeywordsPhotometric stereoShape from shadingHayakawa procedureLighting estimationOren-Nayar modelLambertian reflector


Forward Electromagnetic Induction Modelling in a Multilayered Half-Space: An Open-Source Software Tool

March 2023

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312 Reads

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10 Citations

Electromagnetic induction (EMI) techniques are widely used in geophysical surveying. Their success is mainly due to their easy and fast data acquisition, but the effectiveness of data inversion is strongly influenced by the quality of sensed data, resulting from suiting the device configuration to the physical features of the survey site. Forward modelling is an essential tool to optimize this aspect and design a successful surveying campaign. In this paper, a new software tool for forward EMI modelling is introduced. It extends and complements an existing open-source package for EMI data inversion, and includes an interactive graphical user interface. Its use is explained by a theoretical introduction and demonstrated through a simulated case study. The nonlinear data inversion issue is briefly discussed and the inversion module of the package is extended by a new regularized minimal-norm algorithm.


Forward electromagnetic induction modelling in a multilayered half-space: An open-source software tool

January 2023

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183 Reads

Electromagnetic induction (EMI) techniques are widely used in geophysical surveying. Their success is mainly due to their easy and fast data acquisition, but the effectiveness of data inversion is strongly influenced from the quality of sensed data, resulting from suiting the device configuration to the physical features of the survey site. Forward modelling is an essential tool to optimize this aspect and design a successful surveying campaign. In this paper, a new software tool for forward EMI modelling is introduced. It extends and complements an existing open-source package for EMI data inversion, and includes an interactive graphical user interface. Its use is motivated by a theoretical introduction and demonstrated through a simulated case study. The nonlinear data inversion issue is briefly discussed and the inversion module of the package is extended by a new regularized minimal-norm algorithm.


Citations (52)


... Some new results on averaged and generalized averaged Gauss quadrature rules can be found in [12,13]. For some recent results on the application of averaged and generalized averaged Gauss quadrature rules to the numerical solution of integral equations, see [3,4]. : ...

Reference:

Error estimates for Gauss-type quadrature rules for variable-sign weight functions
Averaged Nyström interpolants for the solution of Fredholm integral equations of the second kind
  • Citing Article
  • April 2024

Applied Mathematics and Computation

... However, using deep learning technology to identify and classify cultural relics still has limitations. One challenge in using deep learning for cultural relic identification is the limited availability of data sets and the high cost of labeling relic types [14][15][16][17]. This may lead to overfitting issues in classification tasks [18,19]. ...

Ascertaining the Ideality of Photometric Stereo Datasets under Unknown Lighting

Algorithms

... The mathematical formulation of PS assumes that the object surface is Lambertian, meaning that Lambert's cosine law can be employed to describe the object reflectance. This condition implies that the surface is matte and free from specular reflections, which is seldom verified in practice; see [28,29] and [25], where the Oren-Nayar model has been used to preprocess a dataset originated by a non Lambertian object. ...

Photometric Stereo with Non-Lambertian Preprocessing and Hayakawa Lighting Estimation for Highly Detailed Shape Reconstruction
  • Citing Chapter
  • April 2023

... In such cases, a nonlinear approach is preferable, as it provides a greater accuracy. Different numerical techniques for the analysis of such a nonlinear model have been studied in [10,18,19,20,21,38,39], and a Matlab toolbox implementing the algorithms introduced in the papers has been released [16,17]. Since the solution of a nonlinear problem typically reduces to the solution of a sequence of linear problems, the techniques explored in this paper may be relevant also in the nonlinear setting, as they would influence the choice of a regularized solution. ...

Forward Electromagnetic Induction Modelling in a Multilayered Half-Space: An Open-Source Software Tool

... Diffusion is an occurrence that draws interest from a wide range of fields such as heat, mass, and electric charge transport [1][2][3][4]. It is possible to use the Finite Different Method (FDM) alone or with mixed methods [1], analytical [5], or semi-analytical solutions [6]. ...

Thermal diffusivity from Fourier’s inverse problem supervised by an optimization model: Theoretical analysis and experimental validation

Case Studies in Thermal Engineering

... Social networks are often modeled as directed graphs, representing networks with directionality such as social media interactions [1]. The same occurs with transportation networks [2]. Moreover, specific types of digraphs, such as derivable digraphs, are used in wireless sensor networking [3]. ...

Chained structure of directed graphs with applications to social and transportation networks

Applied Network Science

... Therefore matrix functions applied in network analysis generally have the property that 0 ≤ c k+1 ≤ c k for all k ≥ 1. The most common matrix function used in network analysis is the matrix exponential; see [5,6,[8][9][10] for discussions and illustrations. We prefer to use the the modified matrix exponential ...

SoftNet: A Package for the Analysis of Complex Networks

Algorithms

... When dealing with low induction numbers, i.e., low values of the electrical conductivity, this linear integral model is accurate enough to yield acceptable reconstructions. Quite recently, some authors have analyzed this model from both the theoretical and the numerical points of view; see [22,23]. In [23], the continuous problem has been studied in various function spaces and three different collocation methods have been proposed to discretize the problem. ...

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Numerical Algorithms

... This model has been treated in [4,25,45] and later in [20] in order to obtain an optimized numerical approach. More recently, in [21,22] the inversion problem is solved in a reproducing kernel Hilbert space. The method proposed in this paper can be easily extended to this case. ...

Minimal-norm RKHS solution of an integral model in geo-electromagnetism
  • Citing Conference Paper
  • September 2021