G. Allasia | Università degli Studi di Torino (original) (raw)
Papers by G. Allasia
Applied Mathematics and Computation, 2012
A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real funct... more A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard's functions is finally discussed, including some numerical examples.
Mathematics and Computers in Simulation, 2010
Pathological processes cause abnormal regional motions of the heart. Regional wall motion analyse... more Pathological processes cause abnormal regional motions of the heart. Regional wall motion analyses are important to evaluate the success of therapy, especially of cell therapy, since the recovery of the heart in cell therapy proceeds slowly and results in only small changes of ventricular wall motility. The usual ultrasound imaging of heart motion is too inaccurate to be considered as an appropriate method. MRI studies are more accurate, but insufficient to reliably detect small changes in regional ventricular wall motility. We thus aim at a more accurate method of motion analysis. Our approach is based on two imaging modalities, viz. cardiac CT and biplane cineangiography. The epicardial surface represented in the CT data set at the end of the diastole is registered to the three-dimensionally reconstructed epicardial artery tree from the angiograms in end-diastolic position. The motion tracking procedures are carried out by applying thin-plate spline transformations between the epicardial artery trees belonging to consecutive frames of our cineangiographic imagery.
Patient-specific investigations of the heart’s motion and deformation (contractions and expansion... more Patient-specific investigations of the heart’s motion and deformation (contractions and expansions) are highly important to diagnosis and therapy of coronary artery disease. The results of such investigations would enable cardiologists to identifying regions of impaired ventricular motility. During the past decades, much effort has been devoted to achieve an improved understanding of the motility of the heart. Numerous medical-image-based methods have so far been published for tracking and analyzing the motion of the heart wall. In an early stage of development, the methods for the three-dimensional motion tracking of the coronary arteries were solely based on biplane cineangiograms of the epicardial arteries, a method with excellent temporal and spatial resolution, but we have to bear in mind that the epicardial arteries only sparsely cover the ventricular wall; hence, these early methods cannot provide cardiologists with the required information on the motion of the entire outer v...
This paper discusses a particular type of function approximation on scattered data in a general n... more This paper discusses a particular type of function approximation on scattered data in a general number of variables, and its application to surface representation with imposed conditions. If the given function values are subject to errors, it is not appropriate to interpolate the function at the data in the sense of exact matching. As a consequence, we formulate a weakened version of the classical scattered data interpolation problem, and give a simple and efficient procedure to obtain near-interpolation formulas. Near-interpolants enjoy many remarkable properties, which are very useful from both theoretical and practical points of view (shape preserving properties, operator positivity, subdivision techniques, parallel and multistage computation). Applications of near-interpolants to the representation of surfaces, in particular with faults, are discussed in detail (parameter values, localizing weights, etc.).
In coronary artery disease, the contractility and therefore the motility of the myocardium will b... more In coronary artery disease, the contractility and therefore the motility of the myocardium will become reduced. The loss of contractility will differ from one supply territory to the other and gives rise to regionally variable patterns of heart motion. We developed an accurate method for tracking and analyzing the regional motion and deformation of the heart. Our algorithms provide cardiologists with an enormous wealth of numerical data. To be of practical use, the confusing amount of data must be visualized.
Journal of Computational and Applied Mathematics, 2011
Two interpolation operators in inner product spaces for irregularly distributed data are compared... more Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so as to get a partition-of-unity interpolant. Numerical tests and considerations on errors show that the two operators have very different approximation performances, and that by suitable modifications both can provide acceptable results,
International Journal of Computer Mathematics, 2013
In this paper we investigate numerical integration on multivariate scattered data by a class of s... more In this paper we investigate numerical integration on multivariate scattered data by a class of spline functions, called Lobachevsky splines. Starting from their interpolation properties, we focus on the construction of new quadrature and cubature formulas. The use of Lobachevsky splines takes advantages of their feature of being expressible in the multivariate setting as a product of univariate functions. Numerical results using Lobachevsky splines turn out to be interesting and promising for both accuracy and simplicity in computation. Finally, a comparison with radial basis functions (RBFs) confirms the validity of the proposed approach.
Applied Mathematics and Computation, 2011
(AAM) is copyrighted and published by Elsevier. It is posted here by agreement between Elsevier a... more (AAM) is copyrighted and published by Elsevier. It is posted here by agreement between Elsevier and the University of Turin. Changes resulting from the publishing process-such as editing, corrections, structural formatting, and other quality control mechanisms-may not be reflected in this version of the text. The definitive version of the text was subsequently published in [Scattered and track data interpolation using an efficient strip searching procedure,
Mathematical Methods in the Applied Sciences, 2012
A class of spline functions, called Lobachevsky splines, is proposed for landmark-based image reg... more A class of spline functions, called Lobachevsky splines, is proposed for landmark-based image registration. Analytic expressions of Lobachevsky splines and some of their properties are given, reasoning in the context of probability theory. Since these functions have simple analytic expressions and compact support, landmark-based transformations can be advantageously defined using them. Numerical results point out accuracy and stability of Lobachevsky splines, comparing them with Gaussians and thin plate splines. Moreover, an application to a real-life case (cervical X-ray images) shows the effectiveness of the proposed method.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012
In the field of diagnosis and therapy of coronary artery disease, it is highly important to acqui... more In the field of diagnosis and therapy of coronary artery disease, it is highly important to acquire a fair knowledge of the heart wall motion and its regional variations. Unfortunately, the accuracy of all currently applied methods for the acquisition and analysis of the regional heart wall motion is rather limited. We developed a sufficiently accurate technique for tracking and analysing the regional motion of the epicardium throughout the cardiac cycle which is based on cardiac CT and biplane angiography. In the end-diastolic position, the epicardial surface in the 3D CT data is segmented and registered to the skeleton representation of the coronary artery tree obtained from the end-diastolic frame of a biplane cineangiogram. In doing so, a landmark-based approach based on TPS transformations has been chosen. The motion tracking is accomplished by carrying out further landmark-based TPS transformations of the surface to the successive frames of the cineangiogram.
Biofizika, 2010
We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model i... more We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model is parameterized with the use of field observation data. We show that the mysid invasion can lead to an increase in the time-averaged fish population size, and to a decrease in the time-averaged rotifer population size.
Numerische Mathematik, 1986
The trapezoidal rule is applied to the numerical calculation of a known integral representation o... more The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function F(a, x) in the region a<-1 and x>0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.
AIP Conference Proceedings, 2010
ABSTRACT We propose the use of a class of spline functions, called Lobachevsky splines, for landm... more ABSTRACT We propose the use of a class of spline functions, called Lobachevsky splines, for landmark-based registration. We recall the analytic expressions of the Lobachevsky splines and some of their properties, reasoning in the context of probability theory. These functions have simple analytic expressions and compact support. Numerical tests appear to be promising.
Advanced Mathematical and Computational Tools in Metrology VI, 2004
ABSTRACT We consider the problem of approximating a continuous real function known on a set of po... more ABSTRACT We consider the problem of approximating a continuous real function known on a set of points, which are situated on a family of (straight) lines or curves on a plane domain. The most interesting case occurs when the lines or curves are parallel. More generally, it is admitted that some points (possibly all) are not collocated exactly on the lines or curves but close to them, or that the lines or curves are not parallel in a proper sense but roughly parallel. The scheme we propose approximates the data by means of either an interpolation operator or a near-interpolation operator, both based on radial basis functions. These operators enjoy, in particular, two interesting proper-ties: a subdivision technique and a recurrence relation. First, the recurrence relation is applied on each line or curve, so obtaining a set of approximated curves on the considered surface. This can be done simultaneously on all the lines or curves by means of parallel computation. Second, the obtained approximations of the surface curves are composed together by using the subdivision technique. The procedure gives, in general, satisfactory approximations to continuous surfaces, possibly with steep gradients.
Quaderni scientifici del Dipartimento di Matematica, 2009
... 26. Camillo Costantini &amp;amp; Paolo Vitolo, ON A RESULT OF AUMANN AND SHAPLEY ABOU... more ... 26. Camillo Costantini &amp;amp; Paolo Vitolo, ON A RESULT OF AUMANN AND SHAPLEY ABOUT VALUES OF NONATOMIC GAMES 27. ... Quaderni 2006 1. Claudia Chanu &amp;amp; Giovanni Rastelli, FIXED ENERGY R-SEPARATION FOR SCHRÖDINGER EQUATION ...
Journal of Computational and Applied Mathematics, 2011
Two interpolation operators in inner product spaces for irregularly distributed data are compared... more Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so to get a partition-of-unity interpolant. Numerical tests and theoretical considerations on errors show that the second operator gives better approximations, working in particular from IR m to IR n and from C[−π, π] to IR . The operators could represent interesting tools for application to nonlinear system modelling.
Applied Mathematics and Computation, 2012
A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real funct... more A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard's functions is finally discussed, including some numerical examples.
Numerical Functional Analysis and Optimization, 2013
ABSTRACT A class of cardinal basis functions is proposed in order to achieve a generalization to ... more ABSTRACT A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.
Biofizika
We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model i... more We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model is parameterized with the use of field observation data. We show that the mysid invasion can lead to an increase in the time-averaged fish population size, and to a decrease in the time-averaged rotifer population size.
Simulation Modelling Practice and Theory, 2009
We consider a method for the detection and approximation of fault lines of a surface, which is kn... more We consider a method for the detection and approximation of fault lines of a surface, which is known only on a finite number of scattered data. In particular, we present an adaptive approach to detect surface discontinuities, which allows us to give an (accurate) approximation of the detected faults. First, to locate all the nodes close to fault lines, we consider a procedure based on a local interpolation scheme involving a cardinal radial basis formula. Second, we find further sets of points generally closer to the faults than the fault points. Finally, after applying a nearest-neighbor searching procedure and a powerful refinement technique, we outline some different approximation methods. Numerical results highlight the efficiency of our approach.
Applied Mathematics and Computation, 2012
A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real funct... more A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard's functions is finally discussed, including some numerical examples.
Mathematics and Computers in Simulation, 2010
Pathological processes cause abnormal regional motions of the heart. Regional wall motion analyse... more Pathological processes cause abnormal regional motions of the heart. Regional wall motion analyses are important to evaluate the success of therapy, especially of cell therapy, since the recovery of the heart in cell therapy proceeds slowly and results in only small changes of ventricular wall motility. The usual ultrasound imaging of heart motion is too inaccurate to be considered as an appropriate method. MRI studies are more accurate, but insufficient to reliably detect small changes in regional ventricular wall motility. We thus aim at a more accurate method of motion analysis. Our approach is based on two imaging modalities, viz. cardiac CT and biplane cineangiography. The epicardial surface represented in the CT data set at the end of the diastole is registered to the three-dimensionally reconstructed epicardial artery tree from the angiograms in end-diastolic position. The motion tracking procedures are carried out by applying thin-plate spline transformations between the epicardial artery trees belonging to consecutive frames of our cineangiographic imagery.
Patient-specific investigations of the heart’s motion and deformation (contractions and expansion... more Patient-specific investigations of the heart’s motion and deformation (contractions and expansions) are highly important to diagnosis and therapy of coronary artery disease. The results of such investigations would enable cardiologists to identifying regions of impaired ventricular motility. During the past decades, much effort has been devoted to achieve an improved understanding of the motility of the heart. Numerous medical-image-based methods have so far been published for tracking and analyzing the motion of the heart wall. In an early stage of development, the methods for the three-dimensional motion tracking of the coronary arteries were solely based on biplane cineangiograms of the epicardial arteries, a method with excellent temporal and spatial resolution, but we have to bear in mind that the epicardial arteries only sparsely cover the ventricular wall; hence, these early methods cannot provide cardiologists with the required information on the motion of the entire outer v...
This paper discusses a particular type of function approximation on scattered data in a general n... more This paper discusses a particular type of function approximation on scattered data in a general number of variables, and its application to surface representation with imposed conditions. If the given function values are subject to errors, it is not appropriate to interpolate the function at the data in the sense of exact matching. As a consequence, we formulate a weakened version of the classical scattered data interpolation problem, and give a simple and efficient procedure to obtain near-interpolation formulas. Near-interpolants enjoy many remarkable properties, which are very useful from both theoretical and practical points of view (shape preserving properties, operator positivity, subdivision techniques, parallel and multistage computation). Applications of near-interpolants to the representation of surfaces, in particular with faults, are discussed in detail (parameter values, localizing weights, etc.).
In coronary artery disease, the contractility and therefore the motility of the myocardium will b... more In coronary artery disease, the contractility and therefore the motility of the myocardium will become reduced. The loss of contractility will differ from one supply territory to the other and gives rise to regionally variable patterns of heart motion. We developed an accurate method for tracking and analyzing the regional motion and deformation of the heart. Our algorithms provide cardiologists with an enormous wealth of numerical data. To be of practical use, the confusing amount of data must be visualized.
Journal of Computational and Applied Mathematics, 2011
Two interpolation operators in inner product spaces for irregularly distributed data are compared... more Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so as to get a partition-of-unity interpolant. Numerical tests and considerations on errors show that the two operators have very different approximation performances, and that by suitable modifications both can provide acceptable results,
International Journal of Computer Mathematics, 2013
In this paper we investigate numerical integration on multivariate scattered data by a class of s... more In this paper we investigate numerical integration on multivariate scattered data by a class of spline functions, called Lobachevsky splines. Starting from their interpolation properties, we focus on the construction of new quadrature and cubature formulas. The use of Lobachevsky splines takes advantages of their feature of being expressible in the multivariate setting as a product of univariate functions. Numerical results using Lobachevsky splines turn out to be interesting and promising for both accuracy and simplicity in computation. Finally, a comparison with radial basis functions (RBFs) confirms the validity of the proposed approach.
Applied Mathematics and Computation, 2011
(AAM) is copyrighted and published by Elsevier. It is posted here by agreement between Elsevier a... more (AAM) is copyrighted and published by Elsevier. It is posted here by agreement between Elsevier and the University of Turin. Changes resulting from the publishing process-such as editing, corrections, structural formatting, and other quality control mechanisms-may not be reflected in this version of the text. The definitive version of the text was subsequently published in [Scattered and track data interpolation using an efficient strip searching procedure,
Mathematical Methods in the Applied Sciences, 2012
A class of spline functions, called Lobachevsky splines, is proposed for landmark-based image reg... more A class of spline functions, called Lobachevsky splines, is proposed for landmark-based image registration. Analytic expressions of Lobachevsky splines and some of their properties are given, reasoning in the context of probability theory. Since these functions have simple analytic expressions and compact support, landmark-based transformations can be advantageously defined using them. Numerical results point out accuracy and stability of Lobachevsky splines, comparing them with Gaussians and thin plate splines. Moreover, an application to a real-life case (cervical X-ray images) shows the effectiveness of the proposed method.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012
In the field of diagnosis and therapy of coronary artery disease, it is highly important to acqui... more In the field of diagnosis and therapy of coronary artery disease, it is highly important to acquire a fair knowledge of the heart wall motion and its regional variations. Unfortunately, the accuracy of all currently applied methods for the acquisition and analysis of the regional heart wall motion is rather limited. We developed a sufficiently accurate technique for tracking and analysing the regional motion of the epicardium throughout the cardiac cycle which is based on cardiac CT and biplane angiography. In the end-diastolic position, the epicardial surface in the 3D CT data is segmented and registered to the skeleton representation of the coronary artery tree obtained from the end-diastolic frame of a biplane cineangiogram. In doing so, a landmark-based approach based on TPS transformations has been chosen. The motion tracking is accomplished by carrying out further landmark-based TPS transformations of the surface to the successive frames of the cineangiogram.
Biofizika, 2010
We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model i... more We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model is parameterized with the use of field observation data. We show that the mysid invasion can lead to an increase in the time-averaged fish population size, and to a decrease in the time-averaged rotifer population size.
Numerische Mathematik, 1986
The trapezoidal rule is applied to the numerical calculation of a known integral representation o... more The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function F(a, x) in the region a<-1 and x>0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.
AIP Conference Proceedings, 2010
ABSTRACT We propose the use of a class of spline functions, called Lobachevsky splines, for landm... more ABSTRACT We propose the use of a class of spline functions, called Lobachevsky splines, for landmark-based registration. We recall the analytic expressions of the Lobachevsky splines and some of their properties, reasoning in the context of probability theory. These functions have simple analytic expressions and compact support. Numerical tests appear to be promising.
Advanced Mathematical and Computational Tools in Metrology VI, 2004
ABSTRACT We consider the problem of approximating a continuous real function known on a set of po... more ABSTRACT We consider the problem of approximating a continuous real function known on a set of points, which are situated on a family of (straight) lines or curves on a plane domain. The most interesting case occurs when the lines or curves are parallel. More generally, it is admitted that some points (possibly all) are not collocated exactly on the lines or curves but close to them, or that the lines or curves are not parallel in a proper sense but roughly parallel. The scheme we propose approximates the data by means of either an interpolation operator or a near-interpolation operator, both based on radial basis functions. These operators enjoy, in particular, two interesting proper-ties: a subdivision technique and a recurrence relation. First, the recurrence relation is applied on each line or curve, so obtaining a set of approximated curves on the considered surface. This can be done simultaneously on all the lines or curves by means of parallel computation. Second, the obtained approximations of the surface curves are composed together by using the subdivision technique. The procedure gives, in general, satisfactory approximations to continuous surfaces, possibly with steep gradients.
Quaderni scientifici del Dipartimento di Matematica, 2009
... 26. Camillo Costantini &amp;amp; Paolo Vitolo, ON A RESULT OF AUMANN AND SHAPLEY ABOU... more ... 26. Camillo Costantini &amp;amp; Paolo Vitolo, ON A RESULT OF AUMANN AND SHAPLEY ABOUT VALUES OF NONATOMIC GAMES 27. ... Quaderni 2006 1. Claudia Chanu &amp;amp; Giovanni Rastelli, FIXED ENERGY R-SEPARATION FOR SCHRÖDINGER EQUATION ...
Journal of Computational and Applied Mathematics, 2011
Two interpolation operators in inner product spaces for irregularly distributed data are compared... more Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so to get a partition-of-unity interpolant. Numerical tests and theoretical considerations on errors show that the second operator gives better approximations, working in particular from IR m to IR n and from C[−π, π] to IR . The operators could represent interesting tools for application to nonlinear system modelling.
Applied Mathematics and Computation, 2012
A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real funct... more A class of cardinal basis functions for Hermite-Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard's functions is finally discussed, including some numerical examples.
Numerical Functional Analysis and Optimization, 2013
ABSTRACT A class of cardinal basis functions is proposed in order to achieve a generalization to ... more ABSTRACT A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.
Biofizika
We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model i... more We present a mathematical model of the invasion of mysid into the Naroch Lake system. The model is parameterized with the use of field observation data. We show that the mysid invasion can lead to an increase in the time-averaged fish population size, and to a decrease in the time-averaged rotifer population size.
Simulation Modelling Practice and Theory, 2009
We consider a method for the detection and approximation of fault lines of a surface, which is kn... more We consider a method for the detection and approximation of fault lines of a surface, which is known only on a finite number of scattered data. In particular, we present an adaptive approach to detect surface discontinuities, which allows us to give an (accurate) approximation of the detected faults. First, to locate all the nodes close to fault lines, we consider a procedure based on a local interpolation scheme involving a cardinal radial basis formula. Second, we find further sets of points generally closer to the faults than the fault points. Finally, after applying a nearest-neighbor searching procedure and a powerful refinement technique, we outline some different approximation methods. Numerical results highlight the efficiency of our approach.