Giampietro Allasia

Simultaneous Interpolation and Approximation by a Class of Multivariate Positive Operators

The purpose of this paper is to show that the interpolation positive operators of a wide class satisfy also the approximation property. Such a situation of simultaneous interpolation and approximation may be very desirable, but is rather unusual. Our attention is focused on the convergence problem, giving the conditions under which a sequence of operators of the considered class converges to a continuous function in a convex compact set in Rm (m∈N). It must be recalled that many of these operators are very interesting in applications and that suitable algorithms can be devised for parallel, multistage and iterative computation.

Scattered and track data interpolation using an efficient strip searching procedure

Abstract A new local algorithm for bivariate interpolation of large sets of scattered and track data is presented. The method, which changes partially depending on the kind of data, is based on the partition of the interpolation domain in a suitable number of parallel strips, and, starting from these, on the construction for any data point of a square neighbourhood containing a convenient number of data points. Then, the well-known modified Shepard’s formula for surface interpolation is applied with some effective improvements. The proposed algorithm is very fast, owing to the optimal nearest neighbour searching, and achieves good accuracy. Computational cost and storage requirements are analyzed. Moreover, the efficiency and reliability of the algorithm are shown by several numerical tests, also performed by Renka’s algorithm for a comparison.

Geometric modeling and motion analysis of the epicardial surface of the heart

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.

In vitro micronucleus induction by polymethyl methacrylate bone cement in cultured human lymphocytes.

Human lymphocytes cultured in vitro were used to assess the ability of polymethyl methacrylate (PMMA), currently used in orthopaedic surgery as bone cement, to induce micronuclei in binucleated cells. The results of the study show a significant increase in the micronucleus frequency in treated cultures and therefore the genotoxic effect of PMMA bone cement or its ingredients (methyl methacrylate, dimethyl para-toluidine and hydroquinone) usually present in self-curing methacrylate bone cement and released in small quantities after polymerisation. This effect is evident during the stage immediately after the polymerisation process, and after a certain period of time (5 days in our experimental model).

Multivariate Hermite–Birkhoff interpolation by a class of cardinal basis functions

Abstract 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.

Local interpolation schemes for landmark-based image registration

In this paper we focus, from a mathematical point of view, on properties and performances of some local interpolation schemes for landmark-based image registration. Precisely, we consider modified Shepards interpolants, Wendlands functions, and Lobachevsky splines. They are quite unlike each other, but all of them are compactly supported and enjoy interesting theoretical and computational properties. In particular, we point out some unusual forms of the considered functions. Finally, detailed numerical comparisons are given, considering also Gaussians and thin plate splines, which are really globally supported but widely used in applications.

Radial Basis Functions and Splines for Landmark‐Based Registration of Medical Images

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.

Numerical integration on multivariate scattered data by Lobachevsky splines

In this paper, we investigate the numerical integration problem of a real valued function generally known only on multivariate scattered points using Lobachevsky splines, a pioneering version of cardinal B-splines. Starting from their interpolation properties, we focus on the construction of new integration formulas, which are quite flexible requiring no special distribution of nodes. Numerical results using Lobachevsky splines turn out to be interesting and promising for both accuracy and simplicity in computation. Finally, a comparison with integration by radial basis functions confirms the validity of the proposed approach.

Numerical calculation of incomplete gamma functions by the trapezoidal rule

SummaryThe trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function Г (a,x) in the regiona<−1 andx>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.

Adaptive detection and approximation of unknown surface discontinuities from scattered data

Abstract 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 fault points, i.e. the nodes on or 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-neighbour searching procedure and a powerful refinement technique, we outline some different approximation methods. Numerical results highlight the efficiency of our approach.