Alessandra De Rossi

Fast and accurate interpolation of large scattered data sets on the sphere

In this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented. The solution method is local and involves a modified spherical Shepards interpolant, which uses zonal basis functions as local approximants. The associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere. Specifically, this technique is mainly based on the partition of the sphere in a suitable number of spherical zones, the construction of spherical caps as local neighbourhoods for each node, and finally the employment of a spherical zone searching procedure. Computational cost and storage requirements of the spherical algorithm are analyzed. Moreover, several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm.

A TRIVARIATE INTERPOLATION ALGORITHM USING A CUBE-PARTITION SEARCHING PROCEDURE ∗

In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm.

Spherical interpolation using the partition of unity method: An efficient and flexible algorithm

Abstract An efficient and flexible algorithm for the spherical interpolation of large scattered data sets is proposed. It is based on a partition of unity method on the sphere and uses spherical radial basis functions as local approximants. This technique exploits a suitable partition of the sphere into a number of spherical zones, the construction of a certain number of cells such that the sphere is contained in the union of the cells, with some mild overlap among the cells, and finally the employment of an optimized spherical zone searching procedure. Some numerical experiments show the good accuracy of the spherical partition of unity method and the high efficiency of the algorithm.

A meshless interpolation algorithm using a cell-based searching procedure

In this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets. It is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell-based searching procedure. More precisely, we construct a cell structure, which partitions the domain and strictly depends on the dimension of the subdomains, thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in Allasia et al. (2011) and Cavoretto and De Rossi (2010, 2012). In fact, this efficient algorithm and, in particular, the new searching procedure enable us a fast computation also in several applications, where the amount of data to be interpolated is often very large, up to many thousands or even millions of points. Analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm. This is also supported by numerical experiments.

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.

Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

A particular solution of a dynamical system is completely determined by its initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, with the same parameter set, it is possible that several steady states can originate from different initial conditions (multi-stability). In that case the outcome depends on the chosen initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. The algorithms presented here are general and robust enough so as to solve the problem of reconstructing the basin of attraction of each stable equilibrium point. In order to have a graphical representation of the separatrix manifolds, we focus on systems of two and three ordinary differential equations exhibiting bi- or tri-stability. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction and for the reconstruction of such curves and surfaces. We approximate the latter with the implicit partition of unity method using radial basis functions as local approximants. Numerical results, obtained with a Matlab package made available to the scientific community, support our findings.

Approximation of Dynamical System’s Separatrix Curves

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models, like prey‐predator or competition systems. In this paper we construct programs for the detection of points lying on the separatrix curve, i.e. the curve which partitions the domain. Finally, an efficient algorithm, which is based on the Partition of Unity method with local approximants given by Wendlands functions, is used for reconstructing the separatrix curve.

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.