9,474 research outputs found
Partition of Unity Interpolation on Multivariate Convex Domains
In this paper we present a new algorithm for multivariate interpolation of
scattered data sets lying in convex domains \Omega \subseteq \RR^N, for any
. To organize the points in a multidimensional space, we build a
-tree space-partitioning data structure, which is used to efficiently apply
a partition of unity interpolant. This global scheme is combined with local
radial basis function approximants and compactly supported weight functions. A
detailed description of the algorithm for convex domains and a complexity
analysis of the computational procedures are also considered. Several numerical
experiments show the performances of the interpolation algorithm on various
sets of Halton data points contained in , where can be any
convex domain like a 2D polygon or a 3D polyhedron
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
Two-dimensional interpolation using a cell-based searching procedure
In this paper we present an efficient algorithm for bivariate interpolation,
which is based on the use of the partition of unity method for constructing a
global interpolant. It is obtained by combining local radial basis function
interpolants with locally supported weight functions. In particular, this
interpolation scheme is characterized by the construction of a suitable
partition of the domain in cells so that the cell structure strictly depends on
the dimension of its subdomains. This fact allows us to construct an efficient
cell-based searching procedure, which provides a significant reduction of CPU
times. Complexity analysis and numerical results show such improvements on the
algorithm performances
Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields
Spatially referenced data often have autocovariance functions with elliptical
isolevel contours, a property known as geometric anisotropy. The anisotropy
parameters include the tilt of the ellipse (orientation angle) with respect to
a reference axis and the aspect ratio of the principal correlation lengths.
Since these parameters are unknown a priori, sample estimates are needed to
define suitable spatial models for the interpolation of incomplete data. The
distribution of the anisotropy statistics is determined by a non-Gaussian
sampling joint probability density. By means of analytical calculations, we
derive an explicit expression for the joint probability density function of the
anisotropy statistics for Gaussian, stationary and differentiable random
fields. Based on this expression, we obtain an approximate joint density which
we use to formulate a statistical test for isotropy. The approximate joint
density is independent of the autocovariance function and provides conservative
probability and confidence regions for the anisotropy parameters. We validate
the theoretical analysis by means of simulations using synthetic data, and we
illustrate the detection of anisotropy changes with a case study involving
background radiation exposure data. The approximate joint density provides (i)
a stand-alone approximate estimate of the anisotropy statistics distribution
(ii) informed initial values for maximum likelihood estimation, and (iii) a
useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure
Monotonicity preserving approximation of multivariate scattered data
This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /
On Polyharmonic Interpolation
In the present paper we will introduce a new approach to multivariate
interpolation by employing polyharmonic functions as interpolants, i.e. by
solutions of higher order elliptic equations. We assume that the data arise
from or analytic functions in the ball We prove two main
results on the interpolation of or analytic functions in the
ball by polyharmonic functions of a given order of polyharmonicity
$p.
Local interpolation schemes for landmark-based image registration: a comparison
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 Shepard's interpolants,
Wendland's 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
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