Decomposition and reconstruction of multidimensional signals by radial functions with tension parameters
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Decomposition and reconstruction of multidimensional signals by radial functions with tension parameters Mira Bozzini1 · Christophe Rabut2 · Milvia Rossini1
Received: 15 October 2015 / Accepted: 13 November 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The aim of the paper is to construct a multiresolution analysis of L2 (IR d ) based on generalized are fundamental solutions of differential oper kernels which 2 I ). We study its properties and provide a set of (− + κ ators of the form m =1 pre-wavelets associated with it, as well as the filters which are indispensable to perform decomposition and reconstruction of a given signal, being very useful in applied problems thanks to the presence of the tension parameters κ . Keywords Generalized Whittle–Mat´ern kernels · Radial basis functions · Multiresolution analysis · Wavelets · Filters · Tension parameters Mathematics Subject Classification (2010) 41A05 · 41063 · 41065 · 65D05 · 65D15
Communicated by: Robert Schaback Milvia Rossini
[email protected] Mira Bozzini [email protected] Christophe Rabut [email protected] 1
Dipartimento di Matematica e Applicazioni, Universit`a di Milano–Bicocca, via Roberto Cozzi 55, 20125 Milano, Italy
2
INSA Toulouse (Institut National des Sciences Appliqu´ees), 135, Avenue de Rangueil, 31077 Toulouse Cedex 4, France
M. Bozzini, et al.
1 Introduction In [5], the fundamental solutions associated with the differential operator defined for a given integer m > d/2 and m non negative real numbers {κ }m =1 by L=
m =1
(− + κ2 I ),
m>
d 2
were studied. The parameters {κ }m =1 can be considered as tension parameters. Such operators determine a new family of radial functions. The choice κ > 0, = 1, . . . m,provides strictly positive definite functions the so called generalized Whittle–Mat´ern kernels which have a computable analytic expression. Suitable choices of the parameter κ give known classes of radial kernels. For κ = 0, = 1, . . . m,we get the (conditionally positive definite) polyharmonic splines, while if all the parameters are equal to κ > 0 we get a scaled Whittle–Mat´ern–Sobolev radial kernel which is positive definite. When n < m values of the κ are zero, the resulting functions are are linear combinations of generalized Whittle–Mat´ern kernels with polyharmonic functions. In this paper we discuss the properties of this new family of functions that offers the opportunity of choosing an element of the family according to the applied problem to deal with. That is, strictly positive definite functions or conditionally positive definite functions of order 2n−1 < m that are fundamental solutions of known differential operators. Connecting (conditionally) positive definite kernels to fundamental functions, provides an interpretation of native spaces as generalized Sobolev spaces associated to L (see [10]). Here, we mainly consider the case when the parameters κ are all positive. In this case we denote by u the fundamental solution of L
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