Milvia Rossini

Generalized Whittle---Matérn and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator ( − Δ)m and the Whittle–Matérn kernels related to the differential operator ( − Δ + I)m. This is done by allowing general differential operators of the form

Irregularity detection from noisy data in one and two dimensions

\prod_{j=1}^m(-\Delta+\kappa_j^2I)

An anisotropic directional subdivision and multiresolution scheme

with nonzero κj and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle–Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to

Non-regular Surface Approximation

W_2^m(\ensuremath{\mathbb{R}}^d)

Polyharmonic multiresolution analysis: an overview and some new results

. On the side, we prove that generalized inverse multiquadric kernels of the form

Polyharmonic splines: An approximation method for noisy scattered data of extra-large size

\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}

Radial kernels via scale derivatives

are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle–Matérn form with a variable scale κ(r) between κ1,...,κm. We also consider the case where some of the κj vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle–Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration.

Filters for anisotropic wavelet decompositions

In this work we consider the problem of detecting the irregularities of univariate functions from noisy data and its extension to bivariate functions which present lines of points of irregularity.

Edge detection methods based on RBF interpolation

In order to handle directional singularities, standard wavelet approaches have been extended to the concept of discrete shearlets in Kutyniok and Sauer (SIAM J. Math. Anal. 41, 1436–1471, 2009). One disadvantage of this extension, however, is the relatively large determinant of the scaling matrices used there which results in a substantial data complexity. This motivates the question whether some of the features of the discrete shearlets can also be obtained by means of different geometries. In this paper, we give a positive answer by presenting a different approach, based on a matrix with small determinant which therefore offers a larger recursion depth for the same amount of data.

Decomposition and reconstruction of multidimensional signals by radial functions with tension parameters

The aim of the paper is to provide a method for approximating non regular surfaces from a set of scattered data in a faithful way. The method we propose is effective and particularly well-suited for recovering geophysical surfaces with faults or drainage patterns. Some real examples will be presented.