We propose a novel kernel-based method for image reconstruction from
scattered Radon data. To this end, we employ generalized Hermite--Birkhoff
interpolation by positive definite kernel functions. For radial kernels,
however, a straightforward application of the generalized Hermite--Birkhoff
interpolation method fails to work, as we prove in this paper. To
obtain a well-posed reconstruction scheme for scattered Radon data,
we introduce a new class of weighted positive definite kernels, which
are symmetric but not radially symmetric. By our construction, the
resulting weighted kernels are combinations of radial positive definite
kernels and positive weight functions. This yields very flexible
image reconstruction methods, which work for arbitrary distributions
of Radon lines. We develop suitable representations for the weighted
basis functions and the symmetric positive definite kernel matrices
that are resulting from the proposed reconstruction scheme. For the
relevant special case, where Gaussian radial kernels are combined
with Gaussian weights, explicit formulae for the weighted Gaussian
basis functions and the kernel matrices are given. Supporting numerical
examples are finally presented.
%0 Journal Article
%1 demarchi2018image
%A De Marchi, S.
%A Iske, A.
%A Santin, G.
%D 2018
%J Calcolo
%K imported vorlaeufig
%N 1
%P 2
%R 10.1007/s10092-018-0247-6
%T Image reconstruction from scattered Radon data by weighted positive
definite kernel functions
%U https://doi.org/10.1007/s10092-018-0247-6
%V 55
%X We propose a novel kernel-based method for image reconstruction from
scattered Radon data. To this end, we employ generalized Hermite--Birkhoff
interpolation by positive definite kernel functions. For radial kernels,
however, a straightforward application of the generalized Hermite--Birkhoff
interpolation method fails to work, as we prove in this paper. To
obtain a well-posed reconstruction scheme for scattered Radon data,
we introduce a new class of weighted positive definite kernels, which
are symmetric but not radially symmetric. By our construction, the
resulting weighted kernels are combinations of radial positive definite
kernels and positive weight functions. This yields very flexible
image reconstruction methods, which work for arbitrary distributions
of Radon lines. We develop suitable representations for the weighted
basis functions and the symmetric positive definite kernel matrices
that are resulting from the proposed reconstruction scheme. For the
relevant special case, where Gaussian radial kernels are combined
with Gaussian weights, explicit formulae for the weighted Gaussian
basis functions and the kernel matrices are given. Supporting numerical
examples are finally presented.
@article{demarchi2018image,
abstract = {We propose a novel kernel-based method for image reconstruction from
scattered Radon data. To this end, we employ generalized Hermite--Birkhoff
interpolation by positive definite kernel functions. For radial kernels,
however, a straightforward application of the generalized Hermite--Birkhoff
interpolation method fails to work, as we prove in this paper. To
obtain a well-posed reconstruction scheme for scattered Radon data,
we introduce a new class of weighted positive definite kernels, which
are symmetric but not radially symmetric. By our construction, the
resulting weighted kernels are combinations of radial positive definite
kernels and positive weight functions. This yields very flexible
image reconstruction methods, which work for arbitrary distributions
of Radon lines. We develop suitable representations for the weighted
basis functions and the symmetric positive definite kernel matrices
that are resulting from the proposed reconstruction scheme. For the
relevant special case, where Gaussian radial kernels are combined
with Gaussian weights, explicit formulae for the weighted Gaussian
basis functions and the kernel matrices are given. Supporting numerical
examples are finally presented.},
added-at = {2018-07-20T10:54:15.000+0200},
author = {De Marchi, S. and Iske, A. and Santin, G.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2b5298749614346d9e7bf175f78bea9ca/mhartmann},
doi = {10.1007/s10092-018-0247-6},
file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/DeMarchi2017_www_kernel_radon_preprint.pdf:PDF},
interhash = {ce8ce49e6fb13bf3217ebadd1c0376bf},
intrahash = {b5298749614346d9e7bf175f78bea9ca},
issn = {1126-5434},
journal = {Calcolo},
keywords = {imported vorlaeufig},
month = feb,
number = 1,
owner = {santinge},
pages = 2,
timestamp = {2018-07-20T08:54:15.000+0200},
title = {Image reconstruction from scattered Radon data by weighted positive
definite kernel functions},
url = {https://doi.org/10.1007/s10092-018-0247-6},
volume = 55,
year = 2018
}