Image reconstruction from scattered Radon data by weighted positive definite kernel functions

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 wellposed 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.

Greedy algorithms for image approximation from scattered Radon data

“…, c N ) ∈ R N of s in (2) are uniquely determined by the interpolation conditions R L (f ) = R L (s). Moreover, according to [2], the matrix entries in A L,K are well-defined for weighted kernels…”

“…More flexible reconstructions are kernel-based scattered data approximation schemes. Just recently in [2], kernel-based approximation was adapted to the specific requirements of functional approximation from bivariate scattered Radon data. The method in [2] works with weighted kernels for the well-posedness of the reconstruction problem.…”

“…Just recently in [2], kernel-based approximation was adapted to the specific requirements of functional approximation from bivariate scattered Radon data. The method in [2] works with weighted kernels for the well-posedness of the reconstruction problem. Following along the lines of [2], we use greedy algorithms to select suitable approximation spaces.…”

“…The method in [2] works with weighted kernels for the well-posedness of the reconstruction problem. Following along the lines of [2], we use greedy algorithms to select suitable approximation spaces. This leads to significant improvements on both the efficiency and the numerical stability of the kernel-based reconstruction.…”

THB-spline approximations for turbine blade design with local B-spline approximations

“…Scattered data fitting is nowadays of fundamental importance in a variety of fields, ranging from geographic applications to medical imaging and geometric modeling, see for example [8,13,14]. The topic can be addressed in different approximation spaces, either by using spline spaces or radial basis functions which are particularly attractive in high dimension [19].…”