Scattered data interpolation of Radon data

Beatson, R. K.; Castell, Wolfgang zu

doi:10.1007/s10092-010-0024-7

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Scattered data can then be interpolated using suitable methods. For example a radial functions method was recently introduced by Beatson and zu Castell [1] to obtain new values of the Radon transform. This technique can be combined with the methods introduced below to obtain a reconstruction using less initial data.…”

Section: Kernel Based Image Reconstructionmentioning

confidence: 99%

Medical Image Reconstruction Using Kernel Based Methods

Sironi

2011

Preprint

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The image reconstruction problem consists in finding an approximation of a function f starting from its Radon transform Rf . This problem arises in the ambit of medical imaging when one tries to reconstruct the internal structure of the body, starting from its X-ray tomography. The classical approach to this problem is based on the Back-Projection Formula. This formula gives an analytical inversion of the Radon transform, provided that all the values of Rf are known. In applications only a discrete set of values of Rf is given, thus, one can only obtain an approximation of f . Another class of methods, called ART, can be used to solve the reconstruction problem. Following the ideas contained in ART, we try to apply the Hermite-Birkhoff interpolation to the reconstruction problem. It turns out that, since the Radon transform of a kernel basis function can be infinity, a regularization technique is needed. The method we present here is then based on positive definite kernel functions and it is very flexible thanks to the possibility to choose different kernels and parameters. We study the behavior of the methods and compare them with classical algorithms.

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Section: Kernel Based Image Reconstructionmentioning

confidence: 99%

Medical Image Reconstruction Using Kernel Based Methods

Sironi

2011

Preprint

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“…In either case, robust numerical algorithms are required to reconstruct characteristic features of images at sufficiently high accuracy, on the one hand, and at sufficiently small computational costs, on the other hand. For details concerning the acquisition of X-ray scans, their underlying mathematical models, and standard computational methods for medical image reconstruction, we refer to the textbook [5] of Feeman. To describe the mathematical problem of image reconstruction from X-ray scans, we regard the Radon transform Rf of f ∈ L 1 (R 2 ), defined as Rf (t, θ) = R f (t cos θ − s sin θ, t sin θ + s cos θ) ds (1) for (t, θ) ∈ R × [0, π). Note that for any f ∈ L 1 (R 2 ) the stability estimate…”

Section: Introductionmentioning

confidence: 99%

“…c j g j of the basis functions. According to our problem formulation at the outset of this introduction, we require g j ∈ L 1 (R 2 ), for all 1 ≤ j ≤ n, so that the Radon transform Rg j , as in (1), is for any basis function g j well-defined. This then amounts to solving the linear system…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…A first kernel-based approach for scattered data interpolation from Radon data was developed in [1]. The resulting kernel method in [1] transfers the reconstruction problem (6) into one for suitable projective spaces, on which positive definite zonal functions are utilized to accomplish the recovery step. We remark that the straightforward approach taken in this paper is essentially different from that in [1], although [1] relies on positive definite kernels, too.…”

Section: Introductionmentioning

confidence: 99%

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

Marchi

Iske

Santin

2018

Calcolo

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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.

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