Image reconstruction from scattered Radon data by weighted positive definite kernel functions
- 103 Downloads
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.
KeywordsImage reconstruction Kernel-based approximation Generalized Hermite–Birkhoff interpolation Radon transform Positive definite kernels
Mathematics Subject Classification65D05 65D15
The authors thank one anonymous referee for suggesting an improvement concerning Proposition 1. The first author was supported by the project “Multivariate approximation with application to image reconstruction” of the University of Padova, years 2013–2014. The third author thanks the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. The research of the first and the third author has been accomplished within RITA (Rete ITaliana di Approssimazione).
- 7.Guillemard, M., Iske, A.: Interactions between kernels, frames, and persistent homology. In: Pesenson, I., Gia, Q.T.Le, Mayeli, A., Mhaskar, H., Zhou, D.-X. (eds.) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Volume 2: Novel Methods in Harmonic Analysis, pp. 861–888. Birkhäuser, Basel (2017)CrossRefGoogle Scholar
- 10.Iske, A.: Charakterisierung bedingt positiv definiter Funktionen für multivariate Interpolations methoden mit radialen Basisfunktionen. Dissertation, University of Göttingen (1994)Google Scholar
- 11.Iske, A.: Reconstruction of functions from generalized Hermite–Birkhoff data. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory VIII, Vol 1: Approximation and Interpolation, pp. 257–264. World Scientific, Singapore (1995)Google Scholar
- 15.Schaback, R., Wendland, H.: Characterization and construction of radial basis functions. In: Dyn, N., Leviatan, D., Levin, D., Pinkus, A. (eds.) Multivariate Approximation and Applications, pp. 1–24. Cambridge University Press, Cambridge (2001)Google Scholar