Reconstructing a function on the sphere from its means along vertical slices

DOI10.3934/ipi.2016018zbMath1348.65183OpenAlexW2517231363MaRDI QIDQ326412

Ralf Hielscher, Michael Quellmalz

Publication date: 12 October 2016

Published in: Inverse Problems and Imaging (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.3934/ipi.2016018

zbMATH Keywords

Radon transformfast algorithmsspherical harmonicsspherical meansminimax riskasymptotic boundsstatistical inverse problems

Mathematics Subject Classification ID

Nonparametric estimation (62G05) Radon transform (44A12) Numerical methods for trigonometric approximation and interpolation (65T40) Numerical methods for inverse problems for boundary value problems involving PDEs (65N21) Inverse problems for integral equations (45Q05)

Related Items

Funk-Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere ⋮ Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces ⋮ An SVD in Spherical Surface Wave Tomography ⋮ On the spherical slice transform ⋮ Vector fields with zero flux through circles of fixed radius on $\mathbb{H}^2$ ⋮ The vertical slice transform on the unit sphere ⋮ ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR ⋮ Sliced optimal transport on the sphere ⋮ Reconstruction of the Cauchy-Riemann operator by complex integration operators along circles ⋮ Non-central Funk-Radon transforms: single and multiple ⋮ A discrete Funk transform on the cubed sphere ⋮ Inversion of a Convolution Operator Associated with Spherical Means ⋮ The Funk-Radon transform for hyperplane sections through a common point ⋮ On two families of Funk-type transforms ⋮ A generalization of the Funk–Radon transform ⋮ Unnamed Item ⋮ An inversion formula for the horizontal conical Radon transform ⋮ Reconstruction of functions on the sphere from their integrals over hyperplane sections

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A new approach to the reconstruction of images from Radon projections
• The Gibbs- Wilbraham phenomenon: An episode in Fourier analysis
• The Radon transform.
• Probability theory. Translated from the German by Robert B. Burckel
• Fast spherical Fourier algorithms.
• Optimal spherical deconvolution
• Lectures on constructive approximation. Fourier, spline, and wavelet methods on the real line, the sphere, and the ball
• Polynomial interpolation and hyperinterpolation over general regions
• A new linogram algorithm for computerized tomography
• Inversion algorithms for the spherical Radon and cosine transform
• A mollifier method for linear operator equations of the first kind
• Convergence of variational regularization methods for imaging on Riemannian manifolds
• A Remark on Stirling's Formula
• Optimal mollifiers for spherical deconvolution
• Nonparametric statistical inverse problems
• Computerized Tomography: The New Medical X-Ray Technology
• Fast algorithms for discrete polynomial transforms
• Numerical Quadrature in the Presence of a Singularity
• Approximation Theory and Harmonic Analysis on Spheres and Balls
• Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry
• Joint image reconstruction and segmentation using the Potts model
• An Algorithm for the Machine Calculation of Complex Fourier Series
• Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten
• Uniqueness Theory for Laplace Series
• Introduction to nonparametric estimation