Reconstructing Functions on the Sphere from Circular Means

Michael Quellmalz (Autor)

Buch | Softcover

170 Seiten

2020
Universitätsverlag Chemnitz
978-3-96100-116-3 (ISBN)

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The present thesis considers the problem of reconstructing a function f that is defined on the d-dimensional unit sphere from its mean values along hyperplane sections. In case of the two-dimensional sphere, these plane sections are circles. In many tomographic applications, however, only limited data is available. Therefore, one is interested in the reconstruction of the function f from its mean values with respect to only some subfamily of all hyperplane sections of the sphere. Compared with the full data case, the limited data problem is more challenging and raises several questions. The first one is the injectivity, i.e., can any function be uniquely reconstructed from the available data? Further issues are the stability of the reconstruction, which is closely connected with a description of the range, as well as the demand for actual inversion methods or algorithms. We provide a detailed coverage and answers of these questions for different families of hyperplane sections of the sphere such as vertical slices, sections with hyperplanes through a common point and also incomplete great circles. Such reconstruction problems arise in various practical applications like Compton camera imaging, magnetic resonance imaging, photoacoustic tomography, Radar imaging or seismic imaging. Furthermore, we apply our findings about spherical means to the cone-beam transform and prove its singular value decomposition.

Erscheinungsdatum	10.08.2020
Zusatzinfo	Illustrationen, Diagramme
Verlagsort	Chemnitz
Sprache	englisch
Maße	148 x 210 mm
Gewicht	255 g
Themenwelt	Mathematik / Informatik ► Mathematik
Schlagworte	Bilderzeugung • Bildgebung • Funk-Radon-Transformation • Inverse Probleme • Inverses Problem • Kugelflächenfunktionen • Radon-Transformation • Singulärwertzerlegung • Sphärische Mittelwerte • sphärische Tomografie
ISBN-10	3-96100-116-2 / 3961001162
ISBN-13	978-3-96100-116-3 / 9783961001163
Zustand	Neuware