8 research outputs found
Spectral analysis of the truncated Hilbert transform with overlap
We study a restriction of the Hilbert transform as an operator from
to for real numbers . The
operator arises in tomographic reconstruction from limited data, more
precisely in the method of differentiated back-projection (DBP). There, the
reconstruction requires recovering a family of one-dimensional functions
supported on compact intervals from its Hilbert transform measured
on intervals that might only overlap, but not cover . We
show that the inversion of is ill-posed, which is why we investigate the
spectral properties of .
We relate the operator to a self-adjoint two-interval Sturm-Liouville
problem, for which we prove that the spectrum is discrete. The Sturm-Liouville
operator is found to commute with , which then implies that the spectrum
of is discrete. Furthermore, we express the singular value
decomposition of in terms of the solutions to the Sturm-Liouville
problem. The singular values of accumulate at both and , implying
that is not a compact operator. We conclude by illustrating the
properties obtained for numerically.Comment: 24 pages, revised versio
Filtered backprojection inversion of the cone beam transform for a general class of curves
We extend a cone beam transform inversion formula, proposed earlier for helices by one of the authors, to a general class of curves. The inversion formula remains efficient, because filtering is shift-invariant and is performed along a one-parametric family of lines. The conditions that describe the class are very natural. Curves C are smooth, without self-intersections, have positive curvature and torsion, do not bend too much, and do not admit lines which are tangent to C at one point and intersect C at another point. The notions of PI lines and PI segments are generalized, and their properties are studied. The domain U is found, where PI lines are guaranteed to be unique. Results of numerical experiments demonstrate very good image quality
Asymptotic Analysis of The SVD For The Truncated Hilbert Transform With Overlap
The truncated Hilbert transform with overlap H-T is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection. Recent work [R. Al-Aifari and A. Katsevich, SIAM J. Math. Anal., 46 (2014), pp. 192213] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which the singular values approach zero and one. In this paper, we exploit the property that H-T commutes with a second-order differential operator L-S and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of H-T
Abbildungsmethoden für die Brust mit einem 3D-Ultraschall-Computertomographen
In dieser Arbeit wird die Theorie, Implementierung und Evaluierung von Algorithmen der Ultraschall-Transmissionstomographie für den am KIT entwickelten Prototypen 3D-USCT II behandelt. Bisherige Arbeiten gehen von idealen Voraussetzungen aus, diese Arbeit befasst sich hingegen mit der Bildrekonstruktion rauschbehafteter Echtdaten, die in einer klinischen Pilotstudie aufgenommen wurden. Von drei Krebsfällen konnten mit den Methoden dieser Arbeit zwei eindeutig identifiziert werden