700 research outputs found
Stability estimates for the regularized inversion of the truncated Hilbert transform
In limited data computerized tomography, the 2D or 3D problem can be reduced
to a family of 1D problems using the differentiated backprojection (DBP)
method. Each 1D problem consists of recovering a compactly supported function
, where is a finite interval, from its
partial Hilbert transform data. When the Hilbert transform is measured on a
finite interval that only overlaps but does not cover
this inversion problem is known to be severely ill-posed [1].
In this paper, we study the reconstruction of restricted to the overlap
region . We show that with this restriction and by
assuming prior knowledge on the norm or on the variation of , better
stability with H\"older continuity (typical for mildly ill-posed problems) can
be obtained.Comment: added one remark, larger fonts for axis labels in figure
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
Semi-classical analysis and passive imaging
Passive imaging is a new technique which has been proved to be very
efficient, for example in seismology: the correlation of the noisy fields,
computed from the fields recorded at different points, is strongly related to
the Green function of the wave propagation. The aim of this paper is to provide
a mathematical context for this approach and to show, in particular, how the
methods of semi-classical analysis can be be used in order to find the
asymptotic behaviour of the correlations.Comment: Invited paper to appear in NONLINEARITY; Accepted Revised versio
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