13 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
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