4,326 research outputs found
A General Local Reconstruction Approach Based on a Truncated Hilbert Transform
Exact image reconstruction from limited projection data has been a central topic in the computed tomography (CT) field. In this paper, we present a general region-of-interest/volume-of-interest (ROI/VOI) reconstruction approach using a truly truncated
Hilbert transform on a line-segment inside a compactly supported object aided by partial knowledge on one or both neighboring
intervals of that segment. Our approach and associated new data sufficient condition allows the most flexible ROI/VOI image
reconstruction from the minimum account of data in both the fan-beam and cone-beam geometry. We also report primary numerical
simulation results to demonstrate the correctness and merits of our finding. Our work has major theoretical potentials
and innovative practical applications
Differential Phase-contrast Interior Tomography
Differential phase contrast interior tomography allows for reconstruction of
a refractive index distribution over a region of interest (ROI) for
visualization and analysis of internal structures inside a large biological
specimen. In this imaging mode, x-ray beams target the ROI with a narrow beam
aperture, offering more imaging flexibility at less ionizing radiation.
Inspired by recently developed compressive sensing theory, in numerical
analysis framework, we prove that exact interior reconstruction can be achieved
on an ROI via the total variation minimization from truncated differential
projection data through the ROI, assuming a piecewise constant distribution of
the refractive index in the ROI. Then, we develop an iterative algorithm for
the interior reconstruction and perform numerical simulation experiments to
demonstrate the feasibility of our proposed approach
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
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