466 research outputs found
Analysis of view aliasing for the generalized Radon transform in
In this paper we consider the generalized Radon transform in the
plane. Let be a piecewise smooth function, which has a jump across a smooth
curve . We obtain a formula, which accurately describes view
aliasing artifacts away from when is reconstructed from the
data discretized in the view direction. The formula is
asymptotic, it is established in the limit as the sampling rate .
The proposed approach does not require that be band-limited. Numerical
experiments with the classical Radon transform and generalized Radon transform
(which integrates over circles) demonstrate the accuracy of the formula
Histogram Tomography
In many tomographic imaging problems the data consist of integrals along
lines or curves. Increasingly we encounter "rich tomography" problems where the
quantity imaged is higher dimensional than a scalar per voxel, including
vectors tensors and functions. The data can also be higher dimensional and in
many cases consists of a one or two dimensional spectrum for each ray. In many
such cases the data contain not just integrals along rays but the distribution
of values along the ray. If this is discretized into bins we can think of this
as a histogram. In this paper we introduce the concept of "histogram
tomography". For scalar problems with histogram data this holds the possibility
of reconstruction with fewer rays. In vector and tensor problems it holds the
promise of reconstruction of images that are in the null space of related
integral transforms. For scalar histogram tomography problems we show how bins
in the histogram correspond to reconstructing level sets of function, while
moments of the distribution are the x-ray transform of powers of the unknown
function. In the vector case we give a reconstruction procedure for potential
components of the field. We demonstrate how the histogram longitudinal ray
transform data can be extracted from Bragg edge neutron spectral data and
hence, using moments, a non-linear system of partial differential equations
derived for the strain tensor. In x-ray diffraction tomography of strain the
transverse ray transform can be deduced from the diffraction pattern the full
histogram transverse ray transform cannot. We give an explicit example of
distributions of strain along a line that produce the same diffraction pattern,
and characterize the null space of the relevant transform.Comment: Small corrections from last versio
EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments
We review developments, issues and challenges in Electrical Impedance
Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT,
Manchester 2003. We focus on the necessity for three dimensional data
collection and reconstruction, efficient solution of the forward problem and
present and future reconstruction algorithms. We also suggest common pitfalls
or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of
EIT, Manchester, UK, 200
Tomography: mathematical aspects and applications
In this article we present a review of the Radon transform and the
instability of the tomographic reconstruction process. We show some new
mathematical results in tomography obtained by a variational formulation of the
reconstruction problem based on the minimization of a Mumford-Shah type
functional. Finally, we exhibit a physical interpretation of this new technique
and discuss some possible generalizations.Comment: 11 pages, 5 figure
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