9,442 research outputs found
Inversion of the star transform
We define the star transform as a generalization of the broken ray transform
introduced by us in previous work. The advantages of using the star transform
include the possibility to reconstruct the absorption and the scattering
coefficients of the medium separately and simultaneously (from the same data)
and the possibility to utilize scattered radiation which, in the case of the
conventional X-ray tomography, is discarded. In this paper, we derive the star
transform from physical principles, discuss its mathematical properties and
analyze numerical stability of inversion. In particular, it is shown that
stable inversion of the star transform can be obtained only for configurations
involving odd number of rays. Several computationally-efficient inversion
algorithms are derived and tested numerically.Comment: Accepted to Inverse Problems in this for
Fast Mojette Transform for Discrete Tomography
A new algorithm for reconstructing a two dimensional object from a set of one
dimensional projected views is presented that is both computationally exact and
experimentally practical. The algorithm has a computational complexity of O(n
log2 n) with n = N^2 for an NxN image, is robust in the presence of noise and
produces no artefacts in the reconstruction process, as is the case with
conventional tomographic methods. The reconstruction process is approximation
free because the object is assumed to be discrete and utilizes fully discrete
Radon transforms. Noise in the projection data can be suppressed further by
introducing redundancy in the reconstruction. The number of projections
required for exact reconstruction and the response to noise can be controlled
without comprising the digital nature of the algorithm. The digital projections
are those of the Mojette Transform, a form of discrete linogram. A simple
analytical mapping is developed that compacts these projections exactly into
symmetric periodic slices within the Discrete Fourier Transform. A new digital
angle set is constructed that allows the periodic slices to completely fill all
of the objects Discrete Fourier space. Techniques are proposed to acquire these
digital projections experimentally to enable fast and robust two dimensional
reconstructions.Comment: 22 pages, 13 figures, Submitted to Elsevier Signal Processin
On Artifacts in Limited Data Spherical Radon Transform: Curved Observation Surface
In this article, we consider the limited data problem for spherical mean
transform. We characterize the generation and strength of the artifacts in a
reconstruction formula. In contrast to the third's author work [Ngu15b], the
observation surface considered in this article is not flat. Our results are
comparable to those obtained in [Ngu15b] for flat observation surface. For the
two dimensional problem, we show that the artifacts are orders smoother
than the original singularities, where is vanishing order of the smoothing
function. Moreover, if the original singularity is conormal, then the artifacts
are order smoother than the original singularity. We provide
some numerical examples and discuss how the smoothing effects the artifacts
visually. For three dimensional case, although the result is similar to that
[Ngu15b], the proof is significantly different. We introduce a new idea of
lifting the space
FASTLens (FAst STatistics for weak Lensing) : Fast method for Weak Lensing Statistics and map making
With increasingly large data sets, weak lensing measurements are able to
measure cosmological parameters with ever greater precision. However this
increased accuracy also places greater demands on the statistical tools used to
extract the available information. To date, the majority of lensing analyses
use the two point-statistics of the cosmic shear field. These can either be
studied directly using the two-point correlation function, or in Fourier space,
using the power spectrum. But analyzing weak lensing data inevitably involves
the masking out of regions or example to remove bright stars from the field.
Masking out the stars is common practice but the gaps in the data need proper
handling. In this paper, we show how an inpainting technique allows us to
properly fill in these gaps with only operations, leading to a new
image from which we can compute straight forwardly and with a very good
accuracy both the pow er spectrum and the bispectrum. We propose then a new
method to compute the bispectrum with a polar FFT algorithm, which has the main
advantage of avoiding any interpolation in the Fourier domain. Finally we
propose a new method for dark matter mass map reconstruction from shear
observations which integrates this new inpainting concept. A range of examples
based on 3D N-body simulations illustrates the results.Comment: Final version accepted by MNRAS. The FASTLens software is available
from the following link : http://irfu.cea.fr/Ast/fastlens.software.ph
- …