1,452 research outputs found
A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems
Classical iterative methods for tomographic reconstruction include the class
of Algebraic Reconstruction Techniques (ART). Convergence of these stationary
linear iterative methods is however notably slow. In this paper we propose the
use of Krylov solvers for tomographic linear inversion problems. These advanced
iterative methods feature fast convergence at the expense of a higher
computational cost per iteration, causing them to be generally uncompetitive
without the inclusion of a suitable preconditioner. Combining elements from
standard multigrid (MG) solvers and the theory of wavelets, a novel
wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to
significantly speed-up Krylov convergence. The performance of the
WMG-preconditioned Krylov method is analyzed through a spectral analysis, and
the approach is compared to existing methods like the classical Simultaneous
Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods
on a 2D tomographic benchmark problem. Numerical experiments are promising,
showing the method to be competitive with the classical Algebraic
Reconstruction Techniques in terms of convergence speed and overall performance
(CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13
figures, 3 table
Four-dimensional tomographic reconstruction by time domain decomposition
Since the beginnings of tomography, the requirement that the sample does not
change during the acquisition of one tomographic rotation is unchanged. We
derived and successfully implemented a tomographic reconstruction method which
relaxes this decades-old requirement of static samples. In the presented
method, dynamic tomographic data sets are decomposed in the temporal domain
using basis functions and deploying an L1 regularization technique where the
penalty factor is taken for spatial and temporal derivatives. We implemented
the iterative algorithm for solving the regularization problem on modern GPU
systems to demonstrate its practical use
Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems
Like many other advanced imaging methods, x-ray phase contrast imaging and
tomography require mathematical inversion of the observed data to obtain
real-space information. While an accurate forward model describing the
generally nonlinear image formation from a given object to the observations is
often available, explicit inversion formulas are typically not known. Moreover,
the measured data might be insufficient for stable image reconstruction, in
which case it has to be complemented by suitable a priori information. In this
work, regularized Newton methods are presented as a general framework for the
solution of such ill-posed nonlinear imaging problems. For a proof of
principle, the approach is applied to x-ray phase contrast imaging in the
near-field propagation regime. Simultaneous recovery of the phase- and
amplitude from a single near-field diffraction pattern without homogeneity
constraints is demonstrated for the first time. The presented methods further
permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval
and tomographic inversion. We demonstrate the potential of this approach by
three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may
be made for personal use only. Systematic reproduction and distribution,
duplication of any material in this paper for a fee or for commercial
purposes, or modifications of the content of this paper are prohibite
GENFIRE: A generalized Fourier iterative reconstruction algorithm for high-resolution 3D imaging
Tomography has made a radical impact on diverse fields ranging from the study
of 3D atomic arrangements in matter to the study of human health in medicine.
Despite its very diverse applications, the core of tomography remains the same,
that is, a mathematical method must be implemented to reconstruct the 3D
structure of an object from a number of 2D projections. In many scientific
applications, however, the number of projections that can be measured is
limited due to geometric constraints, tolerable radiation dose and/or
acquisition speed. Thus it becomes an important problem to obtain the
best-possible reconstruction from a limited number of projections. Here, we
present the mathematical implementation of a tomographic algorithm, termed
GENeralized Fourier Iterative REconstruction (GENFIRE). By iterating between
real and reciprocal space, GENFIRE searches for a global solution that is
concurrently consistent with the measured data and general physical
constraints. The algorithm requires minimal human intervention and also
incorporates angular refinement to reduce the tilt angle error. We demonstrate
that GENFIRE can produce superior results relative to several other popular
tomographic reconstruction techniques by numerical simulations, and by
experimentally by reconstructing the 3D structure of a porous material and a
frozen-hydrated marine cyanobacterium. Equipped with a graphical user
interface, GENFIRE is freely available from our website and is expected to find
broad applications across different disciplines.Comment: 18 pages, 6 figure
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