9 research outputs found
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
Preasymptotic Convergence of Randomized Kaczmarz Method
Kaczmarz method is one popular iterative method for solving inverse problems,
especially in computed tomography. Recently, it was established that a
randomized version of the method enjoys an exponential convergence for
well-posed problems, and the convergence rate is determined by a variant of the
condition number. In this work, we analyze the preasymptotic convergence
behavior of the randomized Kaczmarz method, and show that the low-frequency
error (with respect to the right singular vectors) decays faster during first
iterations than the high-frequency error. Under the assumption that the inverse
solution is smooth (e.g., sourcewise representation), the result explains the
fast empirical convergence behavior, thereby shedding new insights into the
excellent performance of the randomized Kaczmarz method in practice. Further,
we propose a simple strategy to stabilize the asymptotic convergence of the
iteration by means of variance reduction. We provide extensive numerical
experiments to confirm the analysis and to elucidate the behavior of the
algorithms.Comment: 20 page
Sparsity-Homotopy Perturbation Inversion Method with Wavelets and Applications to Black-Scholes Model and Todaro Model
Sparsity regularization method plays an important role in reconstructing parameters. Compared with traditional regularization methods, sparsity regularization method has the ability to obtain better performance for reconstructing sparse parameters. However, sparsity regularization method does not have the ability to reconstruct smooth parameters. For overcoming this difficulty, we combine a sparsity regularization method with a wavelet method in order to transform smooth parameters into sparse parameters. We use a sparsity-homotopy perturbation inversion method to improve the accuracy and stability and apply the proposed method to reconstruct parameters for a Black-Scholes option pricing model and a Todaro model. Numerical experiments show that the proposed method is convergent and stable
An Inexact Newton Regularization in Banach Spaces based on the Nonstationary Iterated Tikhonov Method
We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients
Taxas de convergĂȘncia para mĂ©todos iterativos cĂclicos em problemas mal postos
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de CiĂȘncias FĂsicas e MatemĂĄticas, Programa de PĂłs-Graduação em MatemĂĄtica Pura e Aplicada, FlorianĂłpolis, 2015.Na classe dos mĂ©todos de regularização iterativos, os mĂ©todos tipo Kaczmarz sĂŁo uns dos mĂ©todos mais utilizados para resolver problemas na matemĂĄtica aplicada. No entanto, na literatura a quantidade de resultados sobre convergĂȘncia e as respectivas taxas de convergĂȘncia nĂŁo Ă© abundante. Este trabalho trata da anĂĄlise de convergĂȘncia de algumas versĂ”es do mĂ©todo de Landweber-Kaczmarz, obtendo convergĂȘncia e estabilidade do mĂ©todo modificado com um parĂąmetro de relaxamento, e taxas de convergĂȘncia no mĂ©todo para operadores lineares em bloco nas versĂ”es simĂ©trica e nĂŁo simĂ©trica. Finalmente, compara-se mediante experimentos numĂ©ricos o desempenho dos mĂ©todos estudados com o desempenho de mĂ©todos bem estabelecidos.Abstract : In the class of iterative regularization methods, Kaczmarz type methods are some of that are more often used to solve problems in applied mathematics. However, in the literature the amount of convergence results and their convergence rate is not abundant. This work deals with the analysis of convergence of some versions of the Landweber-Kaczmarz method, obtaining convergence and stability of the modified method with a relaxation parameter, and convergence rates for method for linear block operators in versions symmetrical and non-symmetrical. Finally, the performance of the methods is compared with the performance of well-established methods
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Mathematics and Algorithms in Tomography
This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis
On Inexact Newton Methods for Inverse Problems in Banach Spaces
This self-contained monograph investigates an inexact-Newton algorithm and its adaptation to the Kaczmarz methods for solving nonlinear ill-posed problems in Banach spaces. This Newton-like method linearizes the original problem and then applies a regularization technique to stably solve the resulting linear systems. A complete convergence analysis is carried out and its performance is tested in the Electrical Impedance Tomography to provide the necessary support to the theoretical results