Search results for "Tikhonov regularization"

On 1-Laplacian Elliptic Equations Modeling Magnetic Resonance Image Rician Denoising

2015

Modeling magnitude Magnetic Resonance Images (MRI) rician denoising in a Bayesian or generalized Tikhonov framework using Total Variation (TV) leads naturally to the consideration of nonlinear elliptic equations. These involve the so called $1$-Laplacian operator and special care is needed to properly formulate the problem. The rician statistics of the data are introduced through a singular equation with a reaction term defined in terms of modified first order Bessel functions. An existence theory is provided here together with other qualitative properties of the solutions. Remarkably, each positive global minimum of the associated functional is one of such solutions. Moreover, we directly …

Spline Algorithms for Deconvolution and Inversion of Heat Equation

2014

In this chapter, we present algorithms based on Tikhonov regularization for solving two related problems: deconvolution and inversion of heat equation. The algorithms, which utilize the SHA technique, provide explicit solutions to the problems in one and two dimensions.

Optimization and Inversion

2009

Validation of the solution method using Tikhonov regularization algorithm for spectral line diagnostics of microsize plasma

2014

This paper is devoted obtaining the threshold of the credibility of the solution by means of Tikhonov's regularization method in case of spectral lines, emitted from microsize plasma sources. The reliability of Tikhonov algorithm was verified by means of solving model tasks with different ratio between instrumental function and measured profile and, with different levels of noise.

Regularization Method in Infrared Image Processing

2003

Abstract Infrared images often present distortions induced by the measurement system. Thus, image processing is a vital part of infrared measurements. A distortion model based on a convolution product is presented. Image restoration is an ill-posed problem and its solution can be obtained using regularization methods. In this paper, image restoration is performed using a variation of Tikhonov regularization that makes use of the particular form of the convolution kernel matrix, which is built as a block-circulant matrix that admits a diagonal form in the two-dimensional Fourier space. The restoration procedure is used to restore a knife-edge infrared source image.

New Regularization Method for EXAFS Analysis

2007

As an alternative to the analysis of EXAFS spectra by conventional shell fitting, the Tikhonov regularization method has been proposed. An improved algorithm that utilizes a priori information about the sample has been developed and applied to the analysis of U L3‐edge spectra of soddyite, (UO2)2SiO4⋅2H2O, and of U(VI) sorbed onto kaolinite. The partial radial distribution functions g1(UU), g2(USi), and g3(UO) of soddyite agree with crystallographic values and previous EXAFS results.

Deconvolution of Multiple Spectral Lines Shapes by Means of Tikhonov’s Regularization Method

2013

We present deconvolution of multiple narrow Zeeman split Hg lines, emitted from Hg/Xe micro-size capillary and measured by the Fourier Transform spectrometer. The ill-posed inverse problem was solved using the Tikhonov& rsquo;s regularization method.

Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring

2013

We introduce a new iterative scheme for solving linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization, but with an approximation of the underlying operator to be used for the Tikhonov equations. For image deblurring problems, such an approximation can be a discrete deconvolution that operates entirely in the Fourier domain. We provide a theoretical analysis of the new scheme, using regularization parameters that are chosen by a certain adaptive strategy. The numerical performance of this method turns out to be superior to state-of-the-art iterative methods, including the conjugate gradient iteration for the normal equation, with and without additional precondi…

On the discrete linear ill‐posed problems

1999

An inverse problem of photo‐acoustic spectroscopy of semiconductors is investigated. The main problem is formulated as the integral equation of the first kind. Two different regularization methods are applied, the algorithms for defining regularization parameters are given. Diskrečiųjų blogai sąlygotų uždavinių klausimu Santrauka Darbe nagrinejamas foto‐akustines spektroskopijos puslaidininkiuose uždavinys, kuriame i vertinami nešeju difuzijos ir rekombinacijos procesai. Reikia atstatyti šaltinio funkcija f(x), jei žinoma antrosios eiles difuzijos lygtis ir atitinkamos kraštines salygos. Naudojantis matavimu, atliktu ivairiuose dažniuose, rezultatais sprendžiamas atvirkštinis uždavinys, kel…

Multicomponent line profile restoring by means of ill-posed inverse task solution

2017

The investigation of the criteria of usage of the Tikhonov regularization method for multicomponent overlapping line profiles restoring was done by means of model task solution. The influence of the width and kind of the instrumental function, number of the components of the profile and distance between components are discussed.