Search results for "Tikhonov regularization"
showing 10 items of 21 documents
Focal plane array infrared camera transfer function calculation and image restoration
2004
Infrared images often present distortions induced by the measurement system. Image processing is thus an essential part of infrared measurements. A distortion model based on a convolution product is presented. The analytical form of the convolution kernel has been obtained from an image formation theory, along with an analysis of the sampling of the focal plane array camera detector's matrix. Image restoration is an ill-posed problem, and its solution can be obtained using regularization methods. In this work, 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…
Line Shape Measurement and Modelling for Plasma Diagnostics
2014
In this paper we discuss different methods of narrow spectral line shape measurements for a wide spectral range by means of high-resolution spectrometers such as the Fabry-Perot spectrometer, Zeeman spectrometer and Fourier transform spectrometer as well as a theoretical model for spectral line shape modelling and solving of the inverse task based on Tikhonov's regularization method. Special attention is devoted to the line shape measurements for the optically thin light sources filled with Hg, Ar, Xe, Kr for their use in high precision analysers for detection of heavy metals and benzene.
A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration
1999
We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $|\nabla u|$ in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint,…
Thickness Inhomogeneity Effect in EXAFS Spectroscopy
2005
In many cases x-ray absorption spectra measured in transmission mode are degraded by an inhomogeneity in thickness of the samples. As a result, the EXAFS amplitude is decreased and information about the coordination numbers is distorted. To avoid this influence, it is necessary to prepare a homogeneous sample. But, for powder samples, thick inhomogeneous foils, and sputtered films this is not possible. Absorption spectra on these samples should be corrected for the thickness inhomogeneity effect.To correct an absorption spectrum it is necessary to know the sample thickness distribution function. We propose a method of solving an integral equation for a distribution function as an inverse pr…
COMPUTATION OF LOCAL VOLATILITIES FROM REGULARIZED DUPIRE EQUATIONS
2005
We propose a new method to calibrate the local volatility function of an asset from observed option prices of the underlying. Our method is initialized with a preprocessing step in which the given data are smoothened using cubic splines before they are differentiated numerically. In a second step the Dupire equation is rewritten as a linear equation for a rational expression of the local volatility. This equation is solved with Tikhonov regularization, using some discrete gradient approximation as penalty term. We show that this procedure yields local volatilities which appear to be qualitatively correct.
On the condition number of the antireflective transform
2010
Abstract Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for …
Deconvolution of the spectral line profiles for the plasma temperature estimation
2010
Abstract The Hg 253.7 nm spectral line profiles, emitted from the mercury–argon high-frequency electrodeless discharge lamps (HFEDL) have been measured by means of a high-resolution scanning Fabry–Perrot interferometer at the mercury cold spot temperature value at 20 °C, different discharge current and buffer gas values. The deconvolution procedure by means of the Tikhonov's regularization method was performed to obtain the real spectral line shape. The influence of the instrumental function and absorption, real width of the Hg 253.7 nm resonance line and temperature of the radiating atoms are obtained. The results were compared with the results of the nonlinear multiparameter mathematical …
Data analysis procedures for pulse ELDOR measurements of broad distance distributions
2004
The reliability of procedures for extracting the distance distribution between spins from the dipolar evolution function is studied with particular emphasis on broad distributions. A new numerically stable procedure for fitting distance distributions with polynomial interpolation between sampling points is introduced and compared to Tikhonov regularization in the dipolar frequency and distance domains and to approximate Pake transformation. Distance distributions with only narrow peaks are most reliably extracted by distance-domain Tikhonov regularization, while frequency-domain Tikhonov regularization is favorable for distributions with only broad peaks. For the quantification of distribut…
Iterative Regularization Techniques in Image Reconstruction
2000
In this survey we review recent developments concerning the efficient iterative regularization of image reconstruction problems in atmospheric imaging. We present a number of preconditioners for the minimization of the corresponding Tikhonov functional, and discuss the alternative of terminating the iteration early, rather than adding a stabilizing term in the Tikhonov functional. The methods are examplified for a (synthetic) model problem.
Fourier analysis of periodic Radon transforms
2019
We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.