Abel inversion using Legendre polynomials approximations

An improved Abel inversion method based on Legendre polynomials approximations is presented for reconstructing the original radial distribution of plasma emission coefficients from projected intensities. The method uses the technique of overlapping two near segments for obtaining an excellent approximation of the intensity distribution. The approximated function of the intensity profile is a combination of various shifted Legendre polynomials which in the Abel inverse equation can be integrated exactly to deduce the emission coefficient. It is shown, using simulated intensity data with and without noise inverted for a comparison with those obtained by other methods, that the method is more accurate and has a better property of noise resistance. It is well suited for applying to experimental intensities distorted by noise.

[1]  Hongming Gao,et al.  Abel inversion using Legendre wavelets expansion , 2007 .

[2]  Use of Wavelets for Analyzing Transient Radiative Heat Transfer in an Inhomogeneous Medium , 2004 .

[3]  J. Palop,et al.  A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I‐V probe characteristic , 1995 .

[4]  Liang-Shih Fan,et al.  Inversion of Abel's integral equation by a direct method , 1984 .

[5]  Novel smoothing algorithm , 1975 .

[6]  A. Mihajlov,et al.  The realization of Abel's inversion in the case of discharge with undetermined radius , 2002 .

[7]  Gerald N. Minerbo,et al.  Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials , 1969 .

[8]  K. Bockasten,et al.  Transformation of Observed Radiances into Radial Distribution of the Emission of a Plasma , 1961 .

[9]  Sohrab Ali Yousefi,et al.  Numerical solution of Abel's integral equation by using Legendre wavelets , 2006, Appl. Math. Comput..

[10]  A. Savitzky,et al.  Smoothing and Differentiation of Data by Simplified Least Squares Procedures. , 1964 .

[11]  Oguzhan Guven,et al.  The radiative transfer solution of a rectangular enclosure using angular domain discrete wavelets , 2003 .

[12]  W. Lapatovich,et al.  Iterative method for computing the inverse Abel transform , 1987 .

[13]  S. Katz,et al.  Determination of a Radiance-Coefficient Profile from the Observed Asymmetric Radiance Distribution of an Optically Thin Radiating Medium , 1963 .

[14]  Alexei Ossadtchi,et al.  Reconstruction of Abel-transformable images: The Gaussian basis-set expansion Abel transform method , 2002 .

[15]  I. Powis,et al.  Two-dimensional charged particle image inversion using a polar basis function expansion , 2004 .

[16]  R. Birkebak,et al.  Application of the Abel integral equation to spectrographic data. , 1966, Applied optics.

[17]  Antonio Rodero,et al.  An Abel inversion method for radially resolved measurements in the axial injection torch , 2002 .

[18]  Dennis Keefer,et al.  Abel inversion using transform techniques , 1988 .

[19]  J. Kaiser,et al.  Data smoothing using low‐pass digital filters , 1977 .

[20]  K A Nugent,et al.  Abel inversion using fast Fourier transforms. , 1988, Applied optics.

[21]  J. Boettner,et al.  Abel inversion applied to plasma spectroscopy: a new interactive method. , 1978, Applied Optics.

[22]  M. Deutsch,et al.  Inversion of Abel’s integral equation for experimental data , 1983 .

[23]  M. Vrakking.  An iterative procedure for the inversion of two-dimensional ion/photoelectron imaging experiments , 2001 .

[24]  H. Griem.  Principles of Plasma Spectroscopy , 1997 .