Application of the Abel Integral Equation to Spectrographic Data

Abstract

An improved method is proposed for applying the Abel integral equation to radiance data gathered from rotationally symmetric sources by side-on observation. The method involves dividing the data into a number of segments going out from the center. A least-squares polynomial is then fitted to each segment so that the inverted Abel integral equation can be integrated exactly to yield the emission coefficient. It is shown, with test curves approximating spectral line shapes from nonhomogeneous sources, that the method becomes superior to the other more common numerical methods of reducing such data as the number of data points becomes large. This superiority increases as the data scatter increases.

On the Use of the Inverted Abel Integral for Evaluating Spectroscopic Sources

“…Note that the value A 01 in Equation (16) is the coefficient from Equation (7), while all other A's in the right hand sides of Equations (13)(14)(15)(16) are from Equation (12).…”

“…A method involving multisegmented polynomials in which the segments are overlapped to provide smoothing was developed by Cremers and Birkebak (13). The method has been implemented in this laboratory with two changes:…”

“…The methods investigated were a modification of the technique of Cremers and Birkebak (13), an extension of the method of Barr (12), and a new method, based upon Cubic Spline functions, which was developed by the authors. The methods are compared with regard to their accuracy -3-under varied conditions of data resolution and noise level, and recommendations are developed for which method is most suitable for a given quality of data.…”

Application of inverse Abel techniques in in-line holographic microscopy

“…the SNR ratio depends on the width (FWHM) of the object and roughly stabilizes to a saturation value for large widths. The Noise Level 2, for instance, corresponds to noise level of the power of 13 …”

“…Moreover, the projection function ( , ) P x z , cannot be treated as a continuous function, as its value is known only in certain discrete points [9]. Various approaches that are based on geometrical or numerical methods and on the use of polynomials, have been adopted aiming at the calculation of the inverse Abel transform [9][10][11][12][13][14][15].…”