The observer is assumed to be located infinitely far from the origin so that the limits of integration are ±∞ The function f( r) is represented in gray in this figure.
the integral) of the circularly symmetric function f( r) along the line of sight. What the observer sees is the projection (i.e. An observer (I) looks along a line parallel to the x-axis a distance y above the origin. Among recent most notable extensions of inverse Abel transformation are the Onion Peeling and BAsis Set Expansion (BASEX) methods of photoelectron and photoion image analysis.Ī geometrical interpretation of the Abel transform in two dimensions. In recent years, the inverse Abel transformation (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as Algebraic Reconstruction Technique ( ART), Maximum Likelihood Expectation Maximization ( MLEM), Filtered Back-Projection ( FBP) algorithms should be employed. Abel transform is limited to applications with axially symmetric geometries. In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. a scan or a photograph) of that emission function. In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the reverse Abel transform is used to calculate the emission function given a projection (i.e. The Abel transform of a function f( r) is given by:Īssuming f( r) drops to zero more quickly than 1/ r, the inverse Abel transform is given by In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions.
For summation transformation, see summation by parts.
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