page_title: Abel transform
section_title: Geometrical interpretation
hierarchical_section_title: Abel transform / Geometrical interpretation
caption_attribution_description: English: Geometry of the Abel transform in two dimensions
context_page_description: 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. The Abel transform of a function f is given by
  Assuming that f drops to zero more quickly than 1/r, the inverse Abel transform is given by
  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 of that emission function.
  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. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique, maximum likelihood expectation maximization, filtered back-brojection algorithms should be employed.
context_section_description: In two dimensions, the Abel transform F(y) can be interpreted as the projection of a circularly symmetric function f(r) along a set of parallel lines of sight at a distance y from the origin. Referring to the figure on the right, the observer (I) will see
  where f(r) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at x = ∞, so that the limits of integration are ±∞, and all lines of sight are parallel to the x axis.
  Realizing that the radius r is related to x and y as r² = x² + y², it follows that
  for x > 0. Since f(r) is an even function in x, we may write
  which yields the Abel transform of f(r).
  The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function f(ρ, z), where ρ² = x² + y² is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz plane, so that
  which is just the Abel transform of f(ρ, z) in ρ and y.
  A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r), where r² = x² + y² + z².
  The projection onto, say, the yz plane will then be circularly symmetric and expressible as F(s), where s² = y² + z². Carrying out the integration, we have
  which is again, the Abel transform of f(r) in r and s.
caption: A geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the x axis a distance y above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function f(r) along the line of sight. The function f(r) is represented in gray in this figure. The observer is assumed to be located infinitely far from the origin so that the limits of integration are ±∞.
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