In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Abel transform,
[N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826).] named for
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
, is an
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function ''f''(''r'') is given by
:
Assuming that ''f''(''r'') 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 inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
In
absorption spectroscopy
Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating fi ...
of cylindrical flames or plumes, the forward Abel transform is the integrated
absorbance
Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...
along a ray with closest distance ''y'' from the center of the flame, while the inverse Abel transform gives the local
absorption coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
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 (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed.
In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in
photofragment-ion imaging and
photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
Geometrical interpretation
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
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''r'' is related to ''x'' and ''y'' as ''r''
2 = ''x''
2 + ''y''
2, 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 ''ρ''
2 = ''x''
2 + ''y''
2 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
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, 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 ''r''
2 = ''x''
2 + ''y''
2 + ''z''
2.
The projection onto, say, the ''yz'' plane will then be circularly symmetric and expressible as ''F''(''s''), where ''s''
2 = ''y''
2 + ''z''
2. Carrying out the integration, we have
:
which is again, the Abel transform of ''f''(''r'') in ''r'' and ''s''.
Verification of the inverse Abel transform
Assuming
is continuously differentiable and
,
drop to zero faster than
, we can set
and
. Integration by parts then yields
:
Differentiating
formally,
:
Now substitute this into the inverse Abel transform formula:
:
By
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
, the last integral equals
:
Generalization of the Abel transform to discontinuous ''F''(''y'')
Consider the case where
is discontinuous at
, where it abruptly changes its value by a finite amount
. That is,
and
are defined by