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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 : F(y) = 2 \int_y^\infty \frac \,dr. Assuming that ''f''(''r'') drops to zero more quickly than 1/''r'', the inverse Abel transform is given by : f(r) = -\frac \int_r^\infty \frac \,\frac. 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 : F(y) = \int_^\infty f\left(\sqrt\right) \,dx, 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 : dx = \frac for ''x'' > 0. Since ''f''(''r'') is an even function in ''x'', we may write : F(y) = 2 \int_0^\infty f\left(\sqrt\right) \,dx = 2 \int_^\infty f(r)\,\frac, 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 : F(y, z) = \int_^\infty f(\rho, z) \,dx = 2 \int_y^\infty \frac, 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 : F(s) = \int_^\infty f(r) \,dx = 2 \int_s^\infty \frac, which is again, the Abel transform of ''f''(''r'') in ''r'' and ''s''.


Verification of the inverse Abel transform

Assuming f is continuously differentiable and f, f' drop to zero faster than 1/r, we can set u=f(r) and v=\sqrt . Integration by parts then yields :F(y) = -2 \int_y^\infty f'(r) \sqrt \, dr. Differentiating formally, :F'(y) = 2 y \int_y^\infty \frac \, dr. Now substitute this into the inverse Abel transform formula: :-\frac \int_r^\infty \frac \, dy = \int_r^\infty \int_y^\infty \frac f'(s) \, ds dy. 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 :\int_r^\infty \int_r^s \frac \, dy f'(s) \,ds = \int_r^\infty (-1) f'(s) \, ds = f(r).


Generalization of the Abel transform to discontinuous ''F''(''y'')

Consider the case where F(y) is discontinuous at y=y_\Delta, where it abruptly changes its value by a finite amount \Delta F. That is, y_\Delta and \Delta F are defined by \Delta F \equiv \lim_ F(y_\Delta-\epsilon) - F(y_\Delta+\epsilon) /math>. Such a situation is encountered in tethered polymers (
Polymer brush A polymer brush is the name given to a surface coating consisting of polymers tethered to a surface. The brush may be either in a solvated state, where the tethered polymer layer consists of polymer and solvent, or in a melt state, where the teth ...
) exhibiting a vertical phase separation, where F(y) stands for the polymer density profile and f(r) is related to the spatial distribution of terminal, non-tethered monomers of the polymers. The Abel transform of a function ''f''(''r'') is under these circumstances again given by: :F(y)=2\int_y^\infty \frac. Assuming ''f''(''r'') drops to zero more quickly than 1/''r'', the inverse Abel transform is however given by : f(r)=\left \frac\delta(r-y_\Delta)\sqrt - \frac \frac \right\Delta F-\frac\int_r^\infty\frac\frac. where \delta is the Dirac delta function and H(x) the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous F(y), and it reduces to the classical Abel transform when \Delta F=0. If F(y) has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains ''n'' additional terms, each of them corresponding to one of the ''n'' discontinuities.


Relationship to other integral transforms


Relationship to the Fourier and Hankel transforms

The Abel transform is one member of the FHA cycle of integral operators. For example, in two dimensions, if we define ''A'' as the Abel transform operator, ''F'' as the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
operator and ''H'' as the zeroth-order
Hankel transform In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
operator, then the special case of the
projection-slice theorem In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimensional function ''f''(r), project (e.g. using the ...
for circularly symmetric functions states that : FA = H. In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.


Relationship to the Radon transform

Abel transform can be viewed as the
Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
of an isotropic 2D function ''f''(''r''). As ''f''(''r'') is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.


See also

*
GPS radio occultation Radio occultation (RO) is a remote sensing technique used for measuring the physical properties of a planetary atmosphere or ring system. Atmospheric radio occultation Atmospheric radio occultation relies on the detection of a change in a radio ...


References

* {{Authority control Integral transforms Image processing Niels Henrik Abel