Abel transform

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 is given by : F(y) = 2 \int_y^\infty \frac{f(r)r}{\sqrt{r^2 - y^2}} \,dr. Assuming that drops to zero more quickly than , the inverse Abel transform is given by : f(r) = -\frac{1}{\pi} \int_r^\infty \frac{dF}{dy} \,\frac{dy}{\sqrt{y^2 - r^2}}. 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 of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance 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 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 is given by F(y) = 2 \int_y^\infty \frac{f(r)r}{\sqrt{r^2 - y^2}} \,dr. Assuming that drops to zero more quickly than , the inverse Abel transform is given by f(r) = -\frac{1}{\pi} \int_r^\infty \frac{dF}{dy} \,\frac{dy}{\sqrt{y^2 - r^2}}. 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 of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance 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. F(y) = \int_{-\infty}^\infty f\left(\sqrt{x^2 + y^2}\right) \,dx, where is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at , so that the limits of integration are , and all lines of sight are parallel to the axis. Realizing that the radius is related to and as , it follows that dx = \frac{r\,dr}{\sqrt{r^2 - y^2}} for . Since is an even function in , we may write F(y) = 2 \int_0^\infty f\left(\sqrt{x^2 + y^2}\right) \,dx = 2 \int_ ^\infty f(r)\,\frac{r\,dr}{\sqrt{r^2 - y^2}}, which yields the Abel transform of . 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 , where 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 plane, so that F(y, z) = \int_{-\infty}^\infty f(\rho, z) \,dx = 2 \int_y^\infty \frac{f(\rho, z) \rho \,d\rho}{\sqrt{\rho^2 - y^2}}, which is just the Abel transform of in and . A particular type of axial symmetry is spherical symmetry. In this case, we have a function , where . The projection onto, say, the plane will then be circularly symmetric and expressible as , where . Carrying out the integration, we have F(s) = \int_{-\infty}^\infty f(r) \,dx = 2 \int_s^\infty \frac{f(r) r \,dr}{\sqrt{r^2 - s^2}}, which is again, the Abel transform of in and . u=f(r) ,\quad v' = \frac{r}\sqrt{r^2-y^2}, to find F(y) = -2 \int_y^\infty f'(r) \sqrt{r^2-y^2} \, dr. Differentiating formally, F'(y) = 2 y \int_y^\infty \frac{f'(r)}{\sqrt{r^2-y^2}} \, dr. Now substitute this into the inverse Abel transform formula: -\frac{1}{\pi} \int_r^\infty \frac{F'(y)}{\sqrt{y^2-r^2}} \, dy = \int_r^\infty \int_y^\infty \frac{-2 y}{\pi \sqrt{\left(y^2-r^2\right) \left(s^2-y^2\right)}} f'(s) \, ds dy. By Fubini's theorem, the last integral equals \int_r^\infty \int_r^s \frac{-2 y}{\pi \sqrt{\left(y^2-r^2\right) \left(s^2-y^2\right)}} \, dy f'(s) \,ds = \int_r^\infty (-1) f'(s) \, ds = f(r). F(y)=2\int_y^\infty \frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. Assuming drops to zero more quickly than , the inverse Abel transform is however given by f(r)=\left( \frac{1}{2}\delta\left(r-y_\Delta\right)\sqrt{1-\left(\frac{y_\Delta}{r}\right)^2} - \frac{1}{\pi} \frac{H\left(y_\Delta-r\right)}{\sqrt{y_\Delta^2-r^2}} \right) \Delta F-\frac{1}{\pi}\int_r^\infty\frac{d F}{dy}\frac{dy}{\sqrt{y^2-r^2}}. where is the Dirac delta function and the Heaviside step function. The extended version of the Abel transform for discontinuous is proven upon applying the Abel transform to shifted, continuous , and it reduces to the classical Abel transform when . If 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 additional terms, each of them corresponding to one of the discontinuities. FA = H. In other words, applying the Abel transform to a one-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.

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