Abel transform

In two dimensions, the Abel transform can be interpreted as the projection of a circularly symmetric function along a set of parallel lines of sight at a distance from the origin. Referring to the figure on the right, the observer () will see : 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 . == Verification of the inverse Abel transform ==

Assuming is continuously differentiable, and , drop to zero faster than , we can integrate by parts by setting :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). == Generalization of the Abel transform to discontinuous ==

Consider the case where is discontinuous at , where it abruptly changes its value by a finite amount . That is, and are defined by :\Delta F \equiv \lim_{\varepsilon\rightarrow 0} \bigl( F(y_\Delta-\varepsilon) - F(y_\Delta+\varepsilon) \bigr). Such a situation is encountered in tethered polymers (Polymer brush) exhibiting a vertical phase separation, where 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 is under these circumstances again given by: :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. == 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 as the Abel transform operator, as the Fourier transform operator and as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that : 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. Relationship to the Radon transform Abel transform can be viewed as the Radon transform of an isotropic 2D function . As 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 ==