Sombrero function explained

A sombrero function (sometimes called besinc function or jinc function) is the 2-dimensional polar coordinate analog of the sinc function, and is so-called because it is shaped like a sombrero hat. This function is frequently used in image processing. It can be defined through the Bessel function of the first kind (

J _{1} ) where .$$\backslash operatorname\; (\backslash rho)\; =\; \backslash frac.$$

The normalization factor makes . Sometimes the factor is omitted, giving the following alternative definition:$$\backslash operatorname\; (\backslash rho)\; =\; \backslash frac.$$

The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:$$\backslash operatorname\; (\backslash rho)\; =\; \backslash frac.$$

The Fourier transform of the 2D circle function (

\operatorname{circ}(\rho) ) is a sombrero function. Thus a sombrero function also appears in the intensity profile of far-field diffraction through a circular aperture, known as an Airy disk