Sombrero function

A sombrero function (sometimes called besinc function or jinc function[1]) 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.[2] It can be defined through the Bessel function of the first kind where ρ² = x² + y².

${\displaystyle \operatorname {somb} (\rho )={\frac {2J_{1}(\pi \rho )}{\pi \rho }}}$.

Sombrero function 3D

The normalization factor 2 makes somb(0) = 1. Sometimes the π factor is omitted, giving the following alternative definition:

${\displaystyle \operatorname {somb} (\rho )={\frac {2J_{1}(\rho )}{\rho }}}$.

The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:[3]

${\displaystyle \operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}}$.