Sombrero function

File:Sombrero function 3d.png

Sombrero function 3D

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

$\operatorname{somb} (\rho) = \frac{2 J_1(\pi \rho)}{\pi \rho}$.

where ρ² = x² + y². The normalization factor 2 makes somb(0) = 1. Sometimes the π factor is omitted, giving the following alternative definition:

$\operatorname{somb}' (\rho) = \frac{2 J_1(\rho)}{\rho}$.