Sombrero function

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 ρ² = x² + y². $$\operatorname {somb} (\rho )={\frac {2J_{1}(\pi \rho )}{\pi \rho }}.$$

Sombrero function 3D

The normalization factor 2 makes somb(0) = 1. Sometimes the π factor is omitted, giving the following alternative definition: $$\operatorname {somb} (\rho )={\frac {2J_{1}(\rho )}{\rho }}.$$

The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5: $$\operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}.$$

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.