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 (${\displaystyle J_{1}}$) where ρ² = x² + y².

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

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]

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

The Fourier transform of the 2D circle function (${\displaystyle \operatorname {circ} (\rho )}$) 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.

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