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 {\displaystyle J_{1}} ) where ρ2 = x2 + y2.
somb ( ρ ) = 2 J 1 ( π ρ ) π ρ . {\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:
somb ( ρ ) = 2 J 1 ( ρ ) ρ . {\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:
somb ( ρ ) = J 1 ( ρ ) ρ . {\displaystyle \operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}.}
The Fourier transform of the 2D circle function ( circ ( ρ ) {\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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