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 (formula_1) where "ρ"2 = "x"2 + "y"2.
formula_2
The normalization factor 2 makes somb(0) = 1. Sometimes the "π" factor is omitted, giving the following alternative definition:
formula_3
The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:
formula_4
The Fourier transform of the 2D circle function (formula_5) 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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