Sombrero function
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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 . \operatorname{somb} (\rho) = \frac{2 J_1(\pi \rho)}{\pi \rho}. The normalization factor makes . Sometimes the factor is omitted, giving the following alternative definition: \operatorname{somb} (\rho) = \frac{2 J_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.
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 . \operatorname{somb} (\rho) = \frac{2 J_1(\pi \rho)}{\pi \rho}. The normalization factor makes . Sometimes the factor is omitted, giving the following alternative definition: \operatorname{somb} (\rho) = \frac{2 J_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. ==References==
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