Sombrero function

Sombrero function 3D

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] [ failed verification ]} 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.

References

1. Richard E. Blahut (2004-11-18). Theory of Remote Image Formation. Cambridge University Press. p. 82. ISBN   9781139455305.
2. William R. Hendee, Peter Neil Temple Wells (1997-06-27). The perception of visual information. Springer. p. 204. ISBN   978-0-387-94910-9.
3. Weisstein, Eric W. "Jinc Function". MathWorld--A Wolfram Web Resource. Retrieved 1 Jan 2019.