# 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] It can be defined through the Bessel function of the first kind 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]

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

## 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. p. 204. ISBN 978-0-387-94910-9.
3. ^ Weisstein, Eric W. "Jinc Function". MathWorld--A Wolfram Web Resource. Retrieved 1 Jan 2019.