A sombrero function (sometimes called besinc function or jinc function) is the 2-dimensional

polar coordinate In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from ...

analog of the

sinc function In mathematics, physics and engineering, the sinc function ( ), denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatorname(x) = \frac. Alternatively, ...

, and is so-called because it is shaped like a

sombrero In English, a , ; ) is a type of wide-brimmed Mexican men's hat used to shield the face and eyes from the sun. It usually has a high, pointed crown; an extra-wide brim (broad enough to cast a shadow over the head, neck, and shoulders of the w ...

hat. This function is frequently used in

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

. It can be defined through the

Bessel function of the first kind Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

(

J_1

) where .

\operatorname (\rho) = \frac.

The normalization factor makes . Sometimes the factor is omitted, giving the following alternative definition:

\operatorname (\rho) = \frac.

The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:

\operatorname (\rho) = \frac.

The

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

of the 2D circle function (

\operatorname(\rho)

) is a sombrero function. Thus a sombrero function also appears in the intensity profile of far-field

diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...

through a circular aperture, known as an

Airy disk In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focus (optics), focused Point source#Light, spot of light that a perfect lens (optics), lens with a circular aperture can make, limited by the diffraction of ...

References

{{reflist, refs= {{cite book , title = Theory of Remote Image Formation , author = Richard E. Blahut , page = 82 , date = 2004-11-18 , publisher =

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, isbn = 9781139455305 , url = https://books.google.com/books?id=d8FMlHewYp0C&dq=%22jinc+function%22&pg=PA82 {{cite book , url = https://books.google.com/books?id=T_KNSWU4uz4C&pg=PA204 , title = The perception of visual information , author = William R. Hendee, Peter Neil Temple Wells , page = 204 , date = 1997-06-27 , publisher = Springer , isbn = 978-0-387-94910-9 {{cite web , url=http://mathworld.wolfram.com/JincFunction.html , title= Jinc Function , last= Weisstein , first= Eric W. , website= MathWorld--A Wolfram Web Resource , access-date= 1 Jan 2019 Signal processing