Sombrero function

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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 ( J 1 {\displaystyle J_{1}} {\displaystyle J_{1}}) where ρ2 = x2 + y2. somb ⁡ ( ρ ) = 2 J 1 ( π ρ ) π ρ . {\displaystyle \operatorname {somb} (\rho )={\frac {2J_{1}(\pi \rho )}{\pi \rho }}.} {\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: somb ⁡ ( ρ ) = 2 J 1 ( ρ ) ρ . {\displaystyle \operatorname {somb} (\rho )={\frac {2J_{1}(\rho )}{\rho }}.} {\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] somb ⁡ ( ρ ) = J 1 ( ρ ) ρ . {\displaystyle \operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}.} {\displaystyle \operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}.}

The Fourier transform of the 2D circle function ( circ ⁡ ( ρ ) {\displaystyle \operatorname {circ} (\rho )} {\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.