Sombrero function

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 $ρ 2 = x 2 + y 2$ .

\operatorname {somb} (\rho )={\frac {2J_{1}(\pi \rho )}{\pi \rho }}

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The normalization factor $2$ makes $somb(0) = 1$ . Sometimes the $π$ factor is omitted, giving the following alternative definition:

\operatorname {somb} (\rho )={\frac {2J_{1}(\rho )}{\rho }}

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The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:^[3]

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

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References

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