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 }}

.

Sombrero function 3D

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 }}

.

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 }}

.

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[Blahut-1] Richard E. Blahut (2004-11-18). Theory of Remote Image Formation. Cambridge University Press. p. 82. ISBN 9781139455305.

[Hendee-2] William R. Hendee, Peter Neil Temple Wells (1997-06-27). The perception of visual information. p. 204. ISBN 978-0-387-94910-9.

[Wolfram-3] Weisstein, Eric W. "Jinc Function". MathWorld--A Wolfram Web Resource. Retrieved 1 Jan 2019.