Jinc
A sombrero function (sometimes called besinc function or jinc function) 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. It can be defined through the Bessel function of the first kind (
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 }}.}
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 }}.}
The factor of 2 is also often omitted, giving yet another definition and causing the function maximum to be 0.5:
somb
(
ρ
)
=
J
1
(
ρ
)
ρ
.
{\displaystyle \operatorname {somb} (\rho )={\frac {J_{1}(\rho )}{\rho }}.}
The Fourier transform of the 2D circle function (
circ
(
ρ
)
{\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.
心晴
- 2019-01-11T00:00:00.000000Z
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