Auto-correlation (Wiener-Khinchin's theorem)

Using the auto-correlation function to obtain the power spectrum is preferred over the direct Fourier transform as most of the signals have very narrow bandwidth.
correlation.gif
If P(ω) is independent of the frequency, it is called white noise. If P(ω) is proportional to 1/f ( = 1/ω), the original data is called pink noise, 1/f noise or fractal noise and if P(ω) is proportional to 1/f2, it is called brown noise.
White noise

Pink noise

Brown noise

Application of Fourier Transforms (Computed Tomography and Radon transforms)

The Radon transform is defined as an integral of a two-dimensional density function, f(x, y), along the y′ axis as

R(x′, θ) ≡ ⌠
⌡ 		∞

−∞ 
 f(x, y) dy′.
(1)
where the (x′, y′) coordinate system is a rotation from the (x, y) coordinate system by θ as
⎛
⎜
⎜
⎝
x′
y′
⎞
⎟
⎟
⎠ 		= ⎛
⎜
⎜
⎝
cos θ,
sin θ
− sin θ,
cos θ
⎞
⎟
⎟
⎠ 		⎛
⎜
⎜
⎝
x
y
⎞
⎟
⎟
⎠ 		.
(2)
radon.gif
Note that R(x′, θ) depends on x′ and θ.
The Fourier transform for a two variable function, f(x, y), is defined as
F(ξ, η) ≡ ⌠
⌡ 		∞

−∞ 
 ⌠
⌡ 		∞

−∞ 
 f(x, y) e − i ( ξx + ηy) dx dy.
(3)
If the polar coordinate system, (ρ, θ), is introduced into the (ξ, η) system with
ξ = ρcos θ,     η = ρsin θ,
(4)

ξx + ηy = ρ(x cos θ+ y sin θ) = ρx′.
(5)
So
F(ξ, η)
=
⌠
⌡ 		∞

−∞ 
 ⌠
⌡ 		∞

−∞ 
 f(x, y) e− i ρx′ dx dy
(6)
=
⌠
⌡ 		∞

−∞ 
 ⌠
⌡ 		∞

−∞ 
 f(x, y) e− i ρx′⎢
⎢ 		∂(x, y)
∂(x′, y′)
⎢
⎢ 		dx′d y′
(7)
=
⌠
⌡ 		∞

−∞ 
 ⌠
⌡ 		∞

−∞ 
 f(x, y) e− i ρx′ dx′d y′
(8)
=
⌠
⌡ 		∞

−∞ 
 ⎛
⎝ 		⌠
⌡ 		∞

−∞ 
 f(x, y) dy′⎞
⎠ 		e− i ρx′ dx′
(9)
=
⌠
⌡ 		∞

−∞ 
 R(x′, θ) e− i ρx′ dx′.
(10)
Thus, the Fourier transform of f(x, y) can be obtained by multiplying e− i ρx′ on the Radon transform of f(x, y) and integrating the result with respect to x′. The original image can be restored by the inverse Fourier transform defined as
f(x, y) = 1
(2 π)2
 ⌠
⌡ 		∞

−∞ 
 ⌠
⌡ 		∞

−∞ 
 F(ξ, η)ei (ξx + ηy) dξdη.
(11)
This is the principle of CT (computed tomography).
ctscanner.gif


File translated from TEX by TTH, version 4.03.
On 21 Aug 2022, 22:51.