Properties of the Fourier Transform CS/BIOEN 4640: Image Processing - - PowerPoint PPT Presentation

Properties of the Fourier Transform

CS/BIOEN 4640: Image Processing Basics March 22, 2012

Review: Euler’s Representation

Im Re φ r

◮ A complex number can be

given as an angle φ and a radius r

◮ Think 2D polar coordinates ◮ Exponential form:

reiφ = r cos(φ) + i (r sin(φ))

Review: Fourier Transform

Given a complex-valued function g : R → C, Fourier transform produces a function of frequency ω:

G(ω) = 1 √ 2π ∞

−∞

g(x) ·

• cos(ωx) − i · sin(ωx)
• dx

= 1 √ 2π ∞

−∞

g(x) · e−iωx dx

Inverse Fourier Transform

The Fourier transform is invertible. That is, given the Fourier transform G(ω) we can reconstruct the original function g as

g(x) = 1 √ 2π ∞

−∞

G(ω) · eiωxdω

We use the notation: Fourier transform:

G = F{g}

Inverse Fourier transform:

g = F−1{G}

The Dirac Delta

Definition

The Dirac delta or impulse is defined as

δ(x) = 0 for x = 0,

and

∞

−∞

δ(x) dx = 1

◮ The Dirac delta is not a function ◮ It is undefined at x = 0. ◮ Has the property

∞

−∞

f(x) δ(x) dx = f(0)

for any function f

The Dirac Delta

Even though the Dirac delta is not a function, we will plot it like this:

−1.0 −0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0

x δ(x)

Shifting and Scaling Deltas

−1.0 −0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0

x δ(x − 0.5)

−1.0 −0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0

x 2δ(x)

)

A shifted impulse is A scaled impulse is given by δ(x − a) given by s · δ(x)

∞

−∞ s · δ(x) dx = s

Fourier Transform Pairs: Cosine

−5 −3 −1 1 3 5 −1.0 −0.5 0.5 1.0 −5 −3 −1 1 3 5 −1.5 −1.0 −0.5 0.5 1.0 1.5

g(x) = cos(ω0 x) G(ω) = π

2 · (δ(ω + ω0) + δ(ω − ω0))

(Here ω0 = 3)

Fourier Transform Pairs: Sine

−5 −3 −1 1 3 5 −1.0 −0.5 0.5 1.0 −5 −3 −1 1 3 5 −1.5i −1.0i −0.5i 0.5i 1.0i 1.5i

g(x) = sin(ω0 x) G(ω) = i π

2 · (δ(ω + ω0) − δ(ω − ω0))

(Here ω0 = 5)

BOOK TYPO

Fourier Transform Pairs: Gaussian

−9 −7 −5 −3 −1 1 3 5 7 9 −1.0 −0.5 0.5 1.0 −9 −7 −5 −3 −1 1 3 5 7 9 −1.0 −0.5 0.5 1.0

g(x) = 1

σ exp

• −x2

2σ2

• G(ω) = exp
• −ω2

2·(1/σ2)

• (Here σ = 3)

Fourier Transform Pairs: Box

−9 −7 −5 −3 −1 1 3 5 7 9 −0.5 0.5 1.0 1.5 −9 −7 −5 −3 −1 1 3 5 7 9 −0.5 0.5 1.0 1.5

g(x) =

• 1

if |x| < b

• therwise

G(ω) = 2 sin(b ω)

√ 2π ω

(Here b = 2) BOOK TYPO

Properties: Linearity

Scaling:

F{c · g(x)} = c · G(ω)

Addition:

F{g1(x) + g2(x)} = G1(ω) + G2(ω)

Properties for Real-Valued Functions

◮ If g is a real-valued function, then

G(ω) = G∗(−ω)

◮ If g is real-valued and even: g(x) = g(−x), then

G(ω) is real-valued and even

◮ If g is real-valued and odd: g(x) = −g(−x), then

G(ω) is purely imaginary and odd

Properties: Similarity

Stretching a signal horizontally leads to a shrinking of the Fourier spectrum:

F{g(s · x)} = 1 |s| · G ω s

• And vice versa, shrinking the signal causes a stretching

in the Fourier spectrum

Properties: Shift

A horizontal shift of the signal results in a phase shift of the Fourier transform:

F{g(x + d)} = e−iωd · G(ω)

◮ Notice magnitude |G(ω)| stays the same ◮ This is a rotation in the complex plane by −ωd

Properties: Convolution

Convolution becomes multiplication in the Fourier domain:

F{g(x) ∗ h(x)} = √ 2π G(ω) · H(ω)

And vice versa, multiplication becomes convolution:

F{g(x) · h(x)} = 1 √ 2π G(ω) ∗ H(ω)