The Fourier Transform
CS/BIOEN 4640: Image Processing Basics
March 20, 2012
Why Study Spectral Methods?
I
Is often more efficient method for filtering than
convolution
I
Helps us understand image sampling, filtering, and
aliasing
I
Used in image compression
I
Important for solving PDEs in image processing
I
Fourier transform used in MRI
Harmonic Analysis
I
I
Decompose a function into
basic waves called “harmonics”
0
1
1/2
1/3
Any signal can be written as a
summation of harmonics
1/4
1/5
I
Think of sound waves and
music: harmonics are pure
tones
1/6
1/7
Sine and Cosine Functions
1.0
Harmonics are given by sine and cosine functions
−1.0
−0.5
0.0
0.5
sin
cos
0
1
2
3
4
5
6
7
1.0
Wave Properties: Frequency
Frequency is how
many times a wave
repeats.
0.0
0.5
sin(t)
sin(2t)
sin(3t)
sin(ωx)
−0.5
cos(ωx) and
−1.0
ω is the frequency
0
1
2
3
4
5
6
3
Wave Properties: Amplitude
2
sin(t)
2 sin(t)
3 sin(t)
0
1
Amplitude is the
height of the wave.
−1
a·cos(x) and a·sin(x)
−3
−2
a is the amplitude
0
1
2
3
4
5
6
1.0
Wave Properties: Phase
sin(t)
sin(t − pi/4)
sin(t − pi/2)
0.0
0.5
Phase is the horizontal shift of the wave.
−0.5
cos(x−φ) and sin(x−φ)
−1.0
φ is the phase shift
0
1
2
3
4
5
6
Who’s This Guy?
While studying heat conduction, Fourier discovered that
functions could be decomposed into summations of
cosine waves with different
amplitudes and frequencies.
Jean Baptiste Joseph de Fourier
(1768 - 1830)
Trivia fact: Fourier also first
described the greenhouse effect!
Fourier Series
Definition
Consider a function g(x) that is periodic on [0, 2π
ω ]
0
It’s Fourier series is given as
g(x) =
∞
X
[ Ak cos(kω0 x) + Bk sin(kω0 x)] ,
k=0
where Ak , Bk are constants called the Fourier
coefficients.
Complex Numbers
Definition
A complex number is an ordered pair of real numbers,
z = (a, b), with a the real part and b the imaginary
part.
Also written as
where i =
√
z = a + ib,
−1 is the imaginary unit.
The set of complex numbers is denoted C.
Complex Numbers as 2D Coordinates
Im
b
a+bi
The real and imaginary parts of
z = a + ib are the coordinates:
Re{z} = a
Im{z} = b
0
a
Re
Complex Number Arithmetic
Take two complex numbers z1 = (a1 , b1 ) and
z2 = (a2 , b2 ).
I
Addition
z1 + z2 = (a1 + a2 , b1 + b2 )
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Multiplication
z1 · z2 = (a1 + ib1 ) · (a2 + ib2 )
= (a1 a2 − b1 b2 ) + i(a1 b2 + a2 b1 )
Conjugation and Absolute Value
Consider a complex number z = (a, b)
I
Conjugation: Simply negate the imaginary part:
z∗ = a − ib
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Absolute value: Same as 2D vector length:
p
|z| = a2 + b2
√
Also given by |z| = z · z∗
Euler’s Representation of Complex Numbers
Im
r
φ
0
Re
I
A complex number can be
given as an angle φ and a
radius r
I
Think 2D polar coordinates
I
Exponential form:
reiφ = r cos(φ) + i (r sin(φ))
Operations in Euler’s Notation
Take z1 = r1 eiθ1 and z2 = r2 eiθ2 .
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Multiplication:
z1 · z2 = r1 r2 ei(θ1 +θ2 )
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Conjugation:
z∗1 = r1 e−iθ1
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Absolute value:
|z1 | = r1
Fourier Integral
For periodic functions we have the Fourier series:
g(x) =
∞
X
[ Ak cos(kω0 x) + Bk sin(kω0 x)] ,
k=0
But for nonperiodic functions we need a continuum of
frequencies. So, our Fourier series becomes an integral:
Z
g(x) =
∞
Aω cos(ωx) + Bω sin(ωx) dω
0
Computing Fourier Coefficients
Fourier coefficients describe how much a particular
frequency ω contributes to the function g.
They are computed by just multiplying and integrating
with cos/sin waves:
Z
1 ∞
g(x) · cos(ωx) dx
Aω = A(ω) =
π −∞
Z
1 ∞
Bω = B(ω) =
g(x) · sin(ωx) dx
π −∞
Fourier Transform
Now, let’s put the B coefficient (the sine part) into the
imaginary part of a complex number
r h
i
π
G(ω) =
A(ω) − i · B(ω)
2
Z ∞
h
i
1
√
=
g(x) · cos(ωx) − i · sin(ωx) dx
2π Z−∞
∞
1
=√
g(x) · e−iωx dx
2π −∞