### The Fourier Transform

```Basis beeldverwerking (8D040)
dr. Andrea Fuster
Prof.dr. Bart ter Haar Romeny
dr. Anna Vilanova
Prof.dr.ir. Marcel Breeuwer
The Fourier Transform II
Contents
• Fourier Transform of sine and cosine
• 2D Fourier Transform
• Properties of the Discrete Fourier Transform
2
Euler’s formula
3
Cosine
Recall
4
Sine
5
Contents
• Fourier Transform of sine and cosine
• 2D Fourier Transform
• Properties of the Discrete Fourier Transform
6
Discrete Fourier Transform
• Forward
• Inverse
7
Formulation in 2D spatial coordinates
• Discrete Fourier Transform (2D)
• Inverse Discrete Transform (2D)
f(x,y) digital image of size M x N
8
Spatial and Frequency intervals
• Inverse proportionality
• Suppose function is sampled M times in x,
with step
, distance
is covered,
which is related to the lowest frequency that
can be measured
• And similarly for y and frequency v
9
Examples
10
Examples
11
Periodicity
• 2D Fourier Transform is periodic in both directions
12
Periodicity
• 2D Inverse Fourier Transform is periodic in both
directions
13
Contents
• Fourier Transform of sine and cosine
• 2D Fourier Transform
• Properties of the Discrete Fourier Transform
14
Properties of the 2D DFT
15
Real
SinSin
(x +(x)
π/2)
Real
Imaginary
16
• Note: translation has no effect on the
magnitude of F(u,v)
17
Symmetry: even and odd
• Any real or complex function w(x,y) can be
expressed as the sum of an even and an odd part
(either real or complex)
18
Properties
• Even function (symmetric)
• Odd function (antisymmetric)
19
Properties - 2
20
FT of even and odd functions
• FT of even function is real
• FT of odd function is imaginary
21
Even
Real
Imaginary
Cos (x)
22
Odd
Real
Imaginary
Sin (x)
23
Even
Real
Imaginary
F(Cos(x+k))
F(Cos(x))
24
Odd
Real
Imaginary
Sin (x)Sin(y)
(x)
25
Consequences for the Fourier Transform
• FT of real function is conjugate symmetric
• FT of imaginary function is conjugate antisymmetric
26
Scaling property
• Scaling t with a
27
• a
Imaginary parts
28
Differentiation and Fourier
• Let
be a signal with Fourier transform
• Differentiating both sides of inverse Fourier
transform equation gives:
29
Examples – horizontal derivative
30
Examples – vertical derivative
31
Examples – hor and vert derivative
32
Thanks and see you next Wednesday!☺
33
```