Fourier Transform (Chapter 4)

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Fourier Transform (Chapter 4)

• CS474/674 Prof. Bebis

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Mathematical BackgroundComplex Numbers

• A complex number x is of the form
• a real part, b imaginary
  part
• Addition
• Multiplication

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Mathematical BackgroundComplex Numbers (contd)

• Magnitude-Phase (i.e., vector) representation
• 
  Magnitude
• Phase
  

f
Magnitude-Phase notation
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Mathematical BackgroundComplex Numbers (contd)

• Multiplication using magnitude-phase
  representation
• Complex conjugate
• Properties

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Mathematical BackgroundComplex Numbers (contd)

• Eulers formula
• Properties

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Mathematical BackgroundSine and Cosine Functions

• Periodic functions
• General form of sine and cosine functions

y(t)Asin(atb) y(t)Acos(atb)
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Mathematical BackgroundSine and Cosine Functions
Special case A1, b0, a1
period2p
p
3p/2
p/2
p
3p/2
p/2
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Mathematical BackgroundSine and Cosine
Functions (contd)

• Changing the phase shift b

Note cosine is a shifted sine function
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Mathematical BackgroundSine and Cosine
Functions (contd)

• Changing the amplitude A

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Mathematical BackgroundSine and Cosine
Functions (contd)

• Changing the period T2p/a
• Asssume A1, b0 ycos(at)

a 4
period 2p/4p/2
shorter period higher frequency (i.e.,
oscillates faster)
frequency is defined as f1/T
Alternative notation cos(at) or cos(2pt/T) or
cos(t/T) or cos(2pft) or cos(ft)
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Basis Functions

• Given a vector space of functions, S, then if any
  f(t) ? S can be expressed as
• the set of functions fk(t) are called the
  expansion set of S.
• If the expansion is unique, the set fk(t) is a
  basis.

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Image Transforms

• Many times, image processing tasks are best
  performed in a domain other than the spatial
  domain.
• Key steps
• (1) Transform the image
• (2) Carry the task(s) in the transformed domain.
• (3) Apply inverse transform to return to the
  spatial domain.

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Transformation Kernels
forward transformation kernel

• Forward Transformation
• Inverse Transformation

inverse transformation kernel
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Kernel Properties

• A kernel is said to be separable if
• A kernel is said to be symmetric if

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Fourier Series Theorem

• Any periodic function f(t) can be expressed as a
  weighted sum (infinite) of sine and cosine
  functions of varying frequency

is called the fundamental frequency
 
 
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Fourier Series (contd)
a1
a2
a3
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Continuous Fourier Transform (FT)

• Transforms a signal (i.e., function) from the
  spatial (x) domain to the frequency (u) domain.

where
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Definitions

• F(u) is a complex function
• Magnitude of FT (spectrum)
• Phase of FT
• Magnitude-Phase representation
• Power of f(x) P(u)F(u)2

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Why is FT Useful?

• Easier to remove undesirable frequencies in the
  frequency domain.
• Faster to perform certain operations in the
  frequency domain than in the spatial domain.

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Example Removing undesirable frequencies
frequencies
noisy signal
remove high frequencies
reconstructed signal
To remove certain frequencies, set
their corresponding F(u) coefficients to zero!
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How do frequencies show up in an image?

• Low frequencies correspond to slowly varying
  pixel intensities (e.g., continuous surface).
• High frequencies correspond to quickly varying
  pixel intensities (e.g., edges)

Original Image
Low-passed
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Example of noise reduction using FT
Input image
Spectrum (frequency domain)
Output image
Band-reject filter
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Frequency Filtering Main Steps

• 1. Take the FT of f(x)
• 2. Remove undesired frequencies
• 3. Convert back to a signal

Well talk more about these steps later .....
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Example rectangular pulse
rect(x) function
sinc(x)sin(x)/x
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Example impulse or delta function

• Definition of delta function
• Properties

 
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Example impulse or delta function (contd)

• FT of delta function

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Example spatial/frequency shifts

Special Cases
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Example sine and cosine functions

• FT of the cosine function

cos(2pu0x)
F(u)
1/2
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Example sine and cosine functions (contd)

• FT of the sine function

 
jF(u)
sin(2pu0x)
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Extending FT in 2D

• Forward FT
• Inverse FT

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Example 2D rectangle function

• FT of 2D rectangle function

2D sinc()
top view
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Discrete Fourier Transform (DFT)
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Discrete Fourier Transform (DFT) (contd)

• Forward DFT
• Inverse DFT

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Example
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Extending DFT to 2D

• Assume that f(x,y) is M x N.
• Forward DFT
• Inverse DFT

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Extending DFT to 2D (contd)

• Special case f(x,y) is N x N.
• Forward DFT
• Inverse DFT

u,v 0,1,2, , N-1
x,y 0,1,2, , N-1
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Extending DFT to 2D (contd)
2D cos/sin functions
Interpretation
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Visualizing DFT

• Typically, we visualize F(u,v)
• The dynamic range of F(u,v) is typically very
  large
• Apply streching
  (c is const)

D(u,v)
F(u,v)
original image
before stretching
after stretching
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DFT Properties (1) Separability

• The 2D DFT can be computed using 1D transforms
  only
• Forward DFT

kernel is separable
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DFT Properties (1) Separability (contd)

• Rewrite F(u,v) as follows
• Lets set
• Then

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DFT Properties (1) Separability (contd)

• How can we compute F(x,v)?
• How can we compute F(u,v)?

N x DFT of rows of f(x,y)
DFT of cols of F(x,v)
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DFT Properties (1) Separability (contd)
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DFT Properties (2) Periodicity

• The DFT and its inverse are periodic with period
  N

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DFT Properties (3) Symmetry
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DFT Properties (4) Translation

• Translation in spatial domain

• Translation in frequency domain

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DFT Properties (4) Translation (contd)

• To show a full period, we need to translate the
  origin of the transform at uN/2 (or at (N/2,N/2)
  in 2D)

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DFT Properties (4) Translation (contd)

• To move F(u,v) at (N/2, N/2), take

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DFT Properties (4) Translation (contd)
sinc
sinc
no translation
after translation
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DFT Properties (5) Rotation

• Rotating f(x,y) by ? rotates F(u,v) by ?

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DFT Properties (6) Addition/Multiplication
but
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DFT Properties (7) Scale
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DFT Properties (8) Average value
Average
F(u,v) at u0, v0
So
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Magnitude and Phase of DFT

• What is more important?
• Hint use the inverse DFT to reconstruct the
  input image using only magnitude or phase
  information

magnitude
phase
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Magnitude and Phase of DFT (contd)
Reconstructed image using magnitude only (i.e.,
magnitude determines the strength of each
component)
Reconstructed image using phase only (i.e.,
phase determines the phase of each component)
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Magnitude and Phase of DFT (contd)
only phase
only magnitude
phase (woman) magnitude (rectangle)
phase (rectangle) magnitude (woman)