EE4328, Section 005 Introduction to Digital Image Processing TwoDimensional Discrete Fourier Transfo

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EE4328, Section 005 Introduction to Digital
Image ProcessingTwo-Dimensional Discrete
Fourier TransformZhou WangDept. of Electrical
EngineeringThe Univ. of Texas at ArlingtonFall
2006
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Summary of FT, FT, DTFT/DSFT, DFS, DFT and FFT
Fourier Transform (FT)
(continuous)
(continuous)
Fourier Series (FS)
(discrete)
(continuous, periodic)
Discrete Time/Space Fourier Transform (DTFT/DSFT)
(continuous, periodic)
(discrete)
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Summary of FT, FT, DTFT/DSFT, DFS, DFT and FFT
Discrete Fourier Series (DFS)
(discrete, periodic)
(discrete, periodic)
Discrete Fourier Transform (DFT)
(discrete, finite)
(discrete, finite)
Fast Fourier Transform (FFT) Fast algorithm for
computing DFT
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Two-Dimensional Discrete Fourier Transform
(2D-DFT)
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2D DFT and Inverse DFT (IDFT)
f(x, y)
F(u, v)
M, N image size
often used short notation
x, y image pixel position
u, v spatial frequency
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The Meaning of DFT and Spatial Frequencies

• Important Concept
• Any signal can be represented as a linear
  combination of a set of basic components
• Fourier components sinusoidal patterns
• Fourier coefficients weighting factors assigned
  to the Fourier components
• Spatial frequency The frequency of Fourier
  component
• Not to confused with electromagnetic frequencies
  (e.g., the frequencies associated with light
  colors)

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Real Part, Imaginary Part, Magnitude, Phase,
Spectrum
Real part
Imaginary part
Magnitude-phase representation
Magnitude (spectrum)
Phase (spectrum)
Power Spectrum
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2D DFT Properties
Mean of image/ DC component
Highest frequency component
Half-shifted Image
Conjugate Symmetry
Magnitude Symmetry
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2D DFT Properties
Spatial domain differentiation
Frequency domain differentiation
Distribution law
Laplacian
Spatial domain Periodicity
Frequency domain periodicity
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Computation of 2D-DFT
Fourier transform matrices remember
relationship
In particular, for N 4
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Computation of 2D-DFT

• To compute the 1D-DFT of a 1D signal x (as a
  vector)

To compute the inverse 1D-DFT

• To compute the 2D-DFT of an image X (as a
  matrix)

To compute the inverse 2D-DFT
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Computation of 2D-DFT Example

• A 4x4 image

• Compute its 2D-DFT

MATLAB function fft2
lowest frequency component
highest frequency component
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Computation of 2D-DFT Example
Real part
Imaginary part
Magnitude
Phase
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Computation of 2D-DFT Example

• Compute the inverse 2D-DFT

MATLAB function ifft2
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Centered Representation
MATLAB function fftshift
From Prof. Al Bovik
Example
From Gonzalez Woods
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Log-Magnitude Visualization
2D-DFT
centered
From Gonzalez Woods
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Apply to Images
2D-DFT ? centered ? log intensity transformation
From Gonzalez Woods
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2D-DFT (Frequency) Domain Filtering
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Convolution Theorem
f (x,y)
g(x,y)
h(x,y)
input image
impulse response (filter)
output image
DFT
IDFT
DFT
IDFT
DFT
IDFT
G(u,v)
H(u,v)
F(u,v)

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Frequency Domain Filtering
Filter design design H(u,v)
From Gonzalez Woods
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2D-DFT Domain Filter Design

• Ideal lowpass, bandpass and highpass

From Prof. Al Bovik
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2D-DFT Domain Filter Design

• Ideal lowpass, bandpass and highpass

From Gonzalez Woods
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2D-DFT Domain Filter Design
Ideal lowpass filtering with cutoff frequencies
set at radii values of 5, 15, 30, 80, and 230,
respectively
From Gonzalez Woods
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2D-DFT Domain Filter Design

• Gaussian lowpass

From Gonzalez Woods
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2D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
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2D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
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2D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
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2D-DFT Domain Filter Design
Gaussian lowpass filtering
Gaussian highpass filtering
From Gonzalez Woods
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2D-DFT Domain Filter Design

• Choices of highpass filters

Butterworth
Gaussian
Ideal
From Gonzalez Woods
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2D-DFT Domain Filter Design
Ideal
Butterworth
Gaussian
Obtained by applying inverse 2D-DFT to the
corresponding frequency domain filters
From Gonzalez Woods
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2D-DFT Domain Filter Design
Ideal
Butterworth
Gaussian
From Gonzalez Woods
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2D-DFT Domain Filter Design
Gaussian filter with different width
From Gonzalez Woods
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2D-DFT Domain Filter Design

• Orientation selective filters

From Prof. Al Bovik
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2D-DFT Domain Filter Design

• Narrowband Filtering

by combining radial and orientation selection

From Prof. Al Bovik