Title: EE4328, Section 005 Introduction to Digital Image Processing TwoDimensional Discrete Fourier Transfo
1EE4328, Section 005 Introduction to Digital
Image ProcessingTwo-Dimensional Discrete
Fourier TransformZhou WangDept. of Electrical
EngineeringThe Univ. of Texas at ArlingtonFall
2006
2Summary 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)
3Summary 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
4Two-Dimensional Discrete Fourier Transform
(2D-DFT)
52D 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
6The 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)
7Real Part, Imaginary Part, Magnitude, Phase,
Spectrum
Real part
Imaginary part
Magnitude-phase representation
Magnitude (spectrum)
Phase (spectrum)
Power Spectrum
82D DFT Properties
Mean of image/ DC component
Highest frequency component
Half-shifted Image
Conjugate Symmetry
Magnitude Symmetry
92D DFT Properties
Spatial domain differentiation
Frequency domain differentiation
Distribution law
Laplacian
Spatial domain Periodicity
Frequency domain periodicity
10Computation of 2D-DFT
Fourier transform matrices remember
relationship
In particular, for N 4
11Computation 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
12Computation of 2D-DFT Example
MATLAB function fft2
lowest frequency component
highest frequency component
13Computation of 2D-DFT Example
Real part
Imaginary part
Magnitude
Phase
14Computation of 2D-DFT Example
- Compute the inverse 2D-DFT
MATLAB function ifft2
15Centered Representation
MATLAB function fftshift
From Prof. Al Bovik
Example
From Gonzalez Woods
16Log-Magnitude Visualization
2D-DFT
centered
From Gonzalez Woods
17Apply to Images
2D-DFT ? centered ? log intensity transformation
From Gonzalez Woods
182D-DFT (Frequency) Domain Filtering
19Convolution 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)
20Frequency Domain Filtering
Filter design design H(u,v)
From Gonzalez Woods
212D-DFT Domain Filter Design
- Ideal lowpass, bandpass and highpass
From Prof. Al Bovik
222D-DFT Domain Filter Design
- Ideal lowpass, bandpass and highpass
From Gonzalez Woods
232D-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
242D-DFT Domain Filter Design
From Gonzalez Woods
252D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
262D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
272D-DFT Domain Filter Design
Effect of Gaussian lowpass filter
From Gonzalez Woods
282D-DFT Domain Filter Design
Gaussian lowpass filtering
Gaussian highpass filtering
From Gonzalez Woods
292D-DFT Domain Filter Design
- Choices of highpass filters
Butterworth
Gaussian
Ideal
From Gonzalez Woods
302D-DFT Domain Filter Design
Ideal
Butterworth
Gaussian
Obtained by applying inverse 2D-DFT to the
corresponding frequency domain filters
From Gonzalez Woods
312D-DFT Domain Filter Design
Ideal
Butterworth
Gaussian
From Gonzalez Woods
322D-DFT Domain Filter Design
Gaussian filter with different width
From Gonzalez Woods
332D-DFT Domain Filter Design
- Orientation selective filters
From Prof. Al Bovik
342D-DFT Domain Filter Design
by combining radial and orientation selection
From Prof. Al Bovik