EE-583: Digital Image Processing Prepared By: Dr. Hasan Demirel, PhD Image Enhancement in the Frequency Domain Fourier Transform •Fourier Series: Any function that periodically repeats itself can be expressed as the sum of sines/cosines of different frequencies, each multiplied with a different coefficient. •Fourier Transform: Functions that are not periodic, whose area under the curve is finite, can be expressed as the integral of sines and/cosines multiplied by a weighting function.
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EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
Fourier Transform
•Fourier Series: Any function that
periodically repeats itself can be expressed
as the sum of sines/cosines of different
frequencies, each multiplied with a different
coefficient.
•Fourier Transform: Functions that are not
periodic, whose area under the curve is
finite, can be expressed as the integral of
sines and/cosines multiplied by a weighting
function.
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Continuous Fourier Transform
•The Fourier Transform is an important tool in Image Processing, and is
directly related to filter theory, since a filter, which is a convolution in the
spatial domain, is a simple multiplication in the frequency domain.
•1-D Continuous Fourier Transform
The Fourier transform, F(u), of a single variable continuous function, f(x), is defined by:
2( ) ( ) j uxF u f x e dx
Given Fourier transform of a function F(u), the inverse Fourier transform can be used to
obtain, f(x), by:
2( ) ( ) j uxf x F u e du
Note: F(u), is the frequency spectrum, where, u represents the frequency, and f(x) is the
signal where x represents time/space. 1j
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Continuous Fourier Transform
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 20 40 60 80 100 1200
50
100
150
200
250
300
Sine wave
Delta function
Time/Space domain
Frequency domain
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Continuous Fourier Transform
Gaussian
Gaussian
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 2500
1
2
3
4
5
6
Time/Space domain
Frequency domain
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
-100 -50 0 50 1000
1
2
3
4
5
6
Image Enhancement in the Frequency Domain
1D Continuous Fourier Transform
Sinc function
Square wave Frequency domain
Time/Space domain
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
Fourier Transform pairs (spatial versus Frequency)
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Discrete Fourier Transform
•1-D Discrete Fourier Transform
The Fourier transform, F(u), of a discrete function of one variable, f(x), x=0, 1, 2, …, M-1,
is given by: 12 /
0
1( ) ( )
Mj ux M
x
F u f x eM
Note: F(u), which is the Fourier transform of f(x) contains discrete complex quantities and
it has the same number of components as f(x).
12 /
0
( ) ( )M
j ux M
u
f x F u e
cos sinje j
The inverse Discrete Fourier Transform can be used to calculate f(x), by:
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Discrete Fourier Transform
•1-D Discrete Fourier Transform
Then; 1
0
1( ) ( )[cos2 / sin 2 / ]
M
x
F u f x ux M j ux MM
1/ 22 2( ) ( ) ( )F u R u I u
•The Fourier Transform generates complex quantities. The magnitude or the spectrum of
the Fourier transform is given by:
•The Phase Spectrum of the transform refers to the angles between the real and imaginary
components and it is denoted by:
R(u) is the Real Part and
I(u) is the Imaginary Part
1 ( )( ) tan
( )
I uu
R u
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Discrete Fourier Transform
•1-D Discrete Fourier Transform
•The Power Spectrum/spectral density is defined as the square of the Fourier spectrum and
denoted by
2 2 2( ) ( ) ( ) ( )P u F u R u I u
•The power spectrum can be used, for example to separate a portion of a specified
frequency (i.e. low frequency) power from the power spectrum and monitor the effect.
Typically used to define the cut off frequencies used in lowpass and highpass filtering.
• We primarily use the Fourier Spectrum for image enhancement applications.
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
1D Discrete Fourier Transform
Double the area under the
curve in spatial domain
Double the area under the
curve in the frequency domain
Double the amplitude in the
frequency domain
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
2D Discrete Fourier Transform
The Fourier transform of a 2D discrete function (image) f(x,y) of size MxN is given by:
1 12 ( / / )
0 0
1( , ) ( , )
M Nj ux M vy N
x y
F u v f x y eMN
u=0, 1, 2, …, M-1, and v=0, 1, 2, …, N-1 and the inverse 2D Discrete Fourier Transform
can be calculated by:
x=0, 1, 2, …, M-1, and y=0, 1, 2, …, N-1.
1
0
1
0
)//(2),(),(
M
u
N
v
NvyMuxjevuFyxf
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
2D Discrete Fourier Transform
The 2D Fourier Spectrum, Phase Spectrum and Power Spectrum can be respectively
denoted by:
2/122 ),(),(),( vuIvuRvuF
),(
),(tan),( 1
vuR
vuIvu
),(),(),(),( 222vuIvuRvuFvuP
Magnitude/Fourier
Spectrum
Phase Spectrum
Power Spectrum
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
2D Discrete Fourier Transform
The Periodicity property: F(u) in 1D DFT has a period of M
The Symmetry property: The magnitude of the transform is centered on the origin.
The F(u) not centered
The centered F(u)
EE-583: Digital Image Processing
Prepared By: Dr. Hasan Demirel, PhD
Image Enhancement in the Frequency Domain
2D Discrete Fourier Transform
The Periodicity property: F(u,v) in 2D DFT has a period of N in horizontal and M in
vertical directions
The Symmetry property: The magnitude of the transform is centered on the origin.