Frequency Domain Processing Lecture: 3. In image processing, linear systems are at the heart of many...
36
Frequency Domain Processing Lecture: 3
-
Upload
ralph-booker -
Category
Documents
-
view
218 -
download
0
Transcript of Frequency Domain Processing Lecture: 3. In image processing, linear systems are at the heart of many...
Frequency Domain Processing
Lecture: 3
In image processing, linear systems are at the heart of many filtering operations, and they provide the basis for analyzing complex problems in areas such as image restoration.
In this section we give a short review of linear systems
0. Overview of Linear Systems
0.1 Definitions
0.1 Definitions
0.1 Definitions
0.1 Definitions
0.1 Definitions
0.2 Convolutions
0.2 Convolutions
0.2 Convolutions
0.2 Convolutions
0.2 Convolutions
Frequency Domain Processing
1 The 2D Discrete Fourier Transform
Let f(x,y) for x=0,1,2, ..., M-1 and y=1,2, ..., N-1 denote an M×N image. The 2D DFT of f is given by
The frequency domain is simply the coordinate system spanned by F(u,v) with u and v as frequency variables.
1...,,2,1,0 and 1...,,2,1,0for
),(),(1
0
1
0
)//(2
NvMu
eyxfvuFM
x
N
y
NvyMuxj
1 The 2D Discrete Fourier Transform
The inverse DFT is given by
Thus, given F(u,v), we can obtain f(x,y) back by means of the inverse DFT.
1...,,2,1,0 and 1...,,2,1,0for
),(1
),(1
0
1
0
)//(2
NyMx
evuFMN
yxfM
u
N
v
NvyMuxj
1 The 2D Discrete Fourier Transform
The value of the transform at the origin of the frequency domain that is F(0,0) is called the dc component of the Fourier transform.
Even if f(x,y) is real, its transform is complex. The principal method of visually analyzing a transform is to
compute its spectrum that is the magnitude of F(u,v).
1 The 2D Discrete Fourier Transform
The Fourier spectrum is defined as
The phase angle is defined as
2/122 )],(),([),( vuIvuRvuF
]),(
),([tan),( 1
vuR
vuIvu
1 The 2D Discrete Fourier Transform
The polar representation of F(u,v) is defined by
The power spectrum is defined as
),(),(),( vujevuFvuF
),(),(
),(),(22
2
vuIvuR
vuFvuP
1 The 2D Discrete Fourier Transform
It can be shown that
The DFT is infinitely periodic in both u and v directions, with
the periodicity determined by M and N.
),(),(),(),( NvMuFNvuFvMuFvuF
1 The 2D Discrete Fourier Transform
Periodicity is also a property of the inverse DFT.
An image obtained by taking the inverse DFT is also infinitely
periodic in both u and v directions, with the periodicity
determined by M and N.
),(),(),(),( NyMxfNyxfyMxfyxf
1 The 2D Discrete FourierTransform
1 The 2D Discrete FourierTransform
2 Computing and Vizualizing the 2D DFT
The FFT of an M × N image array is obtained by the syntax
F=fft2(f)
and with padding by the syntax
F=fft2(f, P,Q)
The Fourier spectrum is obtained as
S=abs(F)
2 Computing and Vizualizing the 2D DFT
The funcion F=fft2 (f) moves the origin of transform to the center of the frequency rectangle.
Fc=fftshift (F)
Log transformation:
S2=log(1+abs(Fc))
Function ifftshift reverses the centering:
F=ifftshift(Fc)
2 Computing and Vizualizing the 2D DFT
To compute inverse FFT
f=ifft2 (F)
If the imput used to compute F is real then the inverse should also be real. However fft2 often has small imaginary components resulting from round-off errors. It is good practice to extract the real part of the result:
f=real(ifft2(F))
2 Computing and Vizualizing the 2D DFT
3 Filtering in the frequency domain
The convolution theorem:
Linear spatial convolution is by convolving f(x,y) and h(x,y). The same result is obtained in the frequency domain by multiplying F(u,v) and H(u,v).
The basic idea in frequency domain is to select a filter transfer function that modifies F(u,v) in a specified manner. A transfer function is multiplied by a centered F(u,v).
A filter is called low pass filter if it attenuates the high frequency components to F(u,v) while leaving the low frequencies relatively unchanged.
),(),(),(),( vuFvuHyxhyxf
3 Filtering in the frequency domain
3 Filtering in the frequency domain
Example: Linear spatial filtering.
f =
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
w = 1 2 3 4 5 6 7 8 9
g=imfilter(f, w, filtering_mode, boundary_options, size_options)
3 Filtering in the frequency domain
g=imfilter(f, w, filtering_mode, boundary_options, size_options)
>> g=imfilter(f,w,'corr',0,'full')
g =
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 8 7 0 0 0 0 6 5 4 0 0 0 0 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 Filtering in the frequency domain
With specified values of P and Q we use the following syntax to compute the FFT using zero padding.
>> F=fft2(f, 7,7)
F = Columns 1 through 2
1.00000000000000 -0.22252093395631 - 0.97492791218182i -0.22252093395631 - 0.97492791218182i -0.90096886790242 + 0.43388373911756i -0.90096886790242 + 0.43388373911756i 0.62348980185873 + 0.78183148246803i 0.62348980185873 + 0.78183148246803i 0.62348980185873 - 0.78183148246803i 0.62348980185873 - 0.78183148246803i -0.90096886790242 - 0.43388373911756i -0.90096886790242 - 0.43388373911756i -0.22252093395631 + 0.97492791218182i -0.22252093395631 + 0.97492791218182i 1.00000000000000 . . .
3.2 Basic steps in DFT filtering
1. Obtain the padding parameters P and Q.2. Obtain the Fourier transform with padding: F=fft2(f, P,Q);
3. Generate a filter function, H of size P×Q. If the filter is centered then use : H=fftshift(H) before using the filter.
4. Multiply the transform by the filter: G=H.*F;
5. Obtain the real part of G: g=real(ifft2(G));
6. Crop the top,left rectangle to the original size: g=g(1:size(f,1), 1:size(f,2));
4. Lowpass frequency domain filters
Ideal lowpass filter (ILPF):
where D(u,v) is the distance from point (u,v) to the center of the filter.
0
0
),( if0
),( if1),(
DvuD
DvuDvuH
4. Lowpass frequency domain filters
Butterworth lowpass filter (ILPF):
where D(u,v) is the distance from point (u,v) to the center of the filter.
nDvuDvuH
2
0 ]/),([1
1),(
4. Lowpass frequency domain filters
Gaussian lowpass filter (ILPF):
where D(u,v) is the distance from point (u,v) to the center of the filter.
)2/(),(exp{),( 22 vuDvuH
1 The 2D Discrete FourierTransform
