Image processing in the frequency domain
• The use of image transformation to the frequency domain
– The image filter design in the frequency domain and the
implementation of fast methods for image filtering
– Emphasize some features in the image invisible in the spatial
domain (e.g. areas that appear periodically as well as periodical
disturbances or noise)
– Pattern detection in the frequency domain
– Obtain more compact (memory efficient) methods of storing images
(used in image compression techniques, e.g. in JPEG, MJPEG,
MPEG, etc.)
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Image processing in the frequency domain
• The Fourier transform (FT) of continuous function from the time domain
to the frequency
– FT
– Inverse FT
Euler’s formula:
• The Fourier transform (DFT) of discrete function
– DFT
– Inverse DFT
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Image processing in the frequency domain
• The two-dimensional discrete Fourier transform (2-D DFT)
– 2-D DFT
• Magnitude spectrum of Fourier transform
• Phase spectrum of Fourier transform
– Inverse 2-D DFT
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Image processing in the frequency domain
• Magnitude spectrum of the image
F=fft2(double(i));
F1=log(abs(F)+1);
F1=F1/max(max(F1));
imshow(F1)
figure;imshow(F1); colormap(jet); colorbar
v
u
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Image processing in the frequency domain
F1=fftshift(log(abs(F)+1));
mesh(F1);
v
u
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Image processing in the frequency domain
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Image processing in the frequency domain
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Image processing in the frequency domain
• The properties of the Fourier transform
– Symmetry
For real function Fourier transform is conjugate symmetric
* - indicates conjugate operation
on a complex number
Symmetry of magnitude spectrum
– Periodicity
For the transform
with size
is true that
It is assumed that the image is a periodic function with period
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Image processing in the frequency domain
– Separability
2-D Fourier transform can be expressed in the separable form and
calculated using 1-D transformations. First, 1-D DFT can be calculated along
the rows and then for the result, along the columns of the image.
Transformation
of rows
Transformation
of columns
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Image processing in the frequency domain
Transformation
of rows
Transformation
of columns
Transformation
of columns
Transformation
of rows
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Image processing in the frequency domain
– Translation in frequency and spatial domain
For
Similarly
fftshift
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Image processing in the frequency domain
– The average value of the image (DC component of the spectrum)
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Image processing in the frequency domain
– Rotation in the spatial and frequency domain of angle
For the polar coordinates
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Image processing in the frequency domain
• The Fourier transform of the product and convolution of 2-D function
– The second relationship is used in the filtering of images in the frequency
domain
• Filtering in the frequency domain
1. Compute the Fourier transform of the image
spectrum)
2. Multiply F-image
by a filter function
(centering of the
(array multiplication)
3. Compute the inverse transform of
(decentering of the spectrum and calculation of real part of the result)
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Image processing in the frequency domain
• Ideal filters
– Lowpass filters
is the distance form point
to the center of F-image
- cutoff frequency
v
u
v
u
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Image processing in the frequency domain
Image of ideal lowpass filter in frequency domain and corresponding spatial filter
x
v
y
u
x
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Image processing in the frequency domain
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Image processing in the frequency domain
– Ideal highpass filters
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Image processing in the frequency domain
– Filtering of periodic disturbances
256x256
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Image processing in the frequency domain
– Gaussian filter in the frequency domain
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Image processing in the frequency domain
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Image processing in the frequency domain
– Butterworth filters
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Image processing in the frequency domain
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Image processing in the frequency domain
• Pattern recognition in frequency domain using correlation
C = real(ifft2(fft2(bw).* fft2(rot90(a,2),256,256)));
figure, imshow(C,[])
mx=max(C(:));
thresh = mx-10;
figure, imshow(C > thresh)
Correlation
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Cosine transform in image compression
• Cosine transform (DCT)
– DCT
– Inverse DCT
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Cosine transform in image compression
figure, imshow(i)
c=dct2(i);
imshow(log(abs(c)),[]), colormap(jet), colorbar
c(abs(c)< 0.05) = 0;
j = idct2(c);
The number of image pixels: 65536
figure, imshow(j)
The number of coefficients reset to zero: 51079
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Cosine transform in image compression
– Basis functions for DCT
– DCT matrix
• In Matlab dctmtx
• The matrix allows to determine the one-dimensional DCT in image
columns
• The two-dimensional DCT for image
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Cosine transform in image compression
– Cosine transform in JPEG compression
Zigzag pattern is used in the
last step (entropy encoder)
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