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IMAGE PROCESSING IN FREQUENCY SPACE 19.4.2015Erkki Rämö1

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19.4.2015Lauri Toivio2

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Images frequency domain 2D spatial domain image can be altered into frequency domain by applying Fourier transformation Frequency image has the same dimensions as the original, but the components are complex numbers Frequency image is a map of image frequencies in the spatial image 19.4.20153

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Images frequency domain Components of frequency image are complex numbers Consists of magnitude and phase components Frequency image is visualized by showing its magnitude components Calculated from spatial images first by rows then by columns 19.4.20154

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Example of frequency images 5 Low frequencies are near origin Frequency is symmetrical in relation to the coordinate axis

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Numeral scope of frequency image Complex number consists of magnitude and phase components Magnitude components differencies of samples are so big that a logarithmic scaling is needed to visualize the frequency image 19.4.20156

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Visualization of frequency image 7 Original Magnitude component Logarithmic scaling

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Directional dependency of frequency image 19.4.20158

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Lauri Toivio 9

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Directional dependency – application Straightening of scanned text 19.4.201510 Threshold FFT

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Some hardcore mathematics 19.4.201511

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Fourier-transform Fourier –transform in one dimension: 19.4.201512 Fourier –counter transform:

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Fourier-transform If using angular frequen instead of oscillation frequency, the formulas are: 19.4.201513

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Discrete Fourier trasform X(k) and its counter transform x(n): 19.4.201514

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2D Fourier-transform 19.4.201515 = =

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DFT - 2D 19.4.201516

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Euler formula Lauri Toivio17 Example: for (i=0;i

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Fast Fourier Transform - FFT Speed up calculation by decreasing values to be calculated 19.4.201518 where

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Single-frequency images frequency domain In image, only one vertical frequency Shows as a dot in frequency image 19.4.201519

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19.4.2015Lauri Toivio 20

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19.4.2015Lauri Toivio 21

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Threshold 200 2 pixel wide vertical lines FFT

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Frequency filtering Chosen frequencies are masked off of frequency image 19.4.201523

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FFT-filtering Low-pass filtering High-pass filtering

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19.4.2015Lauri Toivio 25

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19.4.2015Lauri Toivio27

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Image restoration by Photoshop 19.4.2015Lauri Toivio 30

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Group discussion Discuss application areas for frequency based image processing 19.4.2015Lauri Toivio31

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Fourier-transform in Matlab >> load trees >> I=ind2gray(X,map); >> FI=fft2(I); >> SFI=fftshift(FI); >> abs(SFI); >> max(max(abs(SFI))) ans = 3.7987e+004 >> m=3.7987e+004 >> imshow(abs(SFI)/m,64) 19.4.201532

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More information: http://www.dai.ed.ac.uk/HIPR2/fourier.htm http://www.dai.ed.ac.uk/HIPR2/fourier.htm 19.4.201533

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