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IMAGE PROCESSING IN FREQUENCY SPACE 19.4.2015Erkki Rämö1.

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Presentation on theme: "IMAGE PROCESSING IN FREQUENCY SPACE 19.4.2015Erkki Rämö1."— Presentation transcript:

1 IMAGE PROCESSING IN FREQUENCY SPACE Erkki Rämö1

2 Lauri Toivio2

3 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

4 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

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

6 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

7 Visualization of frequency image 7 Original Magnitude component Logarithmic scaling

8 Directional dependency of frequency image

9 Lauri Toivio 9

10 Directional dependency – application  Straightening of scanned text Threshold FFT

11 Some hardcore mathematics

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

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

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

15 2D Fourier-transform = =

16 DFT - 2D

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

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

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

20 Lauri Toivio 20

21 Lauri Toivio 21

22 Threshold pixel wide vertical lines FFT

23 Frequency filtering  Chosen frequencies are masked off of frequency image

24 FFT-filtering Low-pass filtering High-pass filtering

25 Lauri Toivio 25

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27 Lauri Toivio27

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30 Image restoration by Photoshop Lauri Toivio 30

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

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

33 More information: 


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