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


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 { "@context": "", "@type": "ImageObject", "contentUrl": "", "name": "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


27 Lauri Toivio27



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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