IMAGE PROCESSING IN FREQUENCY SPACE Erkki Rämö1

Lauri Toivio2

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

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

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

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

Visualization of frequency image 7 Original Magnitude component Logarithmic scaling

Directional dependency of frequency image

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Directional dependency – application  Straightening of scanned text Threshold FFT

Some hardcore mathematics

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

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

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

2D Fourier-transform = =

DFT - 2D

Euler formula Lauri Toivio17  Example: for (i=0;i<m;i++) { x2[i] = 0; y2[i] = 0; arg = - dir * 2.0 * * (double)i / (double)m; for (k=0;k<m;k++) { cosarg = cos(k * arg); sinarg = sin(k * arg); x2[i] += (x1[k] * cosarg - y1[k] * sinarg); y2[i] += (x1[k] * sinarg + y1[k] * cosarg); }

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

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

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Threshold pixel wide vertical lines FFT

Frequency filtering  Chosen frequencies are masked off of frequency image

FFT-filtering Low-pass filtering High-pass filtering

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Image restoration by Photoshop Lauri Toivio 30

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

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)

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