image processing in frequency space 19.4.2015erkki rämö1
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IMAGE PROCESSING IN FREQUENCY SPACE
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Images frequency domain2D 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
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Images frequency domainComponents of frequency image are complex numbersConsists of magnitude and phase components
Frequency image is visualized by showing its magnitude components
Calculated from spatial images first by rows then by columns
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Example of frequency images
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Low frequencies are near origin
Frequency is symmetrical in relation to the coordinate axis
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Numeral scope of frequency imageComplex number consists of magnitude and phase
componentsMagnitude components differencies of samples are so big
that a logarithmic scaling is needed to visualize the frequency image
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Visualization of frequency image
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Original
Magnitude component
Logarithmic scaling
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Directional dependency of frequency image
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Directional dependency – applicationStraightening of scanned text
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ThresholdFFT
FFT
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Some hardcore mathematics
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Fourier-transform
Fourier –transform in one dimension:
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Fourier –counter transform:
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2D Fourier-transform
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=
=
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DFT - 2D
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Fast Fourier Transform - FFT
Speed up calculation by decreasing values to be calculated
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where
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Single-frequency images frequency domainIn image, only one vertical frequencyShows as a dot in frequency image
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Frequency filteringChosen frequencies are masked off of frequency image
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FFT-filtering
Low-passfiltering
High-passfiltering
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Image restoration by Photoshop
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More information:http://www.dai.ed.ac.uk/HIPR2/fourier.htm
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