IMAGE PROCESSING IN FREQUENCY SPACE

18.04.23 Erkki Rämö 1

18.04.23 Lauri Toivio 2

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

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

Directional dependency of frequency image

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Directional dependency – applicationStraightening of scanned text

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ThresholdFFT

FFT

Some hardcore mathematics

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Fourier-transform

Fourier –transform in one dimension:

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Fourier –counter transform:

2D Fourier-transform

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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

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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