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ImageJ Bandpass filter

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Joachim Walter
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The Fourier transform of an image produces a representation in frequency space: i.e. separated according to spatial frequency (effectively scale). The 2D amplitude map of the different spatial frequencies is symmetrical, and is commonly displayed with low spatial frequencies (large features) in the centre, highest spatial frequencies (small features) at the edges.

Fourier filtering involves suppressing or enhancing features in the Fourier domain before carrying out an inverse Fourier transform to obtain a filtered real-space image. ImageJ's Process > FFT > Bandpass Filter implements two common Fourier-filtering functions: 1. filtering for specific sizes of feature in an image by selecting minimum and maximum feature sizes (selecting a radial band of frequencies in Fourier space); 2. filtering out repetitive horizontal or vertical stripes by cutting out a zero-frequency stripe in the orthogonal direction in frequency space.

The example image above shows the effect of filtering for 2 feature size ranges: 0-8 pixels, and 8-256 pixels; where the former appears "flattened" or washed-out, and the latter very blurred. The small images displayed to the lower-right of each filtered image correspond to the mask applied to the Fourier transform. Such filtering can be useful prior to global thresholding, for noise suppression, etc.