Do J

where #LowPass(w, v) is the value of the filter at location (u, v) in the spatial frequency plane, d0 is the cutoff frequency, and y is the order of the filter, taking positive values of l, 2, 3, etc. The low-pass Butterworth filter transmits 100% at the center (low spatial frequencies), 50% at distance d0 from the center (middle spatial frequencies), and 0% at the outer edge (the highest spatial frequencies in the image). The rapidity of the transition from 100% to 0% is controlled by the order, y. The higher the filter order, the more abrupt is the filter cutoff.

To use the filter, each element in the Fourier transform array is multiplied by the corresponding value in the filter:

where S\u, v) is the modified Fourier transform and B(u, v) is the filter.

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