The two-dimensional discrete Fourier transform and applications: image filtering in the frequency domain

Frequency domain filtering modifies the brightness values of the pixels in the image depending on their periodic patterns and on the spatial distributions of brightness variations in the image. The high pass filters tend to preserve the sharp brightness variation patterns and to remove the smooth variations in the image (along with the DC component, i.e. with the average brightness which is set to zero). The low pass filters, on the opposite, preserve the smooth brightness variations in the image, and remove the sharp edges (sharp brightness variations) in the image. To apply a frequency domain filtering procedure, the most typical approach is to first apply a Fourier transform to the image, to obtain its frequency domain representation; the frequency domain representation of the image through its Fourier transform provides information about the spectral components of the image. The frequency domain filtering process simply modifies certain spectral components in the image and afterwards, by applying the inverse Fourier transform on the frequency domain filtered image represented in the frequency domain, provides the filtered image representation in the spatial domain.