Filtering

The operation of filtering consists in selecting some frequencies in the images. Filtering an image is performed through convolution of the image with a PSF that represents the filter. Similarly, it can be achieved by multiplying the DFT of the image and the PSF. In the example below, the first row in the images is in the spatial domain: the filtered image is the convolution of the original image and the kernel (PSF). The second row shows the same information in the Fourier domain, thus the DFT of the filtered image is the multiplication (point to point) of the DFT of the original image and the kernel.

Especially, we distinguish low-pass filtering which preserves only the low frequencies (Fig. 34) and high-pass filtering which preserves only the high frequencies (Fig. 35).

Fig. 34 Example of low-pass filtering: only the low frequencies are kept. Top row: spatial domain, bottom row: amplitude of the DFT (the phase is not shown here).

Fig. 35 Example of high-pass filtering: only the high frequencies (i.e. the sudden changes in the intensities) are kept.