Two dimensional convolution follows along the lines of one dimensional convolution. The convolution of two functions is given by
In the Fourier transformed domain, we have
In light of this it is easy to see that convolution with the Gaussian function is the equivalent of low pass filtering. In fact, with the convolution kernel
Since
to see that the larger the value of
the greater the low pass characteristic of the filter.
In what follows, we will show the filtering of images with progressively more low pass filters and a band pass filter. First Figure 1 is the frequency domain depiction of an ideal low pass filter
Figure 1: Showing a low pass filter
Figure 2 is the photograph of Aaron, a former denizen of our robotics laboratory, Figure 3 is a low pass filtered version of him and Figure 4 is even more low pass filtered.
Figure 2: Image to be low pass filtered
Figure 3: Low pass filtered version of Aaron
Figure 4: Even narrower low pass filtered version of Aaron
The matlab code for this is available at
http://robotics.eecs.berkeley.edu/ mcenk/Page1.html