Convolution and its Fourier Transform



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Convolution and its Fourier Transform

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



S Sastry
Sun May 4 22:28:26 PDT 1997