, where 
and n are constants. Graph of this
filter is here.
filtering function, and multiplying the other quarters by the corresponding
symmetries of this filter with respect to the center of the image. Butterworth
function is :
, where
and n are constants. Graph of this
filter is here.
Similar procedure is followed for sharpening, but with a different
filtering function. The modified Butterworth filtering function used for
sharpening is
The coefficient 
is taken from the textbook.
In the implementation part, two extra command line arguments are added (optional) for changing D0 and n values in the filtering functions in order to obtain better results for
different kinds of images. After the standard --sharpen or --blur argument, one or two numbers first of which is the new D0 value and second is the new n value can be entered.
While designing the blurring filter, we tried to decrease the magnitudes of high frequency information in the image while preserving the low frequency information. A
smooth pass is obtained by using Butterworth function.
Sharpening filter must amplify the high frequency information while keeping the low frequency information, so we added 1 to Butterworth highpass filter.