$3\mathrm{.}15$ The implementation of linear spatial filters requires moving the center of a

mask throughout an image and, at each location, computing the sum of products

of the mask coefficients with the corresponding pixels at that location $($see

Section $3\mathrm{.}4).$ A lowpass filter can be implemented by setting all coefficients to $1,$

allowing use of a so$-$called box$-$filter or moving$-$average algorithm, which con$-$

sists of updating only the part of the computation that changes from one loca$-$

tion to the next.

$***($a$)$ Formulate such an algorithm for an $nn$ filter, showing the nature of the

computations involved and the scanning sequence used for moving the

mask around the image.

$($b$)$ The ratio of the number of computations performed by a brute$-$force imple$-$

mentation to the number of computations performed by the box$-$filter algo$-$

rithm is called the computational advantage. Obtain the computational

advantage in this case and plot it as a function of $n$ for $n>1.$ The $\frac{1}{n^2}$ scal$-$

ing factor is common to both approaches, so you need not consider it in ob$-$

taining the computational advantage. Assume that the image has an outer

border of zeros that is wide enough to allow you to ignore border effects in

your analysis.