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 3.4). 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 consists of updating only the part of the computation that changes from one location to the next.
(a) Formulate such an algorithm for an n × n 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 implementation to the number of computations performed by the box-filter algorithm is called the computational advantage. Obtain the computational advantage in this case and plot it as a function of n for n >
1. The 1/n2 scaling factor is common to both approaches, so you need not consider it in obtaining 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.
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