10.13 Suppose that an image f(x, y) is convolved with a mask of size nxn (with...

 

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10.13 Suppose that an image f(x, y) is convolved with a mask of size nxn (with co- efficients 1/n) to produce a smoothed image F(x, y). *(a) Derive an expression for edge strength (edge magnitude) of the smoothed image as a function of mask size. Assume for simplicity that n is odd and that edges are obtained using the partial derivatives af/ax = F(x+1, y) - F(x, y) and af/ay = F(x, y + 1) - F(x, y). (b) Show that the ratio of the maximum edge strength of the smoothed image to the maximum edge strength of the original image is 1/n. In other words, edge strength is inversely proportional to the size of the smoothing mask. 10.13 Suppose that an image f(x, y) is convolved with a mask of size nxn (with co- efficients 1/n) to produce a smoothed image F(x, y). *(a) Derive an expression for edge strength (edge magnitude) of the smoothed image as a function of mask size. Assume for simplicity that n is odd and that edges are obtained using the partial derivatives af/ax = F(x+1, y) - F(x, y) and af/ay = F(x, y + 1) - F(x, y). (b) Show that the ratio of the maximum edge strength of the smoothed image to the maximum edge strength of the original image is 1/n. In other words, edge strength is inversely proportional to the size of the smoothing mask.