The results obtained by a single pass through an image of some 2-D masks can be achieved a...

The results obtained by a single pass through an image of some 2-D masks can be achieved also by two passes using 1-D masks. For example, the same result of using a 3 × 3 smoothing mask with coefficients 1/9 can be obtained by a pass of the mask [1 1 1] through an image. The result of this pass is then followed by a pass of the mask

The final result is then scaled by 1/9. Show that the response of Sobel masks (Fig. 10.14) can be implemented similarly by one pass of the differencing mask [−1 0 1] (or its vertical counterpart) followed by the smoothing mask [1 2 1] (or its vertical counterpart).

FIGURE 10.14 A 3 ×3 region of an image (the z’s are intensity values) and various masks used to compute the gradient at the point labeled z5.