We often in slides and homework questions have neglected to induce the normaliza

Question

We often in slides and homework questions have neglected to induce the normalization constants that are needed to make a smoothing filter's coefficients add up to one, or that allow a derivative filter to compute the correct gradient magnitude (which would indicate the actual change in intensity value from dark to bright across an edge). For example, a 3 times 3 box filter 1 1 1 1 1 1 1 1 1 should really have each coefficient divided by 9 (to become 1/9). Otherwise, it simply adds up the 9 numbers in a local neighborhood instead than computing their average. Similarly, a 3 times 3 Gaussian smoothing filter 1 2 1 2 4 2 1 2 11 2 1 2 4 2 1 2 1 should be divided by 16, for the same reason. Finally, note that we often neglect to divide the central finite difference filter 1 0 -1 by 2, as Taylor series says we should. In lecture slides, we justify that it is OK pedagogically to do this because: 1) it is easier to apply filters with integer coefficients (certainly true when computing convolution by hand): and 2) we can apply the normalization at the end by dividing the image of results by the normalizing constant, for example (1/9 times boxfilter)' image = 1/9 times (boxfilter ' image), where x means multiplication, and ' means convolution. This is due to convolution being a linear operator, one of the nice mathematical properties listed on our convolution properties slide. In this question, determine what the normalization constants should be for two common filters that perform smoothed derivative compulsions. A) As discussed in lecture, the Prewitt edge operator 1 0 -1 1 0 -1 1 0 -1 combines a first derivative filter across rows with a box smoothing filter down columns. What should the normalization constant be so that this filter computes the correct gradient magnitude values across an intensity edge? B) The Sobel edge operator 1 0 -1 2 0 -2 1 0 -1 combines a first derivative filter across rows with a Gaussian smoothing filter down columns. What should the normalization constant be so that this filter computes the correct gradient magnitude values across an intensity edge?

Explanation / Answer

A) Normalization constant for Prewitt edge operator is 1/6

B) Normalization constant for Sobel edge operator is 1/8

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