The discrete Haar wavelet transform (HWT) introduced in Chapter 6 serves as an excellent introduction to the theory and construction of wavelet transforms. We were also able to see in Section 6.4 how the transform can be utilized in applications such as image compression and edge detection. There are some limitations to the HWT. The fact that the filters h = (h0, h1) = (images,images) and g = (g0, g1) = (images, − images) are short leads to a fast algorithm for computation. However, there is a disadvantage with such short filters.

Consider the vector v = [100, 102, 200, 202]T. Applying the HWT to v gives y = images [101, 201, 1, 1]T. Now think about the application to edge detection in Section 6.4. We were looking for large values in the highpass (difference) portions of the output and possibly designating them as edges. If we were to employ the same method for designating boundary coefficients as in Example 6.16, then the threshold value would be = 1, and since no values ...

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