The HAAR Expansion Revisited

From the transfer function and taking into account the sambsapling by operation, the first two samples of the sequence are given by

Figure (4.10): Tree-Structure Filter bank

This is nothing but the action of the third and fourth rows of the 8 x 8 Haar tansform on the input vector. If we now carry on the splitting one step further, as in Figure (4.10), it is straightforward to show by repeating the preceding arguments that

Figure(4.11) Magnitude response of the Haar filters H₀ and H₁

These equations are the actions of the second and first rows of the Haar tansform on the input vector. The structure of Figure (4.10) is known as a (three-level) tree-structured filter bank generated by the filters and . Figure (4.11) shows the frequency responses of these two filters. One is a high-pass filter and the other a low-pass filter. Here in lies the importance of the filter band interpretation of the Haar transform. The input sequence is first split into two versions of lower resolution with respect to the original one: a low-pass (average) coarser resolution version and a high-pass (difference) detailed resolution one. In the sequel the coarser resolution version is further split into two versions, and so on. This leads to a number of versions with a hierarchy of resolutions. This decomposition is known as multiresolution decomposition .