The tree as a data algorithm

A fast algorithm for wavelet transformations uses the Mallat tree as seen in Figure . The algorithm involves two wavelets, low-pass and high-pass, with the one derived from the other by changing the sign of the even coefficients.

In Figure , the input data series is represented by the circle marked `input'. The result referred to as Lo-1 is from an application of the low-pass filter with a decimation. The low-pass filter for the Haar transform is the 2-term low-pass operator in Figure , and the low-pass filter for the D₄ transform is 4-term low-pass operator in Figure . Hi-1 comes from filtering with the corresponding high-pass filters. These filters are illustrated by the 2-term high-pass operator in Figure , and the 4-term high-pass operator in Figure . Hi-1 is now set aside, and the data in Lo-1 becomes the input for the next step. Elsewhere in this report, Schwab expands on how the filters are calculated (Schwab, 1992).

At each step, the high-pass output is saved and becomes part of the transformed output, and the final lo-pass output makes up the rest of the output.