# Wavelet packet decomposition

Originally known as Optimal Subband Tree Structuring (SB-TS) also called Wavelet Packet Decomposition (WPD) (sometimes known as just Wavelet Packets or Subband Tree) is a wavelet transform where the discrete-time (sampled) signal is passed through more filters than the discrete wavelet transform (DWT).

## Introduction

In the DWT, each level is calculated by passing only the previous wavelet approximation coefficients (cAj) through discrete-time low and high pass quadrature mirror filters. However, in the WPD, both the detail (cDj (in the 1-D case), cHj, cVj, cDj (in the 2-D case)) and approximation coefficients are decomposed to create the full binary tree. Wavelet Packet decomposition over 3 levels. g[n] is the low-pass approximation coefficients, h[n] is the high-pass detail coefficients

For n levels of decomposition the WPD produces 2n different sets of coefficients (or nodes) as opposed to (n + 1) sets for the DWT. However, due to the downsampling process the overall number of coefficients is still the same and there is no redundancy.

From the point of view of compression, the standard wavelet transform may not produce the best result, since it is limited to wavelet bases that increase by a power of two towards the low frequencies. It could be that another combination of bases produce a more desirable representation for a particular signal. The best basis algorithm by Coifman and Wickerhauser finds a set of bases that provide the most desirable representation of the data relative to a particular cost function (e.g. entropy).

There were relevant studies in signal processing and communications fields to address the selection of subband trees (orthogonal basis) of various kinds, e.g. regular, dyadic, irregular, with respect to performance metrics of interest including energy compaction (entropy), subband correlations and others.    

Discrete wavelet transform theory (continuous in the variable(s)) offers an approximation to transform discrete (sampled) signals. In contrast, the discrete subband transform theory provides a perfect representation of discrete signals.

## Applications

Wavelet packets were successfully applied in preclinical diagnosis.