WaveletTrees

This package contains data types I use for modelling wavelet trees. A WaveletTree consists of a lowpass subband and (multiple) highpass subband(s); the highpass subband(s) are arranged in a tree structure.

For 1D signals the coefficients in the highpass subbands are arranged in a binary tree, i.e., a directed acyclic graph where each node has either zero or two children. A binary tree with three levels are seen below.

For 2D signals the highpass subbands within each direction are arranged in a quad tree, i.e., a tree where each node have either zero or four children.

Types

The generic type available is WaveletTree{D}, where D denotes the dimension of the data prior to the wavelet transform and can be 1 or 2. A WaveletTree has entries lowpass and highpass whose types depend on D.

A 1D WaveletTree with L levels and a lowpass/coarsest highpass subband with size coefficients are constructed with

WaveletTree(L, size)

and its entries are

• lowpass: A vector with size lowpass coefficients.
• highpass: A cell of L vectors with coefficients from each level, where W.highpass[1] is of length size.

A 2D WaveletTree with L levels, D directional subbands within each level and a lowpass subband (coarsest highpass) subband with x-by-y coefficients are constructed with

WaveletTree2D(L, (x,y), D)

(with default value D = 3) and its entries are

• lowpass: A matrix with x-by-y lowpass coefficients.
• highpass: A cell of L cells. Each inner cell contains D matrices with coefficients from the directional subbands, where the matrices in W.highpass[1] are all x-by-y matrices.

Functions

For WaveletTrees there are the functions

• size: Returns the number of coefficients in the lowpass subband, highpass subband or the two combined (which is the default). The result is either a vector or an levels-by-2 matrix, depending on the dimension of WaveletTree.
• levels: The number of levels in the tree.

For 2D WaveletTrees there is furthermore

• vec: Vectorizes a single directional subband according to the order induced by the parents ρ.
• tree2mat: The directional subbands on the same level are collected into one matrix.

Acknowledgements

The layout with nested cells is inspired by the contourlet toolbox for Matlab.