UFWT - Undecimated Fast Wavelet Transform

Usage

c = ufwt(f,w,J);
[c,info] = ufwt(...);

Input parameters

Output parameters

Description

ufwt(f,w,J) computes redundant time (or shift) invariant wavelet representation of the input signal f using wavelet filters defined by w in the "a-trous" algorithm.

For all accepted formats of the parameter w see the fwtinit function.

[c,info]=ufwt(f,w,J) additionally returns the info struct. containing the transform parameters. It can be conviniently used for the inverse transform iufwt e.g. fhat = iufwt(c,info). It is also required by the plotwavelets function.

The coefficents c are so called undecimated Discrete Wavelet transform of the input signal f, if w defines two-channel wavelet filterbank. Other names for this version of the wavelet transform are: the time-invariant wavelet transform, the stationary wavelet transform, maximal overlap discrete wavelet transform or even the "continuous" wavelet transform (as the time step is one sample). However, the function accepts any number filters (referred to as \(M\)) in the basic wavelet filterbank and the number of columns of c is then \(J(M-1)+1\).

For one-dimensional input f of length L, the coefficients c are stored as columns of a matrix. The columns are ordered with inceasing central frequency of the respective subbands.

If the input f is \(L \times W\) matrix, the transform is applied to each column and the outputs are stacked along third dimension in the \(L \times J(M-1)+1 \times W\) data cube.

Filter scaling

When compared to fwt, ufwt subbands are gradually more and more redundant with increasing level of the subband. If no scaling of the filters is introduced, the energy of subbands tends to grow with increasing level. There are 3 flags defining filter scaling:

'sqrt'

Each filter is scaled by 1/sqrt(a), where a is the hop factor associated with it. If the original filterbank is orthonormal, the overall undecimated transform is a tight frame. This is the default.

'noscale'

Uses filters without scaling.

'scale'

Each filter is scaled by 1/a.

If 'noscale' is used, 'scale' has to be used in iufwt (and vice versa) in order to obtain a perfect reconstruction.

Boundary handling:

c=ufwt(f,w,J) uses periodic boundary extension. The extensions are done internally at each level of the transform, rather than doing the prior explicit padding.

Examples:

A simple example of calling the ufwt function:

[f,fs] = greasy;
J = 8;
[c,info] = ufwt(f,'db8',J);
plotwavelets(c,info,fs,'dynrange',90);