## Wavelet Indicator

Purpose
Description
Macro Synopsis
Modules
Related Functions
References

### Purpose

Evaluation of 2-dimensional signals, especially wavelet transformations

### Description

Here a bundle of functions for evaluating full continuous wavelet transform results or other 2-dimensional signals are described. Assume is the wavelet transform of the 1-dimensional signal using the wavelet , hence depends on a time (respectively space) and a scale parameter.

A common interpretation of the transform consists of reading the scale and time parameter from the local extrema of the transform to state matching of wavelet pattern and signal. This bases on the fact that the wavelet transform correlates the signal and the dilated as well as the translated wavelet. I. e., extrema mark signal clippings with size corresponding to the scale parameter located at dates corresponding to the time parameter. Usually it is sufficient to compute a thresholded representation of the absolute values of the transform.

The implemented indicators are often able to reveal unexpected events by averaging over one parameter of the wavelet transform chosen by the option Projection On: x-axis does averaging over the scale, y-axis over the time (spatial) parameter.

The indicator is chosen by the Norm option: Power Density, Entropy, Absolute Max.

The Absolute Max indicator detects the absolute maximum at each computed scale or date (depending on the value of Projection on):

respectively

The Power Density indicators average the wavelet transform over the scale (Projection on X-Axis chosen) or the time (Projection on Y-Axis) parameter of the transform:

respectively

These indicators return information about the power distribution for one transform parameter while all information about the other one is lost.

Their integrals are calculated by the Midpoint rule.

The Entropy indicators

respectively

have to be read pointwise, because of wavelet coefficients are normalized pointwise (with respect to the variable of the indicator) to sum up to one. This complies to the definition of entropy. For instance, a small value of the indicator at any point corresponds to nearly constant transform values for that parameter value.

Further parameters of the indicator function are

• nscale, the number of scale samples
• ntime, the number of time samples
• scale, the distance between adjacent scale samples in case of projection on x-axis respectively between time samples in case of projection on y-axis

### Macro Synopsis

y = Indicator(x,norm,projectionOn,nscale,ntime,scale);
signal x,y;
option norm, projectionOn;
int nscale, ntime;
float scale;

Note that x is a 2D signal.

Wavelet

### Related Functions

Full continuous wavelet transform, Define wavelet, Load wavelet, Save wavelet, Decompose, Reconstruct, Wavelet packet decomposition.

### References

Ende et al. [32], Louis/Maass/Rieder [33], Mallat [34].