## 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.

### Modules

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].