Detecting Self-Similarity

The purpose of this example is to show how analysis by wavelets can detect a self-similar, or fractal, signal. The work of many authors and the trials that they have carried out suggest that wavelet decomposition is very well adapted to the study of the fractal properties of signals and images. When the characteristics of a fractal evolve with time and become local, the signal is called a multifractal. The wavelets then are an especially suitable tool for practical analysis and generation.

Loading the Signal

The signal here is the Koch curve -- a synthetic signal that is built recursively. Let's visualize the signal and zoom in one section.

load vonkoch;
xlim([1 length(vonkoch)]);
title('Analyzed signal - Koch curve');
xlabel('Time (or Space)');
xlim([3250 4120]);
title('Analyzed signal - Koch curve');
xlabel('Time (or Space)');

Wavelet Coefficients and Self-Similarity

From an intuitive point of view, the wavelet decomposition consists of calculating a "resemblance index" between the signal and the wavelet. If the index is large, the resemblance is strong, otherwise it is slight. The indices are the wavelet coefficients. If a signal is similar to itself at different scales, then the "resemblance index" or wavelet coefficients also will be similar at different scales. In the coefficients plot, which shows scale on the vertical axis, this self-similarity generates a characteristic pattern.

The command waveinfo displays the main properties of a wavelet family.

 Information on coiflets.
    Coiflets Wavelets
    General characteristics: Compactly supported 
    wavelets with highest number of vanishing 
    moments for both phi and psi for a given 
    support width.
    Family                  Coiflets
    Short name              coif
    Order N                 N = 1, 2, ..., 5
    Examples                coif2, coif4
    Orthogonal              yes
    Biorthogonal            yes
    Compact support         yes
    DWT                     possible
    CWT                     possible
    Filters length          6N
    Number of vanishing 
    moments for psi         2N
    Number of vanishing 
    moments for phi         2N
    Reference: I. Daubechies, 
    Ten lectures on wavelets, 
    CBMS, SIAM, 61, 1994, 258-261.

Let's compute the continuous wavelet transform (CWT) of the Koch curve:

scales = 2:2:128;
wname = 'coif3';
xlim([3250 4120]);

A repeating pattern in the wavelet coefficients plot is characteristic of a signal that looks similar on many scales.

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