Wavelet Toolbox™

Build wavelet from pattern

Syntax

• [PSI,XVAL,NC] = pat2cwav(YPAT,METHOD,POLDEGREE,REGULARITY)
  

Description

[PSI,XVAL,NC] = pat2cwav(YPAT,METHOD,POLDEGREE,REGULARITY) computes an admissible wavelet for CWT (given by XVAL and PSI) adapted to the pattern defined by the vector YPAT, and of norm equal to 1.

The underlying x-values pattern is set to

• xpat = linspace(0,1,length(YPAT))
  

The constant NC is such that NC*PSI approximates YPAT on the interval [0,1] by least squares fitting using

• a polynomial of degree POLDEGREE when METHOD is equal to 'polynomial'
• a projection on the space of functions orthogonal to constants when METHOD is equal to 'othconst'

The REGULARITY parameter defines the boundary constraints at the points 0 and 1. Allowable values are 'continuous', 'differentiable', and 'none'.

When METHOD is equal to 'polynomial'

• if REGULARITY is equal to 'continuous', POLDEGREE must be 3.
• if REGULARITY is equal to 'differentiable', POLDEGREE must be 5.

Examples

The principle for designing a new wavelet for CWT is to approximate a given pattern using least squares optimization under constraints leading to an admissible wavelet well suited for the pattern detection using the continuous wavelet transform (see Misiti et al.).

• % Example: Generate a new wavelet starting from a pattern.
  
  % Load original pattern: a pseudo sine one.
  load ptpssin1; 
  
  % Variables X and Y contain the pattern.
  whos
  
    Name          Size                   Bytes  Class
  
    IntVAL        1x1                        8  double array
    X             1x256                   2048  double array
    Y             1x256                   2048  double array
    caption       1x35                      70  char array
  
  Grand total is 548 elements using 4174 bytes
  
  % This example is a demo-example, so we have the value of the
  % integral of the pattern as well as the details about its
  % construction in the caption variable.
  
  IntVAL
  
  IntVAL =
  
      0.1592
  
  % The pattern defined on the interval [0,1] is of integral 0.1592. 
  % So it is not a wavelet but it is a good candidate since it 
  % oscillates like a wavelet.
  plot(X,Y), title('Original Pattern')
  
  
  % To synthesize a new wavelet adapted to the given pattern, let
  % us use a least squares polynomial approximation of degree 6 with
  % constraints of continuity at the beginning and the end of the
  % pattern.
  [psi,xval,nc] = pat2cwav(Y, 'polynomial',6, 'continuous') ;
  
  % The new wavelet is given by xval and nc*psi.
  plot(X,Y,'-',xval,nc*psi,'--'), 
  title('Original Pattern and Adapted Wavelet (dashed line)')
  
  
  % Note that the version of the wavelet is correctly 
  % defined in order to be used in the CWT algorithm must be of
  % square norm equal to 1. It is simply given by xval and psi.
  

References

Misiti, M.; Y. Misiti, G. Oppenheim, J.-M. Poggi (2003), "Les ondelettes et leurs applications," Hermes.

Provide feedback about this page

orthfilt

plot

© 1984-2008- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS