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wavelets - vanishing moments - what are they?

Started by Ben November 20, 2005
Hello,

Can anyone give a _very_ simple explanation / definition of what
vanishing moments are in the field of wavelets please? I'm reading
Hubbard's the World According to Wavelets and the book keeps using the
phrase vanishing moments but annoyingly hasn't really introduced nor
explained them.

Thanks, Ben.
In article <201120051154385015%z@z.z>, Ben <z@z.z> wrote:

> Hello, > > Can anyone give a _very_ simple explanation / definition of what > vanishing moments are in the field of wavelets please? I'm reading > Hubbard's the World According to Wavelets and the book keeps using the > phrase vanishing moments but annoyingly hasn't really introduced nor > explained them. > > Thanks, Ben.
Just to add to this: I know vanishing moments are features of wavelets. One thing that would very much help is to know this incredibly simple thing: Are vanishing moments something that you can point out in a wavelet, and say "there's one"? If so, what do they look like? That alone won't explain what a vanishing moments are but'll certainly help and be a good start. Thanks, Ben.
Ben wrote:

> In article <201120051154385015%z@z.z>, Ben <z@z.z> wrote: > >> Hello, >> >> Can anyone give a _very_ simple explanation / definition of what >> vanishing moments are in the field of wavelets please? I'm reading >> Hubbard's the World According to Wavelets and the book keeps using the >> phrase vanishing moments but annoyingly hasn't really introduced nor >> explained them. >> >> Thanks, Ben. > > Just to add to this: I know vanishing moments are features of wavelets. > One thing that would very much help is to know this incredibly simple > thing: Are vanishing moments something that you can point out in a > wavelet, and say "there's one"? If so, what do they look like? That > alone won't explain what a vanishing moments are but'll certainly help > and be a good start. > > Thanks, Ben.
When wavelet-functions coefficients are expressed as z-transform, then the number of zeros at pi correspond to the number of vanishing moments. So p zeros in pi means p vanishing moments. Having p vanishing moments means that wavelet-coefficients for pth order polynomial will be zero. That is, any polynomial signal up to order p-1 can be represented completely in scaling space. In theory, more vanishing moments means that scaling function can represent more complex signals accurately. p is also called the accuracy of the wavelet (Wavelets and Filter Banks, Strang&Nguyen). In practise when filter has p zeros in pi, it means that is calculates the pth order difference. So you can represent the system as x --> |Dp| --> |F| --> y Where Dp calculates pth order difference and F is filter with no zeros in pi. The block |Dp| = |1 - z^-1| --> ... --> |1 - z^-1|. Where |1-z^-1| is the difference between two samples. The block Dp is composed of p differences. Now when you put any polynomial with order p-1 or smaller through the Dp out should come zeros. For example when p=2 first order polynomial P(x)=x+1: - First difference P1(x)=P(x-1)-P(x)=(x-1)+1-(x)-1=-1 - Second diff P2(x)=P1(x-1)-P(x)=-1 - -1=0 Here's some more info: http://cnx.rice.edu/content/m11156/latest/?format=pdf -- Jani Huhtanen Tampere University of Technology, Pori