On shift variance and order of Daubechies wavelet

[Devyani Bedekar (bedekard@bme.ri.ccf.org)]

Hi wavelet community,

I am unable to understand the significance of the order of the daubichies wavelet used for decomposition of biomedical signals.
Higher order provides better frequency decomposition and less interference between scales but also causes shift variance. How serious is shift variance. Is it the same as phase lag or time delay.
Also I wanted to know what difference does it make if we use even ordered mother wavelet (ex. db8) or odd ordered mother wavelet (ex. db11 or db15).

Thanks

[Sumit Kumar (snath@site.uottawa.ca)]

HI Devyani,
Shift Variance has nothing to do with the order of the wavelet transform being used.
The signals , after decomposition, are not shift invariant in any DECIMATED wavelet transform.
If you remember, there is a downsampling factor of 2 after u convolve with the analysis and low pass filters.
You can get rid of it if your filters, in some way, can get rid of this alisaing components. The only possible wavelet filter is the HAAR wavelet filter ( aka the Db1) wavelet filter.
May I ask you what you are doing with your signal ?
Are you trying to detect some features from them ? e.g irregular heart beats (from an e.c.g signal ?)
or are you trying to denoise them
or are you trying to compress those signals.

Based on the specifics of your application, there are wavelets/wavelet transforms associated with them.

[Devyani Bedekar (bedekard@bme.ri.ccf.org)]

Hi


I am currently trying to find a best wavelet for RF ultrasound signal for tissue characterization.

Thanks Sumit , for throwing some light that - shift variance is independant of the order of the mother wavelet being used.

Also how does it matter if the order N is odd or even. Am I right that, for N = odd (as for db15) the filters are non symmetric and shift variance is high.
So is it that N=even is better.

Thanks a lot
Devyani

[Sumit Kumar (snath@site.uottawa.ca)]

Hi Devyani,
If my knowledge of Bio-med image processing is not bad, then in mathematical terms you are trying to segment some regions of an image. Your malignant and benign tissues will have different characteristics...hence while the healthy tissue may absorb the ultrasound signals, the malignant portion will give you scattering. It is this feature that you may have to detect.
If this all that you are trying to achieve, then do not go for a decimated wavelet transform. Instead go for the undecimated wavelet transform.
It is the same as its decimated counteraprt, except you do not have the downsampling factor. This removes the problems of subband aliasing associated with the decimated transform. By doing that you can isolate the noise (aka the regions of malignancy )....
Why dont you try using the Morlett wavelet also, in addition to a Daubechies wavelet !
Of'cos you cannot compress these images

[Sumit Kumar (snath@site.uottawa.ca)]

Whoa Elijah,
What I meant was, the Undecimated wavelet transform is not the tool to be used for compression (so my statement " u cannot compress the images" was in that context only). In compression you seek to reduce the number of wavelet coefficients at each succeding step. For that you need a critically sampled wavelet transformed..aka the Decimated wavelet transform

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