Discrete Wavelet Transform (DWT) Filter Bank

Question

I have some stumbling block in my thesis writing.

Do we use the same filter pair while implementing DWT filter bank with downsampling of the filter output, or filters do change also from level to level? Though on the Wiki page and in the book "Biosignal and Medical Image Processing, 3rd edition" by J. L. Semmlow and B. Griffel there are same labelling of filters at each decomposition level on the filter bank schemes: , the description in Wikipedia states: "The filter output of the low-pass filter g in the diagram above is then subsampled by 2 and further processed by passing it again through a NEW low- pass filter g and a high- pass filter h with half the cut-off frequency of the previous one.."(c)

This resulted in big confusion for me. Logically, as I have been thinking before, for DWT we "travel" in frequency domain by signal downsampling and leaving the filters untouched (while e.g. in Stationary Wavelet Transform it is achieved through upsampling the filters themselves without signal modification). The word "new" in Wiki frustrated me a lot.

Please, help me to resolve this issue.

With respect,

-Andrey.

Answer 1

In the DWT scheme, whether it is the classical $2$-band or the $M$-band wavelet setting, the very same analysis filter bank (lowpass/highpass + subsampling) is used at each level. Under this condition, one can derive the cascade algorithm that provides the spectrum of the scaling function: $$\Phi(\omega)= \prod_{k=1}^\infty \frac {1} {\sqrt 2} H\left( \frac {\omega} {2^k}\right) \Phi^{(\infty)}(0)$$ where iterated half-cut-off frequencies are apparent.

However, this is especially important when one wants to address the properties of the underlying wavelet, or cascade many wavelet levels. In practice, you can easily choose a different set of perfect reconstruction filter bank at each level, for several reasons:

• actual filters down the scales tend to become larger, sometimes too much with respect to the signal features, and the signal length. When a convolutive filter becomes longer than the data itself, you run into troubles, especially in regard of boundary handling (symmetry, padding). The consequence can be high details coefficients caused mostly by wrapping around border artifacts.
• filtered/down-sampled data don't have the same characteristics at each scale, so different filters can do a better job
• do we really need multicale theory after all on 3 levels? In practice, it can be useful to use longer filters on the first levels, for better frequency separation, and shorter ones on lower levels, for the multiscale effect.

This is not mainstream, as one has to choose filters wisely, sometimes without many hint. However, there were works with DWT or GenLOT filters on the first scales and DCT or Haar on the lower resolutions. JPEG XR standard somehow behaves in that philosophy. One of the papers I have met on this direction is: On the frame bounds of iterated filter banks, 2009.