E.g., construct FIR filter approximating a ideal low-pass filter ( IIR), such as Equiripple method, Weighted least squares method (eigenfilters) and various window methods.
The answer is YES
Here we only consider approximating a given filter with orthogonal wavelet filters.
Problem:
Given: a filter H(ω)( low-pass or high-pass).
[6] A. Kirac and P. P. Vaidyanathan,(1998) ”On existence of FIR principal component filter banks”
[7] B. Xuan and R. H. Bamberger, (1996) ”2D factorable FIR principal component filter banks”
Pass-band, Stop-band, phase, ripples, etc
Motivation(2):
Filters have been used long long before wavelet appears. Many useful filters are using even now and in the future, they have good physical properties.
[2] M. K. Tsatsanis and G. B. Giannakis,(1995) ”Principal component filter banks for optimal multiresolution analysis”
[3] T. Greiner,(1996) ”Orthogonal and biorthogonal texture-matched wavelet filter banks for hierarchical texture analysis”
Sometime, we know that we need a filter with special properties, we want to make the used wavelet filter possesses these properties, e.g., directional filters.
Motivation(5):
Answer: some of the traditional filter can be a wavelet filter (orthogonal or bi-orthogonal), but most of them are not wavelet filters.
Question: can we approximate a given filter with a wavelet filter?
Filter approximation: approximate a filter by another filter (bank)
Motivation(1):
Filter approximation is a old problem in digital signal processing, which is called filter design, can be found in any text book on DSP.
[8] P. P. Vaidyanathan and S. Akkarakaran,(2001) ”A review of the theory and applications of optimal subband and transform coders,” Appl. Comput. Harmonic Analysis.
It is a big problem for a engineer who is only familiar with classical filters.
Motivation(3):
Signal-adaptive filter designing is to find a optimal( in some senses) for a given signal and application, e.g., compression, denoising, superresolution, etc.
conference 13 Nov 2003, Singapore
Thanks!
Outline 0. Concepts 1. Motivation 2. 1d and examples 3. 2d and examples 4. Open problems and remarks.
”Some concepts:”
and the reference therein.
........
Common method leads to optimal problem:
min J(h) such that h satisfies orthogonal conditions.
several problems:
1. The subject function is difficult to define,
A good approximation is not simple truncation which is mathematically optimal, but not physically suitable.
Some physical properties are important in filter design:
Let H(ω) = f0(z2) + zf1(z2), and Hw(ω) = f0w(z2)+zf1w(z2), than Hw satisfies (2) is equivalent to
|f0w(z)|2 + |f1(z)w|2 = 1 f or all z on unit cycle (3)
Bad news: we have too many wavelets to choose.
For a arbitrary application, which wavelet is best? Daubechies orthogonal? biorthogonal? multi-band filter? vector valued filter? Basis or Frame?
0
(1)
where Hw satisfies the orthogonal wavelet con-
ditቤተ መጻሕፍቲ ባይዱons:
|Hw(ω)|2 + |Hw(ω + π)|2 = 2
(2)
and √
Hw(0) = 2
It is a nonlinear optimal problem.
Stepwise Approximation Method:
Motivation(4):
How to find a proper wavelet filter with the desired properties? (beside vanishing moments and approximation order)
For texture analysis, the usually used wavelet filters, such as 9/7, 5/3 and Daubechies orthogonal wavelet filters with maximal vanishing moments, are not suitable because of the non-smoothness of texture image.
Unfortunately, for real time signal processing, it is difficult to find an optimal filter bank in real time, especially wavelet filter bank,
What is the relationship between wavelet filter banks (especially orthogonal wavelet filters) and traditional filters?
Different signal may need different filter ( wavelet), different application may need different wavelet too.
As a example, Energy compaction wavelet filter bank is very useful in signal compression, that is, given a signal, find a proper wavelet filter with arbitrary length to ensure the lowpass band has maximal energy.
[4] F.Ade, (1983) ”Characterization of texture by ’eignfilters’ ”,
[5] P. Moulin, M. Anitescu, K. Kortanek, and F. Potra,(1996) ”Design of signal-adapted FIR paraunitary filter banks”
Wavelet are widely used. Many many signal processing (image processing) methods are now based on wavelet.