The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
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The overall output signal X(z) of the analysis-synthesis
filterbank can be expressed as
where
X(z) = ( 1 z“')H r (z 2 )H(z~ 2 )x(z 2 ).
Furthermore, to meet perfect reconstruction (PR) condition
X(z)-X(z), we require that H r (z)H(z~') = I .
H r (z)H(z -1 ) = I
H(z) =
Xo (z)
H l0 (z)
n 2 , 0 (z)
Hv(zŸ
H u (z) .
(4)
Selesnick (2004), on the other hand, shows that a three-band
tight frame filterbank PR conditions can be expressed in terms
of the Z -transforms of the filters h 0 , /z, , and h 2 . Moreover,
the PR conditions can be easily extended to N filters
downsampled by 2:
f j H i (z)Hfz-') = 2
i=0
Y j H i (-z)Hfz-') = 0
i=0
Last, Chui et al. (2000) show a necessary condition for the
filterbank to exist. That is, the filters Hfz), / = 0,...,iV-l must
each satisfy the following inequality:
|//,.(z)| 2 +|//,.(-z)| 2 <2, |z| = 1.
Also, if h 0 (n) is compactly supported, then a solution
{ hfn), hfn) } to Eq. (4) exists if and only if
|// 0 (z)| 2 +|/f 0 (-z)| 2 <2, |z| = l. (5)
A wavelet tight frame with only two symmetric or anti
symmetric wavelets is generally impossible to obtain with a
compactly supported symmetric scaling function, f(t).
However, Petukhov (2003) states a condition that the lowpass
filter hfn) must satisfy so that this becomes possible.
Therefore, if hfn) is symmetric, compactly supported, and
satisfies Eq.(5), then an (anti-)symmetric solution
{ hfn), ^(n) } to Eq. (4) exists if and only if all the roots of
Notice that the equality reduces to the traditional case of a two-
band orthogonal filterbank.
2.3 A Symmetric Tight Wavelet Frame with Two
Generators
In this section, we introduce the construction of a symmetric
tight wavelet frame with two generators based on a three-band
tight frame filterbank and provide a result and an example
(Selesnick, 2004).
Figure 1. A three-band PR filterbank
2.3.1 PR Conditions and Symmetry Condition: The PR
conditions for the three-band filterbank, which are illustrated in
Fig. 1, can be obtained by setting the N in the section 2.2. to a
value of 3. That is, we have the following two equations:
Y J Hfz)H i (z-') = 2 (2)
¡=o
¿//,.(-z)//,.(z-') = 0. (3)
i=0
2-H 0 (z)H 0 (z-')-H Q (-z)H 0 (-z-') (6)
have even multiplicity.
2.3.2 Case Hfz) = Hf—z) : The goal is to design a set of
three filters that satisfy the PR conditions in which the lowpass
filter, hfn), is symmetric and the filters hfn) and hfn) are
each either symmetric or anti-symmetric. There are two cases.
Case I denotes the case where hfn) is symmetric and hfn) is
anti-symmetric. Case II denotes the case where h^in) and
hfn) are both anti-symmetric. The symmetry condition for
h 0 (n) is
h 0 (n) = h 0 (N-l-n), (7)
where N is the length of the filter h 0 (n).
We dealt only with Case I of even-length filters. Solutions for
Case I can be obtained from solutions where /^(m) is a time-
reversed version of /z,(w) (and where neither filter is (anti-)
symmetric).
The PR conditions can also be written in matrix form as