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