Design of 6-Band Tight Frame Wavelets With
Limited Redundancy
A. Farras Abdelnour
Medical Physics
Memorial Sloan-Kettering Cancer Center
New York, NY 10021
Email: abdelnoa@mskcc.org
Telephone: (212) 639 - 2268
Abstract— In this paper we explore the design of tight frame
wavelets with a dilation factor M = 4. The resulting limit
functions are significantly smoother than their orthogonal coun-
terparts. An advantage of the proposed filters over the dyadic
filterbanks is that the proposed filterbanks result in a reduced
redundancy when compared with dyadic tight frames, while
maintaining smoothness. The proposed filterbanks generate five
wavelets and a scaling function with the underlying filters related
as follows: H5-i (z)= Hi (-z),i =0 ... 5.
I. I NTRODUCTION
Dyadic (M =2) wavelets have been extensively studied
and analyzed [5], [20], and wavelets based on critically sam-
pled FIR filterbanks have witnessed various applications [14],
[21]. The tight frame variation allows for additional design
degrees of freedom and smooth limit functions. Dyadic tight
frames have been well documented now. Some of the earliest
papers discussing compactly supported dyadic tight frames
are [7], [17]. Tight frames with wavelets possessing more
than one vanishing moment each were developed in [18]. The
design of 4-channel dyadic and symmetric limit functions has
been addressed in [2], [3], [6], [15], [16]. Symmetric tight
frames with two generators have been addressed in [12], [19]
where symmetry and compact support require the lowpass
filter H
0
(z) to satisfy an additional constraint, namely the
roots of 2 - H
0
(z)H
0
(1/z) - H
0
(-z)H
0
(-1/z) must be
of even multiplicity [16]. Tight frames have been shown to
exhibit a nearly shift-invariant behavior, which is important
for applications such as denoising. Orthogonal symmetric limit
functions have been obtained under M =4 [10], but the
resulting limit functions lack smoothness.
In this paper we seek 6-band tight frame FIR filters
{h
0
,h
1
.h
2
,h
3
,h
4
,h
5
} with dilation factor M =4. The filters
are related as follows: H
i
(z)= H
5-i
(-z),i =0 ... 5. The
decimation rate of 4 limits the throughput redundancy of the
resulting filterbanks, while taking advantage of the design
flexibility of frames. The filterbank design is accomplished
using Gr¨ obner basis [8], [9], [13]. The resulting filterbanks
enjoy a limited redundancy when compared with their dyadic
tight frame counterparts, 1.66 for M = 4 versus a re-
dundancy of 3 (for three dyadic wavelets) and 2 (for two
dyadic wavelets), while maintaining smooth limit functions.
We present examples of frames with nonsymmetric, symmet-
ric, and interpolating symmetric scaling functions.
II. BACKGROUND THEORY
In this section, we discuss some definitions and properties
pertaining to the case M =4 tight frame filterbanks generating
five wavelets. A set of wavelets {ψ
l
(4
m
·-n),l =1 ... 5}
constitutes a frame when for 0 ≤ A ≤ B< ∞ and any
function f ∈ L
2
we have [11]
A‖f ‖
2
≤
5
i=1
m,n
|〈f,ψ
i
(4
m
·-n)〉|
2
≤ B‖f ‖
2
,
where A and B are known as frame bounds. The special case
of A = B is known as tight frame. Moreover, when A = B =
1, and ‖ψ
l,j,k
‖ =1 ∀l, j, k, ψ
l,j,k
constitutes an orthogonal
basis. In general, a wavelet system consisting of one scaling
function and 5 wavelets with dilation 4 defines the following
spaces:
V
j
= Span
n
{φ(4
j
t - n)},
W
i,j
= Span
n
{ψ
i
(4
j
t - n)}, 1 ≤ i ≤ 5
with
V
j
= V
j-1
∪W
1,j-1
∪W
2,j-1
∪ ... ∪W
5,j-1
and the corresponding scaling function and wavelets satisfy
the following multiresolution equations:
φ(t) = 2
n
h
0
(n)φ(4t - n),
ψ
i
(t) = 2
n
h
i
(n)φ(4t - n), 1 ≤ i ≤ 5.
The bounds A and B now take on the value [4]
A = B =
1
4
5
n=0
‖h
n
‖
2
.
A function f ∈ L
2
can then be expanded as follows:
f (t)=
k
c(k)φ
k
(t)+
5
l=1
∞
j=0
k
d
l
(j, k)ψ
l,j,k
(t), (1)
2006 IEEE International
Symposium on Signal Processing
and Information Technology
0-7803-9754-1/06/$20.00©2006 IEEE 133