Design of Odd Number Rational Coefficients Symmetric Compactly-supported
Biorthogonal Wavelet Filters
Han Fang-fang
Duan Fa-jieˈDuan Xiao-jieˈZhang Chao
State Key Lab of Precision Measuring Technology &
Instruments, Tianjin University
Tianjin, 300072, China
School of Electrical Engineering, Tianjin University of
Technology
Tianjin, 300382, China
fangfanghan2004@163.com
State Key Lab of Precision Measuring Technology &
Instruments, Tianjin University
Tianjin, 300072, China
fajieduan@126.com
Abstract—Most wavelet filters that constructed based on
Daubechies compactly-supported wavelet construction theory
have nonlinear phase and irrational number coefficients.
Nonlinear phase may cause distortion in image processing, and
irrational filter coefficients will cause much inconvenience for
wavelet applications on computer, especially for embedded
processor applications. With thought of the theory of
perfect-reconstruction filter banks, this paper does a study for
linear phase biorthogonal perfect-reconstruction wavelet filter
banks, and through of detail derivation, makes a conclusion
for the procedure of odd number rational coefficients
symmetric compactly-supported biorthogonal wavelet filter
construction, which can be achieved by adding some vanishing
moments conditions. With the example of length 7-5
biorthogonal wavelet filters design, this paper makes a detail
discussion on the value selection for free variable appears in
design equations, and then deduces the length 7-5 rational
biorthogonal wavelet filters models.
On the other hand, most biorthogonal wavelet filters’
coefficients obtained based on Daubechies methods are
irrational numbers [1, 2], which cause much inconvenience in
wavelet applications especially on computer hardware
operation. Therefore, since wavelet transform can be seen as
a special structure of perfect-reconstruction filter bank
operation, we can add some proper conditions, such as
vanishing moment constraints, to construct a rational
coefficients biorthogonal wavelet filter bank.
II. CONSTRUCTION OF BIORTHOGONAL
PERFECT-RECONSTRUCTION WAVELET FILTER BANK
Figure 1 shows a group of biorthogonal perfectreconstruction wavelet filters [3, 4]. The upper branch is
constructed by two low-pass filters, where H (z ) is
~
decomposition low-pass filter, and H ( z ) is reconstruction
low-pass filter. The lower branch is constructed by two
high-pass filters, where G (z ) is decomposition high-pass
~
filter, and G ( z ) is reconstruction high-pass filter. p 2 Keywords-wavelet filter construction; rational coefficients
wavelet filter; odd number symmetric filter; biorthogonal wavelet;
wavelet transform
I.
represents down-sampling by 2, and
up-sampling by 2.
INTRODUCTION
Wavelet multi-resolution analysis is performed under the
orthogonal conditions. But when to solve the practical
engineering problems, only orthogonal condition is
inadequate. For example, in order to avoid distortion in
image transformation processing, the response functions of
filters came from wavelet two-scale functions preferably
have linear phases. The necessary and sufficient condition
for filter’s linear phase is that the impulse response of filter
must be symmetric about its center point.
But according to the orthogonal wavelet transform
theory and its construction methods under orthogonal
multi-resolution thoughts, except Haar wavelet, the other
wavelets’ low-pass filter coefficients are not symmetric, i.e.,
have no linear phase [1, 2]. In order to overcome this
limitation, in virtue of the perfect-reconstruction filter banks
thoughts, the filters for signal decomposition and
reconstruction can be not the same one, and then a
biorthogonal wavelet transformation system based on
biorthogonal structure be constructed.
H (z )
x(z )
p 2
n 2
n 2
represents
~
H ( z)
y (z )
†
G (z )
p 2
n 2
~
G( z)
Figure 1. Structure of biorthogonal perfect-reconstruction wavelet filter
bank
If there is no the middle procedure of signal processing,
the output of perfect-reconstruction filter bank should be a
perfect reconstruction of the input, or a perfect
reconstruction with some time delay. Then the four filters
~
~
of H (z) , H ( z ) , G (z ) and G ( z ) should meet the following
relationship, where F0 ( z ) is called distortion term, and
F1 ( z ) is called alias term[3, 4].
76
F0 ( z )
F1 ( z )
~
~
H ( z ) H ( z ) G ( z )G ( z ) 2 z l
~
~
H ( z ) H ( z ) G ( z )G ( z ) 0
Assign < as follow equation:
(1)
2
(2)
<
As long as the four filters meet the requirements
represented by equation (1) and equation (2), they can
construct a perfect-reconstruction filter bank. Therefore,
filters can be designed freely according to those
relationships.
Here, this paper proposes the following filter design
way:
~
~
­°G ( z ) zH
( z )ˈviz. G ( z ) zH ( z )
(3)
®~
°̄G ( z ) z 1 H ( z )
N
1· 2
§
K 1 ¨ < ¸ Q(< )
2¹
©
H (< )
(7)
~
N
1· 2
~
§
K 2 ¨ < ¸ Q (< )
2¹
©
~
H (< )
(8)
Therefore:
~
H (< ) H (< )
Y (< )
~
NN
1· 2
§
¨< ¸
2¹
©
~
Q(< )Q (< )
(9)
Y (< ) should meets two requirements: ķ the
perfect-reconstruction condition represented by equation (5);
ĸDaubechies compactly-supported condition.
First, for the perfect-reconstruction condition, thinking
combination with equation (5) and equation (6), and
considering with the filter symmetric feature, Y (< )
should meets the follow equation:
Then:
2
(6)
Then:
Take equation (3) into equation (1) and equation (2), and
order the time delay l 0 , then F1 ( z ) 0 , F0 ( z ) 2 .
Assign:
~
Y ( z) H ( z) H ( z)
(4)
Y ( z ) Y ( z )
§ 1 z 1 ·
¸ 1
z¨
¨ 2 ¸
2
©
¹
(5)
Therefore, for the four perfect-reconstruction filters, we
can firstly design the two low-pass filters H (z ) and
~
H ( z ) , and consequently deduce the two high-pass filters
~
G (z ) and G ( z ) according to equation (3). Hence, in the
following, this paper will concentrate on the symmetric
~
compactly-supported filters design for H (z) and H ( z ) .
Y (< ) Y (< ) 1
(10)
Second, for the compactly-supported condition, suppose:
~
P(< ) Q(< )Q (< )
iZ
Since z e , and combination with Euler's formula
and Trigonometric Formulas, P (< ) can be represented by
III. CONSTRUCTION OF ODD NUMBER SYMMETRIC
COMPACTLY-SUPPORTED BIORTHOGONAL WAVELET
FILTERS
symbol x , where x sin 2 (Z / 2) :
The original intention to construct biothogonal wavelet
structure is to make the wavelet filter not only form a
perfect-reconstruction system but also have linear phase.
The necessary and sufficient condition for filter’s linear
phase is that the impulse response of the filter must be
symmetric
about
its
center
point,
that
is
i 2OZ
H (Z ) e
H (Z ) , or represents by coefficients
~
P( x) Q( x)Q ( x),
x sin 2
Z
2
With Taylor methods to get a group of special solution:
P( x)
relationship hk h2O k . When hk  R , coefficients can
be regarded as symmetric about the point of O .
This paper’s purpose is to design odd number filter, so
this paper proposes the rule that the designed filter is
symmetric about its axis 0, that is, the filter’s coefficients
meet equations of hk hk . Therefore, according to the
filter design methods proposed from references [2]~[10], the
two low-pass filters for odd number symmetric biorthogonal
wavelet can be designed as follow procedure [2, 5-10].
Suppose the decomposition low-pass filteris represented
by H (z) , and the reconstruction low-pass filter is
~
represented by H ( z ) . The vanishing moments of the two
~
filters are N and N respectively.
~
( N N ) / 21
¦
n 0
C (nN N~ ) / 21 n x n ~
N N
x 2 r ( x)
(11)
r (x) in equation (11) should meet the equation:
r ( x) r (1 x) 0
(12)
What’s more, P(x) should meet the Daubechies
compactly-supported condition:
max
x[ 0,1], r ( x ) r (1 x ) 0
77
P( x) ~·
§ N N
¸ 1
2˜¨¨
¸
2
¹
2 ©
(13)
IV. DESIGN FOR LENGTH 7-5 RATIONAL COEFFICIENTS
SYMMETRIC COMPACTLY-SUPPORTED BIORTHOGONAL
WAVELET FILTER BANK
To construct orthogonal compactly-supported wavelets
based on Daubechies methods, if the vanishing moments
takes the highest order value, that is, Q(1) z 0 and as
~
same as Q(1) z 0 , the constructed wavelet function and
scale function will have a rapid attenuation speed, but the
coefficients of these filters can not be rational. To get the
rational filter coefficients, a method is to reduce the order of
vanishing moments, and then make the designed equation
appear one or more free variables. Select suitable value for
free variables, thus obtain the rational coefficients wavelet
filter [2, 5-10]. So the key problem for rational coefficients
wavelet filter design is how to select free variables’ value in
Daubechies compactly-supported restriction range.
A 4 B 4 At 4 Bt t <
B At 4 Bt < Bt ]
To meet the equation (10), coefficients of < 4 and
< in equation (16) should be zero, and constant term
should be 1/2. Therefore, there is the equation group:
­
°4 A 4t 4 0
°
(17)
® A 4 B 4 At 4 Bt t 0
°K K
1
° 1 2 Bt
2
¯ 4
The solution for the equation group (17) with free
variable t is:
­
° A t 1
°
4t 2 4t 1
°
(18)
®B
4t 4
°
8t 8
°
° K1 K 2
3
4t 4t 2 t
¯
Now discusses the compactly-supported condition. For
P (< ) :
~
P(< ) Q(< )Q (< )
~
H (< )
1·
§
K 2 ¨ < ¸< t 2¹
©
2
P(< )
4t 4t t
3
2
[4(t 1)< 3 4(t 1)< 2
(19)
(4t 3 4t 2 1)< t (2t 1) 2 ]
Take
then:
According to equation (7) and equation (8), and
~
N 2, N 2 , there is:
K 1 K 2 (< 2 A< B)(< t )
Take solution equations (18) into P (< ) :
B. Solving for P(x)
H (< )
(16)
2
2
A. Selection for Wavelet Filter’s Vanishing Moment
Suppose the low-pass filter of length-7 is H (z) , and its
vanishing moment is N ; and the low-pass filter of length-5
~
~
is H ( z ) , and its vanishing moment is N . Because the
vanishing moment for odd-length and symmetric
biorthogonal filter must be even number, for the
biorthogonal wavelet of length 7-5, the vanishing moments
~
~
N and N are both even number, and N and N
should be in such range:
~ 75
NN d
6
2
~
Therefore, values for N and N have the following
~
~
several probabilities: (1) N 2, N 4 ; (2) N 4, N 2 ;
~
(3) N 2, N 2 . For the first two combination probability
values, there will be no free variables in wavelet
construction equations. But for rational coefficients wavelet
filter design, we need the free variables in design equations,
so this paper prefers the third probability for vanishing
~
moment, that is N 2, N 2 .
1·
§
K 1 ¨ < ¸ < 2 A< B
2¹
©
~
H (< ) H (< )
K1 K 2
[4< 5 4 A 4t 4 < 4
4
4 A 4 B 4 At 4t 1< 3
Y (< )
z
e iZ
cos Z i sin Z
2
into equation (6),
1
§ 1 z 1 ·
1
¸ 1 zz
z ¨¨
cos Z
¸
2
2
4
2
©
¹
Take upper < into equation (19),
x sin 2 (Z / 2) , then achieve the equation:
<
(14)
(15)
P( x) 1 2 x Then:
4(t 1)
t (2t 1) 2
x2 1§
2 Z·
¨1 2 sin
¸
2©
2¹
and assign
8(t 1)
t (2t 1) 2
x3
(20)
Because P(x) resolved by equation (20) must meet the
Daubechies compactly-supported condition represented by
equation (13), the value range for free variable t should be
discussed. Try to select different value for t in its
78
D. Solving for Coefficients of Filters
Combined equation (6) and equation (18), and
rewritten equation (14) and equation (15), then:
K1 3
8t 2 3t 1
H ( z)
z
[ z (4t 2) z 2 t 1
64
24t 2 20t 4
t 1
8t 2 3t 1 1
z (4t 2) z 2 z 3 ]
t 1
K2 2
~
H ( z)
[ z (4t 2) z
16
(8t 2) (4t 2) z 1 z 2 ]
permission range, we can do the rational coefficients
wavelet filter design.
C. Determination of Value Range for Free Variable t
Rewritten equation (20) as the form as equation (11),
then:
1
¦ C1n n x n x 2 r ( x)
P( x)
n 0
ª 4(t 1)
8(t 1)
1 2x x 2 «
2
t (2t 1) 2
«¬ t (2t 1)
º
x»
»¼
4(t 1)
8(t 1)
x , and it
t (2t 1) 2 t (2t 1) 2
satisfies the condition of r ( x) r (1 x) 0 shown in
equation (12).
To meet Daubechies compactly-supported condition,
P( x) should meet the requirement of equation (13), that is:
r ( x)
Therefore,
max
x[ 0,1], r ( x ) r (1 x )
ª 4(t 1)
8(t 1)
1 2x x 2 «
2
0
t (2t 1) 2
¬« t (2t 1)
In the value range of t , select different value for t ,
K 1 , K 2 , then we can construct different wavelet filters.
When select a suitable value for t , it is possible to
construct rational coefficients wavelet filters.
Here gives an example. When t 3 / 2 , according to
filter impulse response:
º
x» 23
¼»
H (Z )
(1 / 2) ¦ hk e ikZ , z
e iZ
kZ
According to equation (20), plot the relationship curve
figure about t l max P( x) . For every t , there is an equation
And wavelet filter coefficients condition:
¦ hk
about x l P(x) . What’s more, for x sin (Z / 2) , the value
of x was limited in [0,1] . In the range of x  [0,1] ,
calculate the max value of P(x) .Then for every t , exist
2
2
kZ
It can be calculated that K 1 2 / 3 , K 2 1 .
Then coefficients of the two low-pass filters are:
an max P( x) , thus we can get the Figure 2. Noted that for
equation (20), the denominator should not be zero,
t (2t 1) z 0 . So the value of t is composed of three sects,
that is (f,0.5) , (0.5,0) and (0,f) , shown as Figure
2.
h k : h 3
h0
~ ~
hk : h 2
1
1
25
, h 2 , h1
,
48
12
48
14
25
1
1
, h1
, h2 , h3 12
48
12
48
1 ~
1 ~
5 ~ 1 ~
1
, h0
, h1
, h2 , h1
8
2
4
2
8
And according to equation (12), the high-pass filters are:
g k : g 1
g~ k : g~ 4
Figure 2. Relationship of free variable
t
and
g~ 1
max P( x)
max
The
value
range
(f,0.755) ‰ (0.216,f) .
P ( x) 8
for
t
1
8
The scaling function and wavelet function constructed
by this group of filters [4, 11] is shown in Figure 3.
The amplitude-frequency characteristic and phasefrequency characteristic is shown in Figure 4. In figure 4,
H1 represents the filter of H (z ) , H 2 represents the
~
filter of H ( z ) , G1 represents the filter of G (z ) , and
According to Figure 2, when:
x[0,1],r ( x ) r (1 x ) 0
1
1
5
1
, g 0 , g1
, g2 , g3
8
2
4
2
1 ~
1
25
, g 3 , g~ 2 ,
48
12
48
14 ~
25 ~
1 ~
1
, g 0 , g1 , g 2
12
48
12
48
is
79
~
G 2 represents the filter of G ( z ) . From Figure 4, we can
see that the four filters all have linear phase.
avoid image distortion in image transformation, so it is quite
compatible for image processing.
The difficulty for the rational wavelet filter design
method proposed by this paper is solving for higher order
and multivariable functions, and determination of the value
range for the free variable under the effect of multiple
parameters. This paper makes a discussion for length 7-5
biorthogonal wavelet filters, but when filter length is
increased, the calculation difficulty will be increased more,
which is need a much deeply study and discussion.
REFERENCES
[1]
Daubechies I. Orthonormal basis of compactly supported wavelets.
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applications. Science Press, 2006.10: 100-116
[3] Sweldens W. The lifting scheme: A custom-design of biorthogonal
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Lihua. Tutorials for wavelet basis and application. Machinery
Industry Press, 2006.4: 146-153
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Figure 3. Scaling function and wavelet function constructed by the four
rational filters
Figure 4. Amplitude-frequency characteristic and phase-frequency
characteristic of the four rational filters
V.
CONCLUSION
This paper for rational wavelet filter design adopts the
method of reducing vanishing moment order, but it also
decreases the wavelet orthogonal compactly-supported
performance, which is an unpleasant phenomenon in image
compression field. But this kind of wavelet filter also has
two main advantages: (1) the rational coefficients of wavelet
filter can avoid truncation operation when it is operated on
computer, and it can completely meet the requirements of
perfect-reconstruction condition and vanishing moment
condition. It is especially suit for embedded processor
applications, for rational wavelet transformation can be
achieved by shift operation and multiply-add operation with
fast speed and high precision; (2) more information exists in
high frequency domain is not favorable for image
compression, but in the edge detection and morphological
analysis fields, it can provide more information.
This paper adopts the thoughts of biorthogonal wavelet
filter design, and then the four filters all have linear phase,
which is shown in Figure 4. The linear phase of filter, which
is just the advantage of biorthogonal wavelet structure, can
80