COMPUTATION OF MULTIWAVELET
TRANSFORM USING DISCRETE LINEAR
CONVOLUTION
Dr. Waleed A. AL-Jauhar
Matheel E. AL-Dargazli
Dr. Jasim AL-Sammarai
Received On:
Accepted On:
ABSTRACT:
Multiwavelets are a new addition to the body of wavelet theory.
Realizable as matrix-valued filter banks leading to wavelet bases,
multiwavelets offer simultaneous orthogonality, symmetry, and short
support, which is not possible with scalar two-channel wavelet systems. This
paper describes a new approach for computing the discrete multiwavelet
transforms using discrete linear convolution by table look-up method, and
comparing this method with the others in terms of design, analysis, and
computing complexity in time and space.
Index terms – multiwavelets, wavelet transforms, convolution, and complexity.
‫حساب نقل متعدد المويجات باستخدام التلفيف الخطي المتقطع‬
:‫اخلالصة‬
‫ وحتقيقها كقيم مصفوفة لصفوف املرشحات توصلنا‬.‫متعدد املوجيات هو اضافة جديدة اىل هيكل نظرية املوجية‬
‫ هذا البحث يصف اسلوب جديد حلساب نقل متعدد‬.‫ واليت تكون غري ممكنة مع انظمة املوجية السابقة‬،‫اىل اساسيات املوجية‬
‫املوجيات املتقطع ابستخدام الناقل اخلطي املتقطع وبواسطة طريقة اجلدول ومقارنة هذه الطريقة مع بقية الطرق املقرتحة من‬
.‫ وحساب التعقيد‬،‫ التحليل‬،‫انحية التصميم‬
I-INTRODUCTION:
Wavelets are a useful tool
for signal processing applications
such as image compression and
denoising. Until recently, only
scalar wavelets were known:
wavelets generated by one scaling
function. But one can imagine a
situation when there is more than
one scaling function [2]. This leads
to the notion of multiwavelets,
which have several advantages in
comparison to scalar wavelets [1].
Such features as short support,
orthogonality, symmetry, and
vanishing moments are known to
be important in signal processing.
A scalar wavelet cannot possess all
these properties at the same time
[1]. On the other hand, a
multiwavelet
system
can
simultaneously provide perfect
reconstruction while preserving
length
(orthogonality),
good
performance at the boundaries (via
linear-phase symmetry), and a high
order of approximation (vanishing
moments). Thus multiwavelets
offer the possibility of superior
performance for image processing
applications, compared with scalar
wavelets [5].
We describe here a new
technique for computing the
discrete multiwavelet transform,
and present experimental results
for this method comparing with
another ones.
This paper is organized as
follows. Section II reviews the
definition and construction of
continuous-time
multiwavelet
systems, and Section III describes
the methods of computing
multiwavelet. In Section IV, we
introduce
briefly
the
new
technique for computing the
multiwavelet transform. Finally, in
Section V we describe the
computing
complexity
of
computing
multiwavelet
transforms in all methods.
multiresolution analysis (MRA).
The
difference
is
that
multiwavelets have several scaling
functions.
The
standard
multiresolution has one scaling
function Ø(t).
 The translates Ø(t  k) are
linearly independent and
produce a basis of the
subspace V0.
 The dilates Ø(2j tk)
generate subspaces Vj, j Z,
such that
….  V-1  V0  V1  ….
 Vj  ….

V j  L2 ( R),
j  

v
j
 {0}
j  
 There is one wavelet w(t).
Its translates w(t  k)
produce a basis of the
“detail” subspace W0 to give
V1:
V1=V0  W0
For
multiwavelets,
the
notion of MRA is the same
except that now a basis for V0
is generated by translates of N
scaling functions Ø1(t  k), Ø2(t
 k), …, ØN(t  k). The vector
Ø(t) = [Ø1(t), …, ØN(t),]T, will
satisfy a matrix dilation
equation (analogous to the
scalar case)
(t )   C[k ](2t  k )
II-PRELIMINARIES OF
MULTIWAVELETS:
As in the scalar wavelet
case, the theory of multiwavelets is
based
on
the
idea
of
k
The coefficients C[k] are N by
N matrices instead of scalars.
2
 1

1  2
H0  
10  1
 2
 9
1  2
H2  
10  9
 2
Associated
with
these
scaling functions are N wavelets
w1(t), …, wN(t), satisfying the
matrix wavelet equation
W (t )   D[k ](2t  k )
3 
10 
 9



1
2 ,H   2
2 
1
9
10 
3 
0 
2



3 
1




0
1 
2 
H3   2

10  1 0
3 

2



k
Again, W(t)= [w1(t), …, wN(t)] T is
a vector and the D[k] are N by N
matrices [4].
There are four remarkable
properties of the GHM scaling
functions, as follows:
 They each have short
support (the intervals [0,1]
and [0,2]).
 Both scaling functions are
symmetric, and the wavelets
form
a
symmetric/antisymmetric
pair.
 All integers translates of the
scaling
functions
are
orthogonal.
 The system has second order
of approximation.
Another example of symmetric
orthogonal multiwavelets with
approximation order 2 is due to
Chui and Lian [4]. Here both
scaling functions are supported on
[0,2], which is slightly longer than
GHM. For the CL system, only
three coefficients matrices are
required, but it is less smooth than
GHM ones.
Biorthogonal GHM, BiHermite
and another types of multiwavelets
were proposed depending on the
application that uses it [5].
1.CHOICE OF THE MULTIFILTERS:
In practice multiscaling and
wavelet functions often have
multiplicity r=2. An important
example was constructed by
Geronimo, Hardin and Massopust
[11], which we shall refer to as the
GHM system. For the GHM
multiscaling functions there are
two scaling functions 1(t), 2(t)
shown in Fig. 1 and the two
wavelets w1(t), w2(t) shown in Fig.
2. The dilation and wavelet
equations for this system have four
coefficients, The scaling matrices
G0 ,G1, G2 and G3 are
4 
 3
 3
5 2


5
G0  
 , G1   5 2
 1  3 
 9
10 2 
 20
 20
0 
0
 0
G2   9  3 
G3   1
 20
 20
10 2 


0 

1 
2 
0

0

And the wavelet matrices H0 ,H1,
H2 and H3 are:
3
than zero. For the GHM case,
we get =1/ 2 , since
2. PREPROCESSING:
Before the operation of
decomposition is applied to the
input data, the preprocessing
operation must be done. The aim
of preprocessing is to associate the
given scalar input signal of length
N to a sequence of length-2 vectors
{v0,k} in order to start the analysis
algorithm. Here N is assumed to be
a power of 2, and so is of even
length.
After
the
wavelet
reconstruction (synthesis) step a
postfilter is applied. Clearly,
prefiltering, wavelet transform,
inverse transform, and postfiltering
should recover the input signal
exactly if nothing else has been
done. A different type of
preprocessing was suggested such
as repeated row (oversampling),
matrix approximation (critical
sampling) and so on [8].
C 
[ H 0  H 1  H 2  H 3 ] C  
 2
 
16   C 

0 
1 8 

C



2
 
10 0
 2  0 
0




The output from the low-pass
multifilter is simply a scaled of
the input
[G0

1
5

In the case of the CL, BiGHM
and BiHermite systems =0
[6].
2.1 Repeated Row
Preprocessing: Oversampling:
In this case the input length-2
vectors are formed from the
original time series via
X 
v0 , k   k 
X k 
C 
 G1  G 2  G3 ] C  
 2
 
6

C 
4  C 
C


C 

2
2
 
 2
4
2   2 
 
2.2 Matrix (Approximation)
Preprocessing: Critical
Sampling:
Another way to preprocess the
input signal {Xk} is to brake it
in a sequence of 21
vectors{[X2(m+k)
X2(m+k)+1]T}
and apply the matrix prefilter P
k  0,..., N  1
Here  is a constant; it is
typically chosen so that if
Xk=C=const, for all k, then the
output from the high-pass
multifilter is zero. This can
always be done if the system
has approximation order higher
 X 2( m k ) 
v0,k   Pm 
,
X
m 0
 2( m k )1 
M
where P0, P1, …, PM are 22
matrix coefficients of P. For
4
the GHM system the following
prefilter with two coefficients
is often used. It preserves two
order of approximation [5].
 3
P0   8 2

 0
10 
8 2 ;
0 
 3
P1   8 2

 1
 3
Q0   8 6

 0

0

0

0

0

III- THE SCALAR MULTIWAVELET OMPUTATION
METHODS:
Strela [1] propose a method
depending on matrix multiplication
principle. The resulting two
channel, 22 matrix filter bank
operates on two input data streams,
filtering them into four output
streams, each of which is
downsampled by a factor of 2.
This is shown in Figure 3. Each
row of the multifilter is a
combination of two ordinary
filters, one operating on the first
data stream and the other operating
on the second. In the time domain,
an infinite lowpass matrix with
double shifts describes filtering
followed by downsampling:
1 0
Q0  

0 1 
The Xia Prefilter
2 

10 
3
2 
20 
2
The Minimal Matrix Prefilter
2 2
Q0  
 1
 3

Q1   8 6
 1
 13
The Minimal Repeated Signal
Prefilter is different from the
definition above. It is defined
by [10]:
 2
S 0,k  f k 1  
 1 
3. PREFILTERS:
A different type of prefilters
and postfilters was used. The
postfilter P that accompanies
the prefilter Q satisfies PQ=I,
where I is the identity matrix.
Therefore, if we apply a
prefilter, DMWT, inverse
DMWT, a postfilter to any
sequence, the output will be
identical to the input. The
commonly used prefilters are
as the following:
The Identity Prefilter

2
 2
10
Q0  
 2 3

20
5 
4 6 ,
0 
 .......

 c[3] c[2] c[1] c[0] 0 0

L
 0 0 c[3] c[2] c[1] c[0]


...... 

 2

0 
The Interpolation Prefilter
Each of the filter taps C[k] is a 2 
2 matrix. The eigenvalues of the
matrix L are critical. The solution
5
to the matrix dilation equation (1)
is a two-element vector of scaling
functions [7]. Any continuous-time
function f(t) inV0 can be expanded
as a linear combination
( 0)
on a multiplexer facility with down
sampling as shown in Fig. 6.
IV. PROPOSED COMPUTATION METHOD:
Several techniques of digital
signal processing (DSP) have been
proposed. In fact a great attention
was
given
to
convolution
operation, while it can be
applicable for different DSP
processes. DSP operations require
only simple arithmetic operations
of multiply, add/subtract and shifts
to carry out. We should notice the
simplicity between most of DSP
operations. The basic DSP
operations are convolution, which
is after specific modification, can
be altered to correlation or filtering
process [3].
The
term
convolution
describes how the input to a
system interacts with the system to
produce the output. Generally, the
system output will be delayed and
attenuation or amplified version of
the input [2].
This type of convolution is
known as a linear convolution. The
word linear refers to each term of
S1(n) and S2(n) that take part only
time in calculation of every term of
S3(n). In the table-lookup method
[3] the terms S1(n) and S2(n) are
arranged as a table by placing
S1(n) in principal row and S2(n) as
principal column such as Table (1).
This table is completed as a
multiplication table. Then by
adding the secondary diagonals,
( 0)
f (t )   v1,n1 (t  n)  v 2,n  2 (t  n)
n
The superscript (0) denotes an
expansion at scale level 0. Given
such a pair of sequences, their
coarse approximation is computed
with the lowpass part of the
multiwavelet filter bank:
Also, Martin [6] depends on
the Strela ideas and propose a new
analysis
and
synthesis
of
multiwavelet called Mult-Input
Multi-Output (MIMO) filterbank,
as shown in Fig. 4 for the r=2 case.
Each filterblock in Fig. 4 is really a
2-input, 2-ouitput system as shown
in Fig. 5.
Also, Tham, and others [9]
proposed a new approach for
analysis and application of discrete
multiwavelet transforms depending
6
S3(n) can be obtained. If S1(n) and
S2(n) have a number of terms N1
and N2 respectively, then S3(n) has
(M)terms such that M=(N1+N2)1.
This method is used in a
wavelet case [2], but in a
multiwavelet the case is different.
The vectors are not a single input
but a set of matrices (r=2). The
decomposition and reconstruction
structure are shown in Fig. 7
The algorithm of this method is as
follows in Figure 8.
The previous algorithm is added to
the two-dimensional signal (an
image) as in Fig. 9.
A new technique for
computing multiwavelet transform
is proposed. Its simple structure
and computing results in (50%)
reduction
of
multiplication
operation and greater than (50%)
reduction of addition operation.
The
reduction
computation
requirement
for transforming
makes
multiwavelet
more
comparable with scalar wavelet
transform with greater properties.
REFERENCES
1. V. Strela, “Multiwavelets:
Theory and applications”,
Ph.D. Thesis, June 1996.
2. J. C. Goswami, A. K. Chan,
“Fundamentals of wavelets,
theory,
algorithms,
and
applications”, John Willy and
Sons 1999.
3. C. S. Burrus, R. A. Gopinath,
and H. Guo, “Introduction to
wavelets
and
wavelet
transforms”, Prentice Hall Inc.
1998.
4. V. Strela and A. T. Walden,
“Orthogonal and Biorthogonal
multiwavelets
for
signal
denoising
and
image
compression”, Dep. Of Math.,
Dartmouth College, 6188
Bradley Hall, Hanover, NH
03755, U.S.A, 1999.
5. V. Strela, P. N Heller, G.
Strang, P.Topiwala and C.
Heil, “The application of
multiwavelet filterbanks to
image processing”, IEEE
transforms
om
image
V. COMPUTING COMPLEXITY:
The computation complexity
is an important criterion in
computer science and digital signal
processing. In spite of being the
convolution method lead to fully
reconstruction computation, it
gives also a less complexity. The
scalar
methods
have
(64)
multiplication and (48) addition for
transforming a signal of four input
data, while the proposed method
give a (32) multiplication and (21)
addition operation for the same
input size. So it is a well-suited
method for transforming large
signals and images in the
multiwavelet transform with less
computation complexity. Table (2)
show a comparition between other
methods.
VI. CONCLUSION:
7
processing, Vol. 8, No.4, April
1999.
6. M. B. Martin, “Applications of
multiwavelets
to
image
compression”,
M.Sc.
Thesis,Blacksburg
Virginia,
June 1999.
7. K.Cheung,
L.
Po,
“Preprocessing for discrete
multiwavelet transform of
two-dimensional signals”, City
Univ. of Hong Kong, Hong
Kong, 2002.
8. K.Cheung, L. Po, “Low
complexity 2-D preprocessing
for discrete
multiwavelet
transform of image signals”,
City Univ. of Hong Kong,
Hong Kong, 2001.
9. J.Y. Tham, L.Shen, S.L. Lee
and H.H. Tan, “A General
10. approach for analysis and
application
of
discrete
multiwavelet
transforms”,
IEEE transforms on signal
processing, Vol. 48, No.2,
Feb. 2000.
11. S. Li and Y. Wang, “Texture
classification using discrete
multiwavelet
transform”,
Hunan
Univ.,
Changsha,
410082, P.R. China, 2000.
12. G. Chen, “Applications of
wavelet transforms in pattern
recognition and de-noising”,
M.Sc.
Thesis,
Concordia
Univ., Jan. 1999.
8
9
Figure 7: A multiwavelet filterbank using convolution
1. The table lookup depends on variables represents the columns (N×1) of the signal input
(as a columns) and variables that represents a matrices (N×N) of the input filters.
Compute the result of the table in terms of its variables as an equation (for example
the column C1 multiplied by the matrix H3 give the result C1×H3).
2. Down sample the output columns that results from the table look up method.
3. The remaining variables are computed by matrix multiplication to find the
decomposition values.
4. In the reconstruction operation, up sampling operation adds a stream of zero columns
to the result and another lookup table for H and G filters is computed.
5. Add the results from H and G filters to have the original signal X(n).
Figure 8: Proposed algorithm
10
Figure 9: Image transform using discrete multiwavelet transform
S2(0)
S2(1)
S2(2)
….
S2(N1-1)
S1(0)
S1(0) S2(0)
S1(0) S2(0)
S1(0) S2(0)
…..
S1(0) S2(0)
S1(1)
S1(0) S2(0)
S1(0) S2(0)
S1(0) S2(0)
…..
S1(0) S2(0)
S1(2)
S1(0) S2(0)
S1(0) S2(0)
S1(0) S2(0)
…..
S1(0) S2(0)
…..
…..
…..
…..
…..
…..
S1(N1-1)
S1(0) S2(0)
S1(0) S2(0)
S1(0) S2(0)
…..
S1(0) S2(0)
Table (1): Linear convolution table
Computation method
Addition computation
Strela
Martin
Tham
Proposed
76
72
64
21
Multiplication
computation
56
51
48
32
Table (2): Computation complexity comparition
11