# A General Approach for Orthogonal 4-Tap Integer Multiwavelet Transforms

Authors: Mingli Jing; Hua Huang; Wuling Liu; Chun Qi
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

An algorithm for orthogonal 4-tap integer multiwavelet transforms is proposed. We compute the singular value decomposition (SVD) of block recursive matrices of transform matrix, and then transform matrix can be rewritten in a product of two block diagonal matrices and a permutation matrix. Furthermore, we factorize the block matrix of block diagonal matrices into triangular elementary reversible matrices (TERMs), which map integers to integers by rounding arithmetic. The cost of factorizing block matrix into TERMs does not increase with the increase of the dimension of transform matrix, and the proposed algorithm is in-place calculation and without allocating auxiliary memory. Examples of integer multiwavelet transform using DGHM and CL are given, which verify that the proposed algorithm is an executable algorithm and outperforms the existing algorithm for orthogonal 4-tap integer multiwavelet transform.


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## Body

## 1. Introduction

In many applications of image processing, the given data are integer valued. To compress digital image losslessly by means of transformation, the transform must map integers to integers and be perfectly invertible. Calderbank et al. [1] presented two wavelet transforms approaches that map integers to integers, and the proposed lifting scheme has been a popular method for mapping integers to integers. Using lifting scheme, Deever and Hemami [2] presented a projection-based technique for decreasing the first-order entropy of transform coefficients. However, Multiwavelets have several advantages in comparison to scalar wavelets. 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 time.Unlike scalar wavelet, little literature is available on integer multiwavelet transform. Cheung et al. [3] presented an integer multiwavelet transform and its associated integer prefilter based on box-and–slope multiscaling system. Van Fleet [4] developed a factorization of 4-tap multiwavelets with multiplicity 2. In particular, the author in [4] derived a factorization of the DGHM multiwavelet and applied DGHM multiwavelet processing to data compression. In order to implement the DGHM integer transform, the author computed scaling value α under certain conditions [4].The scaling value α is not convenient to compute and larger than 1, which turns out to be bad for lossless compression [1].This paper is to present an algorithm of 4-tap orthogonal multiwavelets that map integers to integers and be perfectly invertible using SVD and TERMs. Unlike [4], the proposed algorithm does not compute scaling value α and calculates in-place and without allocating auxiliary memory.


## 2. Integer Transform and Multiwavelet

### 2.1. Integer Transform

Hao and Shi [5] proved that there exists an implementation of integer mapping when a linear transform is invertible and in finite-dimensional space, as follows.Theorem 2.1.
MatrixA has a TERM factorization of A=PV1V2⋯VMDR if and only if |det(A)|=1, where M is finite, Vk(k=1,2…M) are unit TERMs, P is a permutation matrix, and DR is a rotator for only one complex number. If all the diagonal elements of a TERM are equal to 1, the TERM will be a unit triangular matrix. If A is a real matrix, then DR is an identity matrix.Unless otherwise stated, we will concentrate on the nonsingular real matrixA, thus reversible integer implementation for general linear transforms Y=AX can be derived as follows [5], where Y=[y1,y2,…,yn]T,  X=[x1,x2,…,xn]T.LetL=(li,j)n×n,U=(ui,j)n×n, and Pn×n be a lower TERM, an upper TERM, and a permutation matrix, respectively. If A=L is a lower TERM, the computational ordering of linear transform Y=AX can be arranged to be top-down:(2.1)yi={li,ixi,i=1,li,ixi+[∑k=1i-1li,kxk],2≤i≤n.
Its inverse ordering is reversed:(2.2)xi={yili,i,i=1,(1li,i)(yi-[∑k=1i-1li,kxk]),2≤i≤n.Likewise, ifA=U is an upper TERM, the computational ordering of linear transform Y=AX can be arranged to be top-down:(2.3)yi={ui,ixi,i=n,ui,ixi+[∑k=i+1nui,kxk],1≤i≤n-1.
Its inverse ordering is reversed:(2.4)xi={yiui,i,i=n,(1ui,i)(yi-[∑k=i+1nui,kxk]),1≤i≤n-1,
where [] denotes rounding arithmetic. Since permutation matrix Pn×n is an orthogonal matrix, transform and the inverse transform can be denoted by Y=Pn×nX and X=Pn×nTY, respectively.


### 2.2. Properties of Transform Matrix

While many researchers have investigated multiwavelet of multiplicityr [6–9], little work has been published on integer multiwavelet transforms. We will discuss integer transform on 4-tap multiwavelet of multiplicity r. Let Φ(t)=[ϕ1,…ϕr]T,  Ψ(t)=[ψ1,…ψr]T∈L2(R)r be a multiscaling function and a multiwavelet function, respectively, and then both Φ(t) and Ψ(t) satisfy the following two-scale relation: (2.5)Φ(t)=∑k=03PkΦ(2t-k),Ψ(t)=∑k=03QkΦ(2t-k),
where Pk and Qk are r×r matrices, respectively. For more details about multiwavelets, see, for example, [7–9]. As is the case in [4], the matrix representation Hn×n of multiwavelet transformation is(2.6)Hn×n=[H0H1H0H1⋱⋱H0H1H1H0],
where Hi=[P2iP2i+1Q2iQ2i+1],  i=0,1,Pj,Qj,j=0,1,2,3, are r×r matrices, respectively.In order to find integer multiwavelet transform schemes, inspired by [1], we will prove some remarkable properties of orthogonal matrix H and then present the approach that maps integers to integers.Property 1.
Suppose thatrank(H0)=m, rank(H1)=s, thus m+s=2r.Proof.
From the condition of orthogonality,HTH=HHT=In×n. It follows that
(2.7)H0H0T+H1H1T=I2r×2r,H1H0T=H0H1T=0,2r=rank(I)=rank(H0H0T+H1H1T)≤rank(H0)+rank(H1),0=rank(H0H1T)≥rank(H0)+rank(H1)-2r.
Thus, rank(H1)+rank(H0)=2r. The proof is completed.LetU0Σ0V0T and U1Σ1V1T be the singular value decomposition of H0,H1, respectively. By Property 1, since rank(H1)=rank(Σ1), rank(H0)=rank(Σ0), we see that rank(Σ0)+rank(Σ1)=2r.Property 2.
If  theλi are the singular values of H0 and the vectors u0i and v0i are the ith left singular vector and ith right singular vector, respectively, and the μi are the singular values of H1 and the vectors u1j and v1j are the jth left singular vector and ith right singular vector, respectively, then u0i and v0i are orthogonal to the vectors u1j and v1j, respectively, i=1,2,…m,  m=rank(H0),  j=1,2…s,s=rank(H1).Proof.
From (2.7), we have
(2.8)H1H0T=U1Σ1V1TV0Σ0U0T=0.
Since U0,U1 are two orthogonal matrices, we have Σ1V1TV0Σ0=0,(2.9)Σ1V1TV0Σ0=Σ1[v11,…,v12r]T[v01,…,v02r]Σ0=[λ1⋱λm0][(v11)T(v12)T⋮(v12r)T][v01v02,…,v02r][μ1⋱μs0]=[λ1(v11)T⋮λm(v1m)T0][μ1v01,…,μsv0s0]=[λ1μ1〈v11,v01〉⋯λ1μs〈v11,v0s〉0⋮⋮⋮λmμ1〈v1m,v01〉⋯λmμs〈v1m,v0s〉00⋯00]=0.
From (2.9), we have 〈v0i,v1j〉=0. Likewise, 〈u0i,u1j〉=0. The proof is completed.By Property2, we can rewrite the matrices H0 and H1 in this case as(2.10)H0=U0Σ0V0T=[u01,u02,…,u0m,u11,u12,…,u1s]×diag(λ1,…,λm,0⋯0︸s)[v01,v02,…,v0m,v11,v12,…,v1s]T=U̅Σ0V̅T,(2.11)H1=U1Σ1V1T=[u01,u02,…,u0m,u11,u12,…,u1s]diag(0,…,0︸m,μ1,…,μs)×[v01,v02,…,v0m,v11,v12,…,v1s]T=U̅Σ̅1V̅T,
where U̅=[u01,u02,…,u0m,u11,u12,…,u1s],  V̅=[v01,v02,…,v0m,v11,v12,…,v1s]∑0=diag(λ1,…,λm,0⋯0︸s)  ∑̅1=diag(0⋯0︸,mμ1,…,μs).  From Property 2, we see that U̅ and V̅ are two orthogonal matrices.Property 3.
If theλi and μj are the nonzero singular values of H0 and H1, respectively, then the nonzero singular values of H0 and H1 are 1.Proof.
From (2.7), we can derive
(2.12)I=H0H0T+H1H1T=U̅Σ0V̅TV̅Σ0U̅T+U̅Σ̅1V̅TV̅Σ̅1U̅T=U̅(Σ02+Σ̅12)U̅T.
Left multiplying (2.12) with U̅-1and right multiplying (2.11) with(U̅T)-1, we have I=Σ02+Σ̅12=diag(λ12,…,λm2,μ12,…,μs2).  The proof is completed.By Properties1 and 2, we can rewrite the matrix Hn×n in this case as(2.13)Hn×n=[H0H1H0H1⋱H1H0]=[U̅U̅⋱U̅][Σ0Σ̅1Σ0Σ̅1Σ̅1Σ0][V̅TV̅T⋱V̅T]=UP̅VT,
where Un×n and Vn×nT are block diagonal with the same 2r×2r orthogonal matrices U̅,V̅T comprising the diagonal, respectively, and P̅n×n is a permutation matrix. The advantages of the re-representation of transform Hn×n are as follows:(1)
the TERMs factorization ofHn×n directly requires more calculation than U̅ and V̅, because the TERMs factorization of Hn×n directly involves o(n2) operations [5], and the TERMs factorization of U̅(V̅) involves o((2r)2) operations; in general, U̅(V̅) is 4 by 4 matrices with the case r=2;(2)
integer multiwavelet transform can map integers to integers in-place and without allocating auxiliary memory (see Section2.1).


## 2.1. Integer Transform

Hao and Shi [5] proved that there exists an implementation of integer mapping when a linear transform is invertible and in finite-dimensional space, as follows.Theorem 2.1.
MatrixA has a TERM factorization of A=PV1V2⋯VMDR if and only if |det(A)|=1, where M is finite, Vk(k=1,2…M) are unit TERMs, P is a permutation matrix, and DR is a rotator for only one complex number. If all the diagonal elements of a TERM are equal to 1, the TERM will be a unit triangular matrix. If A is a real matrix, then DR is an identity matrix.Unless otherwise stated, we will concentrate on the nonsingular real matrixA, thus reversible integer implementation for general linear transforms Y=AX can be derived as follows [5], where Y=[y1,y2,…,yn]T,  X=[x1,x2,…,xn]T.LetL=(li,j)n×n,U=(ui,j)n×n, and Pn×n be a lower TERM, an upper TERM, and a permutation matrix, respectively. If A=L is a lower TERM, the computational ordering of linear transform Y=AX can be arranged to be top-down:(2.1)yi={li,ixi,i=1,li,ixi+[∑k=1i-1li,kxk],2≤i≤n.
Its inverse ordering is reversed:(2.2)xi={yili,i,i=1,(1li,i)(yi-[∑k=1i-1li,kxk]),2≤i≤n.Likewise, ifA=U is an upper TERM, the computational ordering of linear transform Y=AX can be arranged to be top-down:(2.3)yi={ui,ixi,i=n,ui,ixi+[∑k=i+1nui,kxk],1≤i≤n-1.
Its inverse ordering is reversed:(2.4)xi={yiui,i,i=n,(1ui,i)(yi-[∑k=i+1nui,kxk]),1≤i≤n-1,
where [] denotes rounding arithmetic. Since permutation matrix Pn×n is an orthogonal matrix, transform and the inverse transform can be denoted by Y=Pn×nX and X=Pn×nTY, respectively.


## 2.2. Properties of Transform Matrix

While many researchers have investigated multiwavelet of multiplicityr [6–9], little work has been published on integer multiwavelet transforms. We will discuss integer transform on 4-tap multiwavelet of multiplicity r. Let Φ(t)=[ϕ1,…ϕr]T,  Ψ(t)=[ψ1,…ψr]T∈L2(R)r be a multiscaling function and a multiwavelet function, respectively, and then both Φ(t) and Ψ(t) satisfy the following two-scale relation: (2.5)Φ(t)=∑k=03PkΦ(2t-k),Ψ(t)=∑k=03QkΦ(2t-k),
where Pk and Qk are r×r matrices, respectively. For more details about multiwavelets, see, for example, [7–9]. As is the case in [4], the matrix representation Hn×n of multiwavelet transformation is(2.6)Hn×n=[H0H1H0H1⋱⋱H0H1H1H0],
where Hi=[P2iP2i+1Q2iQ2i+1],  i=0,1,Pj,Qj,j=0,1,2,3, are r×r matrices, respectively.In order to find integer multiwavelet transform schemes, inspired by [1], we will prove some remarkable properties of orthogonal matrix H and then present the approach that maps integers to integers.Property 1.
Suppose thatrank(H0)=m, rank(H1)=s, thus m+s=2r.Proof.
From the condition of orthogonality,HTH=HHT=In×n. It follows that
(2.7)H0H0T+H1H1T=I2r×2r,H1H0T=H0H1T=0,2r=rank(I)=rank(H0H0T+H1H1T)≤rank(H0)+rank(H1),0=rank(H0H1T)≥rank(H0)+rank(H1)-2r.
Thus, rank(H1)+rank(H0)=2r. The proof is completed.LetU0Σ0V0T and U1Σ1V1T be the singular value decomposition of H0,H1, respectively. By Property 1, since rank(H1)=rank(Σ1), rank(H0)=rank(Σ0), we see that rank(Σ0)+rank(Σ1)=2r.Property 2.
If  theλi are the singular values of H0 and the vectors u0i and v0i are the ith left singular vector and ith right singular vector, respectively, and the μi are the singular values of H1 and the vectors u1j and v1j are the jth left singular vector and ith right singular vector, respectively, then u0i and v0i are orthogonal to the vectors u1j and v1j, respectively, i=1,2,…m,  m=rank(H0),  j=1,2…s,s=rank(H1).Proof.
From (2.7), we have
(2.8)H1H0T=U1Σ1V1TV0Σ0U0T=0.
Since U0,U1 are two orthogonal matrices, we have Σ1V1TV0Σ0=0,(2.9)Σ1V1TV0Σ0=Σ1[v11,…,v12r]T[v01,…,v02r]Σ0=[λ1⋱λm0][(v11)T(v12)T⋮(v12r)T][v01v02,…,v02r][μ1⋱μs0]=[λ1(v11)T⋮λm(v1m)T0][μ1v01,…,μsv0s0]=[λ1μ1〈v11,v01〉⋯λ1μs〈v11,v0s〉0⋮⋮⋮λmμ1〈v1m,v01〉⋯λmμs〈v1m,v0s〉00⋯00]=0.
From (2.9), we have 〈v0i,v1j〉=0. Likewise, 〈u0i,u1j〉=0. The proof is completed.By Property2, we can rewrite the matrices H0 and H1 in this case as(2.10)H0=U0Σ0V0T=[u01,u02,…,u0m,u11,u12,…,u1s]×diag(λ1,…,λm,0⋯0︸s)[v01,v02,…,v0m,v11,v12,…,v1s]T=U̅Σ0V̅T,(2.11)H1=U1Σ1V1T=[u01,u02,…,u0m,u11,u12,…,u1s]diag(0,…,0︸m,μ1,…,μs)×[v01,v02,…,v0m,v11,v12,…,v1s]T=U̅Σ̅1V̅T,
where U̅=[u01,u02,…,u0m,u11,u12,…,u1s],  V̅=[v01,v02,…,v0m,v11,v12,…,v1s]∑0=diag(λ1,…,λm,0⋯0︸s)  ∑̅1=diag(0⋯0︸,mμ1,…,μs).  From Property 2, we see that U̅ and V̅ are two orthogonal matrices.Property 3.
If theλi and μj are the nonzero singular values of H0 and H1, respectively, then the nonzero singular values of H0 and H1 are 1.Proof.
From (2.7), we can derive
(2.12)I=H0H0T+H1H1T=U̅Σ0V̅TV̅Σ0U̅T+U̅Σ̅1V̅TV̅Σ̅1U̅T=U̅(Σ02+Σ̅12)U̅T.
Left multiplying (2.12) with U̅-1and right multiplying (2.11) with(U̅T)-1, we have I=Σ02+Σ̅12=diag(λ12,…,λm2,μ12,…,μs2).  The proof is completed.By Properties1 and 2, we can rewrite the matrix Hn×n in this case as(2.13)Hn×n=[H0H1H0H1⋱H1H0]=[U̅U̅⋱U̅][Σ0Σ̅1Σ0Σ̅1Σ̅1Σ0][V̅TV̅T⋱V̅T]=UP̅VT,
where Un×n and Vn×nT are block diagonal with the same 2r×2r orthogonal matrices U̅,V̅T comprising the diagonal, respectively, and P̅n×n is a permutation matrix. The advantages of the re-representation of transform Hn×n are as follows:(1)
the TERMs factorization ofHn×n directly requires more calculation than U̅ and V̅, because the TERMs factorization of Hn×n directly involves o(n2) operations [5], and the TERMs factorization of U̅(V̅) involves o((2r)2) operations; in general, U̅(V̅) is 4 by 4 matrices with the case r=2;(2)
integer multiwavelet transform can map integers to integers in-place and without allocating auxiliary memory (see Section2.1).


## 3. Multiwavelet Integer Transform

SinceU̅,V̅ are two 2r×2r orthogonal matrices, we suppose that the input data {x1,x2,…,xn} is denoted by {[x11,j,x21,j,…x2r1,j]T,[x12,j,x22,j,…x2r2,j]T,…,[x1L,j,x2L,j,…x2rL,j]T}, that is, {X1,j,X2,j,…XL,j}, where L is a positive integer, the superscript j denotes the jth stages of the transform of Xk,j,  k=1,2,…L, and k is the kth sequence of input data. When {Xk,j}k=1L is an initial input data,  j equals 0. We can state an algorithm for 4-tap orthogonal integer multiwavelet transform as follows.(1)
Compute the singular value decomposition ofH0, H1, and find U̅,  P̅=[Σ0|Σ̅1]2r×4r,  V̅T.(2)
Find the factorization ofU̅, V̅T, such as U̅=P0V1V2,…,VM,  VT=P1W1W2,…,WM, respectively.(3)
Combine the data sequence{Xk,j}k=1Lwith the matrices WM,…W1,P1,P̅,VM,…V1,P0 by corresponding equations in Section 2.1, sequentially, according to transform matrix form (upper, lower, and permutation matrix).Specially speaking, whenj equals M+1,  P̅ is a 2r×4r matrix, so the ordering of transform can be carried through(3.1)Xk,j+1={Σ0Xk,j+Σ̅1Xk+1,j,k=1,2…(L-1),Σ0Xk,j+Σ̅1X1,j,k=L,
where Xk,j is a 2r×1 matrix.Thejth stage of the transform is a product of Xk,j and the jth matrix of the matrices WM,…W1,P1,P̅,VM,…V1,P0 from left to right by rounding arithmetic (see Section 2.1). Since Xk,j is integer valued, Xk,j+1 is also integer valued. After j=M+M+3 stage of the transform, each stage of this process can map integers to integers, therefore transform results of the initial data sequence {Xk,j}k=1L are integer valued.The inverse transform must be the backward running of the forward transformation.


## 4. Experimental Results

In order to demonstrate the availability of the proposed scheme, DGHM [6] and CL [7] multiwavelets are used as examples. The preprocessing is given by balancing the nonbalanced multiwavelet. The way to achieve this is to find the orthogonal matrix R, such that RTPiR=P̅i and RTQiR=Q̅i [10]. Since R is an orthogonal matrix, the new matrices H̅0=DTH0D and H̅1=DTH1D inherit all of the properties of H0 and H1 in Section 2.2, where D=diag(R,R). We can compute the SVD of matrices H̅0 and H̅1, respectively, thus we will have two matrices U̅ and V̅T.For the DGHM case, the corresponding matrices in (2.5) are (4.1)P0=[321045-120-3220],P1=[00920-3220],P2=[3210092022],P3=[00-1200],Q0=[320362000],Q1=[-9320660-33],Q2=[3320-6203610-35],Q3=[-3600-6300].
By orthogonal matrix R [11], (4.2)R=16[1-21+21+2-1+2],
we can find the matrices U̅,  V̅T, and P̅ as follows: (4.3)U̅=[0.01579120.1188553-0.989179-0.084551-0.6383476-0.5786332-0.1218390.49279930.7295663-0.6631722-0.08055460.14644660.24494070.4596299-0.0138210.8535535],V̅T=[00.3445041-0.8684815-0.35645020-0.07195550.3541483-0.932417-0.2841407-0.8974428-0.3325729-0.05706050.9587826-0.2659623-0.0985599-0.0169102],P̅=[Σ0|Σ̅1], where Σ0=diag(1,1,1,0), Σ̅1=diag(0,0,0,1), and then compute the TERMs factorization of U̅,  V̅T as follows: (4.4)U̅=P0V1V2,…,V8=[0010010010000001][1000-0.5042912100-0.0209632-0.1149459100.1175034-0.58873660.27063071][1100010000110001]×[1000-0.2704337100001000-0.14644671][1-1.3706774000100001-1.17157280001]×[100001100-2.1827521100001][1000011000100001][100001000-1.8454857100001]×[1-0.9089951-0.11041440.2007310011.3487059-0.51665130010.01619230001],V̅T=P1W1W2,…,W8=[0010000101001000][1000-0.2533664100-0.0209632-0.3528807100.00316850.0737051-0.30980481][1100010000110001]×[1000-0.0412174100001000-0.0675831][1-1.0429893000100001-1.07248150001]×[100001000-2.0683494100001][1000011000100001][100001000-1.9360232100001]×[1-0.2773958-0.1027969-0.0176372011.43892790.46935910010.37981750001].
Likewise, for the CL case, the corresponding matrices in (2.5) are(4.5)P0=[10-3104056-2154056-315405-31040],P1=[30+3104056-21540-56+7154015-31040],P2=[30+31040-56-2154056+7154015-31040],P3=[10-31040-56-21540-56-315405-31040],Q0=[56-21540-10-31040-5-3104056-31540],Q1=[-56-2154030+3104015-3104056+71540],Q2=[-56-21540-30+31040-15-3104056+71540],Q3=[56-2154010-310405-3104056-31540].
By matrix R [11],(4.6)R=22[1-111],
matrices U̅,  V̅T, and P̅ can be represented as(4.7)U̅=[-0.7104132-0.6924131-0.09023430.0879480.087948-0.0902343-0.6924132-0.71041330.09023430.087948-0.71041320.69241320.6924132-0.71041330.0879480.0902343],V̅T=[0.11326590-0.993564700-0.11326590-0.9935648-0.9935647-0.0000001-0.11326590.00000010.0000001-0.993564700.1132659],P̅=[Σ0|Σ̅1], where Σ0=diag(1,1,0,0), Σ̅1=diag(0,0,1,1), and then compute the TERMs factorization of U̅, V̅T as follows: (4.8)U̅=P0V1V2⋯V8=[1000000100100100][1000-3.3822942100-0.1270166010-0.42960760.1270166-1.43296871][1100010000110001]×[1000-1.7104132100001000-1.71041321][11.40763150001000011.40763140001][1000010000.0161332100001]×[1000011000100001][100001000-0.015877100001][10.97466250.1270166-0.123798401-1.01613320.8633704001-0.97466270001],V̅T=P1W1W2,…,W8=[0010000110000100][1000-2.0064769100-0.1139995010-0.22873740.1139995-2.00647691][1100010000110001]×[1000-1.9935647100001000-1.99356481][11.0064770001000011.00647690001][1000011000100001]×[100001000-0.0128291100001][100.1139995001-1.012996-0.1139994001-0.00000010001][1000010000.0129958100001].
In order to test the effectiveness of the proposed scheme, the effectiveness for lossless compression is measured using the weighted entropy given by [1]. For comparison, we will focus on 4-tap multiwavelets, such as CL, DGHM, and two scalar wavelets: the Daubechies four-tap orthogonal wavelet D4 and the Daubechies biorthogonal wavelet 9-7. We use seven test images, which are 512×512 pixels in size, and we decompose each image one level with each transform. The weighted entropies are tabulated in Table 1.Table 1
Weighted entropies of transformed images.
LenaBoatBaboonManCouplePlanePeppersDGHM5.22285.30936.49265.79005.63044.92205.4094CL5.18225.25506.45815.74155.60824.86895.3871DGHM [4]8.01837.84428.36868.24708.06047.48408.1571D4 [1]5.73885.77546.80126.24166.06855.44165.83749-7 [1]5.07315.15456.39755.68225.50984.70995.3542These results suggest that our method outperforms the method of [4], because the method of [4] enlarges the dynamic range of the output data by scaling value α. It is also worth noting that the performance of CL is very close to that of 9-7. Since preprocessing of the input data is a crucial point in nonbalanced multiwavelet applications [8], we used order-one balanced multiwavelets by balancing the nonbalanced multiwavelet [10]. The focus of this paper is to develop an algorithm of integer multiwavelet transform rather than construct several balanced multiwavelets by orthogonal matrix R, and then a further improvement in the balanced order is needed.


## 5. Conclusions

This paper presented a novel algorithm for orthogonal 4-tap integer multiwavelet transforms. Unlike the existing algorithm, transform matrix can be rewritten in a product of two block diagonal matrices and a permutation matrix, and then we factorize block matrix of block diagonal matrices into TERMs, which is lower than the TERMs factorization ofHn×n directly in computational cost. In addition, the proposed algorithm computes in-place and without allocating auxiliary memory. Better results could perhaps be achieved with higher balanced order and also with other multiwavelet bases.



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*Source: 163758-2010-07-06.xml*