A General Approach for Orthogonal 4-Tap Integer Multiwavelet Transforms

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.


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.

Theorem 2.1. Matrix A has a TERM factorization of
where M is finite, V k k 1, 2 . . .M are unit TERMs, P is a permutation matrix, and D R 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 D R is an identity matrix.
Unless otherwise stated, we will concentrate on the nonsingular real matrix A, thus reversible integer implementation for general linear transforms Y AX can be derived as follows 5 , where Y y 1 , y 2 , . . ., y n T , X x 1 , x 2 , . . ., x n T .
Let L l i,j n×n , U u i,j n×n , and P n×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
Its inverse ordering is reversed:

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Likewise, if A U is an upper TERM, the computational ordering of linear transform Y AX can be arranged to be top-down: Its inverse ordering is reversed: where denotes rounding arithmetic.Since permutation matrix P n×n is an orthogonal matrix, transform and the inverse transform can be denoted by Y P n×n X and X P T n×n Y , respectively.

Properties of Transform Matrix
While many researchers have investigated multiwavelet of multiplicity r 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 φ 1 , . . .φ r T , Ψ t ψ 1 , . . .ψ r T ∈ L 2 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: where P k and Q k are r × r matrices, respectively.For more details about multiwavelets, see, for example, 7-9 .As is the case in 4 , the matrix representation H n×n of multiwavelet transformation is where , i 0, 1,P j , Q j , 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.

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Property 1. Suppose that rank H 0 m, rank H 1 s, thus m s 2r.
Proof.From the condition of orthogonality, H T H HH T I n×n .It follows that

2.7
Thus, rank H 1 rank H 0 2r.The proof is completed.Proof.From 2.7 , we have Since U 0 , U 1 are two orthogonal matrices, we have 2.9 From 2.9 , we have 0. The proof is completed.
By Property 2, we can rewrite the matrices H 0 and H 1 in this case as 10 where where U n×n and V T n×n are block diagonal with the same 2r × 2r orthogonal matrices U, V T comprising the diagonal, respectively, and P n×n is a permutation matrix.The advantages of the re-representation of transform H n×n are as follows: 1 the TERMs factorization of H n×n directly requires more calculation than U and V , because the TERMs factorization of H n×n directly involves o n 2 operations 5 , and the TERMs factorization of U V involves o 2r 2 operations; in general, U V 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 Section 2.1 .

Multiwavelet Integer Transform
Since U, V are two 2r × 2r orthogonal matrices, we suppose that the input data that is, {X 1,j , X 2,j , . . .X L,j }, where L is a positive integer, the superscript j denotes the jth stages of the transform of X k,j , k 1, 2, . . .L, and k is the kth sequence of input data.When {X k,j } L k 1 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 of H 0 , H 1 , and find U, P 3 Combine the data sequence {X k,j } L k 1 with the matrices W M ,. . .W 1 ,P 1 ,P ,V M ,. . .V 1 ,P 0 by corresponding equations in Section 2.1, sequentially, according to transform matrix form upper, lower, and permutation matrix .Specially speaking, when j equals M 1, P is a 2r × 4r matrix, so the ordering of transform can be carried through where X k,j is a 2r × 1 matrix.
The jth stage of the transform is a product of X k,j and the jth matrix of the matrices W M , . . .W 1 , P 1 , P, V M , . . .V 1 , P 0 from left to right by rounding arithmetic see Section 2.1 .Since X k,j is integer valued, X k,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 {X k,j } L k 1 are integer valued.The inverse transform must be the backward running of the forward transformation.

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 R T P i R P i and R T Q i R Q i 10 .Since R is an orthogonal matrix, the new matrices all of the properties of H 0 and H 1 in Section 2.2, where D diag R, R .We can compute the SVD of matrices H 0 and H 1 , respectively, thus we will have two matrices U and V T .For the DGHM case, the corresponding matrices in 2.5 are

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By orthogonal matrix R 11 , we can find the matrices U, V T , and P as follows:

4.4
Likewise, for the CL case, the corresponding matrices in 2.5 are

4.8
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.These 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 orderone 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.

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 of H n×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.
If the λ i and μ j are the nonzero singular values of H 0 and H 1 , respectively, then the nonzero singular values of H 0 and H 1 are 1.
μ s .From Property 2, we see that U and V are two orthogonal matrices.Property 3.

Table 1 :
Weighted entropies of transformed images.