`Mathematical Problems in EngineeringVolume 2010, Article ID 163758, 12 pageshttp://dx.doi.org/10.1155/2010/163758`
Research Article

A General Approach for Orthogonal 4-Tap Integer Multiwavelet Transforms

1School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an 710049, China
2School of Statistics, Xi'an University of Finance and Economics, Xi'an 710061, China
3School of Information Science and Engineering, Southeast University, Nanjing 211189, China

Received 12 December 2009; Accepted 11 May 2010

Copyright © 2010 Mingli Jing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 under certain conditions [4].The scaling value 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 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. Matrix has a TERM factorization of if and only if , where is finite, are unit TERMs, is a permutation matrix, and 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 is a real matrix, then is an identity matrix.

Unless otherwise stated, we will concentrate on the nonsingular real matrix , thus reversible integer implementation for general linear transforms can be derived as follows [5], where ,  .

Let , and be a lower TERM, an upper TERM, and a permutation matrix, respectively. If is a lower TERM, the computational ordering of linear transform can be arranged to be top-down: Its inverse ordering is reversed:

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

2.2. Properties of Transform Matrix

While many researchers have investigated multiwavelet of multiplicity [69], little work has been published on integer multiwavelet transforms. We will discuss integer transform on 4-tap multiwavelet of multiplicity . Let ,   be a multiscaling function and a multiwavelet function, respectively, and then both and satisfy the following two-scale relation: where and are matrices, respectively. For more details about multiwavelets, see, for example, [79]. As is the case in [4], the matrix representation of multiwavelet transformation is where ,  , are matrices, respectively.

In order to find integer multiwavelet transform schemes, inspired by [1], we will prove some remarkable properties of orthogonal matrix and then present the approach that maps integers to integers.

Property 1. Suppose that , , thus .

Proof. From the condition of orthogonality, . It follows that Thus, . The proof is completed.

Let and be the singular value decomposition of ,, respectively. By Property 1, since , , we see that .

Property 2. If the are the singular values of and the vectors and are the th left singular vector and th right singular vector, respectively, and the are the singular values of and the vectors and are the th left singular vector and th right singular vector, respectively, then and are orthogonal to the vectors and , respectively,   ,  .

Proof. From (2.7), we have Since are two orthogonal matrices, we have From (2.9), we have . Likewise, . The proof is completed.

By Property 2, we can rewrite the matrices and in this case as where ,    . From Property 2, we see that and are two orthogonal matrices.

Property 3. If the and are the nonzero singular values of and , respectively, then the nonzero singular values of and are 1.

Proof. From (2.7), we can derive Left multiplying (2.12) with and right multiplying (2.11) with, we have . The proof is completed.

By Properties 1 and 2, we can rewrite the matrix in this case as where and are block diagonal with the same orthogonal matrices comprising the diagonal, respectively, and is a permutation matrix. The advantages of the re-representation of transform are as follows:(1)the TERMs factorization of directly requires more calculation than and , because the TERMs factorization of directly involves operations [5], and the TERMs factorization of involves operations; in general, is 4 by 4 matrices with the case (2)integer multiwavelet transform can map integers to integers in-place and without allocating auxiliary memory (see Section 2.1).

3. Multiwavelet Integer Transform

Since are two orthogonal matrices, we suppose that the input data is denoted by that is, , where is a positive integer, the superscript denotes the th stages of the transform of , , and is the th sequence of input data. When is an initial input data, equals 0. We can state an algorithm for 4-tap orthogonal integer multiwavelet transform as follows.(1)Compute the singular value decomposition of , , and find ,  ,  .(2)Find the factorization of , , such as ,,, ,,, respectively.(3)Combine the data sequence with the matrices ,,,,,, by corresponding equations in Section 2.1, sequentially, according to transform matrix form (upper, lower, and permutation matrix).

Specially speaking, when equals , is a matrix, so the ordering of transform can be carried through where is a matrix.

The th stage of the transform is a product of and the th matrix of the matrices from left to right by rounding arithmetic (see Section 2.1). Since is integer valued, is also integer valued. After stage of the transform, each stage of this process can map integers to integers, therefore transform results of the initial data sequence 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 , such that and [10]. Since is an orthogonal matrix, the new matrices and inherit all of the properties of and in Section 2.2, where . We can compute the SVD of matrices and respectively, thus we will have two matrices and .

For the DGHM case, the corresponding matrices in (2.5) are By orthogonal matrix [11], we can find the matrices ,  , and as follows: , where , , and then compute the TERMs factorization of ,   as follows: Likewise, for the CL case, the corresponding matrices in (2.5) are By matrix [11], matrices ,   and can be represented as, where , , and then compute the TERMs factorization of , as follows: 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 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.

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 . 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 , 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 of 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 60972142) and 973 program (Project no. 2010CB327904).

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