art%3a10.1007%2fs11042-006-0006-6

The application of symmetric orthogonal multiwaveletsand prefilter technique for image compression

Jiazhong Chen & Xun Ouyang & Wu Zheng & Jun Xu &

Jingli Zhou & Shengsheng Yu

Published online: 19 May 2006# Springer Science + Business Media, LLC 2006

Abstract Multiwavelets are the new addition to the body of wavelet theory. Thereare many types of symmetric multiwavelets such as GHM and CL. However, thematrix filters generating the GHM system multiwavelets do not satisfy thesymmetric property. Apparently, GHM cannot solve the edge problem accurately.For this reason, this paper presents some formulas for constructing the symmetricorthogonal matrix filters, which leads the symmetric orthogonal multiwavelets(SOM). Moreover, we analyze the frequency property by vanishing moments andprefilter technology to get a good combining frequency property. To prove the goodproperty of SOM in image compression application, we compared the compressioneffect with other writers’ work, which was in published literature. Extensiveexperimental results demonstrate that our new symmetric orthogonal matrix filterscombining with the prefilter technology and coefficient reorganization exhibitperformance equal to, or in several cases superior to the GHM and CL symmetricmultiwavelets.

Keywords Image compression . Multiwavelets . Prefilter technique

1 Introduction

Recently, multiwavelets have been introduced as a more general context in thewavelet theory, which is led by the notion that there is more than one scalingfunction. Multiwavelets have several advantages in comparison to scalar wavelets.

Multimed Tools Appl (2006) 29: 175–189DOI 10.1007/s11042-006-0006-6

J. Chen (*) :W. Zheng : J. Xu : J. Zhou : S. YuComputer School, Huazhong University of Science & Technology,Wuhan, Hubei 430074, P.R. Chinae-mail: [email protected]

X. OuyangFaculty of Business, Law and Computing, University of Derby, Derby, UKe-mail: [email protected]

The features such as compact support, orthogonality, symmetry, and high ordervanishing moments are known to be important in signal processing. A scalar waveletcannot possess all these properties at the same time but multiwavelets can.

The study of multiwavelets was initiated from Goodman, Lee and Tang in [7].Then Goodman and Lee in [6] discovered the characterization of scaling functionsand wavelets. In [10] Jia et al. constructed a class of continuous orthogonal doublewavelets with symmetry, short support, and orthogonality. The special case of [10]with multiplicity 2 and support [0,2], was studied by Chui and Lian [2]. In [9], Hongand Wu constructed a class of multiwavelets with multiplicity 4 and support [0,2].Generally, after the presenting of prefilter technique, multiwavelets with multiplicity2 can be applied in image compression application successfully [3, 15, 16].

The matrix filters of GHM were constructed by Geronimo, Hardin, andMassopust, and cannot solve the edge problem accurately in image coding for thematrix filters are not symmetric [5]. Though the matrix filters generating CL aresymmetric, the construction of matrix filters suffer from the problem of a lack ofuniversality. So we present some formulas for constructing a symmetric lowpassmatrix filters at first. For scalar wavelets, the highpass filters are determined by anautomatic way from the lowpass filters but it often fails here. The reason is that thelowpass filters are represented by matrices and they do not commute [14]. So wegive some simple formulas to construct the highpass filters. For the complexity ofconstructing the lowpass and highpass filters, we focus only on the orthogonality butomit the good frequency property, which means that the highpass filters are equal tozero at w = 0. So a prefilter technique should be used to get a good combiningfrequency property.

In order to evaluate the performance of multiwavelets for image coding at low bitrate, efficient SPIHT coding of multiwavelet coefficients has been realized, ac-complished with a suitable scanning strategy across scales and inside each detailsubband. Multiwavelet coefficients are more suitable for performing compressionthan scalar wavelet coefficients, for multiwavelet transforms are able to pack a largerproportion of the image energy into the lower frequency subbands than scalarwavelet. Extensive experimental results demonstrate that the performance of ourtechniques is equal to, and in several cases even outperform that of the existing matrixfilters.

2 Overview of symmetric orthogonal multiwavelets

Let 81,82,...,8r be multiscaling functions, and y1,y2,...yr be multiwavelet functions.We suppose that 8 = {81,82,...8r}

T and y = {y1,y2,...yr}T r2N satisfy the following

equations, respectively

8 xð Þ ¼X

k

Hk8 2x� kð Þ ð1Þ

y xð Þ ¼X

k

Gk8 2x� kð Þ ð2Þ

176 Multimed Tools Appl (2006) 29: 175–189

Hk and Gk are matrix conjugate quadrature filter (CQF). The orthogonal con-ditions now can be expressed as following

X

k

HkHTkþ2j ¼ 2�j0Ir; j 2 Z ð3Þ

X

k

GkGTkþ2j ¼ 2�j0Ir; j 2 Z ð4Þ

X

k

HkGTkþ2j ¼ 0; j 2 Z ð5Þ

In [10] Jia et al. constructed a class of continuous orthogonal basis for L2(R) withsymmetric–antisymmetric properties. In this subsection, we will consider matrixfilters that generate orthogonal multiscaling functions 8 and multiwavelets y withsymmetry. Here we discuss the case r = 2 and consider the situation where the firstcomponent of 8 and y are symmetric, whereas the second components are anti-symmetric, then 8(x) = S8(Ljx) where S = diag (1,j1). Using this in the refinementfunction (1), we obtain

8 ¼X

k

Hk8 2� � kð Þ ¼ S8 L� �ð Þ ¼X

k

SHk8 2L� 2� � kð Þ ¼X

k

SHkS8 2� þ k� Lð Þ

¼X

k

SHL�kS8 2� � kð Þ

If we assume L = 2Nj1, N2Z, then 8 will have the desired symmetric propertyprovided that Hk = SH2Nj1jkS. In this paper, we consider the case of r = 2, andN = 2. We can suppose the lowpass matrix filter Hk for scaling function is given by

H0 ¼1

2

h01 h02

h11 h12

� �H1 ¼

1

2

h03 h04

h13 h14

� �H2 ¼ SH1S H3 ¼ SH0S

For example, the following Hk satisfies the orthogonal conditions (3).

H0 ¼1

2

1þ sin� cos�1þ sin� � cos�

� �H1 ¼

1

2

1� sin� cos�sin� cos�

� �ð6Þ

By symmetry, we have H2 = SH1S, H3 = SH0S. If we define Gk = (j1)k

H2Nj1jkP, P = antidiag (1,1), then we can check that Hk and Gk satisfy theorthogonal conditions (4) and (5). When a = arcsin (j0.96), the figures ofmultiscaling functions 81(t) and 82(t), and multiwavelet functions y1(t) and y2(t)are illustrated in figure 1.

The multiscaling and multiwavelet functions of GHM and CL are illustrated infigures 2 and 3, respectively. For they are not smooth enough compared to SOM,their frequency spectrum is less of convergence and good time-frequency localproperties that are emphasized by wavelet theory, even though combined with aprefilter as shown in figure 6. In general, good time-frequency local properties arevery useful for image coding, this motivated us to adopt SOM for the followingsimulation in Section 4.

Multimed Tools Appl (2006) 29: 175–189 177

3 Multiwavelet transform and prefilter technique

3.1 Prefilter technique

For the complexity of constructing the matrix filters we only consider theorthogonality but omit the good frequency property at first. By Eq. 7, G(w) is nota good highpass matrix filters for G(0) is not a 0 matrix. The decomposition resultshown in figure 4b is not suitable for image compression. So we should present aprefilter technique to get a good lowpass and higpass frequency property. For GHMand CL, much research has been done [8, 17, 18]. As illustrated in figure 4c, withprefilter technique, we can get a good energy distribution after decomposition. Aframework of decomposition and reconstruction combining with SOM and thecorresponding prefilter is shown in figure 5.

HSOM 0ð Þ ¼ 1 00 0

� �;GSOM 0ð Þ ¼ 0 0

0 �1

� �ð7Þ

Fig. 1 Multiscaling and multiwavelet functions of SOM


In figure 5, A0(w) and A1(w) are two components of the prefilter. To get goodlowpass and highpass responses, when w = 0, we wish the components of H(w) andG(w) satisfy

H00 0ð ÞA0 0ð Þ þH01 0ð ÞA1 0ð Þ ¼ 1 ð8Þ

G10 0ð ÞA0 0ð Þ þG11 0ð ÞA1 0ð Þ ¼ 0 ð9Þ

By Eqs. 8 and 9, we have A0 (0) = 1 and A1(0) = 1. We can also find this result notonly suitable for SOM but also suitable for GHM and CL. To ensure reconstruct adecomposition signal completely, A0(w) and A1(w) also should satisfy theorthogonal condition

A0 wð ÞA0* wð Þ þA1 wð ÞA1

* wð Þ ¼ 1 ð10Þ

For many scalar wavelets satisfy Eq. 10, so they can be the candidates as aprefilter. There are four combining frequency components as shown in Eqs. 11 and12. According to Eqs. 11 and 12, the frequency responses H0(w) and G0(w) of GHM,CL and SOM are illustrated in figure 6.

HC wð Þ ¼ HC0 wð ÞA0 wð Þ þHC1 wð ÞA1 wð Þ C ¼ 0; 1 ð11Þ

GC wð Þ ¼ GC0 wð ÞA0 wð Þ þGC1 wð ÞA1 wð Þ C ¼ 0; 1 ð12Þ

Fig. 2 Multiscaling and multiwavelet functions of GHM


3.2 Multiwavelet coefficient reorganization

During a single level of decomposition using a scalar wavelet transform, a 2-Dimage data is replaced with four blocks corresponding to the subbands representingeither lowpass or highpass filtering in each direction. The multiwavelets used herehave two channels, so here will be two sets of scaling matrix coefficients andwavelets matrix coefficients. The first level multiwavelet decomposition subbandsare shown in figure 7a.

Fig. 3 Multiscaling and multiwavelet functions of CL

Fig. 4 Prefiltering can compact the energy in lowerpass subband


Scalar wavelets transforms give a single quarter-sized lowpass subband fromthe original larger one. In previous literature, multiwavelet decompositions areperformed in the same way. The multiwavelet decomposition iterate on the lowpasscoefficients from the previous decomposition, the LiLj subbands in figure 7a. Inthe case of scalar wavelets, the lowpass quarter image is a single subbands.However, when the multiwavelet transform is used, the lowpass coefficients isactually a 2 � 2 block of subbands, one lowpass and three bandpasses. The nextdecomposition step will decompose the lowpass subbands L = {L1L1, L1L2, L2L1,L2L2}. In this case, 2-level multiwavelet decomposition of a 2-D signal will produce4(3 � 2 + 1) subbands as illustrated in figure 7b.

The quantization method used to generate the result in this paper is the SPIHT[12]. SPIHT and other type of zero tree quantizers, such as SLCCA [1] and MRWD[13], achieve good performance by exploiting the spatial dependency of the pixels,which in different subbands after scalar wavelet transform. The assumptions that theSPIHT quantizer makes about the relationship between subbands hold wellfor scalar wavelet, but they do not hold for multiwavelets for the multiwavelettransform destroying the parent–children relationship that SPIHT presumes asillustrated in figure 7c.

Examinations of the coefficients in a single level multiwavelet decompositionreveal that there generally exists a large amount of similarity in each 2 � 2 blocks.To take advantage of the similarity, [11] presented a method to construct all themultiwavelet coefficients into several hierachical trees of a quarter sized pyramid ineach 2 � 2 blocks, called as multiwavelet hierarchical trees (MHT). It is obvious that

Fig. 5 A framework of decomposition combining with SOM and prefilter

Fig. 6 Combining frequency response with Haar prefilter. (a) H0(w). (b) G1(w)


three spatial orientations include 12 MHT and are performed by SPIHT algorithmfor each MHT independently. This method resolves the problem of destroying theparent–children relationship efficiently. Meanwhile, the problem of destroying therelationship of each scale 2 � 2 block still exists.

Then we present a new scan and quantization method that allows multiwaveletdecomposition to receive most of the benefits from a SPIHT quantizer. The basicidea is to try to restore the spatial features that SPIHT requires for optimalperformance. This observation suggests the following procedure: riffle the coef-ficients in each 2 � 2 blocks so that coefficients corresponding to the same spa-tial locations are placed together (see coefficients ABCD and EFGH in figure 8aand b). This reorganization procedure restores some of the spatial dependencies ofpixels by moving those pixels that correspond to a particular part of an image tothe position that they would have been located if a scalar wavelet decompositionhad been performed. As a QMF-pyramid subband structure, the dependenciesbetween ancestors and offspring are show in figures 8b and 9b. Then we obtain apyramid liking structure of a four-level scalar wavelet decomposition. Moreover, itis compatible with a four-level scalar wavelet coder for a fair comparison.

(b) (a)

L1L1 L1L2

L2H1L2L1

L1H1 L1H2

L2L2 L2H2

H1L1 H1H2H1H1H1L2

H2L1 H2H2H2H1H2L2

(c)

Fig. 7 (a) 16 subbands after first decomposition. (b) 28 subbands after second decomposition. (c) Thetraditional scan and quantization order among the 28 subbands

E F

G H

E FGH

AB CD

A B

C D

(a) (b)

Fig. 8 Reorganizing the subband structure. (a) Before reorganization. (b) After reorganization, ascalar wavelet scan and quantization can be performed


3.3 Energy comparison between scalar wavelet and multiwavelet transform

Though there are still a few areas of multiwavelet research that require furtherinvestigation for their successful applications, typically in image compression.Intuitively, it is observed that a multiwavelet transform can give higher energycompaction than scalar wavelet transform, despite Barbara image contains moredetail texture as shown in figure 9. We know that the more energy is compacted intothe lower frequency subbands, the more efficiency can be gotten by SPIHT coding.

SPIHT uses a regular tree structure or set-partitioned tree structure toapproximate insignificant fields across subbands, zerotree can be effectivelyrepresented by its root symbol. It is apparent that each zerotree can be effectivelyrepresented by its root symbol. But a root node whether or not to establish azerotree is dependent on its offspring. To a predefined threshold, if the root and itsoffspring are insignificant, then they can be included by a zerotree.

Table 1 shows the energy distribution of Lenna image after multiwavelettransform. In Table 1, subband 1 is the lowest frequency subband, and subbands11, 12, and 13 are the highest frequency subbands. In the lowest frequency subband,the Mean of multiwavelet coefficients is about twice of that of scalar waveletcoefficients. But in the highest frequency subands 11, 12, and 13, the Mean ofmultiwavelet coefficients is about half of the Mean of scalar wavelet coefficients.SPIHT is proceeded followed multiwavelet transform, to a predefined threshold,many children nodes will be treated as an insignificant symbol. It is useful toestablish a zerotree to enhance the coding efficiency. This reveals that the energydistribution coming from multiwavelet transform is more suitable for SPIHT codingthan that coming from scalar wavelet transform.

Fig. 9 Energy comparison. (a) Scalar wavelet transform. (b) Multiwavelet transform after coefficientreorganization

Table 1 Mean of each subband after scalar wavelet and multiwavelet transform

Subbands 1 2 3 4 5 6 7 8 9 10 11 12 13

Multiwavelets 992 64.5 128 49.8 12.9 22 8.8 4.9 7.5 3.5 1.35 1.8 1.0

Scalar wavelets 554 44.1 97.9 43.2 16.7 32.2 16.2 6.5 10.6 5.8 2.7 3.8 2.1


Fig. 10 The frequency spectrum of multiscaling functions. (a) 81(t). (b) 82(t)

Fig. 11 Experiment of symmetric multiwavelet compression of images Barbara and Goodhill


The reason of this phenomena can be explained with high order vanishingmoments which the symmetric orthogonal multiwavelets possess. To some extent,we can say, the higher order vanishing moments are equal to the better frequencyproperties. The frequency spectrum of 8(x) is illustration in figure 10. If thevanishing moments of 8(x) are of degree N, then we can verify the followingstatement is true when N Y V [4].

H00 wð ÞH*00 wð Þ þH01 wð ÞH*

01 wð Þ ! HI wð Þ ¼1; jwj < �

212 ; jwj ¼ �

20; �

2 < jwj < �

8<

: ð13Þ

4 Simulation experiments

4.1 Simulation results

The new symmetric orthogonal multiwavelets, prefilter and scan order are evaluatedon four natural 512 � 512 grayscale images, i.e., Lenna, Barbara, Boat and Goodhill.For an accurate comparison, we have chosen a scalar wavelet for our experiment, anorthogonal and symmetric basis. Particularly, the following multiwavelet bases havebeen considered: GHM with the orthgonal approximation preserving prefilter, andCL with Haar transform matrix prefilter. Usually, the distortion is measured by peaksignal to noise ratio. Computer simulations are presented for the four sample imagesin figure 11.

Table 2 shows the PSNR comparison on Lenna, Boat, Goldhill and Barbaraimage at different bit rates. Our coder consistently outperforms Bi9/7. Compared toBi9/7, our coder gains 0.15 to 0.25 dB in PSNR on average. Then we first comparethe results that come from comparing our SOM coder with GHM. For Barbara

Table 2 PSNR for compression of Lenna, Barbara, Boat, and Goodhill (dB)

Image Filter 8:1 16:1 32:1

Barbara Bi9/7 35.84 31.67 27.37

GHM 36.07 31.64 27.97

CL 36.36 31.71 28.08

SOM 36.54 32.209 28.33

Boats Bi9/7 – – –

GHM 37.75 33.87 29.73

CL 37.71 33.95 29.86

SOM 38.06 34.301 30.238

Goodhill Bi9/7 35.24 32.20 29.46

GHM 35.80 32.35 29.87

CL – 33.12 30.58

SOM 36.52 33.211 30.473

Lenna Bi9/7 39.55 36.62 33.31

GHM 40.14 37.13 33.30

CL 40.46 37.70 34.278

SOM 40.64 37.791 34.478


image, our coder gains more than 0.5 dB at compression ratio (CR) 16:1, and gains0.3 dB at compression ratio 8:1. For boats image, our coder gains 0.25 dB at CR 8:1,0.45 dB at CR 16:1, and gains 0.4 dB at CR 32:1. Moreover, For the Goodhill andLenna, which are relatively smooth images, the performance between our SOMcoder and CL gets closer. These preliminary results suggest that the SOM andprefilter for our coder are worthy of further investigation as a technique for complextextured image compression.

It is also noteworthy that symmetric multiwavelets can achieve good compressionperformance even though it has lower approximation order than both GHM andCL. So we can draw a conclusion that the approximation order and regularity arevery important for some applications such as digital signal processing applications,but in image compression, the effect of approximation order and regularity is stillunknown.

5 Computational expense

Moreover, there is one more critical problem of the work. What is the com-putational expense of applying multiwavelet and prefilter? Besides the PSNR, Weshould also compare the running time of image compression and decompression,because it is impractical to compress an image using 1 or 2 min. As shown inTable 3, the compression and decompression time is listed at different bit rates. Wealso should point out that the prefilter is needed only for the first level multiwaveletdecomposition. We find the time cost is very closer between multiwavelet and scalarwavelet transform coding.

6 Conclusions

Multiwavelets is an important development of wavelet theory for it solved theconflict between orthogonality and linear phase. Multiwavelets offer the advantageof combining symmetry, orthogonality, and short support, properties not mutuallyachievable with scalar wavelet system. For the special frequency response of thematrix filters, we introduce a new prefilter technique. In order to evaluate theperformance of multiwavelets for image coding at low bit rate, efficient SPIHTcoding of multiwavelets coefficients has been realized, accomplished with a suitablescanning strategy across scales and inside each detail subimage. Extensiveexperimental results demonstrate that the parent–children relationship is muchnatural between finer and coarser scales multiwavelet coefficients, and our

Wavelet type Ratio Compression

time (s)

Decompression

time (s)

Scalar wavelet

coding

8:1 5.548 4.567

16:1 4.346 3.455

32:1 3.665 2.924

Multiwavelet

coding

8:1 5.969 4.967

16:1 4.767 3.865

32:1 4.106 3.305

Table 3 Comparison of timecost


techniques exhibit performance equal to, or in several cases superior to theconventional scan and quantization methods. The further work should make aqualitative analysis why there is a strong dependency between parent and itschildren by their statistical characteristics, and why the property of image energybeing compacted into lower frequency subbands caused by multiwavelet transformis better than that caused by scalar wavelet transform.

References

1. Chai B, Vass J, Zhuang X (1999) Significance-linked connected component analysis for waveletimage coding. IEEE Trans Image Process 8(6):774–783

2. Chui CK, Lian J (1996) A study on orthonormal multiwavelets. Appl Numer Math 20:273–2983. Cotronei M, Lazzaro D, Montefusco LB, Puccio L (2000) Image compression through em-

bedded multiwavelet transform coding. IEEE Trans Image Process 9(2):184–1894. Drouiche K, Kateb D (1999, August) New filter banks and more regular wavelets. IEEE Trans

Signal Process 47(8):2220–22275. Geronimo JS, Hardin DP, Massopust PR (1994) Fractal functions and wavelet expansions based

on several scaling functions. J Approx Theory 78:373–4016. Goodman TNT, Lee SL (1994) Wavelets of multiplicity r. Trans Amer Math Soc 342(1):307–3247. Goodman TNT, Lee SL, Tang (1993) Wavelets in wandering subspaces. Trans Amer Math Soc

338(1):639–6548. Hardin DP, Roach DW (1998) Multiwavelet prefilter I: orthogonal prefilters preserving approx-

imation order p e 2. IEEE Trans Circuits Syst II 45(8):1106–11129. Hong D, Wu A (2000) Orthgonal multiwavelets of multiplicity four. Elsevier Comput Math

40:1153–116910. Jia RQ, Remeschneider S, Zhou DX (1998) Vector subdivision schemes and multiple wavelet.

Math Comp 67:1533–156311. Lin G, Liu Z (2000) The application of multiwavelet transform to image coding. IEEE Trans

Image Process 9(2):270–27312. Said A, Pearlman WA (1996) A new fast and efficient image codec based on set partitioning in

hierachical trees. IEEE Trans Circuits Syst Video Technol 6(3):243–25013. Servetto SD, Ramchandran K, Orchard MT (1999) Image coding based on a morphological

representation of wavelet data. IEEE Trans Image Process 8(9):1161–117414. Strang G, Strela V (1995) Short wavelets and matrix dilation equations. IEEE Trans Signal

Process 43:108–11515. Strela V, Heller PN (1999) The application of multiwavelet filterbanks to image processing.

IEEE Trans Image Process 8(4):548–56316. Tham JY, Shen L, Lee SL (2000) A general approch for analysis and application of discrete

multiwavelet transforms. IEEE Trans Signal Process 48(2):457–46417. Xia XG (1998) A new prefilter design for discrete multiwavelet transforms. IEEE Trans Signal

Process 46(4):1558–157018. Xia XG, Geronimo JS, Hardin DP (1996) Design of prefilters for discrete multiwavelet trans-

forms. IEEE Trans Signal Process 44(1):25–35


Jiazhong Chen was born in 1970. He received the M.S. degree in computation mathematics in 1999

and Ph.D. degree in computer system architecture in 2003 from HUST. He is now an associate

professor at School of Computer Science and Technology of HUST. His main research interests

include signal processing and wavelet analysis, image and video coding.

Xun Ouyang was born in 1974. He received the B.E. and M.S. degree in power engineering and

computer science respectively in 1996 and 1999 from HUST. Now he is studying for his Doctorate

in Faculty of Business, Law and Computing in University of Derby, U.K. His main research interests

include virtual reality, virtual learning environment.

Wu Zheng was born in 1971. He received the Ph.D. degree in computer system architecture in 2006

from Huazhong university of science and technology. He will be a lecturer at School of Computer of

Wuhan University of Science and Technology. His main research interests include signal processing

and wavelet analysis, image and video coding.


Jun Xu was born in 1981. He received the B.S. degree in 2003 and M.S degree in 2006 from HUST,

Wuhan, P. R. China. He is now pursuing the doctoral degree in Florida University, USA. His

research interests are wavelet theory, image processing, and video compression.

Jingli Zhou was born in 1946. She received the B.E. degree in 1969. She is a Professor and doctor

advisor at Huazhong University of Science and Technology. She had been a visiting scholar in USA

from 1995 to 1996 and has been honor of the State Department Special Allowance since 1999. Her

main field of research: computer network and multimedia signal processing.

Shengsheng Yu was born in 1944, and received the B.E. degree in 1967. He is a Professor and doctor

advisor at Huazhong University of Science and Technology. He had been a visiting scholar in west

Germany from 1982 to 1983. His main field of research: computer network and storage, discrete

signal processing and communication.


art%3a10.1007%2fs11042-006-0006-6

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