[IEEE 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR) - Xian, China (2012.07.15-2012.07.17)] 2012 International Conference on Wavelet Analysis and

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

A NOVEL VARIABLE-FILTER-BAND DISCRETE WAVELET TRANSFORM

ZHONG ZHANG1, mROSm TODA1, TAKASHI BfAMURA1, TESTUO MIYAKE 1

lToyohashi University of Technology, 1-1 Hibarigaoka Tenpaku-cho, Toyohashi, Aichi, 441-8122, JapanE-MAIL: {zhang.ima.miyake}@is.me.tut.ac.jp

2.1. Characteristics of the DWT

2. Review of discrete wavelet transforms

A wavelet used for the DWT includes two kinds of the orthogonal wavelet and bi-orthogonal wavelet, but here we discuss the bi-orthogonal wavelet (as will be shown later, the orthogonal wavelet can be considered a special case of the biorthogonal wavelet). The bi-orthogonal wavelet is comprisedof two wavelets called the mother wavelet (MW) '¢(t) and thedual-wavelet {;(t). The DWT can be defined as follows:

dj,k =100

j(t),¢j,k(t)dt, (1)

'¢;';(t) = 2j/2{;(2j t - k),

and the inverse transform can be given as follows:

Meyer wavelet has, and proposed a new perfect translation invariance complex discrete wavelet transform (PTI-CDWT) andconfirmed its effectiveness. In order to improve the octave bandanalysis, a best choice basis method has been proposed [1] inwhich each octave band is firstly subdivided using a waveletpacket and then the most suitable basis is selected by evaluating the calculation cost of the wavelet coefficients. In additiona PTI complex wavelet packet transform, which achieved theperfect shift invariance, has been developed [5]. However, theselectable frequency band is limited, and it cannot be necessarily said that the chosen basis is the most suitable.

In this study, we devise a construction method for the bestbasis that forms the basis in accord with the signal and we propose a design method for the variable band filter. We call thediscrete wavelet transform, which uses the designed variableband filters, the Variable-Filter-Band Discrete Wavelet Transform (VBF-DWT). We show details of the theory of the VBFDWT and design method of the VBFs.

(2)f(t) = E E dj,k,¢j,k(t),jEZ kEZ

Keywords:Variable band filter; Band pass wavelet set; Real signal mother

wavelet; Abnormal signal detection.

Abstract:In this study, a new variable-filter-band discrete wavelet trans

form (VBF-DWT) is proposed for detecting and extracting unsteady abnormal signals. The VBF-DWT is achieved by usingthe band pass wavelet set (BP-WS) and band rejection waveletset (BR-WS) instead of traditional mother wavelet (MW). Corresponding to it, in the multi-resolution algorithm, the variableband filters (VBFs) are used instead of traditional low and highpass filters. The VBFs are designed from the BP-WS and BR-WS,which is constructed from a real-signal MW.

It is well known that the DWT has the following characteristics. 1) Calculation speed is fast, and is almost as fast as the fastFourier transform, by using a high-speed calculation algorithmincorporating Multi-resolution analysis (MRA) which was proposed by Mallat[I]. 2) Perfect reconstruction of the analysissignal is possible since a basis that was constructed by an orthogonal (or bi-orthogonal) mother wavelet (MW) is a perfectlyorthogonal basis. 3) The compressibility of data is high sincethe MW has the characteristic of the octave band in the frequency domain and octave analysis has been achieved. However, the high-speed calculation algorithm (hereafter called thefast algorithm) by the MRA has the down-sampling operation,and therefore the DWT lacks shift invariance. In addition, itis possible that the octave band is without necessarily and notmost suitable base in accord with the analysis signal, althoughthe MW affects the efficiency such as data compaction or thenoise deletion to have the characteristic of the octave band.

Many techniques have been suggested so far to improve theDWT shift invariance [2, 3]. In particular, Toda et al. [4] paidattention to the characteristics that the scaling function of the

1. Introduction

978·1-4673·1535·7/121$31.00 ©2012 IEEE371


Figure 1. The DWT by using Mallat's fast algorithms

because the orthogonal wavelet is a self bi-orthogonal waveletand '¢(t)='ljJ(t), ¢(t)=¢(t), Pk=A_k, qk=Jj-k are satisfied. Between the sequence {Pk} and sequence {qk}, the followingequation is satisfied.

(8)

ReconstructionDecomposition

Furthermore, the sequence {qk} has acts as a high-pass filterand the sequence {Pk} acts as a low-pass filter. The decomposition sequences {ak}, {bk} and reconstruction sequences {gk},{hk}, which are necessary for the fast algorithm shown below,can be found as follows:

where dj,k denotes the wavelet coefficient, j the level, and kthe time shift. In the DWT, in order to reconstitute the originalsignal I(t) using expression (2), 'ljJ(t) and '¢(t) should satisfythe following bi-orthogonal condition[6].

100

'l/Jj,k (t)'¢l,n (t)dt = c5j,lc5k,n, (3)

-00 'ljJj,k(t) = 2j/ 2'ljJ(2jt - k),

'¢j,k(t) = 2j/2'¢(2jt - k).

In other words, '¢l,n(t) is called the bi-basis of 'ljJj,k(t) when'ljJj,k (t) and '¢l,n(t) are function sets made by 'ljJ(t) and '¢(t),and equation (3) is established. In the DWT by using the biorthogonal wavelet, 'ljJj,k(t) and '¢j,k(t) are put together andwill guarantee the reconstruction of the original signal. As anexample of the bi-orthogonal wavelet, a Spline wavelet proposed by Chui and Wang can be given [6]. On the other hand,'ljJ(t) is called an orthogonal wavelet when the MW itself is aselfbi-orthogonal wavelet, that is, when 'ljJ(t) ='¢ (t) is satisfied.Notable orthogonal wavelets include the Meyer wavelet [7] andthe Daubechies wavelet [8].

2.2. Mallat's fast algorithm for the DWT

(11)

(10)Cj,n = L a2n-kCj+l,k,kEZ

dj,n = L b2n- kCj+l ,k,kEZ

(6)

In the case of an orthogonal wavelet, all necessary sequencesare demanded by expression (8) and (9) if {Pk} is understood.In addition, the sequence {Pk} of many typical orthogonalwavelets are shown by the proposer.

By inheriting the property of the sequences {qk}, {Pk},the sequences {bk} and {hk} acts as high-pass filters and se

(4) quences {ak} and {gk} acts as low-path filters. Figure 1 showsthe decomposition and reconstruction trees of the fast algorithm

(5) by Mallat's MRA. In the figure the mark ~ 2 denotes the op-eration of down-sampling, and t 2 denotes the operation ofup-sampling. In addition, the decomposition calculation of thesignal along the decomposition tree that is shown in Figure 1 iscarried out recursively by expressions (10) and (11).

(7)¢(t) = L A-k¢(2t - k),kEZ

¢(t) = LPk¢(2t - k),kEZ

'¢(t) = L Jj-k¢(2t - k),kEZ

'ljJ(t) = L qk¢(2t - k),kEZ

The fast algorithm of the DWT makes use of the characteristic of the bi-orthogonal wavelet, and was led to by the theory ofthe MRA [1]. The bi-orthogonal wavelet is constructed by theMW 'ljJ (w) and the dual wavelet '¢(t), and the following waveletequation is established between these.

where {qk} denotes the wavelet sequence, {Jjk} the dualwavelet sequence, and ¢(t), ¢(t) the scaling function and dualscaling function, which correspond to 'ljJ (t), '¢(t), respectively.Furthermore, the following two-scale relation is composed.

where dj,n denotes the wavelet coefficient, Cj,n the scaling cowhere sequence {Pk} is called the two scale sequence, and efficient, subscript j the level (j = -1, -2, ...), and n ex{Ak} is the dual two scale sequence. presses the tum of a coefficient arranged in each level. In the

In this paper, we only discuss the orthogonal wavelet and use case of the DWT, CO,n is assumed to be CO,n = In' where In is ait for the DWT. In the case of an orthogonal wavelet, four equa- digital signal of the target signal, and in the case of the CDWT,tions (4)-(7) can be summarized as two equations (4) and (6), CO,n is demanded by the interpolation process. In addition, the

372


reconstruction of the signal is calculated along the reconstruction tree, which is shown in Figure 1 from Cj,n' dj,n recursivelyby the following expression.

Cj+l,n = E 9n-2kCj,k + E hn - 2k d j ,k o (12)kEZ kEZ

As is shown in expression (11), the wavelet coefficient dj,k ofarbitrary level j is calculated using coefficient Cj+l,k. The coefficient of a deeper level can be obtained recursively by similarprocessing. In addition, the reverse transform becomes the recursive processing in the same way.

3. Proposing Variable Filter Band Discrete WaveletTransform

As is shown in Section 2.1, the desirable analysis result thatwas necessarily correct for the purpose is not always obtainedfrom the DWT, because the wavelet basis, which is constitutedby extending and shifting the MW, has a filter band of the octave interval in the frequency domain. In order to improve thisfault, in this study, a band-pass wavelet set (BP-WS) in whichthe detecting frequency components are passed and a band rejection wavelet set (BR-WS) in which the detecting frequencycomponents are removed, are proposed and used in the DWTinstead of the wavelet basis that is constituted by extending andshifting the MW. Furthermore, the DWT that used the BP-WSand BR-WS is called the Variable Filter Band Discrete WaveletTransform (VFB-DWT). Since the VFB-DWT is built based onthe conventional DWT and the CDWT, they are called the baseDWT. In this section, the design method of the frequency characteristic of the BP-WS and BR-WS, and the design method ofthe variable band filters will be introduced. The characteristicsof the VFB-DWT which uses the variable band filters will alsobe examined.

3.1. Band-pass wavelet set and band reject wavelet set

An example of the desirable frequency properties of the BPWS ;j;BP (w) is shown in Figure 2 and the design method of;j;BP (w) is explained in order as follows:

(1) In consideration of the frequency properties of the analysis signal, the number N of passage bands is set. Thefrequency areas of the number N [w l,2, Wl,3] 1 < l <N, l E Z are set. In the example shown in Figure 2, theBP-WS has three passage areas l = 1, 2, 3 since numberN = 3, where 0 < w l,2 ~ Wl,3 < 71", and each bandarea must be w l,3 < Wl+1,2 so that each band does notoverlap.

Figure 2. An example frequency characteristic of theBP-WS ;j;BP(W)

(2) For the start part and end part of each band area[w l,2, w l,3], the appropriate decrement areas must be established. In other words, as is shown in Figure 2, we putan appropriate interval and set w z.i. w l,4 of each band,so that the neighboring bands do not overlap to becomew l,l < Wl,2 ~ Wl,3 < Wl,4.

(3) The frequency properties ;j;BP(W) of the BP-WS are determined by using the following expression (13) - (15). Inexpression (15), the curve that is made by Daubechies[8]for designing the Meyer wavelet, is used.

N

;j;BP(W) E flBP(W) , (13)l=l

sin[~lI( Wl,l {M_ 1} ) ]2 w l,2 - Wl,l Wl,l '

(w l,l < Iwl < w l,2)

1, (w l 2 ~ Iwl ~ w l 3)tl" (w) " (14)

cos [~1I ( Wl,3 {M -I})]2 Wl,4 - Wl,3 Wl,3 '

(wl,3 < Iwl < w l,4)

0, (otherwize),

lI(x) = x 4 (35 - 84x + 70x 2- 20x 3

) 0 (15)

In practice, the passing band and rejecting band of the;j;BP (w) are established while considering the frequency characteristic of a real analysis signal, although the basic designprocedure of the frequency properties ;j;BP(W) that correspondto the BP-WS is shown as above.

The frequency characteristic ;j;BR(w) of the BR-WS, whichcorresponds to the BP-WS, is demanded from the followingexpression: .

373


Figure 3. Example frequency characteristics of theBP-WS ,(j;BP(W) and the BR-WS ,(j;BR(w)

Fiaure 5. Example of Frequency characteristic ofJt;BP R,BR R d bR

uj,k , uj,k ,a an

Figure 4. Design method of the band-pass filters

(1) The frequency properties ,(j;BP(W) of the BP-WS are defined as a function in area -7r ::; W ::; tt, A functionHfP(w) is demanded by cutting off the frequency component in area - 2j +17r ::; W ::; 2j +17r from the area shownabove, and taking it 1/2j+1 times in the frequency direction and developing it to the area of -7r ::; w ::; tt,

HfP(w) = ,(j;BP(2j +1w),-7r < w < it, (17)

(18)

(2) Since the function HfP(w) expresses the frequency properties of a filter {hf:}, so the filter {hf:} can be demanded as follows from expression (18).

Following the schematic view of the procedure shown in Figure 4, the construction method of the variable band filter is in-troduced with the BP-WS as an example here. Figure 4 shows (3) The {hf:} is divided by using {b~} and the obtainedan example when J = -2, that is, the example is decomposi- {UfkBP} is assumed to be the high-frequency filter oftion unti1level-2. level j. In the case of j = J, the {h~~} can be divided

3.2. Constructing a variable band filter for fast calculation algorithm

Figure 3 shows the graph of the curves ,(j;BP(W), ,(j;BR(w).

374


by using {a~} and the obtained {U~kBPC} is assumed tobe the low-frequency filter of level J. These expressionscan be shown as follows:

cR.BPJ.k

R RC~:~R

C~.kC-2,Jr •• • CJ i U

dR.BP

dR,BI' J.k

dR.BI' -2,k d R•BR

-I.x l.t

d R•BRdR,BR

-2 ,x-u

cR.BP(a) Decomposition tree J.k

~.BR

C~2,k •• • C~-H,tJ,k

R dR.BPC_1,k

d R •BPJ,k

-2.t d R•BR

dR,B1' 1.11.-I.x d R •BR

dR,BRl+U

-I.x(b) Reconstruction tree

Figure 6. Real part of the VFB-DWT tree by usingMallat's fast algorithm

RCO,k

(20)

(19)Eb~-nhf:,nEZ

" R hBPL.-J ak-n J,n 0

nEZ

R,BPuj,k

R,BPcuJ,k

(4) Steps (1) - (3) are carried out for j = -1, -2, 000' J.

For the filters {UfkBR}, {UfkBRC} corresponding to BRWS, the same procedure (1) - (4) shown above can be provided.In the case of step (1), HfR(w) = ¢BR(2j+1w) can be used,but the following expression is arrived at from expressions (16),(17) and the above expression.

(22)

(23)

" R,BP RL.-J uj,2n-k Cj+l,k,

kEZ

" I,BP RL.-J uj,2n-k Cj+l,k o

kEZ

" R,BR RL.-J uj,2n-k Cj+l,k,

kEZ

" I,BR RL.-J uj,2n-k Cj+l,k o

kEZ

d~,BR1,n

d~,BPs-»

part of the decomposition tree and (b) shows the real part of thereconstruction tree. The imaginary parts of the decompositionand reconstruction trees have been omitted since the real andimaginary parts of the decomposition and reconstruction treeshave the same structure.

As is shown in Figure 6(a), variable band filters {U-':kBP} ,1,

{ R BR} { I BP} { I BR} 0Uj,k 'Uj:k 'Uj:k are used for the high-frequencycomponents instead of the conventional decomposition se-quences {b~}, {b~}. In the last level J, variable band fil-t { R,BPC} { R,BRc} { I,BPC} { I,BRc} d s:ers uJ,k ,uJ,k ,uJ,k ,uJ,k are use rornot only the high-frequency components, but also the low-frequency components as needed instead of sequences {a~},

{a~}. The process for calculating cf!,c~ from the analysis digital signal I n uses interpolation, but this processing is not shownin Figure 6. The calculation for the high-frequency componentsin the decomposition tree, which is shown in Figure 6(a) can becarried out by the following equations:

(21)

And the low-frequency components cfn and c3,n can be calculated by using expression (10). However, in decomposition

Figure 6 shows an example of the decomposition and recon- level j = J, the decomposition of the low-frequency compostruction trees using the VBF-DWT, where (a) shows the real nent is performed by a variable-band filter using the following

3.3. Decomposition and reconstruction algorithm by usingVFB-DWT

In addition, HfP(w), HfR(w) have a period of 271", respectively, but expression (21) is established in the whole real domain (w E R). Incidentally, the same procedure can be usedto calculate the imaginary part and {u~:~p} and {u~,,~PC},

{u~:~R} and {u~:~RC} are obtained, although the step mentioned above is for the real part.

Figure 5 shows an example of the filter {UfkBP} and

{UfkBR}, which is made based on the frequency characteristic of the BP-WS and BR-WS shown in Figure 3, and the frequency properties of the traditional decomposition sequences{a~} and {b~} in the positive area (in the negative area, thefrequency property is symmetric around the origin). In Figure 5, (a) shows the high-frequency component of level -1,(b) shows the high-frequency component of level -2, and (c)shows the frequency characteristics of the variable band filterof the low-frequency component level -2. In the Figure, therange is displayed as 0 - 71", but this is because this expressesthe frequency characteristic of the original filter which does notconsider down sampling in the decomposition algorithm. Inother words, when assuming the sampling frequency 18 and dividing the signal by using a filter in each level, the frequencyrange of 0 - 71" shown in Figure 5(a) is equal to 0 - 18/2, in thecase of (b), 0 - 71" is equal to 0 - 18/4, and in the case of (c) isequal to 0 - 18/4.

375


Following the reconstruction tree shown in Figure 6 (b), thereconstruction calculation method is introduced. At first thelow-frequency components C1+1,n, C5+1,n of level J + 1 iscalculated by using the following equation from the high- andlow-frequency components of decomposition level j = J thatare provided by expression (22) - (25).

expression.

R,BPcJ,n

I,BPcJ,n

R,BRcJ,n

I,BRcJ,n

~ R,BPe RL.-J u J,2n-k cJ+l,k'

kEZ~ I,BPe RL.-J u J,2n-k cJ+l,k o

kEZ

~ R,BRe RL.-J u J,2n-k cJ+l,k'

kEZ~ I,BRe RL.-J u J,2n-k cJ+l,k o

kEZ

(24)

(25)

R,iBR R,BR R,iBRe R,BRe (30)Uj,k Uj,_k , UJ,k = UJ,-k

I,iBP I,BP I,iBPe I,BPeUj,k Uj,_k' UJ,k = UJ,-k ,

I,iBR I,BR I,iBRe I,BRe (31)Uj,k Uj,_k' UJ,k = UJ,-k

Incidentally, if there are not frequency components that arenecessary to be detected in the decomposition level, then thecalculation of the decomposition and reconstruction by usingthe traditional DWT can be carried out. In addition, similar toabove, the calculation of the decomposition and reconstructionby using the traditional DWT can be carried out at level J ifnecessary. In other words, the most suitable signal analysisis possible if variable-band filters are used as needed and thecomputational complexity can decreased.

Furthermore, the complete reconstruction of the VFB-DWT,which using the PTI-CDWT as the base DWT, was guaranteedby using the theoretical complete translation invariant theoremthat became basic theory of PTI-CDWT.

(28) Acknowledgements

The low-frequency components c.f+~,~, c~~~ of level j + 1,are then calculated from the low-frequency components cfn'

c3,n of decomposition level j(j=-I,-2,o ° o,J+l) and the highfrequency components provided by expression (22) - (23).

(~ R,iBPe R,BP +~ R,iBRe R,BR)L.-Ju~n-2kc~k L.-Ju~n-2kc~kkEZ kEZ

+(~ R,iBP dR,BP +~ R,iBRdR,BR) (26)L.-J u J,n-2k J,k L.-J u n-2k J,k °

kEZ kEZ

(~ I,iBPe I,BP +~ I,iBRe I,BR)L.-J UJ,n-2k c J,k L.-J U J,n-2k c J,kkEZ kEZ

+(~ I,iBP dI,BP +~ I,iBRdI,BR) (27)L.-J u J,n-2k J,k L.-J u n-2k J,k °

kEZ kEZ

This work was supported in part by a Grant-in-Aid for Scientific Research by the Japan Society for the Promotion of Science.

Conclusion

In this study, we devised a new construction method for themost suitable basis in accord with the analysis signal and proposed a novel variable-band filter discrete wavelet transform(VFB-DWT). The VFB-DWT is achieved by using variableband filters instead of the conventional decomposition and reconstruction sequences. The variable-band filters consist of theband-pass and band-reject filters, which are designed in consideration of the real signal characteristics. In addition, it is confirmed that the technique for creating the variable-band filterwas simple and effective. Furthermore, the VFB-DWT was applied to knocking de-noising and its effectiveness has been confirmed. In the future, the VFB-DWT will be applied to widelydifferent signal processing applications.

4.

(29)

~ R R (~R,iBP dR,BPL.-J 9n - 2k Cj ,k + L.-J u j,n-2k j,kkEZ kEZ

+~ R,iBRdR,BR)L.-J u n-2k j,k ,kEZ

~ I I (~I,iBP dI,BPL.-J 9n - 2k Cj ,k + L.-J U j,n-2k j,kkEZ kEZ

+~ I,iBRdI,BR)L.-J Un-2k j,k °

kEZ

Between the variable band filters of the decomposition and reconstruction, the following equation is formed based on the relations between the conventional decomposition and the reconstruction filter shown in expression (9).

R,iBPUj,k

R,BP R,iBPe R,BPeUj,_k' UJ,k = UJ,-k ,

376


References

[1] S. G. Mallat, "A wavelet tour of signal processing", Academic Press, New York, 1999.

[2] F. C. A. Fernandes, R. van Spaendonck, M. Coates and C.S. Burrus, "Directional complex-wavelet processing", InWavelet Applications VII, Proceedings of SPIE, 2000.

[3] N. Kingsbury, "Complex wavelets for shift invariant analysis and filtering of signals", Journal of Applied and Computational Harmonic Analysis, Vo1.10, No.3, pp.234-253,2001

[4] H. Toda, Z. Zhang, "Perfect Translation Invariance witha Wide Range of Shapes of Hilbert Transform Pairs ofWavelet Bases", Int. J. Wavelets Multiresolut. Inf. Processing, Vo1.8, No.4, pp.501-520, 2010..

[5] H. Toda, Z. Zhang, T. Imamura, ''The Design of ComplexWavelet Packet Transforms based on Perfect TranslationInvariance Theorems", Int. J. Wavelets Multiresolut. Inf.Processing, Vo1.8, No.4, pp.537-558, 2010.

[6] C. K. Chui, "An introduction to wavelets", AcademicPress, New York, 1992.

[7] Y.Meyer, "Wavelets, Algorithms & Applications", SIAM,Philadelphia, 1993.

[8] I. Daubechies, "Ten lectures on wavelets", SIAM,Philadelphia, 1992.

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[IEEE 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR) - Xian, China (2012.07.15-2012.07.17)] 2012 International Conference on Wavelet Analysis and