A Method for Interpolation Wavelet Construction Using Orthogonal Scaling Functions

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1 A Mthod for Intrpoltion Wvlt Contruction Uing Orthogonl Scling Function higuo hng Collg of Autotion Univrity of Elctronic Scinc nd Tchnology of Chin Chngdu, Chin Mr A. Kon Dprtnt of Mthtic nd Sttitic Boton Univrity Boton, USA Abtrct Intrpoltion vlt hv bn idly ud in dicrt ignl procing. In clicl vlt pling thory, intrpoltion vlt r contructd trting ith orthogonl vlt. Sinc th contruction of orthogonl vlt i uully copl, intrpoltion vlt r gnrlly difficult to obtin in prctic. Hnc, th contruction of uch vlt h bco n iportnt topic in ignl procing nd vlt thory. In thi ppr, dcrib n thod for contructing intrpoltion vlt fro orthogonl cling function. Thu, coputtion involving orthogonl vlt r not rquird for thi procdur. Thi in turn iprov fficincy of th contruction. In pl, our thor r pplid to o typicl vlt pc, dontrting th contruction lgorith. Kyord- Wvlt Spling, Intrpoltion Wvlt I. INTRODUCTION In 989, Donoho propod gnrl lgorith for contructing intrpoltion vlt by uing orthogonl vlt trting point []. Sinc th fficincy of th vlt trnfor cn b iprovd grtly ith intrpoltion vlt, Donoho lgorith i idly ud nd h bco n iportnt brnch of dicrt ignl procing [-5]. Hovr, in thi lgorith, intrpoltion vlt r contructd uing utocorrltion of orthogonl vlt. Thrfor, it i difficult to obtin intrpoltion cling function orthogonl to th intrpoltion vlt. For thi ron, Donoho intrpoltion vlt r gnrlly not uitbl for dcopoing ignl orthogonlly in full ultirolution nlyi. To dl ith thi nonorthogonlity, Wltr, Wn Chn nd Jnn hv propod lgorith for contructing intrpoltion vlt vi linr cobintion of orthogonl vlt [6-8]. Sinc intrpoltion vlt cofficint r pl of ignl, thy hv th dvntg of y coputtion of th dcopoition cofficint []. For th ron, th rult of Wltr tc. hv prootd ppliction of intrpoltion vlt in ig procing [3-9], iultion [5], control [6] nd ignl procing []. In ddition, uing intrpoltion vlt, nonbndliitd ignl cn b nlyzd ithin ultirolution nlyi uing thir pl. Thi pproch h lo ld to dvlopnt of dditionl vlt pling thori, including tho for irrgulr pling [, ], pling in hiftinvrint pc [3-5] nd liing tition [6, 7]. Hovr, th typicl lgorith for contructing n intrpoltion vlt gnrlly dpnd on n orthogonl vlt. Sinc coputtion of vlt r oftn pniv, it i unrlitic to intrpoltion vlt through dirct ppliction of vlt b in ll c. In [9], hv hon th rltionhip btn itnc of n intrpoltion vlt nd tht of n intrpoltion cling function. In thi ppr, dcrib n lgorith for contructing intrpoltion vlt dirctly fro orthogonl cling function, lloing or fficint nd ir contruction of uch intrpoltion vlt. II. THE SPECIAL FORM OF A FILTER BANK FOR INTERPOLATION WAVELETS IN THE FREQUENCY DOMAIN To iplify th dicuion, only conidr th c hr intrpoltion point of cling function { S ( } nd intrpoltion vlt { S ( } r rpctivly t {} nd { }, i.. f( f( S ( g( g( / S ( ( hold for ny f ( V or g ( W. Hovr our rult cn b ily tndd to th c hr intrpoltion point r loctd t othr poition, vi th trnfor or coordint trnltion [7]. Au tht { V j } j for ultirolution nlyi (MRA of L (. W j i vlt pc uch tht Vj Vj Wj hr dnot n orthogonl dirct u. Th function ( i cling function for th pproition pc V nd ( i vlt corrponding to (. Th nottion S ( nd S (, rpctivly, dnot intrpoltion function in V. According to vlt thory, if thr it n intrpoltion cling function S ( for th pc V nd n intrpoltion vlt S ( for W, thr ut b filtr bn ( Pˆ (, ˆ Q( corrponding to S ( nd S (, hich tifi

2 ˆ ( ˆ ( ˆ S P S ( / ˆ S Qˆ Sˆ ( / ( hr ˆ ( ( i S S d nd ˆ ( ( i S S d r rpctivly th Fourir trnfor of S ( nd S (. On th othr hnd, { S ( } for n intrpoltion bi in V. Siultnouly, S ( V V nd S ( W V. Hnc, S ( S ( / S ( S ( S ( / S (. (3 Ting th Fourir trnfor on both id in (3 yild ˆ i / ( ( / ˆ S S S ( / i / S S ( / S ( /, (4 hich ipli tht P ˆ ( nd Q ˆ ( in ( r rpctivly th Fourir trnfor of { S ( /} nd { ( /} S. Sinc it follo fro th Poion forul tht i / ( / = ˆ S S ( 4 i / S ( / = Sˆ ( 4, (5 qution (4 cn b furthr prd ˆ ( ˆ ( 4 ˆ S S S ( / Sˆ Sˆ ( 4 Sˆ ( /. (6 Coprion of (6 to ( ho tht ˆ ( ˆ P S ( 4 Qˆ ( ˆ S ( 4. (7 Fro (7, Q ˆ ( nd P ˆ ( hv vry pcil prion in th frquncy doin, i.. thy cn b rpctivly prd th for of ˆ S nd ˆ S. It i hon in th folloing ction tht thi i criticl for contructing n intrpoltion vlt. III. ORTHOGONALITY OF WAVELET SPACES AND PROPERTIES OF Pˆ ( AND Qˆ ( Clrly, fro tndrd vlt thory, V r orthogonl. Sinc { S ( } nd { ( } S rpctiv- ly for Riz b for V, th folloing L ho tht th orthogonlity of V ld to pcil qution for P ˆ ( nd Q ˆ (. L If { S ( } nd { ( } S for Riz b of V rpctivly, thn P ˆ ( nd Q ˆ ( in (7 tify Qˆ Eˆ Pˆ Qˆ ( Eˆ ( Pˆ ( (8 hr ˆ. (9 Eˆ ( S ( Proof. Fro th Prvl idntity, S ( S ( n d ˆ ( ˆ in S S ( d ( ith n. Sinc S ( nd S ( n r orthogonl to in ch othr nd { } for n n orthogonl bi in L [, ], ( ipli tht ˆ ( ˆ S S (. ( On th othr hnd, (6 nd (7 iply tht ˆ ( 4 ˆ ( ˆ S P S (, ( nd iilrly ˆ ( 4 ˆ ( ˆ S Q S (,. (3 Inrting ( nd (3 into ( yild ˆ ( 4 ˆ S S ( 4 ˆ ( 4 ˆ S S ( 4 ˆ ˆ Q ( ˆ S P ˆ ˆ Q ˆ S P Sˆ ( Sˆ ( (. (4 Applying (9 to (4 giv (8. Eqution (8 ri fro orthogonlity of W nd V. Th folloing ction ill ho tht thi qution i iportnt for u to u Pˆ to contruct Qˆ, hich in turn i crucil for contructing intrpoltion vlt.

3 IV. CONSTRUCTING Qˆ ( USING Pˆ ( AND Eˆ ( Bd on th dicuion bov, thi ction ill propo n lgorith for contructing Qˆ ( by uing Pˆ ( nd Eˆ (, ttd th folloing thor. Thor Au tht thr it intrpoltion b { S ( } nd { S ( j } in V. Thn hv i Eˆ ( ˆ P( Q. (5 Pˆ ( Eˆ ( + Pˆ Eˆ Proof. Sinc th intrpoltion point of { S ( } r t { } nd vlt ( i in W, it follo fro th cond idntity in ( tht ( ( / S (. (6 It follo fro th Poion forul nd (6 tht i ˆ ( ( / ˆ S i/ ( ˆ ( Sˆ Fro [8], qution (7 ipli ˆ S ( i/ ˆ ( ˆ ( hr ˆ i th Fourir trnfor of vlt (. By (7 nd (8, i/ i/ ( / ( / (8 Qˆ Qˆ. (9 Eqution (9 ipli tht Qˆ Pˆ ( Eˆ ( Eˆ Pˆ Qˆ Pˆ ( Eˆ ( Qˆ Eˆ Pˆ i/ ˆ ( ˆ ( P E Qˆ ( Pˆ ( Eˆ ( Qˆ Eˆ Pˆ. ( Inrting (8 into ( yild Qˆ Pˆ ( Eˆ ( Eˆ Pˆ ˆ ( ˆ (. ( i/ P E Eqution ( ipli tht (5 hold. Fro Thor, hn thr it n intrpoltion bi { S ( }, ˆ Q ( cn b prd in tr of Pˆ ( nd Eˆ ( in (5. Thi not only uppli n ffctiv thod for contructing th filtr Q ˆ (, but lo otivt u to contruct intrpoltion vlt vi cling function. V. CONSTRUCTION OF INTERPOLATION WAVELETS If Q ˆ ( i obtind, thn th Fourir trnfor of th intrpoltion vlt cn b prd in tr of th cond idntity in (. Inrting (5 into ( giv i ˆ E( P( ( ˆ S S ( /. ( P( E( + P E In fct, uing (, it i trightforrd to contruct n intrpoltion vlt fro n orthogonl cling function (, follo: Thor Au tht { ( } for n orthogonl bi in V. Thn, th intrpoltion vlt S ( in ( cn b rprntd ˆ Bˆ( ( / S ( C ˆ( C ˆ( i/ ˆ (3 hr B ˆ( ˆ ( 4 ˆ ( / nd C ˆ( ˆ( 4 ˆ ( /. Proof. Inrting (7 nd (9 into (, hv i ˆ ( ˆ ˆ A S ( / S ( (4 A ˆ( A ˆ( hr A ˆ( S ˆ ( 4 S ˆ ( Eqution (4 ho tht th Fourir trnfor of th intrpoltion vlt cn b obtind vi ˆ S. On th othr hnd, fro [7], hv ˆ ˆ( S (, (5 ˆ(. hr ˆ( i th Fourir trnfor of (. Inrting (5 into A ˆ( yild ˆ ( 4 ˆ( / ˆ( n n A ˆ ( ˆ ( / n n. (6 Sinc { ( } for n orthogonl bi in V, it follo fro fr thory tht Sinc ˆ( 4. (7 ( / n ( ˆ( A ˆ ˆ n n n ˆ ( / n ˆ ( / n ˆ( / n n,

4 inrting (5 nd (7 into (4 yild (3. Fro Thor, n intrpoltion vlt cn b contructd fro only th orthogonl cling function. Coprd ith clic lgorith for uch contruction, our lgorith void copl coputtion for obtining n orthogonl vlt, o iplifi contruction of intrpoltion vlt. VI. EXAMPLES Sylt nd Coiflt vlt r ud idly in ig nd ignl procing. Hr u our lgorith to contruct intrpoltion vlt for th cond ordr Dubchi, th fifth ordr Sylt, th third nd th cond ordr Coiflt MRA uing MATLAB. A. Contruction Sinc th Fourir trnfor ˆ( / of th cling function r ll tudid for Dubchi, Sylt nd Coiflt vlt, it follo fro (3 tht ˆ S cn b obtind if ˆ( / nd ˆ( 4 r non. Clrly, fro th Poion forul, ˆ( / nd ˆ( 4 cn b obtind by rpctivly ting {} nd { /} cofficint of Fourir ri, i.. ˆ( / ( ˆ( 4 ( / i / i / Sinc th nrgy of ( i ll loclizd in th ti doin, it follo fro (8 tht / ˆ( /.366 i.366 i (9 nd / / ˆ( 4 (.939 i.366 i 4 i3/ i4/ (3 for th cond ordr Dubchi cling function. Inrting (9 nd (3 into (3 dirctly yild ˆ S for th cond ordr Dubchi intrpoltion vlt. By ting th invr Fourir trnfor, obtin th cond ordr Dubchi intrpoltion vlt S (, hich i dpictd in Fig.-. Siilrly, for th fifth ordr Sylt cling function, hv 5 / ˆ( /.5363 i 4 i/ 3 i/ (3 4 i/ 5 i/ 6 i/ i/ 3 8 i/ nd ˆ( 4 ( i/ 5 i/ 4 i3/ i4/ i5/ i6/ i7/ i8/ i9/ i/ i/ i/ i3/ i4/ 4 i5/ 3 i6/ 5 i7/ (3 For th third ordr Coiflt cling function, nd ˆ( /.3 i i 5 i/ 3i i/ 4i 9 i/.3. 5i i/ ˆ( 4 ( i5/ i6/ i7/ i8/ i4/ i9/ i/ i/ i/ i3/ i4/ i5/ i6/ i7/ i8/ 3 i9/ 3 i/ (33 i/ 4 i/ i/ (34 For th cond ordr Coiflt cling function, ˆ( / i/ i3/ i4/ i5/ i6/ i7/ 3 i8/ nd 5 i9/ 6 i/ 9 i/ ˆ( 4 ( i/ 4 i/ 3 i3/ i4/ i5/ i6/ i7/ i8/ i9/ i/ i/ i/ 4 i3/ 3 i4/ i5/ 5 i6/ i7/ 6 i8/ (35 8 i9/ 9 i/ i/ (36 Inrting (3, (3, (33, (34, (35 nd (36 into (3, cn obtin ˆ S rpctivly for th fifth ordr Sylt, th

5 third ordr nd th cond ordr Coiflt vlt pc. Th vlt S ( obtind by ting th invr Fourir trnfor of S ˆ ( r rpctivly dpictd in Fig.-b, c nd d b Figur. Intrpoltion vlt. ( th cond ordr Dubchi intrpoltion vlt. (b th fifth ordr Sylt intrpoltion vlt. (c th third ordr Coiflt intrpoltion vlt. (d th cond ordr Coiflt intrpoltion vlt. B. Illutrtion of vlt rcovry Fro th fr thory, thr it bijctiv linr p btn n intrpoltion vlt bi nd n orthogonl vlt bi. Hnc, for givn vlt (, hv ( ( / S (, (37 inc th intrpoltion point of { S ( } r loctd t / ith. Thi n tht th function in Fig. cn b vrifid intrpoltion vlt if thy cn b pplid to rcovr clic vlt. In our pl, (37 i ud to intrpolt th bov-ntiond Dubchi, Sylt, Coiflt vlt, for th purpo of vlidting our lgorith. Th cond ordr Dubchi, fifth ordr Sylt, third nd cond ordr Coiflt vlt, ll of hich dnot by (, rpctivly hv upport in [,3], [,9] [,7] nd [,]. W u th pl { ( n /} n t n,, for rcovring th cond Dubchi vlt, t n,,8 for th fifth ordr Sylt vlt, t n,,6 for th third Coiflt nd t n,, for th cond Coiflt. Th point r dnotd * in Fig. -, c, nd g. Th Dubchi, Sylt nd third nd cond Coiflt vlt r rpctivly rcovrd by th ri f ( ( / ( n S 8, f ( ( / n n n 6 S (, f ( ( n/ S ( nd f ( n ( n/ S (, hon in Fig.. n Sinc S ( nd ( in our pl r obtind fro nuricl clcultion, our rcovri cnnot void ll nuricl rror, ( hon in Fig. -b, d, f nd h. By clculting th rror in Fig., no tht th iu rror, th n bolut rror nd th root n qur rror for th c d Dubchi vlt r rpctivly ,.33 4 nd Tho for th Sylt r rpctivly , nd Finlly, tho r rpctivly , nd for th third Coiflt nd r rpctivly , nd.46 4 for th cond Coiflt. Such rror lvl r ronbl for th rcovri in our lgorith. Th four rcovri of vlt bov vrify tht th { S ( } obtind by our lgorith r intrpoltion b rltiv to { f ( /} in th vlt pc. f ( f( ( f ( f( ( ( b c d Figur. Rcovri of vlt nd thir rror (. (, b rcovry nd rror for th cond ordr Dubchi vlt. (c, d rcovry nd rror for th fifth ordr Sylt vlt. (c, b rcovry nd rror for th third ordr Coiflt. (, f rcovry nd rror for th cond ordr Coiflt. ( ACKNOWLEDGMENT Thi or i upportd by th Fundntl Rrch Fund for th Cntrl Univriti. VII. CONCLUSION Wvlt pling i n iportnt brnch of th vlt thory nd h bn ud idly in ignl procing. Hovr, th clic contruction of intrpoltion vlt dpnd on orthogonl vlt. Sinc coputtion of orthogonl vlt r gnrlly copl for ny ultirolution nly, ppliction of intrpoltion vlt r liitd in prctic. Bd on th pcil rltion btn intrpoltion vlt nd cling function, n lgorith h bn propod for contructing intrpoltion vlt dirctly fro cling func f g h

6 tion. Sinc orthogonl vlt r not ndd in th contruction procdur, our lgorith h iprovd fficincy of contructing intrpoltion vlt grtly. Finlly, th thor nd pl in thi ppr hv dontrtd our contruction lgorith. REFERENCES [] D.L. Donoho, Intrpolting vlt trnfor, tt.tnford.du/ ~donoho/rport/ 99/ intrpol.pdf, 99. [] S. Mllt, A Wvlt Tour of Signl Procing, 3rd, MA: Burlington, Acdic Pr, 9, pp [3] R.S. Ar, K.M. Bhurchndi nd A.S. Gndhi, Intrpoltion of ig uing dicrt vlt trnfor to iult ig rizing in hun viion, Intrntionl Journl of Autotion nd Coputing, vol. 7, pp. 9-6,. [4] J. F. Rinoo, M. Moncyo, Optil qulity for ig fuion ith intrpoltory prtric filtr, Mthtic nd Coputr in Siultion, vol. 8, pp ,. [5] K. Schnidr, O.V. Vilyv, Wvlt thod in coputtionl fluid dynic, Annul Rvi of Fluid Mchnic, vol. 4, pp ,. [6] S. Sur, C. N. Jon, J. Lygro nd M. Morri, A ultirolution pproition thod for ft plicit odl prdictiv control, IEEE Trnction on Autotic Control, vol. 56, pp ,. [7] G.G. Wltr, A pling thor for vlt ubpc, IEEE Trnction on Infortion Thory, vol. 38, pp , 99. [8] A. Jnn, Th trnfor nd pling thor for vlt ubpc, IEEE Trnction on Signl Procing, vol. 4, pp , 993. [9] Wn Chn nd S. Itoh, On pling in hift invrint pc, IEEE Trnction on Infortion Thory, vol. 48, pp. 8-89,. [] R.S. Ar, K.M. Bhurchndi nd A.S. Gndhi, Intrpoltion of ig uing dicrt vlt trnfor to iult ig rizing in hun viion, Intrntionl Journl of Autotion nd Coputing, vol. 7, pp. 9-6,. [] S. At, K. Ddourin, J. Lindrt, J. Ruiz nd J.C. Trillo, A fily of tbl nonlinr nonprbl ultirolution ch in D, Journl of Coputtionl nd Applid Mthtic, vol. 34, pp.77-9,. [] S. Ericon, Irrgulr pling in hift invrint pc of highr dinion, Intrntionl Journl of Wvlt, Multirolution nd Infortion Procing, vol. 6, pp. -36, 8. [3] P. ho, C. ho nd P. G. Czz, Prturbtion of rgulr pling in hift-invrint pc for fr, IEEE Trnction on Infortion Thory, vol. 5, pp , 6. [4] A. G. Grcí nd G. Pérz-Villlón, Multivrit gnrlizd pling in hift-invrint pc nd it pproition proprti, Journl of Mthticl Anlyi nd Appliction, vol. 355, pp , 9. [5] A.G. Grcí nd M.A. Hrnándz-Mdin nd G. Pérz-Villlón, Ovrpling in hift-invrint pc ith rtionl pling priod, IEEE Trnction on Signl Procing, vol. 57, pp , 9. [6] A.G. Grcí nd G. Pérz-Villlón, Approition fro hift-invrint pc by gnrlizd pling forul, Applid nd Coputtionl Hronic Anlyi, vol. 4, pp , 8. [7] A.G. Grcí, J.M. Ki, K.H. Kon nd Pérz-Villlón, Aliing rror of pling ri in vlt ubpc, Nuricl Functionl Anlyi nd Optiiztion, vol. 9, pp. 6-44, 8. [8] A.G. Grcí, G. Pérz-Villlón, On th liing rror in vlt ubpc, Journl of Coputtionl nd Applid Mthtic, vol. 83, pp , 5. [9] higuo hng, Mr A. Kon, On rlting intrpoltory vlt to intrpoltory cling function in ultirolution nly, Circuit, Syt nd Signl Procing, vol. 34, pp