The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

Transcription

1 Applid ad Coputatioal Mathatics 0; 7(3: doi: 0.6/j.ac ISS: (Prit; ISS: (Oli Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt Fras Basd o FMRA Yuayua Zhag * Zhaofg Li Collg of Scics Chia Thr Gorgs Uivrsity Yichag Chia Thr Gorgs Mathatical Rsarch Ctr Chia Thr Gorgs Uivrsity Yichag Chia Eail addrss: * Corrspodig author To cit this articl: Yuayua Zhag Zhaofg Li. Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt Fras Basd o FMRA. Applid ad Coputatioal Mathatics. Vol. 7 o. 3 0 pp doi: 0.6/j.ac Rcivd: April 0; Accptd: July 3 0; Publishd: August 3 0 Abstract: I 97 J. Morlt raisd th cocpt of wavlt trasfor ad stablishd th ivrsio forula through th xpric of physical ituitio ad sigal procssig. I 96 Y. Mryr cratd a ral sall wav bas ad th wavlt aalysis bga to flourish aftr a ulti scal aalysis of th sa thod of costructig th sall wav bas with S. Mallat. I ordr to aalyz ad dal with o-statioary sigals a sris of w sigal aalysis thoris ar proposd.: Short Ti Fourir Trasfor ti-frqucy aalysis wavlt trasfor ad fractioal Fourir trasfor ad so o. I this papr a xplicit algorith is giv to costruct th iiu-rgy fras basd o fra ultirsolutio aalysis via charactristic vctors of th ask atrix. I sctio w show th structur of iiu-rgy wavlt fras i trs of thir asks (La ad discuss that w should liiat th corrlatio of th rows of th ask atrix by th polyphas dcopositio tchiqu. Basd o FMRA a xplicit algorith is giv to costruct this fras. By this thod all th iiu-rgy wavlt fras ca b obtaid. As a applicatio svral xapls ar showd to xplai this thod i sctio 3. This thod ca also b applid i othr filds of wavlt aalysis. Kywords: Fra Multirsolutio Aalysis Polyphas Dcopositio Miiu-Ergy Fras. Itroductio I 97 a Frch gir J. Morlt raisd th cocpt of wavlt trasfor ad stablishd th ivrsio forula through th xpric of physical ituitio ad sigal procssig. I 96 Y. Mryr cratd a ral sall wav bas ad th wavlt aalysis bga to flourish aftr a ulti scal aalysis of th sa thod of costructig th sall wav bas with S. Mallat. I ordr to aalyz ad dal with o-statioary sigals a sris of w sigal aalysis thoris ar proposd ad dvlopd: Short Ti Fourir Trasfor ti-frqucy aalysis wavlt trasfor ad fractioal Fourir trasfor ad so o. This papr dals with th study of copactly supportd iiu-rgy wavlt fras corrspodig to a sigl rfiabl fuctio with copact support. It is wll kow that th ultirsolutio aalysis (MRA for short is a systatic thod to costruct orthooral wavlt bass for L p (R[-]. MRA rquirs that th rfit ask H ( should satisfy H ( + H( + π [9 p3]. Howvr thr xist ay rfiabl fuctios whos ask H ( posssss H( + H( + π. So w wat to kow whthr ths rfiabl fuctios ca grat tight wavlt fras spcially iiu-rgy wavlt fras. I 99 J. J. Bdtto ad S. Li [0] itroducd th thory of fra ultirsolutio aalysis (FMRA for short. FMRA is a xtsio of th cocpt of MRA. Basd o FMRA w ca costruct tight wavlt fras associatd with a giv rfiabl fuctio. Dfiitio (FMRA A FMRA associatd with a dilatio factor M R M is a squc of clos subspacs of L (R satisfyig th followig coditios: V V j Z. { j} { j+} ;. { 0 } Vj L ( R j Z j Z

2 6 Yuayua Zhag ad Zhaofg Li: Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt Fras Basd o FMRA 3. f ( x Vj if ad oly if f ( Mx V j + ;. Thr xists a fuctio ϕ ( x V0 such that { ( x : Z} ϕ fors a fra i. Th fuctio ϕ (x is calld a fra rfiabl fuctio for th FMRA. It is phasizd that th shifts of ϕ (x for a fra ot cssarily a orthooral or a Risz bas of V0 as MRA. I this papr w fix M. I this cas Charls K. Chui ad Wji H [] discussd ad costructd iiu-rgy fras by th uitary atrix xtsio [s ]. Wh M3 Cui Lihog Chg Zhgxig ad Yag Shouzhi gav a sufficit ad cssary coditio to th tight wavlt fras i [3]. Motivatd by th work of [] ad [3] w costruct th iiu-rgy wavlt fras via charactristic vctors of th ask atrix. Dfiitio (Miiu-rgy wavlt fras [] Lt ϕ(x L (R with ϕˆ L ϕˆ cotiuous at 0 ad ϕ ˆ( 0 b a scalig fuctio that grats a FMRA. Th a fiit faily of fuctios ψ { ψ ψ ψ } V is calld a iiu-rgy wavlt fra associatd with ϕ (x if k Z V 0 f ϕ k f ϕ0k + k Z i k Z f ψ i 0 k All f L (R. Th iiu-rgy wavlt fra ψ is cssarily a tight wavlt fra i L (R with fra boud quals to. O of th advatag of this fra is that it ca avoid th coplicatio of th chag of bass usig th sa wavlts as th orthooral bass [s ]. I this papr a xplicit algorith is giv to costruct this fras. Th papr is orgaizd as follows. I sctio w show th structur of iiu-rgy wavlt fras i trs of thir asks (La ad discuss that w should liiat th corrlatio of th rows of th ask atrix by th polyphas dcopositio tchiqu. Th xplicit algorith is also giv i sctio. Th last sctio is dvotd to so xapls obtaid by this algorith.. Prliiaris ad Mai Rsults.. Prliiaris Lt { j} j Z V grats a FMRA i L (R ad ψ { ψ ψ ψ } V. Sic V V Spa{ ϕ( x : Z} 0 w hav + squcs { } l { g } l l. such that h l ad ϕ(x l ψ (x h ϕ( x l g ϕ( x hr ad throughout l. I this papr th Fourir trasfor of a itgrabl fuctio f (x is dfid as x fˆ ( f(x dx. Takig Fourir trasfor at both sids of ( lads to St H( ϕˆ(x l ψˆ (x h R i h i l g ϕˆ( ϕˆ( Ad l l G ( g th ( is quivalt to ϕˆ( l ψˆ ( H( ˆ( ϕ l G ( ˆ( ϕ Th π -priodic fuctios H( ad G l ( ar calld th rfit ad th wavlt asks rspctivly. With H( ad G l ( a ( + atrix ca b forulatd as M( H( H( + π ( ( (3 G ( G ( ( G ( + π G ( + π Charls K. Chui ad Wji [] gav th structur of th iiu-rgy wavlt fra as follows. La Lt ϕ (x L (R with ϕˆ L ϕˆ cotiuous at 0 ad b a rfiabl fuctio that grats a FMRA i th ss of Dfiitio. Ad Lt H( ad G l ( b th asks cocrig with ϕ ad ψ { ψ ψ ψ }.. Th ψ is a iiu-rgy wavlt fra if ad oly if M( M ( I a.. (5 hr M ( rprsts th coplx cojugat of th traspos of M(. It should b phasizd that Charls K. Chui ad Wji [] hav poitd out that th rfiabl fuctio ϕ (x grats a FMRA tight wavlt fra if ad oly if th rfiabl ask ( H satisfy ( + H( + π. H Fro La th costructio of wavlt fra ca b rducd to

3 Applid ad Coputatioal Mathatics 0; 7(3: th probl of xtdig a vctor atrix ( H( H( + π to a uitary atrix as (. That is w d to sk fuctios G ( G ( G ( such that (5 is satisfid. ot th rows of ( ar corrlativ so w should rov this fatur by usig th polyphas dcopositio tchiqu [s 3 p06 also s 9 p3] first. Siilarly to [3] ad for copltss this tchiqu is itroducd brifly. H( ad G l ( ca b writt i thir polyphas fors rspctivly as Writ ( H( l G ( H ( H( + π (H ( + l (G ( + G ( G ( + π l H ( l G ( (6 G ( (7 G ( + π Sic H ( H( ad G ( G ( ar π -priodic fuctios w hav Th (5 as M( l ( ( ( ( I a.. (9 Rark Th atrix ( is th polyphas dcopositio of M( ad (9 ad (5 look vry alik. W ca xtsiv ( to a uitary atrix to obtai a iiu-rgy wavlt fra. Th diffrc of (9 ad (5 is that th rows of ( ar ot corrlativ. I so applicatios w hop th rfiabl fuctios ad fras hav so spcial proprtis such as sytric or ati-sytric. Sic this algorith ds aihilat th corrlativ of M( ad th polyphas dcopositios do ot kp th sytric or ati-sytric fatur w should rtur to ( ad (5 to obtai a sytric or ati-sytric fra. Charls K. Chui ad Wji [] gav a igious costructiv thod... Mai Rsults I this subsctio w prov that if th rfit ask H( satisfis H ( + H( + π. Th thr xists a xplicit algorith to costruct a iiu-rgy wavlt fra. This thod is otivatd by Cui Lihog Chg Zhgxig ad Yag Shouzhi []. Sic H ( + H ( H( + H( + π ( + H (. w hav H St G G ( ( th (9 is rforulatd as H ( H( or quivaltly G ( I G ( G ( G ( G ( ( H ( H ( + G ( I H ( H ( H ( H ( H( ( H ( H ( H ( H ( H ( (0 ( ( By sipl calculatio th charactristic roots of G ( ar λ λ H ( H ( (3 ad th corrspodig uit charactristic vctors ar α β ( H ( H ( ( H ( H ( T ( T H ( + H (. (5 Sic H ( + H ( by Risz La [9 La 6..3] thr xists a polyoial H3 ( such that H3( H ( H(. (6 ot G ( is a coplx sytrical atrix G ( ca b writt as λ G ( ( α β 0 ( α β 0 ( α β 0 (( α β( 0 0 ( α β λ 0 ( α β H ( H ( 0 ( α β H ( 3 0 g( (( α β H ( g( H ( 3 (7 hr ad throughout g( is a atrix ad satisfis g( g( I. Thrfor (7 as that ( α β 0 0 g(. H ( 3 ( All th abov lads to th followig Thor.

4 6 Yuayua Zhag ad Zhaofg Li: Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt Fras Basd o FMRA Thor Lt ϕ (x L (R with ϕ ˆ L ϕˆ cotiuous at 0 ad ϕ ˆ( 0 b a rfiabl fuctio that grats a FMRA whos ask H( satisfis H( + H( + π a.. (9 ad lt H ( ad H ( b th polyphas copots of H( rspctivly. Th thr xists a iiu-rgy wavlt fra { } ψ ψ ψ ψ V associatd with ϕ (x. Furthror all th iiu-rgy wavlt fras ca b writt i th ss of thir asks as whr H( H ( H ( H ( 0 0 g( H ( 3 H( + H( ad H3 ( satisfis 3 (0 H ( + H ( + H (. ( Rark If th rfiabl fuctio ϕ (x grats a orthooral wavlt bas i L (R th th FMRA is a stadard MRA ad th rfit ask H( satisfis H ( + H (. I this cas H ( 0.. Sic 3 a orthooral wavlt bas also is a iiu-rgy wavlt fra Thor icluds this cas i which th corrspodig orthooral wavlt bas is ( H ( H ( T. Exapls xplais this cas. Rark 3 By Thor if w choos a diffrt g( w ca fid all th copact support iiu-rgy fras cosists of wavlt associatd with a giv ψ ψ ψ copactly support rfiabl fuctio ϕ (x. Wh th iiu-rgy wavlt fra { ψ } ψ [9]. 3. Exapls ψ is obtaid as I this sctio svral xapls associatd with th cardial B-splis ar giv. It is wll kow that i th dvlopt of wavlt aalysis cardial B-splis srv as a caoical xapl of scalig fuctios that grat MRA i L (R. Th ordr cardial B-splis (x is dfid iductig by H ( + Ad w ca s that H ( satisfis H ( H (. + H ( + π + H ( + π. (3 ( Exapl (Haar wavlt This is th spcial cas wh ad kow as th Haar wavlt. Th Haar fuctio is (x ad th rfit ask + H ( satisfis H ( + H ( + π. It is asy to fid th iiu-rgy wavlt fra ask is g(. Exapl (Liar B-splis Wh H ( + ( + + Th polyphas dcopositios ar obtaid as follows Ad H ( H +. (5 (6 (7 (. ( thr xists H3 ( ( such that H ( + H( + H3(. By Thor all th iiu-rgy wavlt fra asks associatd with (x ar H( H ( H ( H ( 0 0 g( H ( 3 (9 (t xdx ( 0 with (x dotig th charactristic fuctio of th uit itrval [0 ] (s [ p]. Th ask of (x is Hr ( 6 + I this cas if w choos g( + as i. (30

5 Applid ad Coputatioal Mathatics 0; 7(3: ad g( ( + ( i i ( (3 i ( + It is asy to fid a iiu-rgy fra i trs of asks Q ( Q ( + i. (3 (33 Furthror it is asy to s th corrspodig wavlt fuctio ψ (x is sytric ad ψ (x is ati-sytric. This rsult was also giv i []. Exapl 3 (Quadratic B-splis Wh 3 ad 3 H ( ( Siilarly to Exapl w obtai 3 H ( ( H ( ( H3 ( ( i. (3 (35 (36 (37 So all th corrspodig iiu-rgy wavlt tight fra asks ar hr 3 3 H ( H ( 0 g( 3 3 H ( H ( 3 H ( 0 3 ow w choos g( 6 ( ( + i i + 3. i 3 ( + i (3 (39 (0 th a ati-sytric iiu-rgy fra is obtaid as ad 3 Q ( ( Q( ( Coclusio 3i. ( ( This papr dals with th study of copactly supportd iiu-rgy wavlt fras corrspodig to a sigl rfiabl fuctio with copact support. I sctio w show th structur of iiu-rgy wavlt fras i trs of thir asks (La ad discuss that w should liiat th corrlatio of th rows of th ask atrix by th polyphas dcopositio tchiqu. Basd o FMRA a xplicit algorith is giv to costruct this fras. I sctio 3 so xapls is giv by this algorith. This thod ca also b applid i othr filds of wavlt aalysis. Rfrcs [] S. Mallat Multirsolutio approxiatios ad orthooral bass of wavlts for L (R Tras. Ar. Math. Soc. vol pp [] P. Stff P.. Hllr R. A. Gopiath ad C. S. Burrus Thory of rgular M-bad wavlt bass IEEE Tras. Sigal Procssig vol. ( [3] C. Chaux ad L. Duval Iag aalysis usig a dual-tr M-bad wavlt trasfor IEEE Trasactios o Iag Procssig vol pp [] M. K. Mihcak I. Kozitsv K. Rachadra ad P. Mouli Low-coplxity iag doisig basd o statistical odlig of wavlt cofficits IEEE Sigal Procssig Lttrs vol pp [5] L. Ga ad K. K. Ma A siplifid lattic factorizatio for liar-phas prfct rcostructio filtr bak IEEE Sigal Procssig Lttrs vol. 00 pp [6] B. Ha Sytric orthooral scalig fuctios ad wavlts with dilatio factor Adv. Coput. Math. Vol. 99 pp. -7. [7] Chao Zhag t al. "Optial scal of crop classificatio usig uad arial vhicl rot ssig iagry basd o wavlt packt trasfor." Trasactios of th Chis Socity of Agricultural Egirig (06. [] Shlyovich M. P. M. V. Mdvdv ad S. A. Lyashva. "Objct dtctio i th iags i idustrial procss cotrol systs basd o salit poits of wavlt trasfor aalysis." Itratioal Cofrc o Idustrial Egirig Applicatios ad Maufacturig IEEE (07:-6. [9] I. Daubchis T lcturs o wavlts CBMF cofrc sris i applid athatics 6 SIAM Philadlphia 99. [0] J. J. Bdtto ad S. Li Th thor of ultirsolutio aalysis fras ad applicatio to filtr baks Appl. Coput. Haro. Aal. vol pp

6 66 Yuayua Zhag ad Zhaofg Li: Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt Fras Basd o FMRA [] C. K. Chui ad W. H Copactly supportd tight fras associatd with rfiabl fuctios Appl. Cop. Haroic Aal. vol. 000 pp [] W. Lawto S. L. L ad Z. Sh A algorith for atrix xtsio ad wavlt costructio Math. Cop. vol pp [3] Cui Lihog Chg Zhgxig ad Yag Shouzhi Explicit Structur of Wavlt Tight Fras Acta Mathatica Scitia vol. ( 00 pp [] Su Qiyu Bi ig ad Huag Dar A itroductio to ultibad wavlts Zhjiag uivrsity prss 00.