The Noval Properties and Construction of Multi-scale Matrix-valued Bivariate Wavelet wraps

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1 Available olie at Phyic Procedia 5 ( ) Iteratioal Coferece o Solid State Device ad Material Sciece The Noval Propertie ad Cotructio of Multi-cale Matrix-valued Bivariate Wavelet wrap ZHANG Hai-mo Departmet of Math. Sciece,Huaghuai Uiverity,Zhumadia 463, Chia Abtract I thi paper, we itroduce matrix-valued multi-reolutio tructure ad matrix-valued bivariate wavelet wrap. A cotructive method of emi-orthogoal matrix-valued bivari-ate wavelet wrap i preeted. Their propertie have bee characterized by uig time-frequecy aalyi method, uitary exteio priciple ad operator theory. The direct decom-poitio relatio i obtaied. Publihed by Elevier Ltd. B.V. Selectio ad/or peer-review uder repoibility of of [ame Garry orgaizer] Lee Ope acce uder CC BY-NC-ND licee. Keyword :bivariate, iterative method, matrix-valued mul-tireolutio tructure, uitary exteio priciple, wavelet wrap.itroductio Wavelet aalyi ha become a popular ubect i ciet-ific reearch for twety year. It ha bee a powerful tool for explorig ad olvig may complicated problem i atural ciece ad egieerig computatio. Samplig theorem play a baic role i digital igal proceig. They eure that cotiuou igal ca be repreeted ad proceed by their dicrete ample.the claical Shao Samplig Theorem a-ert that badlimmited igal ca be exactly repreeted by their uiform ample a log a the amplig rate i ot le tha the Nyquit rate. Thi theorem ha bee proved to be fudametal i may applicatio of igal proceig ad comm-uicatio theory. Whe we ue the wavelet decompoitio i digital igal pro-ceig, the coefficiet i the high level re-preetatio ca be choe tobe ample of the cotiuou igal. However, for the wavelet decompoitio, the Mallat algorithm i ofte ued, ad FIR filter are preferred, although fi-lter with expoetially decayig impule repoe alo provide atifactory reult for may applicatio. Cotructio of wavelet bae i a importat apect of wavelet aalyi, ad multireolutio aalyi method i oe of importmet way of deigig variou wavelet bae. The mai trait of the wavelet tra-form i to hierarchically decompoe Publihed by Elevier B.V. Selectio ad/or peer-review uder repoibility of Garry Lee Ope acce uder CC BY-NC-ND licee. doi:.6/.phpro..3.67

2 Zhag Hai-mo / Phyic Procedia 5 ( ) geeral fuctio, a a igal or a proce, ito a et of approximatio fuctio wi-th differet cale. Recetly, Multiwavelet [,] have bee a-pplied to may apect i techology ad ciece,uch a,image compre, igal proceig [3], olvig Itegral Equatio [4] ad o o, maily becaue of their ability to offer propertie lie ymmetry, orthogoality, hort upport at the ame time.it i oticed that multiwavelet ca be geerated from the compo-et fuctio i multiple vector-valued wavelet. Reearchig ito multiple vector-valued wavelet i ueful i multiwavelet th-eory. Che [5] itroduced the otio of multiple vector-valued wavelet ad tudied the exitece ad cotructio of orthog-oal multiple vector-valued wavelet. Fowler ad Li [6] imple-meted orthogoal multiple vector-valued wavelet traform to tudy fluid flow i oceaography ad aerodyamic. However, multiwavelet ad multiple vector-valued wavelet are di-fferet i the followig ee. Prefilterig i uually required for dicrete multi-wavelet raform but ot eceary for dic-rete multiple vector-valued wavelet traform. Therefore it i eceary to tudy multiple vector-valued wavelet. However, a yet there ha ot bee a geeral method to obtai orthogo-al multiple vector-valued wavelet. I the igal deoig met-hod, wavelet aa-lyi i a ew aalytical tool which develop o the bai of the Fourier aalyi.the locatio ad multicale reolutio fuctio imultaeouly cotaied i time domai ad frequecy domai. I order to ehace the applicatio of wavelet aalyi i the igal proce-ig, ad to improve the accuracy of igal proceig a far a poible, predeceor have doe a large umber of practice ad exploratio, reul-tig i a lot of igal proceig method baed o wavelet tra-form. They eure that coti-uou igal ca be repreet-ed ad proceed by their dicrete ample.we hall preet a algorithm for deigig a ort of fiitely upported orthogoal multiple vector-valued calig fuctio ad wavelet. We alo tudy the trait of multiple vector-valued wavelet pac..matrix-valued multireolutio aalyi The multiple vector-valued multireolutio aalyi i i--troduced ad the defiitio for orthogoal multiple vector-va-lued wavelet i give. Moreover, we are ow ready to dicu the cotructio of orthogoal multiple vector -valued wavele--t. Let ad deote all complex ad all real umber,repectively. ad tad for, repectively, all iter ad o-egative iteger. Set a, be a cotat ad a,. The igal pace L (, ) i defied to be the et of all multiple vector-valued fuctio, i.e., where L ( ), l,,,. Example of matrix-valued igal are video image i l, which l, deote the pixel at the time t the th row ad the l th colum. For L (, ), dy, ad the itgratio of multiple vector-valued,, fuctio i defied to be dy:= ( ydy ), l, l

3 494 Zhag Hai-mo / Phyic Procedia 5 ( ) i.e., the matrix of the itegral of every calar fuctio ( x), l,,,,. l, For arbitrary, L (, ), their y-mbol ier product i defied by, : dy, where mea the trapoe ad the complex cougate. Defiitio. A family of mul-tiple vector -valued fuctio { } u L (, ) i called a u orthoormal bai, if the followig coditio i atified:, : I l,, 3 ad ( x) L (, ), there i a equece of l l, cotat matrice M u uch that M u, 4 u u where I deote the idetity matrix ad l, = whe l ad l, whe l. Defiitio. A biary multiple vector-valued multireolutio aalyi of ( L, ) i a eted equece of cloed ubp-ace U, Z of L (, ) uch that it follow, (i) U U, Z ; Z a. a (ii) U ( a y) U, Z; (iii) U { O}; U i dee i L (, ) Z Z (iv) There i a Ft () X uch that it tralate { Hv: H( yv), vz } form a Riez bai for U. Sice Ht () U U, by defiitio ad (), there i a fiite-term equece of matrice { P v } v uch that H a PH v ( ayv). (5) Equatio (5) i called a refiemet equatio ad H( y ) i cal-led a vector calig fuctio. W, deote the complemetary ubpace of U i U ad there exit a multiple vector- / () t L (, ) uch that G ( x) a G ( a x),, form a Riez bai of,, W, where {,, a }. It i clear that G W U.Hece there exit a equece of matrice { B } Z {,, a }, uch that G () t a Bv H( ay). (6) Let, valued fuctio We call H( x ) a orthogoal multiple vector-valued ca-lig fuctio, if it atifie H(), H( ) I. (7) v, We ay that G L (, ) are orthogoal multiple vector-valued wavelet aociated with orthogoal multiple vector-valued calig fuctio H( y ), if (), ( ),, ; v, v, H G O (8) G (), G ( ) I,,, v. (9) The, we have the followig reult from (4)-(7). Theorem. Suppoe that H( y ) defied by (5), i a ortho-goal multiple vector -valued calig fuctio. The, for ay u Z, we have * au( ), u. () a P P I

4 Zhag Hai-mo / Phyic Procedia 5 ( ) Proof. Subtitutig equatio (5) ito the orthogoality relat-io formula (7), we have, ui H(), H( ) 4 * * a P H ( ay ) H( ay a ) ( P) dt, R, a P H H au P ( ), ( ) ( ) a P ( P ) * a Theorem []. Aume that G are multiple vector-valued fuctio i L (, ). The G are orthogoal multi-ple vector-valued wavelet fuctio aociated with orthogo-al vector-valued calig fuctio H( y ), the we have * P ( B mu ) O, u Z, Z () a B ( B ) I,. * a, () Thu, both Theorem ad ()-() provided a approach to deig orthogoal multiple vector-valued wavelet with a. 3.Cotructio of vector-valued wavelet I what follow, we ivetigate the cotructio of compactly upported orthogoal matrix-valued wavelet fuctio a--d preet a algorithm for cotructig them Theorem 3 [7]. Let H L (, ) be a 5-cocfficiet compactly upported orthogoal multiple vector-valued calig fuctio aifyig the followig equatio: Ht () 4 PH 4 PH(y) 4 PH(y4). (3) 4 Aume that there exit a iteger v,v,uch that the matrix M defied i the followig equatio, i ot oly a i--vertible matrix but alo a Hermitia matrix: * * M ((/ 4) I PP v v ) PP v v. (4) Defie B MP, v, v, {,,,3, 4}. (5) B M P, v, The Gy ( ), defied i (7), i a orthogoal matrix-valued wavelet aociated with H( y ): G 4{ B H BH(y) B H(y4)} (6) 4.The trait of a ort of wavelet wrap 4. We will coider wavelet wrap i L (, ), we et () H, G, () P, B, v. v v v v *

5 496 Zhag Hai-mo / Phyic Procedia 5 ( ) Defiitio 3. A et {,,,,,,,,3} i called multiple vector-valued wavelet pac cocerig the orthogoal multiple vector -valued calig fuctio H( y ), where ( ) y y Taig the Fourier traform for the both ide of (7) give Lemma [6]. Aume y,,,,3. (7) ( ) 4 ( ) ( ) ( ) ( ) ( ),,,,,3, (8) ( ) ( ) ( ) exp{ i},. (9) ( ) L (, ) are orthogoal multiple vector-valued wavelet fuctio aociated with. The, for,, we have ( ) ( ) * (( ) / a) (( ) / a) I., () Theorem 4 [6]. Aume {,,,, } i multiple vector-valued wavelet wrap cocerig the orthogoal multiple vector -valued wrap calig fuctio H( y ),The (), ( ) O,,. () Theorem 5. If {,,,, } i a multiple vector -valued wavelet wrap cocerig the orthogoal multiple vec- tor -valued calig fuctio H( y ), the, we have (), ( ) I, m,,. () {} m m,, Proof. Formula () hold for m by Theorem. For the cae of m, without lo of geerality, we aume m. We ca write ma, m[ m ], [ ], where,,. (i) If [m ]=[ ], the. By (9), () ad (), we get that Theorem 5. If {, } (), ( ) m ( ) [ m ] [ ] ( ) e i d O. (ii) For [m ] [ ]. Let [ m ] [[ m ] ], [ ] [[ ] ],where,,. (), ( ) m ( ) ( ) O O ad {, } are biorthogoal iary wavelet pac accordig to cocerig to a pair of biorthogoal vector-valued calig fuctio ad, the for ay a 4,,, we have,, I (), ( ),. (3)

6 Zhag Hai-mo / Phyic Procedia 5 ( ) Proof. Whe, (8) follow by Lemma. a ad,, it follow from Lemma that (8) hold, too. Aumig that i ot equal to, a well a at leat oe of {, } doe t belog to, we rewrite a, 4, 4, where a 4,,. Cae. If, the. (3) follow by virtue of (7) a well a Lemma, i.e., ( ) (), ( ) * 4 ( ) 4 ( ) exp{ i } d R I exp{ i} d O. [, ], Cae If, order 4, 4, where, Z, ad,. Provided that, the. Similar to Cae, (8) ca be etablihed. Whe, order 4 3 3, 4 3 3, where 3, 3 Z, 3, 3. Thu, after taig fiite tep (de-oted by ), we obtai, ad,. If,the. (8) hold. If, the we obtai * i (), ( ) ( ) ( ) e d ( ) R ( ) ( ) ( / 4) ( /6) ( /6) ( ) R ( ) ( ) * * * i ( /6) ( /6) ( /4) e d O ( l ) l { ( /4 )} ( ) ([,4 ] l ( l ) * l l { ( )} exp{ i } d O 4 Therefore, for ay,, (8) i etablihed. Theorem 6. Let hy ( ), hy ( ), ad, be fuctio i L ( R ) defied by (5), (6), (9) ad (), repectively. Aume that coditio i Theorem are atified. The, for ay fuctio Furthermore, for ay f y ( ) L ( R ), f y ( ) L ( R ), ad ay iteger, 4. (4) f, h h y f, y,, : v, : v, Z v Z 4 f y f y v Z. (5), :, v :, v Proof. (i) Coider, for, Z, the operator E : L ( R ) L ( R ) uch that E y y, h, h., The the operator E are well defied ad uiformly bouded i the orm o L ( R ). To how that Z

7 498 Zhag Hai-mo / Phyic Procedia 5 ( ) E a, it i ufficiet to how that, for all g () i ay dee ubpace of bad-limited 6 L ( R ), fuctio i gh,, Z h a., I particular, we may chooe the dee et of fuctio g (), whoe Fourier traform have compact upport, i cotiuou, ad vaihe i a eighborhood of.,,, (6) gh, h B gh, where B i the Beel boud of { h, }. Implemetig tadard calculatio of the right-had ide of (3), we have gh,, where B i the Beel boud of { h, } with compact upport, the term 4 4 fiite itegral g we get that 4 v v u g u h u h g d B g u d (4 ( 4 ) ) 4 hv 4 g u. Followig the lead of [8] ad ice g C t dt. Moreover, ice f i cotiuou, beig a Riema um to the g vaihe i a eighborhood of for all Z,, gh, ( B ) C 4 h (4 ) g ( ) d B C g ( 4 h(4 ) d) Note that the lat itegral at the right-had ide ted to a. Thi prove the firt part of the theorem ice, by uig (8) recurively, we have 4, h, h, y, :, v :, v. (ii) Sice V L ( R ), for ay L ( R ) ad ay g v Z there exit, ad for ay gv V uch that g () g, h v, h () Moreover, for C BB, Z g C. Now, by (), for all, we have v, 4, :, v :, v v Z g

8 Zhag Hai-mo / Phyic Procedia 5 ( ) g C g g C. Referece [] L. Teleca, et al.. Multireolutio wavelet aalyi of earthquae, Chao, Solito & Fractal, Vol.,No.3, PP , 4. [] H. He, S. Cheg. Home etwor power-lie commui-catio igal proceig baed o wavelet pacet aalyi, IEEE Tra Power Delivery, Vol., No.3, PP ,5. [3] N. Zhag, X.Wu X. Lole Compreio of Color Moaic Image, IEEE Tra Image proceig Vol. 5, No. 6, PP , 6. [4] Q. Che, A. Huo. The reearch of a cla of biorth-ogoal compactly upported vector-valued wavelet, Chao, Solito & Fractal, 4 ( ), pp , Augut 9 [5] X. G. Xia, B. W.Suter. Vector-valued wavelet ad vector filter ba, IEEE Tra. igal Proceig, 996,44(): [6] Q. Che, etal. A tudy o compactly upported orthogo-al vector-valued wavelet ad wavelet pacet, Chao, Solito & Fractal. Vol.3, No.4, PP. 4-34,7. [7] S. Li, et al, Peudo-dual of frame with appli-catio Appl.Comput. Harmo. Aal., ( ), pp ,.