GENERALIZED TWO DIMENSIONAL CANONICAL TRANSFORM

Transcription

1 IOSR Joural o Egieerig (IOSRJEN) ISSN: Volume, Iue 6 (Jue 0), PP GENERALIZED TWO DIMENSIONAL CANONICAL TRANSFORM S.B.Chavha Yehawat Mahavidhalaya Naded (Idia) Abtract: The two-dimeioal caoical traorm ca be ued i optical ytem aalyi, image proceig ad patter recogitio. I thi paper two-dimeioal traorm aoical i exteded to the ditributio o compact upport. Aalyticity theorem, iverio theorem, i proved or thi traorm. Latly propertie o kerel are dicued. eyword:- Caoical traorm, two-dimeioal caoical traorm Geeralized uctio, igal proceig. I. Itroductio: Now a day ractioal itegral traorm play a importat role i igal proceig, image recotructio, patter recogitio, acoutic igal proceig,[], [].The ractioal Fourier traorm[3], [4], which i the geeralizatio o the oe-dimeioal Fourier traorm i deied a i cot. i cot cott icott F e e e tdt... () It ha the ollowig additivity property o. o o... () F F F I act, the ractioal Fourier traorm i the pecial cae o the caoical traorm [5],[6]. The caoical traorm i deied a i d i a i t,,. t b b b DCT t x w e e e tdt b 0...(3) ib The oe-dimeioal caoical traorm ca be exteded i to two-dimeioal caoical traorm a ollow. DCT t, x, w i d i d w i a i a w i t i x x t b b b b b b e e e e e e t, xdxdt b 0...(4) ib ib Notatio ad termiology a per Zemaia [7]. Thi paper i orgaized a ollow: Sectio the deiitio two- dimeioal caoical traorm, ad tetig uctio pace. Sectio 3 iverio ad Aalyticity theorem, are proved. Sectio 4 propertie o kerel are dicued. II. Deiitio two dimeioal (D) caoical traorm: Where we have, give the deiitio o two dimeioal (D) geeralized caoical traorm.. Two-dimeioal Geeralized caoical traorm E R R : It ca be eaily proved that the uctio t, ad x, w which are the uctio o t ad x are member o E R R where,. ISSN: P a g e

2 i d i a t i t t, e e. e ib b b b ad That i up i d i a w w x i x b b b x, w e e. e ib t, x, w t D D t, x, w E, k t x x let E R R deote the dual o E R R t, xe R R ca be deied a DCT t, x, w t, x, t, x, w DCT t, x, w Where,. Thereore the geeralized caoical traorm o i d i d w i a i a w i t i x x t b b b b b b e e e. e. e e t, xdxdt ib ib i d i a i t t b b b t e e e whe b 0 ib i cd de t d whe b = 0 i d w i a w i x x b b b x, w e e e whe b 0 ib i cdw d e x d. w whe b = 0. Deiitio o tetig uctio pace: I a, J b A iiitely dieretiable complex valued uctio o R belog to E(R ), i or each compact et. where : a t t R, t a, a 0, : b x x R, x b, b 0 ad or I R, J R, up E, k t, x Dt Dx ( t, x) k=0,,,3..ad l=0,,,3... t t Thu ER will deote the pace o all t, x E R i complete ad a Frechet pace, let III. with upport cotaied i a ad b. Note that pace E E deote the dual pace o E. Iverio ad Aalyticity o Two Dimeioal caoical traorm: 3. Ivere o Two Dimeioal caoical traorm: I DCT t, x, w i caoical traorm o, t x the ivere o traorm i give by i a i a w i d i d i i, t x i t i t w b b b b b b t x e e e e e e DCT t, x, wddw b b 3. Aalyticity theorem: E ( R ) ad let it two caoical traorm be deied by, ISSN: P a g e

3 i d i d i a i a w w t x i t i x b b b b b b DCT ( t, x) (, w) e e e e e e ( t, x) dxdt ib ib the DC T ( t, x) (, w ) i aalytic o a :,, 0, : b x x R, x b, b 0 t t R t a a C,i the a, b, up p a ad b where moreover D D DCT ( t, x) (, w) ( t, x), D D ( t, ) ( x, w) dieretiable ad w w Proo:, : {,, } C ad w: {w, w, w w } We irt prove that, { DCT ( t, x)}(, w) exit, w C D C T(, t) x( i, ) w { DCT ( t, x)}(, w) ( t, x), ( t, ) (, ) x w w w we prove the reult =, the geeral reult ollowig by iductio. For ixed 0 chooe two cocetric circle C ad C with cetre ad radii r ad r repectively uch that 0<r<r <. be a complex icremet atiyig r cocetric circle C ad C with cetre w ad radii r ad Coider, w be a complex icremet atiyig 0. Alo or ixed w 0.Agai chooe two 0 w r repectively uch that 0< r ( DCT) (, w ) ( DCT)(, w ) ( DCT) (, w w ) ( DCT)(, w ) ( t, x), ( t) w ( x)...(5) ( t, x), ( t, ) (, ) x w w w where ( t) w ( x) ( t,, ) ( t, ) r < r < w. ( x, w, w w w w ) ( x, w) w For ay ixed t, x R ad ay ixed iteger. w ( t, ) ( x, w) k ( k, k k) N0 ad l ( l, l l) N0 D D ( t, ) ( x, w ) i aalytic iide ad o C ad t x We have, by Cauchy itegral ormula. C. ISSN: P a g e

4 D D w ( t, x) D D t, x, w t x t x 4 i ( ) c c z z z where, w y w w y w ( y w ) = ( -, z, + ) ad w = (w w -, y,w + w ). w D D ( t, ) ( x, w) dzdy 4 ( ) ( ) ( ) ( ) t x z c c z y w w y w dzdy But or all z C ad ad, y C tx retricted to a compact ubet o R, D D ( t, ) ( x, w) i bouded by cotatq. t x w Q Dt Dx w ( t, x) dz dy 4 ( r r) ( r )( r r) ( r ) c c w Q 4 ( r r) ( r )( r r) ( r ) Thu a 0, ad w 0, D D w ( t, x) t x R.Thereore it ollow that w ( t, x) coverge i E(R ) to zero ice ted to zero. Uiormly o the compact ubet o E, we cocluded (5) ted to zero. Thereore DCT ( t, x) (, w ) i dieretiable with repective ad w. But thi i true or all i, =,,.Hece DCT ( t, x) (, w ) i aalytic o C ad, D D DCT ( t, x) (, w) ( t, x), D D ( t, ) ( x, w) w w IV. Propertie o kerel: i d i d w i a i a DCT t x w e e e e e e t x dxdt ib ib I w i t i x,, t x b b b b b b, i deiitio two dimeioal caoical traorm o t, x Where, i d i d w i a i a w i t i x t x b b b b b b k t,, k, x w e e e e e e whe b 0 ib ib i i cd cdw d e t d d e x dw whe b= 0 kerel o D caoical traorm atiied ollowig property 4.) k t, k,,, x w k t k x w 4.) k t, k,,, x w k t k x w 4.3) k t k x w k t k w x 4.4),,,, i a,,,, i a=b k t k x w k t k w x b Five propertie o kerel, tated above are imple to prove, hece the proo omitted. ISSN: P a g e

5 V. Cocluio: The two-dimeioal caoical traorm i geeralized i the ditributioal ee. It iverio ad Aalyticity theorem i proved. Some propertie o kerel are dicued. It ca be ued optical ytem aalyi. Reerece: [] Tatiaa Alieva ad Batiaa Marti J., O Fractioal Fourier traorm momet, IEEE igal proceig letter, Vol. 7, No., Nov [] Tatiaa Alieva ad Batiaa Martia J., Wiger Ditributio ad Fractioal Fourier Traorm or -Dimetioal Symmetric Beam, JOSA A, Vol.7, No., P.39 33, Dec.000. [3] L. B. Almeida. The ractioal Fourier traorm ad time requecy, repreetatio IEEE tra. Sigal Proceig, Vol. 4, No., Nov [4] V. Namia., The ractioal order Fourier traorm ad it applicatio to quatum mechaic, J. It. Math. Appticat., Vol. 5, pp (980). [5] M. Mohiky ad C.Quee.,Liear caoical traorm ad their uitary repreetatio, J. math, Phy., Vol., No. 8, P , 97. [6] S.Abe ad J.T.Sherida., Optical operatio o wave uctio a the Abelia ubgroup o the pecial aie Fourier traormatio, Opt.lett.,vol.9,o.,pp ,994. [7] Zemaia A.H., Geeralized itegral traorm, Iter Sciece Publiher New York, 968. ISSN: P a g e