【正文】 ?????????????????????????????????????????jjZkjkkjkkjZkjkjZkjkjjjjjjjjjjjjjjjZkjkZkjkjkjkjfw)kx()aa()kx(ψ)aa(/))kx(ψ)kx((a/))kx()kx(ψ(a( x )f/))kx(ψk)x2()kx(/))kx(k)x2ψ()kx()kx(kxx ) ) / 2ψ ( 2x)(2()( 2 xx ) ) / 2(2 x)2( ψx)(2)kx(a)kx(a( x )f)kx(a( x )f??????????????Wj1ponent Vj1ponent ? Theorem (Haar Deposition) 2 2 22 wh e r e as d e c o m p o s e d bec a n T h e n V 2002122111122112211111111111jf. ..fwwfwfaaaaabw i t hV)kx(afW)kx(ψbw,fwff)kx(a( x )fjjjjjjjjjkjkjkjkjkjkZkjjjkjZkjjjkjjjjjZkjjkj??????????????????????????????????????????????????????????? Example ????????????1208888880018221202.c o m p o n e n t s V,W, Wi n t o f d e c o m p o s e 1,x0 813, F i g u r ekkk)()/kf()x(fk),/kf(a,j?V8ponent V7ponent V6ponent V4ponent W7ponent Reconstruction jjjlZljjl39。jk39。j/)l(x/laf)lx(af ( x )b.bWW39。0f o r c o m p o n e n t s Wa n dV i n t o f d e c o m p o s e d H a v i n gj0????????????????General reconstruction scheme jlljjlklllklZkkllja)lx(a)x(fj 1l 0W)kx(ψbwV)kx(a)x(fw h e r eWww)x(w)x(ff ( x ) c o n s t a n t s ofn c o m p u t a t i o f o r t h e a l g o r i t h m a f i n da n d 2 r e wr i t e To: Go a lf o r 2 a n d ( x )000100?????????????????????????General reconstruction scheme ????????????????????????????????????odd is 1 if e v e n is if ? ( 4 . 1 6 ) )2(?)( so ))122()22(( )()(h a v e we,by r e p l a c e d wi t h 4 . 1 2 U s i n g( 4 . 1 5 ) 1222( 4 . 1 4 ) 1222( 4 . 1 3 ) 122( 4 . 1 2 ) 122001100000112kla2klaaw h e r elxaxfkxakxakxaxfx kx)x(x)(x)ψ()x(x)(x)()x(x)(ψ ( x ))x(x)(( x )kklZklZkkkZkkjjjjjj??????????????????????????????????????????????odd is 1 if e v e n is if ?? ))( ( )2()()(odd is 1 if e v e n is if ? ( 4 . 1 7 ) )2(?)(as wr i t t e n bec a n )( ,0000111110000110002klba2klbabaaw h e r exflxaxwxf2klb2klbbw h e r elxbxwkxψawS i mi l a r l ykkkklllZllkklZllkk??m a n n e r . r e c u r s i v e ai n on so a n dt s ,c o e f f i c i e n t h ed e t e r m i n e tsc o e f f i c i e n,t s .c o e f f i c i e n t h ed e t e r m i n e tsc o e f f i c i e n,odd is 1 if, e v e n is if, ) ( )2()()()(s u m t h e t o)2( A d d211100111122221011llllllkkkklZllokkabaaba2klba2klbaaw h e r eflxaxwxwxfkxψbw????????????????????????? Theorem (Haar Reconstruction) ?????????????????????????????????????????odd is 1 if, e v e n is if, a l g o r i t h m b y t h e j, u n t i lon so a n d 2,t h e n 1,f o r y r e c u r s i v e l d e t e r m i n e d a r e t h e 2 f o r 2 a n d 11110001002klba2klbaa39。j39。j 0W)kx(ψb)x(wV)kx(a)x(fw i t hwwffS u p p o s e39。jk39。jk39。jlZljjjlZk39。j39。jZkkj??? Example s i g n a l . t t h er e c o n s t r u ct h e n n ) ,c o m p r e s s i o ( 8 0 % z e r o t oe q u a l 80%s m a l l e s t s e t t h e s i z e ,by t h eo r d e r i n gAf t e r .70f o r )2()( a n d )( wi t h120),2/( 1,x0 ,819, F i g u r e39。39。39。0f o r c o m p o n e n t s a n d i n t o f set s .d e c o m p oc o e f f i c i e n wa v e l e t a n d s a c l i n g 1j l e v e l g e t t h e t o tsc o e f f i c i e n s c a l i n g jl e v e l f r o m Go122112211139。jk39。jk39。jl111j21j11j01j11111139。139。39。 a n d s e t weso c h o o s e , t oo u r s a r e 39。 t h e0, a r e 39。 t h a t a s s u m e d h a v e eAl t h o u g h wodd is 1 if, e v e n is if, 4 . 1 4 . T h e o r e mi n g i v e n f o r m u l at i n r e c o n s t r u c f o r t h ep a t t e r n a l m o s t t h e is T h i s)~()~(us g i v e s t h e n ~ a n d ~ s e q u e n c e s t wo t h eAd d i n g?????????????????????????????????????????jj jk2kjk2k2k2k2k2kjkjkjkjkjl2k2k2 k 2 k llabyxbbb baybxsysxsysx2klba2klbaa1 i s o d d2k lxy2 k i s e v e n lyxyxhyxh??1j1jj1j1jUbHUaLaUayUbx????????~~ a nd so op e ra t or, up s a m pl i ng t hede no t e t o Uus e Weupsampling operator Summary w a ve l e ta ndfun c t i on s c a l i ngH a a r t hebe a nd L e t , s i g na lG i ve n ??f (t )y ?????????????ZkJJkJJJJJkJk)xφ ( 2a( x )ff)2k 1 ( o r 12k1 , 0t0kkfa2J j i o n t oa p p r o x i m a t l e v e lh i g h e s ts t e p . t h i ss k i p : s i g n a l d i s c r e t e e x .s i g n a l . t h eofd u r a t i o n b y t h e d e t e r m i n e d i n t e r v a l f i n i t e a : )2/(L e t r a t e .N y q u i s t t h en l a r g e r t h a is a n d l e v e l t o pc h o o s e : s i g n a l c o n t i n u o u sS