陽(yáng)明大學(xué)放射醫(yī)學(xué)科學(xué)研究所(2)-展示頁(yè)
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【正文】 p o r t s s i n c e ,0k ) d x(xψ ( x ),0 01 / 21 / 21dx1dxk ) d x(xψ ( x ),0)Vψ :( 1 s t 1)( 2 x ( 2 x )1 / 2 ) )( 2 ( x ( 2 x )ψ ( x )012/12/101?????????????????????????kk)12()2()( ??? xxx ???001100k020122123011WV VVi n V of c o m p l e m e n t o r t h o g o n a l t h eis Wn o n z e r o ) . a r e a ofn u m b e r f i n i t e ao n l y ( a s s u m e , ),(f o r m t h eof f u n c t i o n s a l l of s p a c e t h ebe l e t W So)( f o r m t h eof isi t ifo n l y a n d if V t oo r t h o g o n a l is Vi n f u n c t i o n a s,o t h e r wo r dI n )())122()22((c a s e , I n t h i s . .. .ifo n l y a n d if )),( of nsc o m b i n a t i ol i n e a r ( V t oo r t h o g o n a l is )2(f u n c t i o n An y ?????????????????????????????Rakxakxakxakxkxaf,a, aaaZllxVkxafkZkkkkZkkZkk01kk???????? Theorem (extend to Vj) jj1j1jjjjWV V a n d Vi n V of c o m p l e m e n t o r t h o g o n a l t h eis WR 2 f o r m t h eof f u n c t i o n s of s p a c e t h ebe L e t W??????? kjkk a)kx(a ?0dxf ( x )g ( x )fg,s h o wm u s t W t ob e l o n g s t h a t s u p p o s e a n d W t ob e l o n g s )2( t h a t S u p p o s e a r t . W t ob e l o n gm u s t V t oo r t h o g o n a l is t h a t Vi n f u n c t i o n A n y 2.Vi n f u n c t i o n e v e r y t oo r t h o g o n a l is i n Wf u n c t i o n E a c h 1.: f a c t s t wos h o wm u s t w e t h e o r e m ,t h i se s t a b l i s h To : P r o o fLjjjj1jjj2??????????fkxagjZkk?c a s e . V as d e r i v e d bec a n t r e q u i r e m e n s e c o n d T h eV a n y t oo r t h o g o n a l is T h e r e f o r e 2) l e t t i n g( b y 2)V t oo r t h o g o n a l is ( b e c a s u e 0So V t ob e l o n g s f u n c t i o n t h e,V t ob e l o n g s S i n c e1jjj0.fg.yd)yf()y(gx2yyd)yf()ky(2adxx)f ( 2)kx(ax)f ( 2 f ( x )jjZkkjZkkj????????? ?? ????? ??? ????Deposing Vj h s .o t h e r wi d t of s p i k e s of nsc o m b i n a t i ol i n e a r as dr e p r e s e n t e bec a n n o t t h a t 1 / 2 wi d t h of of s p i k e s t h er e p r e s e n t s y,I n t u i t i v e lV t ob e l o n g s a n d 10 , W t ob e l o n g s e a c h wh e r efs u m a asu n i q u e l y d e c o m p o s e d bec a n Vi n e a c h SoV. . .VVV1000021j0202j1j2j2j1j1jjj??????????????????????????????????lllljjfwfjlwffWWWWWW? Theorem: j00002210022 W t ob e l o n g s a n d V t ob e l o n g s w h e r e asu n i q u e l y w r i t t e n bec a n ( R )L e a c h ,p a r t i c u l a rI n .WWWV( R )Ls u md i r e c t o r t h o g o n a l i n f i n i t ea n as d e c o m p o s e d bec a n ( R )L s p a c e T h ejjjwfwfff?????????????? Haar Deposition and Reconstruction Algorithms n o i s e . o u t t h i sf i l t e r t oz e r o t oe q u a l sc o m m p o n e n t s e t t h e s e n o i s e , r e p r e s e n t s 6j w i t h a n y ,n o i s e r e p r e s e n t s t h a n l e s s w i d t h of s p i k e s t h a t S u p p o s e: E x a m p l e76???jw wc o m p o n e n t s i t s i n t o D e c o m p o s e l a r g e )s u i t a b l y ( f o r f u n c t i o n s t e p aby eA p p r o x i m a t p r o b l e m f i l t e r i n g N o i s e110 ll,jjjjjWwwfffjVff????????????Implementation .for )2/( tol e a dsw hi c h ,...,0,1/ 22...,1, 1/a t x s i g na l t heS a m pl e jjZllfa jl ???????Zljlj )lx(a(x )f 2?? Step 1 : Approximate the original signal f by a step function of the form s i g na l . t heofdom a i n on t he de pe nds ra ng e:s i g na l . t heof fe a t ure s e s s e nt i a l t hec a pt ure e noug h t o s m a l l:lj( 4 . 5 ) 1( 4 . 4 ) (.f o r c o m p o n e n t s Wi t s i n t o )(2 De c o m p o s e2 S t e pjψ ( x ) ) / 2( x )()( 2 x( x ) ) / 2 x)( ψ( 2 x )jllxl??????????????( 4 . 7 ) 1( 4 . 6 ) (:Rx a l lf o r h o l d s r e l a t i o n s f o l l o wi n g T h ex 11111x ) ) / 2ψ ( 2x)(2()x(2x ) ) / 2(2 x)2( ψx)(2x2jjjjjjj????????????????? Example 1 0 02 , de c om pose f i nto W ,W ,V c om pon e nts .f ( x) 2 ( 4 ) 2 ( 4 1 ) ( 4 2) ( 4 3 )4 ( 2 2 4 1 2 ( 24 2 4( 1 / 2 ) ( 2( 1 / 2 ) 2( 1 / 2 ) 4 3 4( 1 / 2 1 ) ( 2jx x x x( x ) ( ψ x ) ( x ) ) / 2( x ) ( ( x ) ψ x ) ) / 2( x ) ( x ) ( ψ x ) ( x ) ) / 2( x ) ( x ) (? ? ? ?????? ? ?? ? ??? ? ? ? ? ? ???? ? ?? ? ? ? ? ? ?? ? ? ? ?111 0 0( 1 / 2 2( 1 / 2 ) f ( x) [ ( 2 2 ] [ 2 ( 2 ][ ( 2 1 2 1 ] / 2 [ 2 1 ( 2 1 ] / 2( 2 1 2 2W c om pon e nts : ( 2 1V c om pon e nts : 2 2V V ,Wf ( x) ( 2 1 (x) ψ ( x ) ) / 2ψ x ) ( x ) ( x ) ψ x)ψ x ) ( x ) ( x ) ψ x)ψ x ) ( x )ψ x)( x )ψ x) ψ x ) ( x )???????? ? ?? ? ? ?? ? ? ? ? ? ? ?? ? ???? ? ? ?General deposition scheme 111122112211121121111111111222222222 222( 4 . 1 1 ) 22(122( 4 . 1 0 ) 22(222222( 4 . 7 ) 1 ( 4 . 6 ) (( 4 . 9 ) 12222: t e r m sodd a n de v e n i n t o 2 s u m t h eDi v i d e : 1 S t e p?????????????????????????????????
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