Reconstruction History. Image Reconstruction. Radon Transform. Central Slice Theorem (I)

Transcription

1 Reconstructon Hstory wth Bomedca Acatons EEG-475/675 Prof. Barner Reconstructon methods based on Radon s wor 97 cassc mage reconstructon from roectons aer 97 Hounsfed deveo the frst commerca x-ray CT scanner Hounsfed and Cormac receve the 979 obe Prze for ther CT contrbutons Cassca reconstructon s based on the radon transform Method nown as bacroecton Aternatve aroaches: Fourer transform and teratve seres-exanson methods Statstca estmaton methods Waveet and other mutresouton methods Unversty of Deaware Radon Transform Centra Sce Theorem (I) Radon transform: P(, ) R{ f( x, y) } f( x, y) d s defned aong the ath such that x cos + ysn Convertng the oar (,) system to a rotated coordnate system (,q) x cos + ysn xsn + ycos q Aternatve Radon exresson R{ f( x, y) } J ( ) f( cos qsn, sn + qcos ) dq Images are reconstructed by bacroectng the roectons A mted number of roectons eads to aasng and/or geometrc artfacts Theorem reates Fourer transforms of obect functon and Fourer transform of roecton Fourer transform the of the (-D) roecton: S( ω) J( ) e d ω f ( cos q sn, sn q cos ) e dqd ω Utzng xcos + ysn q xsn + ycos yeds S ( ω) f( x, y) e dxdy F( ω, ) + x y ω( cos sn ) Ths s a -D FT of f(x,y), denoted as F(u,v), u ω cos v ω sn u and v are the frequency comonents n the x- and y-drectons Unversty of Deaware 3 Unversty of Deaware 4

2 Centra Sce Theorem (II) Inverse Transform S (ω) s the Fourer transform of the roecton J () J () s taen that ange n the sace doman wth rotated coordnate system (,q) The sectrum S (ω) s ocated on a ne of ange n the frequency doman F(u,v) S (ω) obtaned at dfferent anges are set aong the corresondng rada nes n the F(u,v) frequency doman Reconstructon can be acheved by nverse Fourer transformaton Samng ssues? Inverse FT reconstructon yeds: (, ) { (, )} (, ) Changng to oar coordnates: ω ( xcos 0 + v ) sn Reacng F(ω,)wth S (ω) yeds: f x y F u v F u v e dudv ( xu+ vy ) f (, r ) F( ω, ) e ω dωd f r ω S ω e dωd J d (, ) 0 ( ) ( ') x v ω ( + ) cos sn 0 ( cos sn ) J ( ') S ( ) e x + v ω ω dω Modfed roectons Reconstructon s the bacroected ntegra of modfed roectons Unversty of Deaware 5 Unversty of Deaware 6 Bacroecton Method (I) Bacroecton Method (II) The modfed roecton can be exressed as: J( ') S( ) e d + x v ω ( cos sn ) ω ω ω ω ω S ( ω) e dω I { ω S ( ω) } I { ω } J ( ) Convouton of roecton and ω fter Genera exresson for reconstructon: f ( xy, ) J ( ') h ( dd ') ' 0 Genera fter functon: h() Infnte bandwdth fters are robematc why? H(ω) s generay bandmted: H(ω) ω B(ω) f ω Ω B( ω) 0 otherwse Bandmted fter n the frequency doman Imuse resonse for samng nterva t sn( / τ) sn( / τ) h ( ) τ / τ 4 τ /τ Ftered roecton: J( ') J( ') h( ') d' Reconstructon: * f ( xy, ) J ( ') Quaty of reconstructon deends Resouton of the roectons umber of roectons Unversty of Deaware 7 Unversty of Deaware 8

3 Bacroecton Method (III) Agebrac Reconstructon Technque (ART) Shar fter cut-off resuts n artfacts Cut-off avods amfcaton of hghfrequency nose Cut-off ntroduces rngng Aternatve fter: generazed Hammng wndow Smoother cut-of o rngng n muse resonse Proecton data s dstrbuted over the mage reconstructon Dstrbuton crtera: mnmze error between Actua roectons Proectons comuted from the reconstructed obect System mode: f vaue of the th mage (sce) xe roecton of a secfc ray assng through the mage w, weght (fracton) of f contrbutng to w, f for,..., M M equatons and unnowns Unversty of Deaware 9 Unversty of Deaware 0 ART Souton Estmaton Methods Souton methods Matrx reresentaton and nverson Comutatona and memory ntensve Dynamc rogrammng Iteratve souton (ART) Iteratve souton stes Intazaton a xes assgned a redetermned vaue (average roecton vaue) Comute the ray sums for the reconstructed mage Comute the true ray sum and roected ray sum errors Dstrbute errors to corresondng reconstructon xes Addtve correcton: correcton vaue s added to the current xe vaue Muty to correcton: correcton vaue scaes current xe vaue th ray sum at the th teraton q f w, Iteraton ste + q f f + w, w, Agorthm varatons Addtve ART Ray contrbuton weghts w, are bnary Yeds an effcent mementaton Mutcatve ART Other varatons desgned to mrove convergence, comutatona cost, and fna resuts Statstca estmaton methods often rovde sueror reconstructons Detector hoton count s Posson dstrbuted defnes the ntegraton ne (hoton ray) Vector of detector measurements: J[J, J,, J ] Rewrte ne ntegra ( xyzd,, ) a attenuaton coeffcent n voxe E[ J] me a contrbuton of voxe to ne Set weght matrx: A{a } ( x, yzd, ) Rewrte measurement vector as [ A] J( ) me [ A] a Unversty of Deaware Unversty of Deaware 3

4 Maxmum ehood Souton Roughness Penaty The M souton s gven by: ˆ arg max ( ); 0 ( ) og P J ; ; Returns the (voxe) attenuaton coeffcent vector that maxmzes the robabty of observng the measured hoton counts Usng the Posson dstrbuton mode for the hoton counts ( ) [ ( )] e P J ; P[ J ; ]! If the measurements are obtaned (ndeendenty) as ray sums ( ) h ([ A ] ) h( ) og( ) me me The souton to the above may not be unque A soutons may not be arorate Addressed through the ncuson of a roughness enaty Genera roughness enaty functon K R( ) ψ([ C] ) [ C] c C s a enaty matrx ψ( ) s a convex, symmetrc, nonnegatve, dfferentabe, otenta functon Exame: ψ(t)w t / (quadratc) Quadratc roughness enaty functon: K R( ) w ([ C] ) Penazed M obectve functon arg max Φ( ) Φ ( ) ( ) βr( ) β contros eve of smoothness n the fna reconstructon Unversty of Deaware 3 Unversty of Deaware 4 Surrogate Functon Exectaton Maxmzaton Soutons c 0 and (n) [A (n) ] ( th ne ntegra at teraton n) Reformuatng the estmaton robem usng the surrogate functon: φ( ; ) Q( ; ) βr( ) a Q e ( M ) ( e ) a ( ; ) n + n ar r r S [ A ] M + me me a ( n ) r S {} r r [ A ] + me me S s the set of xes between the source and xe aong the th ray. M and are the current exected number of hotons enterng and eavng xe aong the th ray. E-Ste: M-Ste: Fnd Q( ; ) and φ ( ; ) arg max ( ; ) ( n+ ) φ Combnng the stes yeds the M-EM agorthm ( n ) + ( M ) ( + ) a M Unversty of Deaware 5 Unversty of Deaware 6 4

5 Reconstructon Exames Eght Proecton Case Orgna Phantom Bacroecton ART Reconstructon MAP Reconstructon Unversty of Deaware 7 5