CoE4TN3 Image Processing

Transcription

1 CoE4T3 Imae Processin Imae Reisraion Imae Reisraion Throuh Transform A B f Imae reisraion provides ransformaion of a source imae space o he are imae space. The are imae ma be of differen modaliies from he source one. 4 Wha is imae reisraion? Imae Reisraion is he process of esimain an opimal ransformaion beeen o imaes. Someimes also knon as Spaial ormaliaion. Applicaions in medical imain Machin PET (meabolic) o MR (anaomical) Imaes Alas-based semenaion/brain srippin fmri Specific Moion Correcion Correcin for Geomeric Disorion in EPI Alinmen of imaes obained a differen imes or ih differen imain parameers Formaion of Composie Funcional Maps Muli-modali brain imae reisraion Eernal Markers and Sereoacic Frames Based Landmark Reisraion. Riid-Bod Transformaion Based Global Reisraion. Imae Feaure Based Reisraion. Boundar and Surface Machin Based Reisraion Imae Landmarks and Feaures Based Reisraion 5 Anaomical Reference (SCA) Reference Sinaures Funcional Reference (FCA) Riid-Bod Transformaion Roaion b φ R+ MR Imae (e Subjec) MR-PET Reisraion PET Imae (e Subjec) Roaion Translaion Translaion of Analsis Translaion of Roaion b θ Translaion of A schemaic diaram of muli-modali MR-PET imae analsis usin compueried alases. Roaion b ω 3 6

2 Riid-Bod Transformaion Riid-Bod Transformaion (a) Translaion alon -ais b p + p (b) Translaion alon -ais b q + q (c) Translaion alon -ais b r + r R+ R Riid-Bod Transformaion Riid-Bod Transformaion: Affine Transform (a) Roaion abou -ais b θ + p cosθ + sin θ + cosω ω sin + sin θ cosθ (b) Roaion abou -ais b ω sin ω cosω (c) Roaion abou -ais b φ + + cosϕ sin ϕ sin ϕ cosϕ Affine ransform: ranslaion + roaion + scalin Scalin: a b c a, b and c are scalin parameers in he hree direcions. Affine ransform can be epressed as A 8 R RRR θ ω ϕ Riid-Bod Transformaion R+ cosθ sinθ 0 cosω 0 sinω 0 0 sinθ cosθ cosϕ sinϕ 0 0 sinω 0 cosω 0 sinϕ cosϕ p q r 9 Riid-Bod Transformaion: Affine Transform A A: The affine ransform mari ha ineraes ranslaion, roaion and scalin effecs. R S 0 A 0 0 here a S b c

3 Riid-Bod Transformaion: Affine Transform The overall mappin can be epressed as 0 0 p cosθ sinθ 0 0 cosω 0 sinω q sinθ cosθ r 0 0 0sinω 0 cosω a cosϕ sinϕ 0 0 b sinϕ cosϕ 00 0 c Riid-Bod Transformaion: Principal Aes Reisraion The ineria mari I is diaonal hen compued ih respec o he principal aes. The cenroid and principal aes can describe he orienaion of a volume. The principal aes reisraion can resolve si derees of freedom of an objec (hree roaion and hree ranslaion). I can compare he orienaions of o binar volumes hrouh roaion, ranslaion and scalin. 3 6 Riid-Bod Transformaion: Principal Aes Reisraion Principal aes reisraion (PAR) is used for lobal machin of binar volumes from CT, MR or PET imaes. (,, ) objec B (,, ) 0 (,, ) objec (,, ) T cenroid of B (,, ) B (,, ) B (,, ) B(,, ) B (,, ) B (,, ) B (,, ) Riid-Bod Transformaion: Principal Aes Reisraion ormalie he principal aes (he eienvecors of I) and define: The roaion mari is R RRR θ ω ϕ e e e E e e e e e e cosθ sinθ 0 cosω 0 sinω 0 0 sinθ cosθ cosϕ sinϕ 0 0 sinω 0 cosω 0 sinϕ cosϕ 4 7 Riid-Bod Transformaion: Principal Aes Reisraion The principal aes of B(,,) are he eienvecor of ineria mari I: I I I I I I I I I I I ( ) + ( ) B (,, ),, I B ( ) + ( ) B (,, ) I B ( ) + ( ) (,, ) I I ( )( ) B (,, ) I I ( )( ) B (,, ) I I ( )( ) B (,, ),, 5 Riid-Bod Transformaion: Principal Aes Reisraion Le E R We can resolve he roaion anles as ω arcsin( e ) 3 θ arcsin( e / cos ω) ϕ arcsin( e / cos ω) 3 8 3

4 Riid-Bod Transformaion: Principal Aes Reisraion The PAR alorihm o reiser V and V:. Compue he cenroid of V and ranslae i o he oriin.. Compue he principal aes (b usin ER) of V and roae V o coincide ih he, and aes. 3. Compue he principal aes of V and roae he, and aes o coincide ih i. 4. Translae he oriin o he cenroid of V Scalin V o mach V b usin facor F V/ V. s Ieraive Principal Aes Reisraion (IPAR) Sequenial slices of MR (middle ros) and PET (boom ros) and he reisered MR-PET brain imaes (op ro) usin he IPAR mehod. 9 Riid-Bod Transformaion: Principal Aes Reisraion Landmarks and Feaures Based Reisraion A 3-D model of brain venricles obained from reiserin MR brain imaes usin he PAR mehod. Roaed vies of he 3-D brain venricle model in he lef imae. 0 Wih he correspondin landmarks/feaures idenified in he source and are imaes, a ransformaion can be compued o reiser he imaes. on-riid ransformaions have been used in landmarks/feaures based reisraion i b eploiin he relaionship of correspondin poins/feaures in source and are imaes. To poin-based alorihms ill be inroduced. 3 Ieraive Principal Aes Reisraion (IPAR) Advanaes over PAR: IPAR can be used ih parial volumes. Assumpion made in IPAR: he field of vie (FOV) of a funcional imae (such as PET)) is less han he full brain volume hile he oher volume (such as MR imae) covers he enire brain. Alorihm: refer o paes 59 ~ 64 of he ebook. Similari Transformaion X: source imae Y: are imae : landmark poins in X : landmark poins in Y T(): non-riid ransformaion T( ) s r + scalin roaion ranslaion 4 4

5 The reisraion error: Similari Transformaion E( ) T( ) sr+ The opimal ransform is o find s, r and o minimie: sr i i + i i : he eihin facor for landmark i : he number of landmark poins Similari Transformaion: Alorihm (coninued) 3. Compue he scalin facor: 4. Compue s r i i i r i i i sr 5 8 Similari Transformaion: Alorihm. Se s.. Find r hrouh he folloin seps: a) Compue he eihed cenroids of and. i i i i i b) Compue he disance of each landmark from he cenroid. i i i i i Ieraive Feaures Based Reisraion (WFBR) X i : a daa se represenin a shape in source imae Y i : he correspondin daa se in are imae : poins in X i : poins in Y i T(): ransform operaor T ( ) s r + Dispari funcion o be minimied: s i ( ) ( ) j dt T s : number of shapes i : number of poins in shape X i 6 9 Similari Transformaion: Alorihm (coninued) c) Compue he eihed co-variance mari. Z i ii Sinular value decomposiion Z UΛ V here d) Compue UU VV Ι Λ dia( λ, λ, λ ) λ λ λ r V dia(,, de( VU)) U 7 WFBR: Ieraive Alorihm. Deermine T b he similari ransformaion mehod.. Le T (0) T and iniialie he opimiaion loop for k as (0) T ( ) () (0) (0) 3. For he poins in shape X i, find he closes poins in Y i as C (, Y) j,,3,..., ( k) ( k) i i i here C i is he operaor o find he closes poin. 30 5

6 WFBR: Ieraive Alorihm (coninued) 4.Compue he ransformaion T (k) beeen { (0) } and { (k) } ih he eihs { } usin he similari ransformaion mehod. 5. Compue T ( ) ( k+ ) ( k) (0) End of Lecure 6. Compue d(t (k) ) d(t (k+) ). If he converence crierion is me, sop; oherise o o sep 3 for he ne ieraion Elasic Deformaion Based Reisraion One of he o volume is considered o be made of elasic maerial hile he oher serves as a riid reference. Elasic machin is o map he elasic volume o he reference volume. The machin sars in a coarse mode folloed b fine adjusmens. The consrains in he opimiaion i include smoohness and incompressibili. The smoohness ensures ha here is coninui in he deformed volume hile he incompressibili uaranee ha here is no chane in he oal volume. For deail alorihm, refer o pp. 69 ~ 7 in he ebook. 3 Resuls of he elasic deformaion based reisraion of 3-D MR brain imaes: The lef column shos hree reference imaes, he middle column shos he imaes o be reisered and he rih column shos he reisered brain imaes. 33 6