Geometrical Transformations
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1 Geometricl Trnsformtions Did Atkinson Compttionl Aspects of MRI
2 References Fole, n Dm, Feiner, Hghes. Compter Grphics: Principles nd Prctice. Chpter 5. Wolfrm MthWorld Compttionl Aspects of MRI
3 Otline Motition Vectors nd coordintes Coordinte sstems Homogeneos coordintes Geometricl trnsforms DICOM Compttionl Aspects of MRI
4 Geometricl Trnsforms in Therp [VisionRT] Compttionl Aspects of MRI
5 nd displ Compttionl Aspects of MRI
6 nd Imge Registrtion Non-Rigid Registrtion etween seline nd follow-p Jcoin mp within rin re Modt et l., Fst FFD g the NMI grdient nd GPU, th Septemer, HP-MICCAI 8 workshop Compttionl Aspects of MRI
7 The XMR site t G s Hospitl (KCL) Compttionl Aspects of MRI Ctheter in hmn hert
8 Registrtion Mtri Clcltion Oerll registrtion trnsform is composed of series of stges Clirtion trcking dring interention M Scnner Spce 3D Imge Spce X-r Tle Spce λt M 2 Dicom imge heder M Clirtion M 2, T, P Trcking M 3 Ptient imging λ, R R*P 2D Imge Spce X-r C-rm Spce M 3 [Kwl Rhode, KCL]
9 Compttionl Aspects of MRI Geometr: Coordintes nd Vectors z r (4,2,) k j i r z r ˆ 2ˆ 4ˆ ˆ 2ˆ 4ˆ ˆ nit ector 2 4 or Norm (length) j r r
10 Sclr Prodct (dot prodct, inner prodct) ĵ r. ˆj r ˆj ( ) r r ˆj length of ector r (nit ector with length ) Sclr prodct is length of projection of r onto nit ector ĵ [MthWorld] Compttionl Aspects of MRI
11 Compttionl Aspects of MRI Sclr prodct in component form ˆ ˆ ˆ ˆ ˆ ˆ i i i k j i k j i
12 Vector (Cross) prodct Cross prodct of two ectors nd is third ector perpendiclr to nd. E.g. z is is cross prodct of nd. Compttionl Aspects of MRI
13 Compttionl Aspects of MRI Vector prodct n k j i k j i k j i )ˆ ( ) ˆ ( ) ˆ( ) ˆ( ˆ ˆ ˆ ˆ ˆ ˆ z z z z
14 Right Hnded Coordinte Sstem thm st finger z 2 nd finger Compttionl Aspects of MRI
15 Emple Scnner nd Ptient Coordinte Sstems Z / H Y / P X / L L Left P Posterior H Hed For spine (fce-p) ptient Hed: sperior (S) Foot: inferior (I) Compttionl Aspects of MRI
16 Rdiogrphic Presenttions Anterior Ail / Trnserse Right Left Hed Coronl Posterior Hed Sgittl Right Left Anterior Posterior Foot View ptient Compttionl from Left Aspects foot of MRI Foot
17 Rottion Trnsformtion Rottion mtri trnsforms coordintes of point rottion. Compttionl Aspects of MRI
18 Rottions nd scles cn e epressed g 22 mtrices for 2D coordintes (33 for 3D). Howeer, trnsltions cnnot so se Homogeneos Coordintes. Add nother dimension with le to the coordintes. Add nother row nd colmn to the trnsform mtri. Compttionl Aspects of MRI
19 Homogeneos Coordintes A conenient w of representing coordintes tht llows for geometric trnsformtions inclding trnsltions. Add n etr dimension. w P [Fole] Compttionl Aspects of MRI
20 Compttionl Aspects of MRI t t t t
21 Compttionl Aspects of MRI Alterntie row-ector mtri representtion [ ] [ ] [ ] [ ] [ ] [ ] t t MATLAB mketform t mkehgtform ses mtri ector conention! Note trnspose!
22 T (MP) P T M T Coords s colmns of P post mltipling M Row ectors pre-mltipling M T Compttionl Aspects of MRI
23 Comments Homogeneos coordintes llow representtion of ffine trnsformtions. Trnsforms cn e conctented (composed) into series of mtri mltiplictions - in the correct order. Tke cre ssemling the mtri is it for row-ector*mtri, or, mtri*colmnector? Compttionl Aspects of MRI
24 Affine Trnsforms An ffine geometricl trnsform tht keeps prllel lines prllel. Incldes: trnsltions, rottions, scle chnges, skews. Compttionl Aspects of MRI
25 Compttionl Aspects of MRI Affine Trnsformtion Mtrices (3D, mtri * colmn-ector formt) 2D skew or sher scle λ λ z s s s trnsltion z t t t
26 Compttionl Aspects of MRI 3D rottions z R R R
27 Appling trnsforms to coordintes trnsform pplied to mltiple coordintes to trnsform n oject. in homogeneos coordintes, mtrices cn e mltiplied to compose trnsforms Compttionl Aspects of MRI
28 Compttionl Aspects of MRI Compose trnsforms into one mtri. Arrnge mltiple coordintes into one mtri L L L L One coordinte Mltiple coordintes Trnsform
29 Coordinte nd Rottion Origins Coordinte origin sll does not coincide with imge centre. Affine R mtri ssmes rottion ot coordinte origin. Recipe: Trnslte imge sch tht origin is t centre, ppl rottion, trnslte ck. These cn e composed, note order is importnt. Compttionl Aspects of MRI
30 Smmr Work in right-hnded coordinte sstem. To compose rottions, trnsltions, or n ffine trnsform, se homogeneos coordintes. The order of trnsformtions is importnt. Compttionl Aspects of MRI
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