Diffeomorphic Shape Momentum, Computational Anatomy, & Neuroinformatics at 1mm Scale

Transcription

1 Diffeomorphic Shape Momenum, Compuaional Anaomy, & Neuroinformaics a 1mm Scale Michael I. Miller Torono, June 2011 Insiue for Compuaional Medicine

2 Neuroinformaics a 1mm Scale

3 Diffeomorphic Mapping Subcorical and Corical Srucures Sereoaxic (MNI and Talairach) Brodmann s maps Whie maer racs fmri bold response neworks Specroscopy

4 Compuaional Anaomy models human anaomy as an orbi of exemplars under he diffeomorphism group. Populaions are sudied via emplaes wih saisics encoded in emplae coordinaes. Diffeomorphic correspondences are used o carry he informaion from populaion coordinae sysems o emplae coordinaes. Compuaional Anaomy: An Emerging Discipline, Quar. Applied Mah. Grenander,Miller, 1997

5 We generae he diffeomorphism group as soluions of he ODE s. ( x) y x v ( y) ( x) 1 ( x ) v ( ( x )) Lagrangian ( x) v ( ( x)), id 0 Eulerian 1 ( ) 1 ( ) ( ), 1 0 x D x v x id D=Jacobian marix j Deformable Templaes Using Large Deformaion Kinemaics, IEEE. Trans. on Med. Imaging. Chrisense, Rabbi, Miler, v x i

6 We require he vecor fields o be spaially smooh. ( x) y v ( y) ( x) 1 x ( x) v ( ( x)) Variaional Problem 1 vv : v( ) 2 V inf v d 0 * Avv smoohness A 2p, p1.5( 3 ) subjec o 1 I ~ groupacion I Variaional Problems on Flows of Diffeomorphisms, Quar. Applied Mah. Dupuis Grenander Miller, 1998

7 Geodesic Shooing for Compuaional Anaomy, J. Mah. Imaging and Vision Miller, Trouve Younes 2006 Diffeomorphic Shape Momenum Conservaion v( ) * 0 V 0 X M inf v d Av v d 0 id, 1 Force Equaion on Geodesics d M * div Dv M DM v v M d 0 D=Jacobian marix v x i j Momenum Conservaion : M Av d d X M * D w 1 0 (inerpre momenum as a funcion in acion on smooh vecor fields w) * M D D M (as a vecor funcion deermined by iniial condiion)

8 One example: Seering he momenum via dense image maching. vv : v( ) X * 1 2 I 1 M X I: group acion inf Av v d I dx Momenum Conservaion Law M 1 1 * 1 ( ) D D M0 1 1 D M ( I ) ( I I ),1,1 smooh emplae momenum smooh (Normal Moion) Normal o level lines of image. Compuing LDDMM via Geodesic Flows of Diffeomorphisms Beg, Miller, Trouve Younes 2005.

9 The momenum is a highly compressed represenaion of shape. Momenum Vecor field

10 We have many mehods of seering anaomical configuraions one ono he oher. We call hese codes LDDMM. - poinses (landmarks, curves, surfaces) - dense images - vecors - ensors

11 MRISudio Regisred Downloads LDDMM Mappings MRISudio hp://liss.mrisudio.org/mailman/lisinfo/mrisudio-users Cumlaive Yearly Jan Feb Mar Apr May Jun july Aug Sep Oc Nov Dec Jan

12 Compuaional Anaomy of Populaions & Saisics on Shape Spaces

13 An Anaomical Model of Subcorical Human Anaomy Daniel Tward Anaomy a 1mm scale is clumpy. Diffeomorphic Shape Momenum provides a massive daa reducion for morphomery and helps wih he curse of dimensionaliy. Muli-Srucure Nework Shape Analysis via Normal Surface Momenum Maps, Neuroimage, Qiu, Miller,2008.

14 Subcorical Anaomy

15 Subcorical Anaomy

16 The Subcorical Random Field Model Dimensionaliy Reducion 1/3 x 3 x 10E7 -> 7 x 2 x 10E3 M0 ( dx ) ( x ) S ( dx ) S R V ( ) K(, x) M ( dx) 0 0 X 3x3Green's 31 x Vecor marix Measures 3

17 We use PCA and Surface Harmonics suppored on anaomical srucures for represening srucure and funcion in curved anaomical coordinaes PCA requires raining daa. F Fk k k srucure-funcion Laplace-Belrami or response-variables PCA Basis The emerging discipline of Compuaional Funcional Anaomy, Neuroimage, Qiu, Miller,2009.

18 Represenaion of Subcorical Neuroanaomy in he Aging Populaion (Daniel Tward) A populaion of 600 whole brain anaomies which have been segmened (published) ino all subcorical srucures (ADNI, OASIS). A emplae alas was generaed from he populaion o which saisics on he Momenum is indexed. LDDMM surface mapping calculaed he iniial momenum carrying he emplae ono all 600 anaomies wih 14 arge surfaces. PCA analysis on he 600 momenum fields indexed over each of he surfaces aking ino accoun he meric.

19 1 example: hippocampus emplae surface mapped o single arge Surfaces mapped for all 600 lef and righ srucures PCA was done on he momenum fields indexed o he emplae Shadow shows he amoun of ransformaion from iniial condiion for one example.

20 Thank-You