1. - 1 Parallel Transport with the Pole Ladder: Application to Deformations of time Series of Images Marco Lorenzi, Xavier Pennec Asclepios research group - INRIA Sophia Antipolis, France GSI 2013

2. - 2 GSI 2013 Paradigms of Deformation-based Morphometry Cross sectional Longitudinal t1t2 Sub A Sub B Different topologies Large deformations Biological interpretation is not obvious Within-subject Subtle changes Biologically meaningful

3. - 3 GSI 2013 Sub B Template Combining longitudinal and cross-sectional t1t2 Sub A t1t2 ?

4. - 4 GSI 2013 Sub B Template Sub A Combining longitudinal and cross-sectional Standard TBM approach Focuses on volume changes only Scalar analysis (statistical power) No modeling Jacobian determinant analysis

5. - 5 GSI 2013 Sub A Template Combining longitudinal and cross-sectional Longitudinal trajectories Sub B Vector transport is not uniquely defined Missing theoretical insights

6. - 6 GSI 2013 Diffeomorphic registration  Stationary Velocity Field setting [Arsigny 2006] v(x) stationary velocity field Lie group Exp(v) is geodesic wrt Cartan connections (non-metric) Geodesic defined by SVF Stationary Velocity Field setting [Arsigny 2006] v(x) stationary velocity field Lie group Exp(v) is geodesic wrt Cartan connections (non-metric) Geodesic defined by SVF LDDMM setting [Trouvé, 1998] v(x,t) time-varying velocity field Riemannian exp id (v) is a metric geodesic wrt Levi-Civita connection Geodesic defined by initial momentum LDDMM setting [Trouvé, 1998] v(x,t) time-varying velocity field Riemannian exp id (v) is a metric geodesic wrt Levi-Civita connection Geodesic defined by initial momentum Transporting trajectories: Parallel transport of initial tangent vectors M id v 

7. - 7 GSI 2013 [Schild, 1970] P0P0 P’ 0 P1P1 A C curve P2P2    P’ 1 A’ From relativity to image processing The Schild’s Ladder

8. - 8 GSI 2013 Schild’s Ladder Intuitive application to images P0P0 P’ 0 T0T0 A T’ 0  SL  A) time Inter-subject registration [Lorenzi et al, IPMI 2011]

9. - 9 GSI 2013 t0 t1 t2 t3

10. - 10 GSI 2013 t0 t1 t2 t3 Evaluation of multiple geodesics for each time-point Parallel transport is not consistently computed among time-points

11. - 11 P0P0 P’ 0 T0T0 A T’ 0  A) The Pole Ladder optimized Schild’s ladder -A’ A’ C geodesic GSI 2013

12. - 12 GSI 2013 Pole Ladder Equivalence to Schild’s ladder Symmetric connection: B is the parallel transport of A Locally linear construction  Pole ladder is the Schild’s ladder

13. - 13 GSI 2013 t1 t2 t3 t0

14. - 14 GSI 2013 t0 t1 t2 t3 Minimize the number of geodesics required Parallel transport consistently computed amongst time-points

15. - 15 GSI 2013 Pole Ladder Application to SVF Setting [Lorenzi et al, IPMI 2011] B A + [ v, A ] + ½ [ v, [ v, A ] ] Baker-Campbell-Hausdorff formula (BCH) (Bossa 2007)

16. - 16 GSI 2013 Pole Ladder Iterative computation [Lorenzi et al, IPMI 2011] B A + [ v, A ] + ½ [ v, [ v, A ] ] A … v/nv/n

17. - 17 baselineTime 1 Time 4 … … ventricles expansion from the real time series Synthetic example GSI 2013

18. - 18 Comparison: Schild’s ladder Vector reorientation Conjugate action Scalar transport GSI 2013 Synthetic example

19. EMETTEUR - NOM DE LA PRESENTATION- 19 Transport consistency Deformation Vector transport Scalar transport Scalar summary ( logJacobian det, …) Vector measure GSI 2013 Synthetic example

20. - 20 GSI 2013 Synthetic example

21. - 21 GSI 2013 Synthetic example Quantitative analysis Pole ladder compares well with respect to scalar transport High variability led by Schild’s ladder

22. - 22 … … Group-wise Statistics Extrapolation Application on Alzheimer’s disease Group-wise analysis of longitudinal trajectories GSI 2013

23. - 23 GSI 2013 Longitudinal changes in Alzheimer’s disease (141 subjects – ADNI data) ContractionExpansion Student’s t statistic

24. - 24 GSI 2013 Longitudinal changes in Alzheimer’s disease (141 subjects – ADNI data) Comparison with standard TBM Student’s t statistic Pole ladder Scalar transport Consistent results Equivalent statistical power

25. - 25 GSI 2013 Conclusions General framework for the parallel transport of deformations (not necessarily requires the choice of a metric) Minimal number of computations for the transport of time series of deformations Efficient solution with the SVF setting Consistent statistical results Multivariate group-wise analysis of longitudinal changes Perspectives Further investigations of numerical issues (step-size) Comparison with other numerical methods for the parallel transport in diffeomorphic registration (Younes, 2007)

26. - 26 Thank you GSI 2013