Parallel Transport with the Pole Ladder: Application to Deformations of time Series of Images

28/08/2013
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Parallel Transport with the Pole Ladder: Application to Deformations of time Series of Images

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- 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 - 2GSI 2013 Paradigms of Deformation-based Morphometry Cross sectional Longitudinal t1 t2Sub A Sub B Different topologies Large deformations Biological interpretation is not obvious Within-subject Subtle changes Biologically meaningful - 3GSI 2013 Sub B Template Combining longitudinal and cross-sectional t1 t2 Sub A t1 t2 ? - 4GSI 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 - 5GSI 2013 Sub A Template Combining longitudinal and cross-sectional Longitudinal trajectories Sub B Vector transport is not uniquely defined Missing theoretical insights - 6GSI 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 expid(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 expid(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  - 7GSI 2013 [Schild, 1970] P0 P’0 P1 A C curve P2     P’1 A’ From relativity to image processing The Schild’s Ladder - 8GSI 2013 Schild’s Ladder Intuitive application to images P0 P’0 T0 A T’0 SLA) time Inter-subjectregistration [Lorenzi et al, IPMI 2011] - 9GSI 2013 t0 t1 t2 t3 - 10GSI 2013 t0 t1 t2 t3 • Evaluation of multiple geodesics for each time-point • Parallel transport is not consistently computed among time-points - 11 P0 P’0 T0 A T’0 A) The Pole Ladder optimized Schild’s ladder -A’ A’ C geodesic GSI 2013 - 12GSI 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 - 13GSI 2013 t1 t2 t3 t0 - 14GSI 2013 t0 t1 t2 t3 • Minimize the number of geodesics required • Parallel transport consistently computed amongst time-points - 15GSI 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) - 16GSI 2013 Pole Ladder Iterative computation [Lorenzi et al, IPMI 2011] B A + [ v , A ] + ½ [ v , [ v , A ] ] A … v/n - 17 baseline Time 1 Time 4 … …ventricles expansion from the real time series Synthetic example GSI 2013 - 18 Comparison: •Schild’s ladder • Vector reorientation • Conjugate action • Scalar transport GSI 2013 Synthetic example EMETTEUR - NOM DE LA PRESENTATION - 19 Transport consistency Deformation Vector transport Scalar transport Scalar summary Scalar summary ( logJacobian det, …) Vector measure GSI 2013 Synthetic example - 20GSI 2013 Synthetic example - 21GSI 2013 Synthetic example Quantitative analysis • Pole ladder compares well with respect to scalar transport • High variability led by Schild’s ladder - 22 … … • Group-wise Statistics • Extrapolation Application on Alzheimer’s disease Group-wise analysis of longitudinal trajectories GSI 2013 - 23GSI 2013 Longitudinal changes in Alzheimer’s disease (141 subjects – ADNI data) ContractionExpansion Student’s t statistic - 24GSI 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 - 25GSI 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 Thank you GSI 2013