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Statistics on Diffeomorphisms in a Log-Euclidean Framework

Mathematical Foundations of Computational Anatomy ( MFCA-2006 ), Copenhagen, October 1st, 2006. Satellite workshop of MICCAI’06. Statistics on Diffeomorphisms in a Log-Euclidean Framework. Vincent Arsigny ¹ ,Olivier Commowick ¹ ² , Xavier Pennec ¹ , Nicholas Ayache ¹ .

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Statistics on Diffeomorphisms in a Log-Euclidean Framework

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  1. Mathematical Foundations of Computational Anatomy (MFCA-2006), Copenhagen, October 1st, 2006. Satellite workshop of MICCAI’06. Statistics on Diffeomorphisms in a Log-Euclidean Framework Vincent Arsigny¹ ,Olivier Commowick¹ ², Xavier Pennec¹, Nicholas Ayache¹. ¹ Research Team ASCLEPIOS, INRIA Sophia, France.² DOSISoft SA, Cachan, France.

  2. Why Statistics on Diffeomorphisms? • Linked to non-rigid registration: • Comparison of algorithms • Introducing constraints[Pennec, MFCA, MICCAI’05], [Commowick, MICCAI’05] • Registration-based morphometry[Lepore, MFCA & MICCAI’06] Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  3. Statistics on Diffeomorphisms • Euclidean statistics:[Charpiat et al., ICCV’05], [Rueckert et al., TMI, 03] • Simple: vectorial on displacement fields (or B-Spline parameters) • Not consistent with invertibility • Space of “initial momentum”[Vaillant et al., NeuroIm, 04] • Remarkable framework of Trouvé et al., widely used • Hard to use for general diffeos (vs. landmarks) Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  4. Log-Euclidean Framework • Idea: • Simple processing • Consistency with group structure(e.g., inversion-invariance) • Previous work: finite-dimensional case Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  5. Outline • Presentation • Finite-Dimensional Case • Case of Diffeomorphisms • Numerical Algorithms • Experimental Results • Conclusions Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  6. Tensor Processing • In recent years : • Need to process symmetric positive-definite matrices (“tensors”) in various contexts • Deformation tensors (e.g., in registration results) • Diffusion tensors (i.e., DT-MRI) • Metric tensors, etc. • Need: • Consistency with manifold and algebraic structures. • Simplicity desirable. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  7. Log-Euclidean Framework • References: [Arsigny, MRM, 06][Arsigny, SIAM, 06], patent pending. • Idea: one-to-one correspondence with symmetric matrices, via matrix logarithm. • Simply process tensors via their (vectorial) logarithm! Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  8. ³ ´ 2 2 ( ) ( ( ) ( ) ) d l l S S T S S i t ¡ s r a c e o g o g = 1 2 1 2 ; : Theoretical Properties • Inversion-invariance • Similarity-invariance, for example with (Frobenius): • No Euclidean defect, exactly as in the affine-invariant case. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  9. Ã ! N X ( ) ( ) l E S S w e x p w o g = L E i i i i ; : i 1 = Log-Euclidean Mean • Log-Euclidean Fréchet meangeneralizes the geometric mean: • Affine-invariant case: implicit equation and iterative solving (20 times slower). Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  10. MedINRIA Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  11. And Linear Transformations? • References: [Arsigny, WBIR’06], [Commowick, ISBI’06], [Alexa, SIGGRAPH’02]. • Idea: linearize geometrical transformationsclose enough to identity via matrix logarithm. • Simply process transformations via their (vectorial) logarithms! • E.g., fuse local linear transformations into global invertible deformations. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  12. Examples: Polyaffine Transformations Fusing two translations Fusing two rotations

  13. Theoretical Properties • Restriction: to data whose logarithm is well-defined (e.g., no negative determinant allowed). • Inversion-invariance • Log-Euclidean mean is: • Affine-invariant (i.e., by affine change of coordinate system) • A geometric mean (determinant is geometric mean of data) Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  14. General Finite-Dimensional Case • References: [Arsigny, PhD, 06] • Data: logarithm must be well-defined(ok near the identity). • Properties: • Inversion-invariance • Log-Euclidean mean: invariant w.r.t. action of adjoint representation. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  15. Outline • Presentation • Finite-Dimensional Case • Case of Diffeomorphisms • Numerical Algorithms • Experimental Results • Conclusions Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  16. ( ) V _ x x = . Generalization to Diffeomorphisms • Diffeomorphisms belong to an infinite-dimensional Lie groups. • Logarithm of a diffeomorphism is a smooth vector field. • Exponential of a smooth vector field V(x): integration during 1 unit of time of the ODE: Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  17. Correspondence between Vector fields and Diffeomorphisms exp log Vector field Diffeomorphism Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  18. ( ) ( ) 8 @ d V V D I i 0 0 ' : e x p e e x p = = V ; . . . Technical Difficulty • Is the exponential locally diffeomorphic? • We have: • Infinite-dimensional case: not sufficient. • For general diffeomorphisms (very large space): not true. • For Banach-Lie groups: true. • Group of A. Trouvé: very close to a Banach-Lie group. Thus excellent candidate. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  19. Outline • Presentation • Finite-Dimensional Case • Case of Diffeomorphisms • Numerical Algorithms • Experimental Results • Conclusions Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  20. ( ( ) ) V t e x p R t 2 : ( ) ( = ) ( = ) V V V 2 2 e x p e x p e x p = : . General Principle • Idea: take advantage of algebraic properties of exp and log. • In particular:is a one-parameter subgroup. • E.g., → Direct generalization of numerical matrix algorithms. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  21. N N N N ¡ ¡ ( ) M 2 2 2 2 e x p : Scaling and Squaring Method Matrix case Choose normalization Compute Square recursively N times Vector field case Choose normalization Compute flow at time Compose recursively N times Deformations double at each recursive step. Vector field Diffeomorphism Numerical precision so far: 0.3% on average. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  22. Scaling and Squaring Method Fusion of two rotations (N=6).

  23. N N N 2 2 2 N N 2 2 Inverse Scaling and Squaring Diffeomorphism case Choose normalization Compute recursively N square roots (gradient descent). Multiply by final displacements Matrix case Choose normalization Compute recursively N square roots. Multiply by final matrix. Numerical precision so far: 3% on average. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006 Inverse Scaling and Squaring Method

  24. Outline • Presentation • Finite-Dimensional Case • Case of Diffeomorphisms • Numerical Algorithms • Experimental Results • Conclusions Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  25. Experimental Setup • Data set: 9 T1 MR images (3D) • Atlas-to-subject registration with 256x256x60 artificial T1 MR image (the ‘atlas’, from the Brainweb) • Robust affine registration followed by non-rigid registration of [Stefanescu, MedIA,04] guaranteeing invertibility of deformations. • → Computation of Euclidean and Log-Euclidean mean deformations. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  26. Experimental Results • Idea: L-E Mean deformation Jacobians Amplitude of def. Euclidean vs. Log-Euclidean • Largest deformations: ventricles, bigger in subjects than atlas. • Euclidean and Log-Euclidean quite close, except in regions of large deformations (then up to 30% of difference).

  27. Outline • Presentation • Finite-Dimensional Case • Case of Diffeomorphisms • Numerical Algorithms • Experimental Results • Conclusions Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  28. Conclusions • Log-Euclidean framework for diffeomorphisms: simple in spite of infinite dimensions. • Nice properties: e.g.,inversion-invariance (compatible with “inverse-consistency”) • Vectorial statistics thus directly generalized to diffeomorphisms. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  29. Perspectives • Addressing technical/mathematical issues • Better numerical algorithms for exp and log, more adapted to geometrical deformations (vs. matrices) • Challenge: finding efficient way of injecting global statistics on deformations in registration algorithms. Vincent Arsigny et al., Log-Euclidean Statistics, MFCA-2006

  30. Any questions? Thank you for your attention!

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