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

2. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 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]

3. October 1st, 2006Vincent 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)

4. October 1st, 2006Vincent 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

5. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 5 Outline 1.Presentation 2.Finite-Dimensional Case 3.Case of Diffeomorphisms 4.Numerical Algorithms 5.Experimental Results 6.Conclusions

6. October 1st, 2006Vincent 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.

7. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 7 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! Log-Euclidean Framework

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

9. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 9 Log-Euclidean Mean Log-Euclidean Fréchet mean generalizes the geometric mean: Affine-invariant case: implicit equation and iterative solving (20 times slower). E LE ( S i ; w i ) = exp à N X i = 1 w i l og ( S i ) ! :

10. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 10 MedINRIA

11. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 11 References: [Arsigny, WBIR’06], [Commowick, ISBI’06], [Alexa, SIGGRAPH’02]. Idea: linearize geometrical transformations close enough to identity via matrix logarithm. Simply process transformations via their (vectorial) logarithms! E.g., fuse local linear transformations into global invertible deformations. And Linear Transformations?

12. Examples: Polyaffine Transformations Fusing two translationsFusing two rotations

13. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 13 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) Theoretical Properties

14. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 14 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. General Finite-Dimensional Case

15. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 15 Outline 1.Presentation 2.Finite-Dimensional Case 3.Case of Diffeomorphisms 4.Numerical Algorithms 5.Experimental Results 6.Conclusions

16. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 16 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: _ x = V ( x ).

17. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 17 Correspondence between Vector fields and Diffeomorphisms exp log Vector field Diffeomorphism

18. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 18 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. 8 V : @ V exp ( 0 ) = V ; i. e. ' D exp ( 0 ) = I d.

19. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 19 Outline 1.Presentation 2.Finite-Dimensional Case 3.Case of Diffeomorphisms 4.Numerical Algorithms 5.Experimental Results 6.Conclusions

20. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 20 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. ( exp ( t : V )) t 2 R exp ( V ) = exp ( V = 2 ) : exp ( V = 2 ).

21. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 21 Scaling and Squaring Method Vector field case 1)Choose normalization 2)Compute flow at time 3)Compose recursively N times Numerical precision so far: 0.3% on average. Vector field Deformations double at each recursive step. Diffeomorphism Matrix case 1)Choose normalization 2)Compute 3)Square recursively N times 2 ¡ N 2 N exp ( 2 ¡ N : M ) 2 N

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

23. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 23 Inverse Scaling and Squaring Inverse Scaling and Squaring Method Numerical precision so far: 3% on average. Matrix case 1)Choose normalization 2)Compute recursively N square roots. 3)Multiply by final matrix. 2 N 2 N Diffeomorphism case 1)Choose normalization 2)Compute recursively N square roots (gradient descent). 3)Multiply by final displacements 2 N 2 N 2 N

24. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 24 Outline 1.Presentation 2.Finite-Dimensional Case 3.Case of Diffeomorphisms 4.Numerical Algorithms 5.Experimental Results 6.Conclusions

25. October 1st, 2006Vincent 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.

26. Experimental Results Idea: L-E Mean deformationJacobiansAmplitude 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. October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA-2006 27 Outline 1.Presentation 2.Finite-Dimensional Case 3.Case of Diffeomorphisms 4.Numerical Algorithms 5.Experimental Results 6.Conclusions

28. October 1st, 2006Vincent 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.

29. October 1st, 2006Vincent 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.

30. Thank you for your attention! Any questions?