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, Satellite workshop of MICCAI’06.

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

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

October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA Log-Euclidean Framework Idea: –Simple processing –Consistency with group structure (e.g., inversion-invariance) –Previous work: finite-dimensional case

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

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

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

October 1st, 2006Vincent Arsigny et al., Log- Euclidean Statistics, MFCA 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 ´ :

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

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

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

Examples: Polyaffine Transformations Fusing two translationsFusing two rotations

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

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

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

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

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

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

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

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

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

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

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

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

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

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).

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

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

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

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