Fast and Simple Calculus on Tensors in the Log-Euclidean Framework Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. Research Project/Team EPIDAURE/ASCLEPIOS INRIA, Sophia-Antipolis, France. 8th International Conference on Medical Image Computing and Computer Assisted Intervention, Oct 26 to 30, 2005.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 2 What are ‘tensors’? In general: all multilinear applications. In this talk: symmetric positive-definite matrices. –Typically : covariance matrices.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 3 Diffusion Tensor MRI Diffusion-weighted MR images Diffusion Tensor: local covariance of diffusion [Basser, 94]. Generalization of vector processing tools (filtering, statistics, etc.) to tensors?

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 4 Outline 1.Presentation 2.Euclidean and Affine-Invariant Calculus 3.Log-Euclidean Framework 4.Experimental Results 5.Conclusions and Perspectives

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 5 Euclidean calculus DTs: 3x3 symmetric matrices, thus belong to a vector space. Simple, but: –unphysical negative eigenvalues appear –‘swelling effect’: more diffusion than originally.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 6 Remedies in the literature First family: 1.process features from tensors 2.propagate processing to tensors. Example: regularization –dominant directions of diffusion [Coulon, IPMI’01] –orientations and eigenvalues separately [Tschumperlé, IJCV, 02, Chefd’hotel JMIV, 04]. Drawback: some information left behind.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 7 Remedies in the literature Second family: specialized procedures –Affine-invariant means [Wang, TMI, 05] –Anisotropic interpolation [Castagno-Moraga, MICCAI’04] –Etc. Drawback: lack of general framework.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 8 A general solution: Riemannian geometry Powerful framework for curved spaces. Statistics [Pennec, JMIV, 98], PDEs [Pennec, IJCV, 05]. Riemannian arithmetic mean: ‘Fréchet mean’. Basic tool: differentiable distance between tensors. /~ wupa4p0/

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 9 Choice of distance? Relevant/natural invariance properties. In 2004: affine-invariant metrics [Fletcher, CVAMIA’04, Lenglet, JMIV, 05, Moakher, SIMAX, 05, Pennec, IJCV, 05]. –invariance w.r.t. any affine change of coordinate system.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 10 Affine-invariant metrics Excellent theoretical properties: no 'swelling effect' non-positive eigenvalues at infinity High computational cost: many algebraic operations d i s t ( S 1 ; S 2 ) = k l og ( S ¡ : S 2 : S ¡ ) k :

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 11 Outline 1.Presentation 2.Euclidean and Affine-Invariant Calculus 3.Log-Euclidean Framework 4.Experimental Results 5.Conclusions and Perspectives

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 12 Surprise: a vector space structure for tensors! Idea: one-to-one correspondence with symmetric matrices, via matrix logarithm. More details: [Arsigny, INRIA RR-5584, 2005]. French patent pending. A novel vector space structure

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 13 A novel vector space structure Tensors: Lie group with 'logarithmic multiplication': Tensors: vector space with 'logarithmic scalar multiplication': S 1 ¯ S 2 = exp ( l og ( S 1 ) + l og ( S 2 )) ¸ ~ S = exp ( ¸ : l og ( S 1 ))

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 14 Log-Euclidean Distances Log-Euclidean metrics: –Euclidean metrics for vector space structure –Bi-invariant Riemannian metrics for Lie group structure ¯¯ ; ~ d i s t ( S 1 ; S 2 ) = k l og ( S 1 ) ¡ l og ( S 2 ) k :

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 15 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 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 16 Log-Euclidean framework in practice Existing Euclidean algorithms readily recycled! Conversion Tensor/Vector with Matrix Logarithm 1 Euclidean Processing on logarithms (filtering, statistics…) 2 Conversion Vector/Tensor with Matrix Exponential 3

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 17 Example: computing the mean Closed form for Log-Euclidean Fréchet 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 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 18 Outline 1.Presentation 2.Euclidean and Affine-Invariant Calculus 3.Log-Euclidean Framework 4.Experimental Results 5.Conclusions and Perspectives

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 19 Interpolation Typical example of bilinear interpolation on synthetic data: 11\ Euclidean Log-Euclidean Affine-invariant

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 20 Interpolation on real DT-MRI Reconstruction by bilinear interpolation of slice in mid-sagital plane: Original slice Euclidean case Log-Euclidean case

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 21 Regularization of tensors Data: clinical DT image 128x128x30 –[a] Raw data –[b] Euclidean reg. –[c] Log-Eucl. reg. –[d] Log-Eucl. vs. affine-inv. (x100!) [a] [b] [c] [d]

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 22 Outline 1.Presentation 2.Euclidean and Affine-Invariant Calculus 3.Log-Euclidean Framework 4.Experimental Results 5.Conclusions and Perspectives

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 23 Conclusions Log-Euclidean Riemannian framework: –Riemannian excellent properties. –Euclidean speed and simplicity –Existing vector algorithms readily recycled. More applications: –Joint estimation and smoothing for DTI: [Fillard, INRIA RR-5607, 2005]. –Statistical priors in non-linear registration [Pennec, MICCAI’05, Post. II-943], [Commowick, Post. II-927].

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 24 Evaluation/validation (phantoms...). Which metric for which application? –Diffusion tensors (statistics, interpolation, estimation, registration…) –Variability tensors [Fillard, IPMI’05] (models of anatomical varibility) –Structure tensors [Fillard, DSSCV’05] (classical image processing) –Metric tensors [Allauzet, INRIA RR-4759, 2003] (anisotropic mesh adaptation for PDE solving) Extension of Log-Euclidean framework to: –Generalized diffusion tensors [Özarslan, MRM, 2003] –Q-balls [Tuch, MRM, 2004]. Perspectives

Thank you for your attention! Any questions?

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 26 Regularization of tensors Effect of anisotropic regularization on Fractional Anisotropy (FA) and gradient: FA Gradient

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 27 Regularization of tensors Anisotropic regularization on synthetic data: Orginal data Data+noiseEuclidean resultLog-Euclidean res.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 28 Very little differences On DT images, Log-Euclidean advantages are: simplicity: Euclidean computations on logarithms! faster computations: computations at least 4 times faster in all situations. Log-Euclidean vs. affine-invariant

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 29 Small difference: larger anisotropy in Log-Euclidean results. (Theoretical) reason: inequality between the 'traces' of the Log-Euclidean and affine-invariant means: T race ( E AI ( S )) < T race ( E LE ( S )) w h enever E AI ( S ) 6 = E LE ( S ) Log-Euclidean vs. affine-invariant

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 30 Geodesics Log-Euclidean case: Affine-invariant case: S : exp ³ t : l og ( S ¡ : S 2 : S ¡ ) ´ : S exp (( 1 ¡ t ) : l og ( S 1 ) + t : l og ( S 2 ))

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 31 Metrics on Tensors Tensor Space Log-Euclidean metrics Homogenous Manifold Structure Vector Space Structure Algebraic structures Affine-invariant metrics Invariant metricEuclidean metric

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 32 with DT images, very similar results. Identical sometimes. Reason: associated means are two different generalizations of the geometric mean. In both cases determinants are interpolated geometrically. Log-Euclidean vs. affine-invariant

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 33 Invariance properties: –Lie group bi-invariance –Similarity-invariance, for example with (Frobenius): –Invariance of the mean w.r.t. S 7! S ¸ Log-Euclidean metrics d i s t ( S 1 ; S 2 ) 2 = T race ³ ( l og ( S 1 ) ¡ l og ( S 2 )) 2 ´

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 34 Variability tensors [Fillard, IPMI'05] Anatomical variability: local covariance matrix of displacement w.r.t. an average anatomy. Variability along sulci on the cortex and their extrapolation.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 35 Use of Tensors Generation of adapted meshes in numerical analysis for faster PDE solving (SMASH project): [Alauzet, RR-4981], GAMMA project. Application to fluid mechanics.

October 27th, 2005Vincent Arsigny et al., Log- Euclidean Framework, MICCAI'05 36 Defects of Euclidean Calculus Typical 'swelling effect' in interpolation: In DT-MRI: physically unacceptable ! Interpolated tensors Interpolated volumes