1. Diffusion Tensor Processing with Log-Euclidean Metrics
Vincent Arsigny, Pierre Fillard,
Xavier Pennec, Nicholas Ayache. Friday, September 23rd, 2005.

2. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
What are ‘Tensors’?
In general: any multilinear mapping. E.g. a vector, a matrix, a tensor products of vectors…
In this talk: a symmetric, positive definite matrix. Typically: a covariance matrix (origin: DT-MRI)
A 3x3 tensor can be visualized with an ellipsoid.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

3. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Use of Tensors
Statistics: covariance matrices. Recently introduced in non-linear registration [Commowick, Miccai'05], [Pennec, Miccai'05].
Image processing (edges, corner dectection, scale-space analysis...) [Fillard, DSSCV'05].
Continuum mechanics : strain and stress tensors, etc.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

4. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

5. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

6. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Diffusion MRI (dMRI)
Water molecules diffuse in biological tissues.
MR images can be weighted with diffusion
[Le Bihan,
Nature rev. in Neurosc.,
2003]
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

7. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Diffusion Tensor MRI
Simple Model: Brownian motion.
Diffusion Tensor: local covariance of diffusion process.
DT images: tensor-valued images.
Typical exemple, from a 1.5 Tesla scanner, 128x128x30,
[Arsigny,RR-5584, 2005]
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

8. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Tensor Processing
Needs: interpolation, extrapolation, regularization, statistics...
Generalization to tensors of classical vector processing tools.
HOW??
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

9. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Outline
Presentation
Euclidean and Affine-Invariant Calculus
Log-Euclidean Framework
Experimental Results
Conclusions and Perspectives
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

10. Defects of Euclidean Calculus
Tensors are symmetric matrices. Euclidean operations can be performed.
simplicity
practically : unphysical negative eigenvalues appear very often
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

11. Defects of Euclidean Calculus
Typical 'swelling effect' in interpolation:
In DT-MRI: physically unacceptable !
Interpolated tensors
Interpolated volumes
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

12. Remedies in the literature
Operations on features of tensors, propagated back to tensors:
dominant directions of diffusion [Coulon, IPMI’01]
orientations and eigenvalues separately [Tschumperlé, IJCV, 02, Chefd’hotel JMIV, 04]
Drawback: some information left behind.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

13. Remedies in the literature
Specialized procedures:
Affine-invariant means based on J-divergence [Wang, TMI, 05]
Interpolation on tensors with structure tensors [Castagno-Moraga, MICCAI’04]
Etc.
Drawback: lack of general framework.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

14. A Solution: Riemannian Geometry
General framework for curved spaces (e.g. rotations, affine transformations, diffeomorphisms, and more).
Allows for the generalization of statistics [Pennec, 98] or PDEs [Pennec, IJCV, 05].
Idea: define a differentiable distance between tensors.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

15. A Solution: Riemannian Geometry
A scalar product for each tangent space of the manifold.
Distance between 2 points: minimum of length of smooth curves joining them.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

16. A Solution: Riemannian Geometry
Each metric induces a generalization of the arithmetic mean, called ‘Fréchet mean’.
The mean point minimizes a ‘metric dispersion’: E ( S i ; w ) = a r g m n T P : d s t 2
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

17. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Choice of metric
Idea: rely on relevant/natural invariance properties
First proposition: affine-invariant metrics [Fletcher (CVAMIA’04), Lenglet (JMIV), Moakher (SIMAX), Pennec (IJCV), 04].
Computations are invariant w.r.t. any (affine) change of coordinate system.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

18. Affine-invariant metrics
Excellent theoretical properties:
no 'swelling effect'
non-positive eigenvalues at infinity
symmetry w.r.t. matrix inversion
High computational cost: lots of inverses, square roots, matrix exponential and logarithms...
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

19. Affine-invariant metrics
Distance between two tensors:
Geodesic between and (parameter ): d ( S 1 ; 2 ) = k l o g : S 1 S 2 t S 1 2 : e x p t l o g ( )
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

20. Beyond affine-invariant metrics
Quotations from [Pennec, RR-5255]:
“The main problem is that the tensor space is a manifold that is not a vector space” (page 5).
“Thus, the structure we obtain is very close to a vector space, except that the space is curved” (page 30).
Not a vector space with usual operations '+' and '.'
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

21. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Outline
Presentation
Euclidean and Affine-Invariant Calculus
Log-Euclidean Framework
Experimental Results
Conclusions and Perspectives
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

22. A novel vector space structure
References: [Arsigny, Miccai’05], [Arsigny, RR-5584]. French patent pending.
The tensor space is a vector space with proper operations.
Idea: use one-to-one correspondence with symmetric matrices, via matrix logarithm and exponential.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

23. A novel vector space structure
New 'addition', called 'logarithmic multiplication':
New 'logarithmic scalar multiplication': S 1 2 = e x p ( l o g ) + ~ S = e x p ( : l o g 1 )
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

24. Metrics on Tensors Tensor Space Algebraic structures Invariant metric
Homogenous Manifold Structure
Vector Space Structure
Algebraic structures
Invariant metric
Euclidean metric
Affine-invariant metrics
Log-Euclidean metrics
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

25. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Distances
Log-Euclidean framework:
Affine-invariant framework: d ( S 1 ; 2 ) = k l o g d ( S 1 ; 2 ) = k l o g :
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

26. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Geodesics
Log-Euclidean case:
Affine-invariant case: e x p ( 1 t ) : l o g S + 2 S 1 2 : e x p t l o g ( )
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

27. Log-Euclidean metrics
Invariance properties:
Lie group bi-invariance
Similarity-invariance, for example with (Frobenius):
Invariance of the mean w.r.t. d i s t ( S 1 ; 2 ) = T r a c e l o g S 7 !
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

28. Log-Euclidean metrics
Exactly like in the affine-invariant case:
no 'swelling effect'
non-positive eigenvalues at infinity
symmetry w.r.t. matrix inversion.
Practically, what differences between the two (families of ) metrics?
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

29. Log-Euclidean vs. affine-invariant
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.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

30. Log-Euclidean vs. affine-invariant
Small difference: larger anisotropy in Log-Euclidean results.
(Theoretical) reason: inequality between the 'traces' of the Log-Euclidean and affine-invariant means: T r a c e ( E A I S )
< L w h n v 6 =
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

31. Log-Euclidean vs. affine-invariant
On usual DT images, the Log-Euclidean framework provides:
simplicity: Euclidean computations on logarithms!
faster computations: means computed 20 times faster, computations at least 4 times faster in all situations.
larger numerical stability.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

32. Log-Euclidean vs. affine-invariant
Log-Euclidean mean (explicit closed form):
Affine-invariant (Fréchet) mean (implicit barycentric equation): E L ( S i ; w ) = e x p P N 1 l o g : P N i = 1 w l o g E A I ( S ; ) 2 :
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

33. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Outline
Presentation
Euclidean and Affine-Invariant Calculus
Log-Euclidean Framework
Experimental Results
Conclusions and Perspectives
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

34. Synthetic interpolation
Typical example of linear interpolation:
Euclidean
Affine-invariant
Log-Euclidean
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

35. Synthetic interpolation
Typical example of synthetic bilinear interpolation:
Euclidean Affine-invariant Log-Euclidean
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

36. Interpolation on real DT-MRI
Reconstruction by bilinear interpolation of a downsampled slice in mid-sagital plane:
Original slice Euclidean Log-Euclidean
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

37. Regularization of tensors
Anisotropic regularization on synthetic data:
Original data Noisy data Euclidean reg. Log-Euc. reg.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

38. Mean reconstruction error
Dissimilarity Measure
Euclidean Regul.
Affine-inv. Regul.
Log-Eucl. Regul.
Mean Eucl. error
0.228
0.080
0.051
Mean Aff-inv. error
0.533
0.142
0.119
Mean Log-Eucl. error
0.532
0.135
0.111
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

39. Regularization of tensors
[b]
[c]
[d]
3D clinical DT image
[a] raw data
[b] Euclidean reg.
[c] Log-Euc. reg.
[d] Abs. difference (x100!) between Log-Euc. And Affine-inv.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

40. Regularization of tensorsFA
Gradient
Effect of anisotropic regularization on Fractional Anisotropy (FA) and gradient:
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

41. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Tensor Estimation
[a]
[b]
[c]
[a] Algebraic Tensor estimation on the logarithm of DWIs
[b] Log-Euclidean Tensor estimation directly on DWIs
[c] Log-Euclidean joint Tensor estimation and smoothing on DWIs
Results from [Fillard, RR-5607]
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

42. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Fiber Tracking
Corticospinal tract reconstructions after classical estimation or Log-Euclidean joint estimation and smoothing [Fillard, RR-5607].
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

43. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Outline
Presentation
Euclidean and Affine-Invariant Calculus
Log-Euclidean Framework
Experimental Results
Conclusions and Perspectives
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

44. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Conclusions
Log-Euclidean Riemannian framework: fast and simple.
Has excellent theoretical properties.
Effective and efficient for all usual types of processing on diffusion tensors.
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

45. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
Perspectives
In-depth evaluation/validation of existing Riemannian frameworks on tensors
Other relevant frameworks?
Log-Euclidean framework allows for straightforward statistics on diffusion tensors
Extension to more sophisticated diffusion models?
September 23rd, 2005
Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

46. Thank you for your attention! Any questions?