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Diffusion Tensor Processing with Log-Euclidean Metrics

Diffusion Tensor Processing with Log-Euclidean Metrics. Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. Friday, September 23 rd , 2005. What are ‘Tensors’?. In general : any multilinear mapping. E.g. a vector, a matrix, a tensor products of vectors…

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Diffusion Tensor Processing with Log-Euclidean Metrics

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

  2. 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. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  3. 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. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  4. [Alauzet, RR-4981], GAMMA project. Application to fluid mechanics. Use of Tensors • Generation of adapted meshes in numerical analysis for faster PDE solving(SMASH project): Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  5. Variability along sulci on the cortex and their extrapolation. Variability tensors • [Fillard, IPMI'05] Anatomical variability: local covariance matrix of displacement w.r.t. an average anatomy. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

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

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

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

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

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

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

  17. 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. 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... Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  19. = = 1 1 1 2 1 1 2 1 ¡ ¡ ³ ´ ¡ ¡ ( ) k ( ) k d l S S S S S ( ) l S S S S S 2 2 2 2 o g t = 1 2 2 e x p o g 1 1 ; : : 2 1 1 1 1 : : : : : t S S 2 1 Affine-invariant metrics • Distance between two tensors: • Geodesic between and (parameter ): 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 '.' Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

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

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

  24. Tensor Space Homogenous ManifoldStructure Vector SpaceStructure Algebraicstructures Metrics on Tensors Invariant metric Euclidean metric Affine-invariant metrics Log-Euclidean metrics Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

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

  26. 1 1 1 1 ³ ´ ¡ ¡ ( ( ) ( ) ( ) ) ( ) l l S S l S S S S S 2 2 2 2 1 t t t ¡ + e x p o g o g e x p o g 1 2 2 1 1 1 1 : : : : : : : Geodesics • Log-Euclidean case: • Affine-invariant case: Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  27. 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 S S ; ! 7 Log-Euclidean metrics • Invariance properties: • Lie group bi-invariance • Similarity-invariance, for example with (Frobenius): • Invariance of the mean w.r.t. 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? 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. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  30. ( ( ) ) ( ( ) ) E E T S T S < r a c e r a c e A I L E ( ) ( ) h 6 E E S S w e n e v e r = A I L E 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: 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. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  32. N = = ³ ´ ¡ ¢ 1 2 1 2 ¡ ¡ P ( ) ( ) N l E E S S S 0 P ( ) ( ) w o g w w l E S S = A I A I i i i i i i i 1 w e x p w o g ; : : ; : = L E i i i i = i 1 ; : = Log-Euclidean vs. affine-invariant • Log-Euclidean mean (explicit closed form): • Affine-invariant (Fréchet) mean (implicit barycentric equation): Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

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

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

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

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

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

  38. Mean reconstruction error Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  39. [b] [a] [d] [c] Regularization of tensors 3D clinical DT image • [a] raw data • [b] Euclidean reg. • [c] Log-Euc. reg. • [d]Abs. difference (x100!) betweenLog-Euc. AndAffine-inv. Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  40. FA Gradient Regularization of tensors • Effect of anisotropicregularization on FractionalAnisotropy (FA)and gradient: Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  41. [a] [b] [c] Results from[Fillard, RR-5607] [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 Tensor Estimation Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

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

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

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

  45. 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? Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.

  46. Thank you for your attention! Any questions?

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