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Fast and Simple Calculus on Tensors in the Log-Euclidean Framework

8th International Conference on M edical I mage C omputing and C omputer A ssisted I ntervention, Oct 26 to 30, 2005. Fast and Simple Calculus on Tensors in the Log-Euclidean Framework. Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache.

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Fast and Simple Calculus on Tensors in the Log-Euclidean Framework

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  1. 8th International Conference on Medical Image Computing and Computer Assisted Intervention, Oct 26 to 30, 2005. Fast and Simple Calculus on Tensors in the Log-Euclidean Framework Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. Research Project/Team EPIDAURE/ASCLEPIOSINRIA, Sophia-Antipolis, France.

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

  3. Diffusion Tensor MRI • Diffusion-weightedMR images • Diffusion Tensor: local covariance of diffusion [Basser, 94]. • Generalizationof vector processing tools (filtering, statistics, etc.) to tensors? Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  4. Outline • Presentation • Euclidean and Affine-Invariant Calculus • Log-Euclidean Framework • Experimental Results • Conclusions and Perspectives Vincent 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. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  6. Remedies in the literature • First family: • process features from tensors • 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. Vincent 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. Vincent 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. http://www.alumni.ca/~wupa4p0/ Vincent 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. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

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

  11. Outline • Presentation • Euclidean and Affine-Invariant Calculus • Log-Euclidean Framework • Experimental Results • Conclusions and Perspectives Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

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

  13. ( ( ) ( ) ) 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 • Tensors: Lie group with 'logarithmic multiplication': • Tensors: vector space with 'logarithmic scalar multiplication': Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  14. ( ) k ( ) ( ) k d l l S S S S i t ¡ s o g o g = 1 2 1 2 ; : ¯ ¯ ~ ; Log-Euclidean Distances • Log-Euclidean metrics: • Euclidean metricsfor vector space structure • Bi-invariant Riemannian metricsfor Lie group structure Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  15. 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 ; : Theoretical properties • Similarity-invariance, for example with (Frobenius): • No Euclidean defect, exactly as in the affine-invariant case. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

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

  17. Ã ! N X ( ) ( ) l E S S w e x p w o g = L E i i i i ; : i 1 = Example: computing the mean • Closed formfor Log-Euclidean Fréchet mean: • Affine-invariant case: implicit equation and iterative solving (20 times slower). Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  18. Outline • Presentation • Euclidean and Affine-Invariant Calculus • Log-Euclidean Framework • Experimental Results • Conclusions and Perspectives Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

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

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

  21. [b] [a] [c] [d] Regularization of tensors Data: clinical DT image128x128x30 • [a] Raw data • [b] Euclidean reg. • [c] Log-Eucl. reg. • [d] Log-Eucl. vs.affine-inv. (x100!) Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  22. Outline • Presentation • Euclidean and Affine-Invariant Calculus • Log-Euclidean Framework • Experimental Results • Conclusions and Perspectives Vincent 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]. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  24. Perspectives • 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]. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  25. Thank you for your attention! Any questions?

  26. FA Gradient Regularization of tensors • Effect of anisotropicregularization on FractionalAnisotropy (FA)and gradient: Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  27. Orginal data Data+noise Euclidean result Log-Euclidean res. Regularization of tensors • Anisotropic regularization on synthetic data: Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  28. Log-Euclidean vs. affine-invariant • 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. Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  29. ( ( ) ) ( ( ) ) 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 et al., Log-Euclidean Framework, MICCAI'05

  30. 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 et al., Log-Euclidean Framework, MICCAI'05

  31. Tensor Space Homogenous ManifoldStructure Vector SpaceStructure Algebraicstructures Metrics on Tensors Invariant metric Euclidean metric Affine-invariant metrics Log-Euclidean metrics Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05

  32. 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 et al., Log-Euclidean Framework, MICCAI'05

  33. 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 et al., Log-Euclidean Framework, MICCAI'05

  34. 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 et al., Log-Euclidean Framework, MICCAI'05

  35. [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 et al., Log-Euclidean Framework, MICCAI'05

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

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