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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.
8th International Conference on Medical Image Computing and Computer Assisted Intervention, Oct 26 to 30, 2005.
October 27th, 2005 Vincent 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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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/
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Affine-invariant metrics
• Excellent theoretical properties:no 'swelling effect'
non-positive eigenvalues at infinity
• High computational cost: many algebraic operations
dist(S1;S2) = klog(S¡ 12
1 :S2:S¡ 1
21 )k:
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• 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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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A novel vector space structure
• Tensors: Lie group with 'logarithmic multiplication':
• Tensors: vector space with 'logarithmic scalar multiplication':
S1 ¯ S2 = exp(log(S1) + log(S2))
¸ ~S = exp(¸:log(S1))
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Log-Euclidean Distances
• Log-Euclidean metrics:– Euclidean metrics
for vector space structure – Bi-invariant Riemannian metrics
for Lie group structure ¯
¯ ; ~
dist(S1;S2) = klog(S1) ¡ log(S2)k:
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• Similarity-invariance, for example with (Frobenius):
• No Euclidean defect, exactly as in the affine-invariant case.
Theoretical properties
dist(S1;S2)2 = Trace³(log(S1) ¡ log(S2))
2´
:
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Log-Euclidean framework in practice
• Existing Euclidean algorithms readily recycled!
Conversion Tensor/Vector
with Matrix Logarithm
1
Euclidean Processingon logarithms
(filtering, statistics…)
2
Conversion Vector/Tensor
with Matrix Exponential
3
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Example: computing the mean
• Closed form for Log-Euclidean Fréchet mean:
• Affine-invariant case: implicit equation and iterative solving (20 times slower).
ELE (Si ;wi ) = exp
ÃNX
i=1
wi log(Si )
!
:
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Interpolation
• Typical example of bilinear interpolation on synthetic data:
11\Euclidean Log-EuclideanAffine-invariant
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Interpolation on real DT-MRI
• Reconstruction by bilinear interpolation of slice in mid-sagital plane:
Original slice Euclidean case Log-Euclidean case
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Regularization of tensors
Data: clinical DT image128x128x30– [a] Raw data– [b] Euclidean reg.– [c] Log-Eucl. reg.– [d] Log-Eucl. vs.
affine-inv. (x100!)
[a]
[b]
[c]
[d]
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
October 27th, 2005 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].
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• 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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Regularization of tensors
• Effect of anisotropicregularization on FractionalAnisotropy (FA)and gradient:
FA
Gradient
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Regularization of tensors
• Anisotropic regularization on synthetic data:
Orginal data Data+noise Euclidean result Log-Euclidean res.
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• 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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• Small difference: larger anisotropy in Log-Euclidean results.
• (Theoretical) reason: inequality between the 'traces' of the Log-Euclidean and affine-invariant means:
Trace(EA I (S)) < Trace(ELE (S))whenever EA I (S) 6= ELE (S)
Log-Euclidean vs. affine-invariant
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Geodesics
• Log-Euclidean case:
• Affine-invariant case:
S121 :exp
³t: log(S¡ 1
21 :S2:S
¡ 12
1 )´
:S121
exp((1¡ t): log(S1) + t: log(S2))
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
31
Metrics on Tensors
Tensor Space
Log-Euclidean metrics
Homogenous ManifoldStructure
Vector SpaceStructure
Algebraicstructures
Affine-invariant metrics
Invariant metric Euclidean metric
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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• 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, 2005 Vincent 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
dist(S1;S2)2 = Trace³(log(S1) ¡ log(S2))
2´
October 27th, 2005 Vincent 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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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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, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Defects of Euclidean Calculus
• Typical 'swelling effect' in interpolation:
• In DT-MRI: physically unacceptable !
Interpolated tensorsInterpolated tensors Interpolated volumes