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

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.


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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?


4

Outline

1. Presentation

2. Euclidean and Affine-Invariant Calculus

3. Log-Euclidean Framework

4. Experimental Results

5. Conclusions and Perspectives


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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.


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.


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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.


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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/


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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.


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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:


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Outline

1. Presentation





SEMIR

combining excellent theoretical properties and simplicity


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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

SEMIR

Big surprise !


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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))


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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:


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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´

:


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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

SEMIR

Conversion Riemannian Euclidean


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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 )

!

:


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Outline

1. Presentation






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Interpolation

• Typical example of bilinear interpolation on synthetic data:

11\Euclidean Log-EuclideanAffine-invariant

SEMIR

Riemannian results very close!


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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


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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]

SEMIR

Affine-inv and Log-Eucl results surprisingly close!


22

Outline

1. Presentation






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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].


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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

SEMIR

Looking forward to using the Log-Euclidean framework for...

Thank you for your

attention!

Any questions?


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• Effect of anisotropicregularization on FractionalAnisotropy (FA)and gradient:

FA

Gradient


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• Anisotropic regularization on synthetic data:

Orginal data Data+noise Euclidean result Log-Euclidean res.


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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


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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)



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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))


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Metrics on Tensors

Tensor Space

Log-Euclidean metrics

Homogenous ManifoldStructure

Vector SpaceStructure

Algebraicstructures

Affine-invariant metrics

Invariant metric Euclidean metric


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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.



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• 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´


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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.


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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.


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Defects of Euclidean Calculus

• Typical 'swelling effect' in interpolation:

• In DT-MRI: physically unacceptable !

Interpolated tensorsInterpolated tensors Interpolated volumes

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