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

Diffusion Tensor Processing with Log-Euclidean Metrics

Vincent Arsigny, Pierre Fillard,Xavier Pennec, Nicholas Ayache.

Friday, September 23rd, 2005.

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

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.


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.


4

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.


5

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.


6

Diffusion MRI (dMRI)

• Water molecules diffuse in biological tissues.

• MR images can be weighted with diffusion

[Le Bihan,Nature rev. in Neurosc.,2003]


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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]


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Tensor Processing

• Needs: interpolation, extrapolation, regularization, statistics...

• Generalization to tensors of classical vector processing tools.

• HOW??


9

Outline

1. Presentation

2. Euclidean and Affine-Invariant Calculus

3. Log-Euclidean Framework

4. Experimental Results

5. Conclusions and Perspectives


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Defects of Euclidean Calculus

• Tensors are symmetric matrices. Euclidean operations can be performed.

• simplicity

• practically : unphysical negative eigenvalues appear very often


11

Defects of Euclidean Calculus

• Typical 'swelling effect' in interpolation:

• In DT-MRI: physically unacceptable !

Interpolated tensorsInterpolated tensors Interpolated volumes


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.


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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.


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.


15


• A scalar product for each tangent space of the manifold.

• Distance between 2 points: minimum of length of smooth curves joining them.


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• Each metric induces a generalization of the arithmetic mean, called ‘Fréchet mean’.

• The mean point minimizes a ‘metric dispersion’:

E(Si ;wi ) = arg minT

Pi wi :dist2(Si ;T)


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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.


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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...


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• Distance between two tensors:

• Geodesic between and (parameter ):t

S1 S2

S121 :exp

³t: log(S¡ 1

21 :S2:S

¡ 12

1 )´

:S121

d(S1;S2) = klog(S¡ 1=21 :S2:S

¡ 1=21 )k


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 '.'


21

Outline

1. Presentation






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• 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.

A novel vector space structure


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A novel vector space structure

• New 'addition', called 'logarithmic multiplication':

• New 'logarithmic scalar multiplication':

S1 ¯ S2 = exp(log(S1) + log(S2))

¸ ~S = exp(¸:log(S1))


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Metrics on Tensors

Tensor Space

Log-Euclidean metrics

Homogenous ManifoldStructure

Vector SpaceStructure

Algebraicstructures


Invariant metric Euclidean metric


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Distances

• Log-Euclidean framework:

• Affine-invariant framework:

d(S1;S2) = klog(S1) ¡ log(S2)k

d(S1;S2) = klog(S¡ 1=21 :S2:S

¡ 1=21 )k


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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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• Invariance properties:– Lie group bi-invariance– Similarity-invariance, for example with

(Frobenius):

– Invariance of the mean w.r.t. S 7! S¸


dist(S1;S2)2 = Trace³(log(S1) ¡ log(S2))

2´


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• 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?


29

• 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


30

• 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)



31

• 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.



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• Log-Euclidean mean (explicit closed form):

• Affine-invariant (Fréchet) mean (implicit barycentric equation):


ELE (Si ;wi ) = exp³ P N

i=1 wi log(Si )´

:

P Ni=1 wi log

¡EA I (Si ;wi )¡ 1=2:Si :EA I (Si ;wi )¡ 1=2

¢= 0:


33

Outline

1. Presentation






34

Synthetic interpolation

• Typical example of linear interpolation:

Euclidean

Affine-invariant

Log-Euclidean


35

Synthetic interpolation

• Typical example of synthetic bilinear interpolation:

Euclidean Affine-invariant Log-Euclidean


36

Interpolation on real DT-MRI

• Reconstruction by bilinear interpolation of a downsampled slice in mid-sagital plane:

Original slice Euclidean Log-Euclidean


37

Regularization of tensors

• Anisotropic regularization on synthetic data:

Original data Noisy data Euclidean reg. Log-Euc. reg.


38

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

Mean reconstruction error


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3D clinical DT image• [a] raw data• [b] Euclidean reg.• [c] Log-Euc. reg.• [d] Abs. difference

(x100!) betweenLog-Euc. AndAffine-inv.

[a] [b]

[c] [d]


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

FA

Gradient


41

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]


42

Fiber Tracking

• Corticospinal tract reconstructions after classical estimation or Log-Euclidean joint estimation and smoothing [Fillard, RR-5607].


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Outline

1. Presentation






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.


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• 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?

Perspectives

Thank you for your attention!

Any questions?

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

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