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.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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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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]
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Tensor Processing
• Needs: interpolation, extrapolation, regularization, statistics...
• Generalization to tensors of classical vector processing tools.
• HOW??
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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
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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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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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A Solution: Riemannian Geometry
• 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.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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...
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Affine-invariant metrics
• 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
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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 '.'
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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))
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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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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¸
Log-Euclidean metrics
dist(S1;S2)2 = Trace³(log(S1) ¡ log(S2))
2´
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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?
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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
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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
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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• 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.
Log-Euclidean vs. affine-invariant
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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• Log-Euclidean mean (explicit closed form):
• Affine-invariant (Fréchet) mean (implicit barycentric equation):
Log-Euclidean vs. affine-invariant
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:
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Synthetic interpolation
• Typical example of linear interpolation:
Euclidean
Affine-invariant
Log-Euclidean
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Synthetic interpolation
• Typical example of synthetic bilinear interpolation:
Euclidean Affine-invariant Log-Euclidean
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Interpolation on real DT-MRI
• Reconstruction by bilinear interpolation of a downsampled slice in mid-sagital plane:
Original slice Euclidean Log-Euclidean
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Regularization of tensors
• Anisotropic regularization on synthetic data:
Original data Noisy data Euclidean reg. Log-Euc. reg.
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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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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.
[a] [b]
[c] [d]
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Regularization of tensors
• Effect of anisotropicregularization on FractionalAnisotropy (FA)and gradient:
FA
Gradient
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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]
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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
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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Conclusions
• Log-Euclidean Riemannian framework: fast and simple.
• Has excellent theoretical properties.
• Effective and efficient for all usual types of processing on diffusion tensors.
September 23rd, 2005 Vincent Arsigny, Séminaire Croisé on Tensors, INRIA.
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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