Estimation of the registration accuracy in the absence of...
Transcript of Estimation of the registration accuracy in the absence of...
Xavier Pennec
Computational Anatomy& Atlases
AGND Traitement d’images médicales:
du voxel aux atlas numériques
Strasbourg, 2 au 6 juin 2008
3 juin 2008 X. Pennec - ANGD traiment d'images médicales 2
Anatomy
Gall (1758-1828) : PhrenologyTalairach (1911-2007)
Antiquity • Animal models • Philosophical physiology
Renaissance:• Dissection, surgery• Descriptive anatomy
Vésale (1514-1564)Paré (1509-1590)
1990-2000:• Explosion of imaging • Computer atlases• Brain decade
2007
Revolution of observation means (1988-2007) :From dissection to in-vivo in-situ imagingFrom representative individual to populationFrom descriptive atlases to interactive and generative models (simulation)
Galien (131-201)
1er cerebral atlas, Vesale, 1543
17-20e century:• Anatomo-physiology• Microscopy, histology
Science that studies the structure and the relationship in space of different organs and tissues in living systems
[Hachette Dictionary]
Visible Human Project, NLM, 1996-2000Voxel-Man, U. Hambourg, 2001
Talairach & Tournoux, 1988
Sylvius (1614-1672)Willis (1621-1675)
Paré, 1585
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The revolution of medical imaging
In vivo observation of living systemsA large number of modalities to image anatomy and function
Growing spatial resolution (molecules to whole body) Multiple temporal scales
Non invasive observations Emergence of large databasesFrom representative individual to population
Extract and structure information 50 to 150 MB for a clinical MRIComputer analysis is necessary From descriptive atlases to interactive and generative models (simulation)
MNI 305 (1993), ICBM 152 (2001), Brain web (1999)Bone Morphing® (Fleute et al, 2001)
200 μm
Mouse colonMicro-vaissels
brain
850 μmheart
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Algorithms to Model and Analyze the AnatomyEstimate representative organ anatomies across species, populations, diseases, aging, ages…Model organ development across timeEstablish normal variability
To understand and to model how life is functioningClassify pathologies from structural deviations (taxonomy)Integrate individual measures at the population level to relate anatomy and function
To detect, understand and correct dysfunctionsFrom generic (atlas-based) to patients-specific models Quantitative and objective measures for diagnosisHelp therapy planning (before), control (during) and follow-up (after)
Computational Anatomy
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Modeling and image analysis: a virtuous loop
Normalization,Interpretation,
Modeling
IdentificationPersonalization
AnatomyPhysicsPhysiology
Images,Signals,Clinics,Genetics,etc.
Individual
Computational models of the human body
PopulationPrevention
Diagnosis
Therapy
Computer assisted medicine
Generative models
Statistical analysis
Knowledge inference
Integrative modelsfor biology and neurosciences
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Methods of computational anatomy
Hierarchy of anatomical manifolds (structural models) Landmarks [0D]: AC, PC [Talairach et Tournoux, Bookstein], functional landmarksCurves [1D]: crest lines, sulcal lines [Mangin, Barillot, Fillard…]Surfaces [2D]: cortex, sulcal ribbons [Thompson, Mangin, Miller…], Images [3D functions]: VBM, Diffusion imagingTransformations: rigid, multi-affine, local deformations (TBM), diffeomorphisms[Asburner, Arsigny, Miller, Trouve, Younes…]
Groupwise correspondances in the population Model observations and its structural variability
Statistical computing on Riemannian manifolds
Structural variability of the cortex
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Outline
Goals and methods of Computational anatomy
Statistical computing on manifoldsThe geometrical and statistical framework
Examples with rigid body transformationsExtending the framework to manifold-valued images
Building a statistical atlas of the hear fibers
Computational neuro-anatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
Conclusion and challenges
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Statistical analyses on manifolds
Medical image analysis: Noisy geometric measuresFeature extracted from images
Lines, oriented pointsExtremal points: semi oriented framesTensors from DTI
Transformations in registrations Rigid, Affine, locally affine, families of deformations
Goal: Deal with noise consistently on these non-Euclidean manifoldsA consistent computing framework
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Probabilités et statistiques simples
Mesure : vecteur aleatoire x de densite
Approximation :
Propagation :
Modèle de bruit : additif, gaussien...
Distance statistique : Mahalanobis et
)x( xxΣx , ~
)(zpx
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
=x
..x
, x)(Thh h ~ h xxΣxy
χ 2
Objets géométriques :• Primitives, Transformations : Variétés différentielles
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Quelques problèmes avec la géométrie
Moyenne de rotations 3D :
Bruit IID sur les rotations :
∑=i
in RR 1 ∑=
iiq
n q 1 ∑=
iirn
r 1
ΣΣ =2
ΣΣ =1
⎥⎦⎤
⎢⎣⎡=
1000100002σΣ
ΣΣ π 28
1' =
ΣΣ 822' π
=
Définition intrinsèque (indep. de la carte)
invariance par un groupe de transfo.
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Riemannian Manifolds: geometrical tools
Riemannian metric :Dot product on tangent space Speed, length of a curveDistance and geodesics
Closed form for simple metrics/manifoldsOptimization for more complex
Exponential chart (Normal coord. syst.) :Development in tangent space along geodesics Geodesics = straight linesDistance = EuclideanStar shape domain limited by the cut-locusCovers all the manifold if geodesically complete
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Computing on Riemannian manifolds
Operation Euclidean space Riemannian manifold
)( ttt CΣ∇−Σ=Σ+ εε
)(log yxy x=xyxy −=
xyxy +=
xyyxdist −=),(x
xyyxdist =),(
)(exp xyy x=
))((exp tt Ct
Σ∇−=Σ Σ+ εε
Subtraction
Addition
Distance
Gradient descent
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Statistical tools on Riemannian manifolds
Metric -> Volume form (measure)
Probability density functions
Expectation of a function φ from M into R :
Definition :
Variance :
Information (neg. entropy):
)x(Md
)M().()(, ydypXxPXX∫=∈∀ x
[ ] ∫=M
M )().(.E ydyp(y)(x) xφφ
[ ] ∫==M
M )().(.),dist()x,dist(E )( 222 zdzpzyyy xxσ
[ ] [ ]))(log(E I xx xp=
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Statistical tools: Moments
Frechet / Karcher mean minimize the variance
Geodesic marching
Covariance et higher moments
[ ]xyE with )(expx x1 ==+ vvtt
( )( )[ ] ( )( )∫==ΣM
M )().(.x.xx.xE TT
zdzpzz xxx xx
[ ] [ ]( ) [ ] [ ]0)( 0)().(.xxE ),dist(E argmin 2 ===⇒= ∫∈
CPzdzpyy MM
MxxxxxΕ
[ Pennec, JMIV06, RR-5093, NSIP’99 ]
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Example with 3D rotations
Exp chart at Id:
Distance:
Frechet mean:
nr . :ectorrotation v θ=
2)1(
121 ),dist( rrRR ο−=
Centered chart:
mean = barycenter
⎟⎠
⎞⎜⎝
⎛= ∑∈ i
),dist(min arg 3
iSORRRR
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Distributions for parametric testsUniform density:
maximal entropy knowing X
Generalization of the Gaussian density:Stochastic heat kernel p(x,y,t) / Wrapped Gaussian Maximal entropy knowing the mean and the covariance
From Dirac to uniform (on compact manifolds)
Mahalanobis D2 distance / test:
Any distribution:
Gaussian:
( ) ( ) ( ) ( )rOkyN / Ric with 2/x..xexp.)( 31)1(T
σεσ ++−=⎟⎠⎞⎜
⎝⎛= −ΣΓxΓx
)Vol(/)(Ind)( Xzzp X=x
yx..yx)y( )1(2 −Σ= xxx
tμ
[ ] n=)(E 2 xxμ
( )rOn /)()( 322 σεσχμ ++∝xx
[ Pennec, JMIV06, NSIP’99 ]
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Validation of the rigid registration accuracy
[ X. Pennec et al., Int. J. Comp. Vis. 25(3) 1997, MICCAI 1998 ]
Comparing two transformations and their Covariance matrix :
Mean: 6, Var: 12KS test
2621
2 ),( χμ ≈TT
Bias estimation: (chemical shift, susceptibility effects)(not significantly different from the identity)(significantly different from the identity)
Inter-echo with bias corrected: , KS test OK62 ≈μ
Intra-echo: , KS test OK62 ≈μInter-echo: , KS test failed, Bias !502 >μ
deg 06.0=rotσmm 2.0=transσ
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Validation using Bronze Standard
Best explanation of the observations (ML) :LSQ criterion
Robust Fréchet mean
Robust initialization and Newton gradient descent
Result
Derive tests on transformations for accuracy / consistency
( )221
221
2 ),,(min),( χμ TTTTd =
transrotjiT σσ ,,,
∑= ij ijij TTdC )ˆ,(2
[ T. Glatard & al, MICCAI 2006,
Int. Journal of HPC Apps, 2006 ]
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Liver puncture guidance using augmented reality3D (CT) / 2D (Video) registration
2D-3D EM-ICP on fiducial markersCertified accuracy in real time
ValidationBronze standard (no gold-standard)Phantom in the operating room (2 mm)10 Patient (passive mode): < 5mm (apnea)
[ S. Nicolau, PhD’04 MICCAI05, ECCV04, IS4TM03, Comp. Anim. & Virtual World 2005 ]
S. Nicolau, IRCAD / INRIAS. Nicolau, IRCAD / INRIA
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Statistical Analysis of the Scoliotic Spine
Database307 Scoliotic patients from the Montreal’s Sainte-Justine Hospital.3D Geometry from multi-planar X-rays
MeanMain translation variability is axial (growth?)Main rot. var. around anterior-posterior axis
[ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ]
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• Mode 1: King’s class I or III• Mode 2: King’s class I, II, III
• Mode 3: King’s class IV + V• Mode 4: King’s class V (+II)
PCA of the Covariance: 4 first variation modes have clinical meaning
Statistical Analysis of the Scoliotic Spine[ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ]
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Outline
Goals and methods of Computational anatomy
Statistical computing on manifoldsThe geometrical and statistical framework
Examples with rigid body transformationsExtending the framework to manifold-valued images
Building a statistical atlas of the heart fibers
Computational neuro-anatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
Conclusion and challenges
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Diffusion Tensor ImagingCovariance of the Brownian motion of water
-> Architecture of axonal fibers
Very noisy data
Tensor image processingRobust estimationFiltering, regularization Interpolation / extrapolation
Information extraction (fibers)
Symmetric positive definite matricesConvex operations are stable
mean, interpolationMore complex operations are not
PDEs, gradient descent… Diffusion Tensor Filed(slice of a 3D volume)
Intrinsic computing on Manifold-valued images?
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A Riemannian Framework on tensors
Affine-invariant Metric (Curved space – Hadamard)
Action of the linear group GLn
Dot product
Geodesics:
Exponential map:
Log map:
Distance
[ Pennec, Fillard, Ayache, IJCV 66(1), 2006, Lenglet JMIV’06, etc]
2/12/12/12/1 )..exp()(exp ΣΣΣΨΣΣ=ΣΨ −−Σ
22/12/12
2)..log(|),(dist
L
−−
ΣΣΨΣ=ΣΨΣΨ=ΨΣ
IdAA
TT WVAWAAVAWV T
2/12/12/12/1 ||| −−−−
ΣΣΣΣΣΣ==
M
ΣΨ
Σ
Ψ
MTΣ
ΣexpΣlog
)exp()(, tWtWId =Γ
2/12/12/12/1 )..log()(log ΣΣΨΣΣ=Ψ=ΣΨ −−Σ
TAAA ..Σ=Σ∗
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Metrics for Tensor computingAffine-invariant Metric (Curved space – Hadamard)
Dot product
Geodesics
Distance
Log-Euclidean similarity invariant metric (vector space)
Transport Euclidean structure through matrix exponential
Dot product
Geodesics
Distance
2/12/12/12/1 )..exp()(exp ΣΣΣΨΣΣ=ΣΨ −−Σ
22/12/12
2)..log(|),(dist
L
−−
ΣΣΨΣ=ΣΨΣΨ=ΨΣ
( ) ( ) ( ) 221
221 loglog,dist Σ−Σ≡ΣΣ
))log()exp(log()(exp Σ∂+Σ=ΣΨΣΨΣ
IdAA
TT WVAWAAVAWV T
2/12/12/12/1 ||| −−−−
ΣΣΣΣΣΣ==
IdWVWV )log(|)log(| Σ∂Σ∂=Σ
[ Pennec, Fillard, Ayache: A Riemannian Framework for Tensor Computing, IJCV 66(1), 2006. ][Fletcher 2004, Moakher SIAM-X 2005, Lenglet, JMIV 2006]
[ Arsigny, Pennec, Fillard, Ayache: Fast and Simple Calculus on Tensors in the Log-Euclidean Framework, MICCAI’05, SIMAX 29(1) 2007, MRM 52(6) 2006 ]
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Euclidean metric
Choleski metric
Affine-invariant / Log-Euclidean metrics
Undefined (non positive)
Geodesic shooting in tensors spacesGeodesics starting from the same tensor with the same tangent vector
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Tensor interpolationGeodesic walking in 1D
Coefficients Riemannian metric
∑ ΣΣ=ΣΣ
2),( )(min)( ii distxwxWeighted mean in general
)(exp)( 211ΣΣ=Σ Σ tt
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Gaussian filtering: Gaussian weighted mean∑=
ΣΣ−=Σn
iii distxxGx
1
2),( )(min)( σ
Raw Coefficients σ=2 Riemann σ=2
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Intrinsic Riemannian ComputingIntegral or sum in M minimize an intrinsic functional
InterpolationLinear between 2 elements: interpolation geodesicBi- or tri-linear in images: weighted means
Gaussian filteringGaussian weighted Mean weighted
Regularization: the exponential map (partially) accounts for curvatureHarmonic
Gradient: Laplace-Beltrami operator
Anisotropic
Perona-Malik 90 / Gerig 92
Robust functions
Trivial intrinsic numerical schemes thanks the the exponential maps!
)xx (exp)(x 21x1tt =
∑ ΣΣ−=Σi iixxGx ),(dist )(min)( 2
σ
∫ ΣΣ∇=Σ dxx
x
2
)()()(Reg
( )∑ ΣΔΣ∂=ΣΔΣu uxuw xxwx )( )()(
)(
( )21 )()()( εεε OuxxxSu
++ΣΣ=ΔΣ ∑ ∈
( )∫ ΣΣ∇Φ=Σ dxx
x
2
)()()(Reg
[ Pennec, Fillard, Arsigny, IJCV 66(1), 2005, ISBI 2006]
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Filtering and anisotropic regularization of DTIRaw Coefficients σ=2
Riemann Gaussian σ=2 Riemann anisotropic
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DTI-based Anatomical modelsDiffusion tensor IRMCovariance of the water Brownian motion
Estimation, filtering, interpolation
[J.M. Peyrat et al, MICCAI’06 and IEEE Trans. Med. Imaging 26(11), 2007 ]
Atlas of the heart fibers7 DTI of dogs heartsFibers and sheets structure
Fiber extraction: architecture of axons tracts
[ Pennec et al, IJCV 66(1) 2006, Fillard et al, ISBI’06 and IEEE TMI, 26(11), 2007 ]
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Cardiac Fiber ArchitectureMyocardial fibers
Vf
VsVn
Laminar sheets
This geometric structure of the myocardiumControls propagation of electrical potentialControls Mechanical contraction Is needed for patient-specific applications
Electromechanical simulations (for therapy planning)Image analysis (motion tracking, strain analysis,…)
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Cardiac Fiber ArchitectureMyocardial fibers
Laminar sheets
Vf
VsVn secondary as orthogonal to
fibers in the sheet plane
primary as fiber orientation
tertiary as normal to sheet plane
Correlation with DTI eigenvectors [Scollan,1998] [Helm,2005]
DTI = Direct 3D description:Build a statistical atlas of fibers from DTI?
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A Computational Framework for the
Statistical Analysis of Cardiac Diffusion tensors
Jean Marc Peyrat (Asclepios, INRIA and Siemens SCR, Princeton)
Use of the whole diffusion tensor?Use a population for atlas building?
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Groupewise Registration
Register the DTI images to an average geometryUse Tensor images => bias in the statistical analysisUse anatomical MRIs = baseline (B0) image of the DT-MRI acquisition (coacquired in the same reference frame)
Shape and intensity average model [Guimond,1999]
Iterative process
Mean deformation (atlas geometry)
Mean intensity (atlas image)
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Tensor reorientation (transforming DT-MRIs)
Eigenvalues are preservedIntrinsic properties of the tissues
Eigenvectors are modifiedLinked to the reference frame that is locally transformed
Transformation of diffusion tensors = rotation of the original diffusion tensor
Two common reorientation strategies [Alexander,2001]Finite Strain (FS) => preferred herePreservation of Principal Direction (PPD)
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Statistics on Diffusion TensorsDiffusion tensor space is not a vector space
Use the Log-Euclidean metric [Arsigny,2005]
Mean and covariance at each voxel [Pennec, IJCV 2006]
Mean tensor gives mean fiber and laminar sheet orientation21x21 covariance matrix measures variability of the mean tensor components: Interpretation in terms of cardiac fiber architecture?
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Mean Geometry and Fiber Orientation
9 normal ex vivocanine hearts [Helm,2005]
High resolution DT-MRI ~256 × 256 × 128 with ~0.3 × 0.3 × 0.9 mm3 per voxel
available at http://www-sop.inria.fr/asclepios/data/heart[J.M. Peyrat et al, MICCAI’06 and IEEE Trans. Med. Imaging 26(11), 2007 ]
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Variability of Diffusion Tensors
Norm of the Covariance Matrix~10%
[J.M. Peyrat et al, MICCAI’06 and IEEE Trans. Med. Imaging 26(11), 2007 ]
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Covariance Matrix Analysis
Projection of the Covariance Matrix
Eigenvalues Variability
Eigenvectors Variability
[J.M. Peyrat et al, MICCAI’06 and IEEE Trans. Med. Imaging 26(11), 2007 ]
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Statistics on Eigenvalues
1st Eigenvalue
2nd Eigenvalue
3rd Eigenvalue
[J.M. Peyrat et al, MICCAI’06 and IEEE Trans. Med. Imaging 26(11), 2007 ]
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Statistics on Eigenvectors
1st & 2nd Eigenvectors (around 3rd)(mode of std dev: 7.9 degrees)
1st & 3rd Eigenvectors (around 2nd)(mode of std dev: 7.7 degrees)
2nd & 3rd Eigenvectors (around 3rd)(mode of std dev: 22.7 degrees)
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Inter-species comparison with a human heart
1st eigenvector
2nd eigenvector
3rd eigenvector
Angular difference Mahalanobis distance
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Outline
Goals and methods of Computational anatomy
Statistical computing on manifoldsThe geometrical and statistical frameworkExamples with rigid body transformations and tensors
Computational neuro-anatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
Conclusion and challenges
3 juin 2008 X. Pennec - ANGD traiment d'images médicales 45
Hierarchy of anatomical manifoldsLandmarks [0D]: AC, PC (Talairach)Curves [1D]: crest lines, sulcal linesSurfaces [2D]: cortex, sulcal ribbonsImages [3D functions]: VBMTransformations: rigid, multi-affine, diffeomorphisms [TBM]
Structural variability of the Cortex
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Morphometry of the Cortex from Sucal Lines
Associated team Brain-Atlas (2001-2006)LONI (UCLA) : P. Thompson et al. ASCLEPIOS (INRIA): V. Arsigny, N. Ayache, P. Fillard, X. Pennec
Neuroanatomical reference:
72 sulcal lines manually extracted and labeled
700 subjects
AlternativeAutomatic extraction
JF. Mangin, D. Rivière, 2003, SHFJ-CEA
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Morphometry of the Cortex from Sucal Lines
Covariance Tensors along Sylvius Fissure
Currently:
80 instances of 72 sulci
About 1250 tensors
Computation of the mean sulci: Alternate minimization of global varianceDynamic programming to match the mean to instancesGradient descent to compute the mean curve position
Extraction of the covariance tensors
Collaborative work between Asclepios (INRIA) Vand LONI (UCLA) P. Thompson [Fillard et al IPMI05, LNCS 3565:27-38, NeuroImage 34(2):639-650, January 2007]
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Compressed Tensor Representation
Representative Tensors (250) Original Tensors (~ 1250)Reconstructed Tensors (1250) (Riemannian Interpolation)
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Extrapolation by Diffusion
Diffusion λ=0.01 Diffusion λ=∞Original tensors
∫ ∫∑Ω Ω
Σ=
Σ∇+ΣΣ−=Σ 2
)(1
2 )(2
)),(()(21)(
x
n
iii xdxxdistxxGC λ
σ
))(()()())((1
xxxxGxCn
iii ΔΣ−ΣΣ−−=Σ∇ ∑
=
λσ
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Full Brain extrapolation of the
variability
∫ ∫∑Ω Ω
Σ=
Σ∇+ΣΣ−=Σ 2
)(1
2
2)),(()(
21)(
x
n
iii dxxdistxxGC λ
σ
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Comparison with cortical surface variability
Consistent low variability in phylogenetical older areas (a) superior frontal gyrus
Consistent high variability in highly specialized and lateralized areas(b) temporo-parietal cortex
AsymmetryMaximale : aire de Broca (langage), cortex pariétal; minimale : aires somatomotrices primaires
P. Thompson at al, HMIP, 2000Average of 15 normal controls by non-linear
registration of surfaces
P. Fillard et al, IPMI 05Extrapolation of our model
(98 subjetcs with 72 sulci)
[ Fillard, Arsigny, Pennec,Thompson, Ayache, IPMI 2005, NeuroImage 34(2), 2007]
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Local and distant structural correlation
[ Fillard, Pennec,Thompson, Thompson, Evaluating Brain Anatomical Correlations via Canonical Correlation Analysis of Sulcal Lines, MICCAI Workshop on stat. Atlases, 2007]
Left inferior temporal sulcus
Localcorrelation
Correlation to the symmetric point
Left central sulcus
Localcorrelation
Unexpected distant correlation
Correlation to the symmetric point
Unexpected distant correlation
Enumeration: Modeling the Green’s function
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Asymmetry Measures
w.r.t the mid-sagittal plane. w.r.t opposite (left-right) sulci
Greatest asymmetry Lowest asymmetryBroca’s speech area and
Wernicke’s language comprehension area
Primary sensorimotor areas
22/1'2/1''2'
2)..log(|),(
Ldist −−
ΣΣΣΣ=ΣΣΣΣ=ΣΣ
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Symmetry of the variability between groups
Men WomenP value corrigée Bonferroni a 5% = 0.0001
[ Fillard, Pennec,Thompson, Thompson, Evaluating Brain Anatomical Correlations via Canonical Correlation Analysis of Sulcal Lines, MICCAI Workshop on stat. Atlases, 2007]
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An alternative approach with diffeomorphisms
Does the method influence results? Matching, then extrapolation[Fillard, NeuroImage 34(2) 2007]Extrapolation of speed vectors and trajectory integration
MethodGlobal space diffeomorphism (integration of time varying vector fields)[Trouve, Younes, Miller, etc]Distance between lines using currents[J. Glaunès, M. Vaillant: IPMI 2005]
AdvantagesGenerative model of deformationsRetrieve some tangential deformation component.
[ S, Durrleman, X. Pennec, A. Trouvé, N. Ayache, MICCAI 2007 ]
Variability at each point
Global variability (2 PGA modes)
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Comparison with a diffeomorphic approach
Courants Points
[ S, Durrleman et al. MICCAI 2007 ] [ Fillard et al., Neuroimage 34(2), 2007]
The aperture problemTangential variability is minimized on purpose with Fillard’s method The global diffeomorphism performs a spatially consistent integration
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Outline
Goals and methods of Computational anatomy
Statistical computing on manifoldsThe geometrical and statistical frameworkExamples with rigid body transformations and tensors
Computational neuro-anatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
Conclusion and challenges
3 juin 2008 X. Pennec - ANGD traiment d'images médicales 58
Use of the variability information?
Learning / modeling phase (anatomy / neurosciences)Goal: analyze and understand the population variabilityMethods can be computationally intensive, experiments relies on good quality observationsFact: Methods have different assumptions
Similar results at some locations, different results at other placesEach method is based on partial observationsEach method is biased by its assumptions
Vary assumptions / data, and discover “truth” by consensus
Personalization of atlases (use in a clinical / medical workflow)Anatomical prior to compensate for incomplete / noisy / abnormal (pathological) observations.Need robust and efficient methodsUse variability statistics as a regularizer to robustify registration?
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Atlas-based segmentation for Radiotherapy planning
Structures to segmentTumor (to maximize irradiation)Organs at risk ( to minimize irradiation)
Registering an atlassegments all structures at once
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One example use of variability information: better constrain the atlas to subject registration
Atlas = artificial MRI (MNI simulator) + segmented structures of interest (P-Y Bondiau, MD, CAL, Nice)Deform the atlas anatomy (without tumor) towards the patient oneUse result as a prior to segment the structures in the patient image
[Pierre-Yves Bondiau. PhD, November 2004. Commowick, et al, MICCAI 2005]
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Introducing local variability and pathologies in non-linear registration
Non stationary regularization: anatomical prior on the deformabilityNon stationary image similarity / regularization tradeoff:takes pathologies into account
[ Runa. Stefanescu et al, PPL 14(2), 2004 & Med. Image Analysis 8(3), 2003]
Patient Stiffness field Atlas to patient With pathology
)(.))(,()( TETJIETE regsim λ+=
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Regularization in dense non-linear registration
Physically based regularizationsElastic [Bajcsy 89] Fluid [Christensen TMI 97]Right-invariant distance [LDDMM, Beg IJCV 05]
Efficient regularization methodsGaussian filtering [Thirion Media 98, Modersitzki 2004]Isotropic but non stationary [Lester IPMI’99]Towards anisotropic non stationary regularization [Stefanescu MedIA 2004]
Observation: Inter-subject: no regularization model is more justified than others Idea: learning statistically the variability from a population[Thompson 2000, Rueckert TMI 2003, Fillard IPMI 2005]
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Statistics on the deformation field• Objective: planning of conformal brain radiotherapy (O. Commowick, Dosisoft)• 30 patients, 2 to 5 time points (P-Y Bondiau, MD, CAL, Nice)
[ Commowick, et al, MICCAI 2005, T2, p. 927-931]
Robust
∑ Φ∇=i iN xxDef )))((log(abs)( 1
∑ Σ=∑i iN xx )))((log(abs)( 1
1))(()( −∑+= xIdxD λ
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Introducing deformation statistics into RUNA
Scalar statistical stiffness Tensor stat. stiffness (FA)Heuristic RUNA stiffness
[ Commowick, et al, MICCAI 2005, T2, p. 927-931]
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Riemannian elasticity : a well posed framework to introduce statistics in non-linear elastic regularization
Gradient descent
Including statistics in RegularizationSt Venant Kirchoff elastic energy
Elasticity is not symmetricStatistics are not easy to include
Idea: Replace the Euclidean by the Log-Euclidean metric
Statistics on strain tensors: Mean, covariance, Mahalanobis computed in Log-space
Using Riemannian Elasticity as a metric (TBM)The mean provides an unbiased atlasBetter constraining the deformation should give better results in TBM
)(Reg),Images(Sim)( Φ+Φ=ΦC
Φ∇Φ∇=Σ .t
( ) ( ) ( )22 Tr2
)(Tr Reg II −Σ+−Σ=Φ ∫λμ
[ Pennec, et al, MICCAI 2005, LNCS 3750:943-950, MFCA’06]
( ) 222 )log(),(dist )(Tr Σ=Σ→−Σ II LE
( ) ( ) ( )∫ −Σ−Σ=Φ − WWg T )log(Vect.Cov.)log(VectRe 1
[ N. Lepore et al, MICCAI’07, C. Brun et al, MICCAI’07 Atlas Workshop, UCLA associated team ]
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Isotropic Riemannian Elasticity Results
Roi 186x124x216 voxels, λ=μ=0.2, 12 PC 2Gh.
Larger computation times 3h vs 1hSlightly larger and better deformation of the right ventriclewithout any statistical information yet…
[ Pennec, et al, MICCAI 2005, T2, p.943-950 ]
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Using Riemannian Elasticity as a metric (TBM)
Register all subjects and controls to template Statistical analysis to find significantly different deformations(at the population level)
Source
Target
22 )log(),(dist )(det Σ=Σ⎯→⎯Σ ILE
[ N. Lepore et al, MICCAI’06]
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Multivariate statistics give additional (more sensitive measures) for genetic studies.
Standard TBM:
Determinant of the Jacobian
Tangent of the Geodesic Anisotropy
Mean absolute difference in regional volume
MZ DZ
)(det Σ
[ N. Lepore et al, MICCAI’07, C. Brun et al, MICCAI’07 Atlas Workshop, UCLA associated team ]
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A valid measure of anatomical resemblance
Minimize this distance to find the unbiased atlas Couple statistics in registration to improve statistical power?
Distance from each of the twins to all others[ N. Lepore et al, ISBI08 ]
[ N. Lepore et al, MICCAI’07 ]
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Statistics on which deformations feature?Local statistics on local deformation (mechanical properties)
Gradient of transformation, strain tensor[Riemannian elasticity, TBM, N. Lepore + C. Brun]
Global statistics on displacement field or B-spline parameters[Rueckert et al., TMI, 03], [Charpiat et al., ICCV’05],[P. Fillard, stats on sulcal lines] Simple vector statistics, but inconsistency with group properties
Space of “initial momentum” [Quantity of motion instead of speed][Vaillant et al., NeuroImage, 04, Durrleman et al, MICCAI’07]Based on left-invariant metrics on diffeos [Trouvé, Younes et al.]Needs theoretically a finite number of point measuresComputationally intensive
An alternative: log-Euclidean statistics on diffeomorphisms?[Arsigny, MICCAI’07][Bossa, MICCAI’07, Vercauteren MICCAI’07, Ashburner NeuroImage 2007]Mathematical problems but efficient numerical methods!
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Outline
Goals and methods of Computational anatomy
Statistical computing on manifoldsThe geometrical and statistical frameworkExamples with rigid body transformations and tensors
Computational neuro-anatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
Conclusion and challenges
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Computing on manifolds: a summaryThe Riemannian metric easily gives
Intrinsic measure and probability density functionsExpectation of a function from M into R (variance, entropy)
Integral or sum in M: minimize an intrinsic functionalFréchet / Karcher mean: minimize the varianceFiltering, convolution: weighted meansGaussian distribution: maximize the conditional entropy
The exponential chart corrects for the curvature at the reference point
Gradient descent: geodesic walkingCovariance and higher order momentsLaplace Beltrami for free
[ Pennec, NSIP’99, JMIV 2006, Pennec et al, IJCV 66(1) 2006, Arsigny, PhD 2006]
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Statistics on geometrical objects
A consistent framework with important applications inMedical Image Analysis (registration evaluation, DTI)Building models of living systems (spine, brain, heart…)
Is the Riemannian metric the minimal structure?No bi-invariant metric but bi-invariant means on Lie groups [V. Arsigny] Change the Riemannian metric for the symmetric Cartan connection?
Computational framework for infinite dimensional manifoldsCurves and surfaces: statistics on currents?Efficient framework for some spaces of diffeomorphisms?
How to chose or estimate the metric? Invariance, reacheability of boundaries, learning the metricFamilies of anatomical deformation metrics (models of the Green’s function)
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Challenges of Computational AnatomyBuild models from multiple sources
Curves, surfaces [cortex, sulcal ribbons]Volume variability [Voxel/Tensor Based Morphometry, Riemannian elasticity]Diffusion tensor imaging [fibers, tracts, atlas]
Compare and combine statistics on anatomical manifoldsEach method is biased by its assumptions (fewer data than unknowns) Validate methods and models by consensus
From modeling to personalized medicineTopological changesEvolution: growth, pathologiesCouple statistical learning / modeling and use of models as prior for inter-subject registration / segmentation
Asclepios (INRIA), Visages (IRISA), Neurospin (CEA + INRIA), LENA (CNRS-CHUPS), CMLA (ENS), LSIS (CNRS).
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Acknowledgements
Image guided therapyBrain surgery (Roboscope): A. Roche and P. CathierDental implantology: S. Granger, AREALLLiver puncture guidance: S. Nicolau and L. Soler, IRCAD Mosaicing confocal microscopic images: T. Vercauteren, MKT
Brain imagingGeometry and statistics for fMRI analysis: G. Flandin, J.-B. Poline, CEAInter-subject non-linear registration: P. Cathier, R. Stefanescu, O. Commowick, T. Vercauteren.
Computational anatomyAssociated team Brain Atlas with LONI: P. Thompson, C. Brun, N. Lepore.Tensor computing: P. Fillard, V. Arsigny.Growth and variability: S. Durrleman.Spine shape: J. Boisvert, F. Cheriet, Ste Justine Hospital, Montreal.Heart fibers: J.-M. Peyrat.ACI Agir / Grid computing: T. Glatard and J. Montagnat, I3S.
Asclepios [Epidaure] teamN. Ayache, G. Malandain, H. Delingette… and all the current and former team members.
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Statistics on ManifoldsX. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. J. of Math. Imaging and Vision, 25(1):127-154, July 2006. Preprint as INRIA RR 5093, Jan. 2004 (and NSIP’99).X. Pennec and N. Ayache. Uniform distribution, distance and expectation problems for geometric features processing. J. of Mathematical Imaging and Vision, 9(1):49-67, July 1998 (and CVPR’96).
Tensor ComputingX. Pennec, P. Fillard, and Nicholas Ayache. A Riemannian Framework for Tensor Computing. Int. Journal of Computer Vision 66(1), January 2006. Preprint as INRIA RR- 5255, July 2004V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Fast and Simple Calculus on Tensors in the Log-Euclidean Framework. Proc. of MICCAI'05, LNCS 3749, p.115-122. MRM 56(2):411-421, August 2006. Preprint as INRIA RR-5584, Mai 2005. P. Fillard, V. Arsigny, X. Pennec, and N. Ayache. Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean Metrics. IEEE. TMI, In press. Also as ISBI’2006 and INRIA RR-5607, June 2005.
Applications in Computational AnatomyP. Fillard, V. Arsigny, X. Pennec, K.M. Hayashi, P. Thompson, and N. Ayache. Measuring brain variability by extrapolating sparse tensor fields measures on sulcal lines. Neuroimage 34(2):639-650, January 2007.X. Pennec, R. Stefanescu, V. Arsigny, P. Fillard, and N. Ayache. Riemannian Elasticity: A statistical regularization framework for non-linear registration. Proc. of MICCAI'05, LNCS 3750, p.943-950, 2005.J. Boisvert, X. Pennec, N. Ayache, H. Labelle and F. Cheriet. 3D Anatomical Assessment of the Scoliotic Spine using Statistics on Lie Groups. ISBI’2006. To appear in IEEE TMIJ.M. Peyrat, M. Sermesant , H. Delingette, X. Pennec, C. Xu, E. McVeigh, N. Ayache, Towards a Statistical Atlas of Cardiac Fibre Structure, MICCAI’06, IEEE TMI dec 2007.
References[ Papers available at http://www-sop.inria.fr/asclepios/Biblio ]