Estimation of the registration accuracy in the absence of ...lsiit-miv.u-strasbg.fr/ecoleTIM/download/Pennec... · • Animal models • Philosophical physiology Renaissance: •

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


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


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


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


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


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


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


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


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.


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


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


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=


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 ]


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


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 ]


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


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 ]


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


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 ]


• 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 ]


Outline


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




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?


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∗


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 ]


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


Tensor interpolationGeodesic walking in 1D

Coefficients Riemannian metric

∑ ΣΣ=ΣΣ

2),( )(min)( ii distxwxWeighted mean in general

)(exp)( 211ΣΣ=Σ Σ tt


Gaussian filtering: Gaussian weighted mean∑=

ΣΣ−=Σn

iii distxxGx

1

2),( )(min)( σ

Raw Coefficients σ=2 Riemann σ=2


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]


Filtering and anisotropic regularization of DTIRaw Coefficients σ=2

Riemann Gaussian σ=2 Riemann anisotropic


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 ]


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,…)


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?


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?


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)


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)


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?


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 ]


Variability of Diffusion Tensors

Norm of the Covariance Matrix~10%



Covariance Matrix Analysis

Projection of the Covariance Matrix

Eigenvalues Variability

Eigenvectors Variability



Statistics on Eigenvalues

1st Eigenvalue

2nd Eigenvalue

3rd Eigenvalue



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)


Inter-species comparison with a human heart

1st eigenvector

2nd eigenvector

3rd eigenvector

Angular difference Mahalanobis distance


Outline


Statistical computing on manifoldsThe geometrical and statistical frameworkExamples with rigid body transformations and tensors




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


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


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]


Compressed Tensor Representation

Representative Tensors (250) Original Tensors (~ 1250)Reconstructed Tensors (1250) (Riemannian Interpolation)


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


Full Brain extrapolation of the

variability

∫ ∫∑Ω Ω

Σ=

Σ∇+ΣΣ−=Σ 2

)(1

2

2)),(()(

21)(

x

n

iii dxxdistxxGC λ

σ


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]


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


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


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]


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)


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


Outline






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?


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


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]


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+=


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]


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


Introducing deformation statistics into RUNA

Scalar statistical stiffness Tensor stat. stiffness (FA)Heuristic RUNA stiffness

[ Commowick, et al, MICCAI 2005, T2, p. 927-931]


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 ]


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 ]


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]


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 ]


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 ]


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!


Outline






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]


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)


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).


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.


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 ]

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