with contributions from v. arsigny, n. ayache, j. boisvert ... · transformations • rigid,...
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September 18, 2006 Mathematics and Image Analysis 2006 1
Statistical Computing on Riemannian manifoldsFrom Riemannian Geometry to Computational Anatomy
With contributions from V. Arsigny, N. Ayache, J. Boisvert,
P. Fillard, et al.
X. Pennec
September 18, 2006 Mathematics and Image Analysis 2006 2
Standard Medical Image Analysis
Methodological / algorithmically axesRegistration SegmentationImage Analysis/Quantification
Measures are geometric and noisyFeature extracted from imagesRegistration = determine a transformations Diffusion tensor imaging
We need:StatistiquesA stable computing framework
September 18, 2006 Mathematics and Image Analysis 2006 3
Historical examples of geometrical features
Transformations
• Rigid, Affine, locally affine, families of deformations
Geometric features
• Lines, oriented points…• Extremal points: semi-oriented frames
How to deal with noise consistently on these features?
September 18, 2006 Mathematics and Image Analysis 2006 4
MR Image Initial USRegistered US
Per-operative registration of MR/US images
Performance Evaluation: statistics on transformations
September 18, 2006 Mathematics and Image Analysis 2006 5
Interpolation, filtering of tensor images
Raw Anisotropic smoothing
Computing on Manifold-valued images
September 18, 2006 Mathematics and Image Analysis 2006 6
Modeling and Analysis of the Human AnatomyEstimate representative / average organ anatomiesModel organ development across timeEstablish normal variabilityTo detect and classify of pathologies from structural deviationsTo adapt generic (atlas-based) to patients-specific models
Statistical analysis on (and of) manifolds
Computational Anatomy
Computational Anatomy, an emerging discipline, P. Thompson, M. Miller, NeuroImage special issue 2004Mathematical Foundations of Computational Anatomy, X. Pennec and S. Joshi, MICCAI workshop, 2006
September 18, 2006 Mathematics and Image Analysis 2006 7
Overview
The geometric computational framework(Geodesically complete) Riemannian manifolds
Statistical tools on pointwise featuresMean, Covariance, Parametric distributions / tests Application examples on rigid body transformations
Manifold-valued images: Tensor ComputingInterpolation, filtering, diffusion PDEsDiffusion tensor imaging
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 8
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
September 18, 2006 Mathematics and Image Analysis 2006 9
Computing on Riemannian manifolds
Riemannian manifoldEuclidean spaceOperation
)( 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
September 18, 2006 Mathematics and Image Analysis 2006 11
Overview
The geometric computational framework(Geodesically complete) Riemannian manifolds
Statistical tools on pointwise featuresMean, Covariance, Parametric distributions / tests Application examples on rigid body transformations
Manifold-valued images: Tensor ComputingInterpolation, filtering, diffusion PDEsDiffusion tensor imaging
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 12
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=
September 18, 2006 Mathematics and Image Analysis 2006 13
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 ]
September 18, 2006 Mathematics and Image Analysis 2006 15
Distributions for parametric testsUniform density:
maximal entropy knowing X
Generalization of the Gaussian density:Stochastic heat kernel p(x,y,t) [complex time dependency] Wrapped Gaussian [Infinite series difficult to compute]Maximal entropy knowing the mean and the covariance
Mahalanobis D2 distance / test:
Any distribution:
Gaussian:
( ) ( ) ⎟⎠⎞⎜
⎝⎛= 2/x..xexp.)(
TxΓxkyN
)Vol(/)(Ind)( Xzzp X=x
( ) ( ) ( )( )rOk n /1.)det(.2 32/12/ σεσπ ++= −− Σ
( ) ( )rO / Ric31)1( σεσ ++−= −ΣΓ
yx..yx)y( )1(2 −Σ= xxx
tμ
[ ] n=)(E 2 xxμ
( )rOn /)()( 322 σεσχμ ++∝xx
[ Pennec, JMIV06, NSIP’99 ]
September 18, 2006 Mathematics and Image Analysis 2006 16
Gaussian on the circle
Exponential chart:
Gaussian: truncated standard Gaussian
[. ; .] rrrx ππθ −∈=
standard Gaussian(Ricci curvature → 0)
uniform pdf with
(compact manifolds)
Dirac
:∞→r
:∞→γ
:0→γ3/).( 22 rπσ =
September 18, 2006 Mathematics and Image Analysis 2006 17
Overview
The geometric computational framework
Statistical tools on pointwise featuresMean, Covariance, Parametric distributions / tests Application examples on rigid body transformations
Manifold-valued images: Tensor ComputingInterpolation, filtering, diffusion PDEsDiffusion tensor imaging
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 20
Validation of the error prediction
[ 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
September 18, 2006 Mathematics and Image Analysis 2006 23
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)
PhD S. Nicolau, MICCAI05, ECCV04, ISMAR04, IS4TM03, Comp. Anim. & Virtual World 2005, IEEE TMI (soumis)
S. Nicolau, IRCAD / INRIAS. Nicolau, IRCAD / INRIA
September 18, 2006 Mathematics and Image Analysis 2006 24
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 rotation var. around anterior-posterior axis
PCA of the Covariance4 first variation modes have clinical meaning
[ J. Boisvert, X. Pennec, N. Ayache, H. Labelle, F. Cheriet,, ISBI’06 ]
September 18, 2006 Mathematics and Image Analysis 2006 25
Statistical Analysis of the Scoliotic Spine
• 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)
September 18, 2006 Mathematics and Image Analysis 2006 26
Overview
The geometric computational framework
Statistical tools on pointwise features
Manifold-valued images: Tensor ComputingInterpolation, filtering, diffusion PDEsDiffusion tensor imaging
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 27
Diffusion tensor imaging
Very noisy data
Preprocessing stepsFilteringRegularizationRobust estimation
Processing stepsInterpolation / extrapolationStatistical comparisons
Can we generalize scalar methods?DTI Tensor field (slice of a 3D volume)
September 18, 2006 Mathematics and Image Analysis 2006 28
Tensor computing
Tensors = space of positive definite matricesLinear convex combinations are stable (mean, interpolation)More complex methods are not (null or negative eigenvalues)(gradient descent, anisotropic filtering and diffusion)
Current methods for DTI regularizationPrinciple direction + eigenvalues [Poupon MICCAI 98, Coulon Media 04]Iso-spectral + eigenvalues [Tschumperlé PhD 02, Chef d’Hotel JMIV04]Choleski decomposition [Wang&Vemuri IPMI03, TMI04]Still an active field…
Riemannian geometric approachesStatistics [Pennec PhD96, JMIV98, NSIP99, IJCV04, Fletcher CVMIA04]Space of Gaussian laws [Skovgaard84, Forstner99,Lenglet04]Geometric means [Moakher SIAM JMAP04, Batchelor MRM05]Several papers at ISBI’06
September 18, 2006 Mathematics and Image Analysis 2006 29
Affine Invariant Metric on TensorsAction of the Linear Group GLn
Invariant distance
Invariant metric
Usual scalar product at identity
Geodesics
Distance
( )2121 | WWTrWW Tdef
Id=
Id
def
WWWW 22/1
12/1
21 ,| ∗Σ∗Σ= −−Σ
),(),( 2121 ΣΣ=Σ∗Σ∗ distAAdist
TAAA ..Σ=Σ∗
[ X Pennec, P.Fillard, N.Ayache, IJCV 66(1), Jan. 2006 / RR-5255, INRIA, 2004 ]
2/12/12/12/1 )..exp()(exp ΣΣΣΨΣΣ=ΣΨ −−Σ
22/12/12
2)..log(|),(
Ldist −−
ΣΣΨΣ=ΣΨΣΨ=ΨΣ
September 18, 2006 Mathematics and Image Analysis 2006 30
Exponential and Logarithmic Maps
Geodesics )exp()(, tWtWId =Γ
M
ΣΨ
Σ
Ψ
MTΣ
ΣexpΣlogLogarithmic Map :
Exponential Map :
2/12/12/12/1 )..exp()(exp ΣΣΣΨΣΣ=ΣΨ −−Σ
2/12/12/12/1 )..log()(log ΣΣΨΣΣ=Ψ=ΣΨ −−Σ
22/12/12
2)..log(|),(
Ldist −−
ΣΣΨΣ=ΣΨΣΨ=ΨΣ
September 18, 2006 Mathematics and Image Analysis 2006 31
Tensor interpolation
Coefficients Riemannian metric
Geodesic walking in 1D
∑ ΣΣ=ΣΣ
2),( )(min)( ii distxwxWeighted mean in general
)(exp)( 211ΣΣ=Σ Σ tt
September 18, 2006 Mathematics and Image Analysis 2006 34
Gaussian filtering: Gaussian weighted mean∑=
ΣΣ−=Σn
iii distxxGx
1
2),( )(min)( σ
Raw Coefficients σ=2 Riemann σ=2
September 18, 2006 Mathematics and Image Analysis 2006 35
PDE for filtering and diffusion
Harmonic regularization
Gradient = manifold Laplacian
Integration scheme = geodesic marching
Anisotropic regularizationPerona-Malik 90 / Gerig 92 Phi functions formalism
( ) ( ) ( )22
)1(2 )()( )( uOu
uxxxu
ii i
ii ++ΣΣ
=Σ∂ΣΣ∂−Σ∂=ΔΣ ∑∑ ∑ −
∫Ω
ΣΣ∇=Σ dxxC
x
2
)()()(
)(2)( xxC ΔΣ−=∇
( )))((exp)( )(1 xCx xt tΣ∇−=Σ Σ+ ε
September 18, 2006 Mathematics and Image Analysis 2006 36
Anisotropic filtering
Initial
NoisyRecovered
September 18, 2006 Mathematics and Image Analysis 2006 37
Anisotropic filtering
Raw Riemann Gaussian Riemann anisotropic
( ) )/exp()( with )( )()( 22 κttwxxwxu
uuw −=ΣΔΣ∂=ΣΔ ∑
September 18, 2006 Mathematics and Image Analysis 2006 38
Log Euclidean Metric on Tensors
Exp/Log: global diffeomorphism Tensors/sym. matricesVector space structure carried from the tangent space to the manifold
Log. product
Log scalar product
Bi-invariant metric
Properties
Invariance by the action of similarity transformations only
Very simple algorithmic framework
( ) ( )( )2121 loglogexp Σ+Σ≡Σ⊗Σ
( )( ) ααα Σ=Σ≡Σ• logexp
( ) ( ) ( ) 221
221 loglog, Σ−Σ≡ΣΣdist
[ Arsigny, Fillard, Pennec, Ayache, MICCAI 2005, T1, p.115-122 ]
September 18, 2006 Mathematics and Image Analysis 2006 39
Log Euclidean vs Affine invariant
Means are geometric (vs arithmetic for Euclidean)Log Euclidean slightly more anisotropicSpeedup ratio: 7 (aniso. filtering) to >50 (interp.)
Euclidean Affine invariantLog-Euclidean
September 18, 2006 Mathematics and Image Analysis 2006 40
Log Euclidean vs Affine invariantReal DTI images: anisotropic filtering
Difference is not significantSpeedup of a factor 7 for Log-Euclidean
Original Euclidean Log-Euclidean Diff. LE/affine (x100)
September 18, 2006 Mathematics and Image Analysis 2006 41
Overview
The geometric computational framework
Statistical tools on pointwise features
Manifold-valued images: Tensor ComputingInterpolation, filtering, diffusion PDEsDiffusion tensor imaging
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 42
Joint Estimation and regularization from DWI
ML Rician MAP RicianStandard
Estimated
tensors
FA
Clinical DTI of the spinal cord
[ Fillard, Arsigny, Pennec, Ayache, RR-5607, June 2005 ]
( )( ) ( )2
)(
20 )( )( exp)(
xi iTii xxbSSC
ΣΣ∇Φ+Σ−−=Σ ∫∑ gg
September 18, 2006 Mathematics and Image Analysis 2006 43
Joint Estimation and regularization from DWI
Clinical DTI of the spinal cord: fiber tracking
MAP RicianStandard
September 18, 2006 Mathematics and Image Analysis 2006 44
Impact on fibers tracking
Euclidean interpolation Riemannian interpolation + anisotropic filtering
From images to anatomy Classify fibers into tracts (anatomo-functional architecture)?Compare fiber tracts between subjects?
September 18, 2006 Mathematics and Image Analysis 2006 45
Towards a Statistical Atlas of Cardiac Fiber Structure
Database7 canine hearts from JHUAnatomical MRI and DTI
MethodNormalization based on aMRIsLog-Euclidean statistics of Tensors
Norm covariance
Eigenvaluescovariance (1st, 2nd, 3rd)
Eigenvectors orientation covariance (around 1st, 2nd, 3rd)
[ J.M. Peyrat, M. Sermesant, H. Delingette, X. Pennec, C. Xu, E. McVeigh, N. Ayache, INRIA RR , 2006, submitted to MICCAI’06 ]
September 18, 2006 Mathematics and Image Analysis 2006 46
Computing on manifolds: a summary
The Riemannian metric easily givesIntrinsic 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 pointGradient descent: geodesic walkingCovariance and higher order momentsLaplace Beltrami for free
Which metric for which problem?
[ Pennec, NSIP’99, JMIV 2006, Pennec et al, IJCV 66(1) 2006]
September 18, 2006 Mathematics and Image Analysis 2006 47
Overview
The geometric computational framework
Statistical tools on pointwise features
Manifold-valued images: Tensor Computing
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 48
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
September 18, 2006 Mathematics and Image Analysis 2006 49
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) V. Arsigny, N. Ayache, P. Fillard, X. Pennec and LONI (UCLA) P. Thompson [Fillard et al IPMI05, LNCS 3565:27-38]
September 18, 2006 Mathematics and Image Analysis 2006 51
Compressed Tensor Representation
Representative Tensors (250) Original Tensors (~ 1250)Reconstructed Tensors (1250) (Riemannian Interpolation)
September 18, 2006 Mathematics and Image Analysis 2006 52
Extrapolation by Diffusion
Diffusion λ=0.01Original tensors Diffusion λ=∞
∫ ∫∑Ω Ω
Σ=
Σ∇+ΣΣ−=Σ 2
)(1
2 )(2
)),(()(21)(
x
n
iii xdxxdistxxGC λ
σ
( )))((exp)( )(1 xCx xt tΣ∇−=Σ Σ+ ε
))(()()())((1
xxxxGxCn
iii ΔΣ−ΣΣ−−=Σ∇ ∑
=
λσ
September 18, 2006 Mathematics and Image Analysis 2006 54
Full Brain extrapolation of the
variability
September 18, 2006 Mathematics and Image Analysis 2006 55
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
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 estimated
from 98 subjects with 72 sulci.
September 18, 2006 Mathematics and Image Analysis 2006 56
Quantitative Evaluation: Leave One Sulcus Out
Original tensors Leave one out reconstructions
Sylvian Fissure
Superior Temporal
Inferior Temporal
• Remove data from one sulcus
• Reconstruct from extrapolation of others
September 18, 2006 Mathematics and Image Analysis 2006 57
Asymmetry Measures
w.r.t the mid-sagittal plane. w.r.t opposite (left-right) sulci
Primary sensorimotor areasBroca’s speech area and Wernicke’s language comprehension area
Lowest asymmetryGreatest asymmetry
22/1'2/1''2'
2)..log(|),(
Ldist −−
ΣΣΣΣ=ΣΣΣΣ=ΣΣ
September 18, 2006 Mathematics and Image Analysis 2006 58
Overview
The geometric computational framework
Statistical tools on pointwise features
Manifold-valued images: Tensor Computing
Metric choices for Computational NeuroanatomyMorphometry of sulcal lines on the brainStatistics of deformations for non-linear registration
September 18, 2006 Mathematics and Image Analysis 2006 59
Statistics on the deformation field• Objective: planning of conformal brain radiotherapy• 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
September 18, 2006 Mathematics and Image Analysis 2006 60
Introducing deformation statistics into RUNA
1))(()( −∑+= xIdxD λ
Scalar statistical stiffness Tensor stat. stiffness (FA)Heuristic RUNA stiffness
RUNA [R. Stefanescu et al, Med. Image Analysis 8(3), 2004]non linear-registration with non-stationary regularizationScalar or tensor stiffness map
September 18, 2006 Mathematics and Image Analysis 2006 61
Riemannian elasticity for Non-linear elastic regularization
Gradient descent
RegularizationLocal deformation measure: Cauchy Green strain tensorId for local rotations, small for local contractions, Large for local expansions
St Venant Kirchoff elastic energy
)(Reg),Images(Sim)( Φ+Φ=ΦC )( 1 ttt C Φ∇−Φ=Φ + κ
Φ∇Φ∇=Σ .t
( ) ( ) ( )22 Tr2
)(Tr Reg II −Σ+−Σ=Φ ∫λμ
[ Pennec, et al, MICCAI 2005, LNCS 3750:943-950]
ProblemsElasticity is not symmetricStatistics are not easy to include
Idea: Replace the Euclidean by the Log-Euclidean metric
Statistics on strain tensorsMean, covariance, Mahalanobis computed in Log-space
Isotropic Riemannian Elasticity
( ) 222 )log(),(dist )(Tr Σ=Σ→−Σ II LE
( ) ( )ΣΣ=Σ ,2d0,d
( ) ( ) ( )∫ −Σ−Σ=Φ − WWg T )log(Vect.Cov.)log(VectRe 1
( ) ( ) ( )222
iso )log(Tr)log(Tr gRe Σ+Σ=Φ ∫ λμ
September 18, 2006 Mathematics and Image Analysis 2006 63
Conclusion : geometry and statisticsA Statistical computing framework on Riemannian manifolds
Mean, Covariance, statistical tests…Interpolation, diffusion, filtering…Which metric for which problem?
Important applications in Medical ImagingMedical Image Analysis
Evaluation of registration performancesDiffusion tensor imaging
Building models of living systems (spine, brain, heart…)
Noise models for real anatomical dataPhysically grounded noise models for measurementsAnatomically acceptable families of deformation metricsSpatial correlation between neighbors… and distant points… and statistics to measure and validate that!
September 18, 2006 Mathematics and Image Analysis 2006 64
Challenges of Computational AnatomyComputing on manifolds
Parametric families of metrics (models of the Green’s function)Topological changesEvolution: growth, pathologies
Build models from multiple sourcesCurves, surfaces [cortex, sulcal ribbons]Volume variability [Voxel Based Morphometry, Riemannian elasticity]Diffusion tensor imaging [fibers, tracts, atlas]
Compare and combine statistics on anatomical manifoldsCompare information from landmarks, courves, surfaces Validate methods and models by consensusIntegrative model (transformations ?)
Couple modeling and statistical learningStatistical estimation of model’s parameters (anatomical + physiological)Use models as a prior for inter-subject registration / segmentation Need large database and distributed processing/algorithms (GRIDS)
September 18, 2006 Mathematics and Image Analysis 2006 65
MFCA-2006: International Workshop on Mathematical Foundations of Computational Anatomy
Geometrical and Statistical Methods for Modelling Biological Shape Variability
October 1st, Copenhagen, in conjunction with MICCAI’06
Goal is to foster interactions between geometry and statistics in non-linear image and surface registration in the context of computational anatomy with a special emphasis on theoretical developments.
Chairs: Xavier Pennec (Asclepios, INRIA), Sarang Joshi (SCI, Univ Utah, USA)
Riemannian and group theoretical methods on non-linear transformation spacesAdvanced statistics on deformations and shapesMetrics for computational anatomyGeometry and statistics of surfaces
www.miccai2006.dk –> Workshops -> MFCA06
www-sop.inria.fr/asclepios/events/MFCA06/
September 18, 2006 Mathematics and Image Analysis 2006 66
Statistics on ManifoldsX. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. To appear in J. of Math. Imaging and Vision, Also 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. Also as INRIA RR- 5255, July 2004P. Fillard, V. Arsigny, X. Pennec, P. Thompson, and N. Ayache. Extrapolation of sparse tensor fields: applications to the modeling of brain variability. Proc of IPMI'05, 2005. LNCS 3750, p. 27-38. 2005.V. 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. To appear in MRM, also as INRIA RR-5584, Mai 2005. P. Fillard, V. Arsigny, X. Pennec, and N. Ayache. Joint Estimation and Smoothing of Clinical DT-MRI with a Log-Euclidean Metric. ISBI’2006 and INRIA RR-5607, June 2005.
Applications in Computational AnatomyX. 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.J.M. Peyrat, M. Sermesant , H. Delingette, X. Pennec, C. Xu, E. McVeigh, N. Ayache, Towards a Statistical Atlas of Cardiac Fibre Structure, MICCAI’06.
References[ Papers available at http://www-sop.inria.fr/asclepios/Biblio ]