Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterrane, - - PowerPoint PPT Presentation
Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterrane, France With contributions from Vincent Arsigny, Pierre Fillard, Marco Lorenzi, Christof Seiler, Jonathan Boisvert, Nicholas Ayache, etc Statistical Computing on
Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis
Asclepios team, INRIA Sophia- Antipolis – Mediterranée, France
With contributions from Vincent Arsigny, Pierre Fillard, Marco Lorenzi, Christof Seiler, Jonathan Boisvert, Nicholas Ayache, etc
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Gall (1758-1828) : Phrenology Talairach (1911-2007)
Antiquity
Renaissance:
Vésale (1514-1564) Paré (1509-1590)
1990-2000:
Science that studies the structure and the relationship in space of different organs and tissues in living systems [Hachette Dictionary]
From dissection to in-vivo in-situ imaging
From representative individual to population
From descriptive atlases to interactive and generative models (simulation)
Galien (131-201)
1st cerebral atlas, Vesale, 1543
17-20e century:
Visible Human Project, NLM, 1996-2000 Voxel-Man, U. Hambourg, 2001 Talairach & Tournoux, 1988
Sylvius (1614-1672) Willis (1621-1675)
Paré, 1585
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Design Mathematical Methods and Algorithms to Model and Analyze the Anatomy
Statistics of organ shapes across species, populations, diseases… Model organ development across time (heart-beat, growth, ageing, ages…) To understand and to model how life is functioning
Classify structural deviations (taxonomy), Relate anatomy and function
To detect, understand and correct dysfunctions
From generic (atlas-based) to patients-specific models
Very active topic in medical image analysis
Keyword of Medic Image Computing and Computer Assisted Intervention (MICCAI)
Tensors, covariance matrices Curves, tracts Surfaces Transformations
Rigid, affine, locally affine, diffeomorphisms
Deal with noise consistently on these non-Euclidean manifolds A consistent statistical (and computing) framework
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Observation = “random” deformation of a reference template Reference template = Mean (atlas) Shape variability encoded by the deformations
Patient 3 Atlas Patient 1 Patient 2 Patient 4 Patient 5 1 2 3 4 5
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Shape of RV in 18 patients
Remodeling of the right ventricle of the heart in tetralogy of Fallot
Mean shape Shape variability Correlation with clinical variables Predicting remodeling effect
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Introduction to computational anatomy The Riemannian manifold computational structure Simple statistics on Riemannian manifolds Applications to the spine shape and registration accuracy
x
xx
T
xx
T
xx
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Locally Euclidean Topological space
Same dimension + differential regularity
Sphere Saddle (hyperbolic space) Surface in 3D space
Projective spaces Rotations of R3 : SO3 ~ P3 Rigid, affine Transformation Diffeomorphisms
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Extrinsic
Embedding in ℝ𝑜
Intrinsic
Coordinates : charts Atlas = consistent set
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Volumes (surfaces) Lengths Straight lines
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t
Bernhard Riemann 1826-1866
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t
p p
p
t
) (
g
p p
Bernhard Riemann 1826-1866
t p
t p
Bernhard Riemann 1826-1866
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t
) (
g
Bernhard Riemann 1826-1866
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Operation Euclidean space Riemannian Subtraction Addition Distance
Gradient descent ) (
t t t
x C x x
) (y Log xy
x
xy x y
x y y x ) , ( dist
x
xy y x ) , ( dist ) (xy Exp y
x
)) ( (
t x t
x C Exp x
t
Vector -> Bi-point (no more equivalence classes)
Expx = geodesic shooting parameterized by the initial tangent Logx = unfolding the manifold in the tangent space along geodesics
Geodesics = straight lines with Euclidean distance Local global domain: star-shaped, limited by the cut-locus Covers all the manifold if geodesically complete
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Introduction to computational anatomy The Riemannian manifold computational structure Simple statistics on Riemannian manifolds Applications to the spine shape and registration accuracy
For every set H
𝐼
Lebesgue’s measure Uniform Riemannian Mesure 𝑒𝑁 𝑧 =
𝑭𝐲 𝜚 = 𝜚 𝑧 𝑞 𝑧 𝑒𝑁 𝑧 𝑁
𝜚 = 𝑒𝑗𝑡𝑢2 (variance) : 𝑭𝐲 𝑒𝑗𝑡𝑢 . , 𝑧 2 = 𝑒𝑗𝑡𝑢 𝑧, 𝑨 2𝑞 𝑨 𝑒𝑁(𝑨)
𝑁
𝜚 = log 𝑞 (information) : 𝑭𝐲 log 𝑞
𝑁
𝜚 = 𝑦 (mean) : 𝑭𝐲 𝐲 = 𝑧 𝑞 𝑧 𝑒𝑁 𝑧
𝑁
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Finite variance at one point
Unique Karcher mean (thus Fréchet) if distribution has support in a
1 2 min 𝑗𝑜𝑘 𝑁 , 𝜌/ 𝜆 (k upper
Empirical mean: a.s. uniqueness [Arnaudon & Miclo 2013]
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y M
2
M
x x
1
y M
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x 1
t
t
M
T T
x xx
2
y M M
x
[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]
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2 ) 1 ( 1 2 1
i
3
i SO R
Convex functions in compact spaces are constant
if the support of x is included in a strongly convex
Difference between barycenters is O()
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maximal entropy knowing X
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
Any distribution: Gaussian:
T
X
x
n
3 2 / 1 2 /
3 1 ) 1 (
) 1 ( 2
xx x t
2 x x
n
3 2 2
x
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2 2
Generative model: Gaussian Find the subspace that best explains the variance
Maximize the squared distance to the mean
Generative model:
Implicit uniform distribution within the subspace Gaussian distribution in the vertical space
Find a low dimensional subspace (geodesic subspaces?) that
Minimize the squared Riemannian distance from the measurements to that sub-manifold (no closed form)
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Introduction to computational anatomy The Riemannian manifold computational structure Simple statistics on Riemannian manifolds Applications to the spine shape and registration accuracy
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Database
307 Scoliotic patients from the Montreal’s Sainte-Justine Hospital.
3D Geometry from multi-planar X-rays
Mean
Main 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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PCA of the Covariance:
4 first variation modes have clinical meaning [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ] AMDO’06 best paper award, Best French-Quebec joint PhD 2009
LSQ criterion Robust Fréchet mean Robust initialization and Newton gradient descent
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2 2 1 2 2 1 2
trans rot j i
,
ij ij ij T
T d C ) ˆ , (
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Per-operative CT “guidance” Respiratory gating
Marker based 3D/2D rigid registration
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2D-3D EM-ICP on fiducial markers Certified accuracy in real time
Bronze standard (no gold-standard) Phantom in the operating room (2 mm) 10 Patient (passive mode): < 5mm (apnea)
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[ S. Nicolau, et al. Comp. Anim. & Virtual World 2005, Medical Image Analysis, 13(3), 2009 ]
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2 pre-op MR (0.9 x 0.9 x 1.1 mm) 3 per-op US (0.63 and 0.95 mm) 3 loops
Success var rot (deg) var trans (mm) MI 29% 0.53 0.25 CR 90% 0.45 0.17 BCR 85% 0.39 0.11
var rot (deg) var trans (mm) var test (mm) Multiple MR 0.06 0.06 0.10 Loop 2.22 0.82 2.33 MR/US 1.57 0.58 1.65 [Roche et al, TMI 20(10), 2001 ] [Pennec et al, Multi-Sensor Image Fusion, Chap. 4, CRC Press, 2005]
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FOV 200x200 µm
Courtesy of Mike Booth, MGH, Boston, MA
FOV 2747x638 µm
[ T. Vercauteren et al., MICCAI 2005, T.1, p.753-760 ]
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Multiple rigid registration Refine with non rigid
Interpolation / approximation
Frame 6 Frame 1 Frame 2 Frame 3 Frame 4 Frame 5
Courtesy of Mike Booth, MGH, Boston, MA
FOV 2747x638 µm
[ T. Vercauteren et al., MICCAI 2005, T.1, p.753-760 ]
Paris, August 28-30 2013, 2013
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Nielsen, F. Barbaresco
Byande, F. Barbaresco, S. Bonnabel, R. Sepulchre,
Angulo, N. Le Bihan, J. Manton, A. Cont, A.Dessein, A.M. Djafari, H. Snoussi, A. Trouvé, S. Durrleman, X. Pennec, J.F. Marcotorchino, M. Petitjean, M. Deza
Nagoya, Japan, September 22 or 26, 2013
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Nielsen, F. Barbaresco
Byande, F. Barbaresco, S. Bonnabel, R. Sepulchre,
Angulo, N. Le Bihan, J. Manton, A. Cont, A.Dessein, A.M. Djafari, H. Snoussi, A. Trouvé, S. Durrleman, X. Pennec, J.F. Marcotorchino, M. Petitjean, M. Deza
http://www-sop.inria.fr/asclepios/events/MFCA11/ http://www-sop.inria.fr/asclepios/events/MFCA08/ http://www-sop.inria.fr/asclepios/events/MFCA06/
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