Centrale OMA MIA -MVA 2017-2018 - · 1 1 Statistic on Riemannian manifolds and Lie groups...

1

1

Statistic on Riemannian manifolds and Lie groups

Computing with diffusion tensor images

X. Pennec

Epione team

2004, route des Lucioles B.P. 93

06902 Sophia Antipolis Cedex

https://team.inria.fr/epione/

Statistical computing on manifolds and data assimilation:

from medical images to anatomical and physiological models

http://www-sop.inria.fr/asclepios/cours/MVA/Module2/index.htm

Centrale OMA MIA - MVA 2017-2018

MVA 01-2018

EPIONE

2

Statistical computing on manifolds and data assimilation: from medical images to anatomical and physiological models

Medical Imaging – Module 2 - MVA 2017-2018

Thu Jan 11: Introduction to Medical Image Analysis and Processing [XP]

Thu Jan 18: Riemannian manifolds and Lie groups [XP]

Thu Jan 25: Biomechanical Modelling [HD]

Thu Feb 1: Statistics & Computing with diffusion tensor images [XP]

Thu Feb 8: Data assimilation methods for physiological modeling [HD]

Thu Mar 1: Diffeomorphic transformations for image registration [XP]

Thu Mar 8: Statistics on deformations for computational anatomy [XP]

Thu Mar 15: Personalization of tumor growth models [XP]

Thu Mar 22: Exam [HD,XP]

MVA 01-2018

3

Bases of Algorithms in Riemannian Manifolds

Operation Euclidean space Riemannian

Subtraction

Addition

Distance

Gradient descent )( ttt xCxx

)(yLogxy xxyxy

xyyx ),(distx

xyyx ),(dist

)(xyExpy x

))( ( txt xCExpxt

xyxy

Reformulate algorithms with expx and logx

Vector -> Bi-point (no more equivalence classes)

Exponential map (Normal coordinate system): 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

MVA 01-2018

2

4

Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds

Mean, Covariance, Parametric distributions / tests

Statistics on the spine

Manifold-Valued Image Processing

Introduction to diffusion tensor imaging

Interpolation, filtering, extrapolation of tensors

Other metrics and applications

MVA 01-2018

5

Statistics on Riemannian manifolds

The geometric framework

Simple statistical tools Mean, Covariance, Parametric distributions / tests

Statistics on the spine

Application examples Evaluation of registration performances

Rigid body transformations

MVA 01-2018

6

Basic probabilities and statistics

Measure: random vector x of pdf

Approximation:

• Mean:

• Covariance:

Propagation:

Noise model: additive, Gaussian...

Principal component analysis

Statistical distance: Mahalanobis and

dzzpz ).(. ) E(x xx

)x( xxΣx , ~

)(zpx

T)x).(x(E xxxx

x

..x

, x)(Thh

h ~ h xxΣxy

2

MVA 01-2018

3

Random variable in a Riemannian Manifold

Intrinsic pdf of x

For every set H

∈

Lebesgue’s measure

Uniform Riemannian Mesure det

Expectation of an observable in M

(variance) : . , ,

log (information) : log log

(mean) :

8MVA 01-2018

9

Fréchet expectation (1944)

Minimizing the variance

Existence

Finite Variance at one point

Variational characterization: exponential barycenter (P(C)=0)

The case of points: classical expectation

Other central primitives

),dist(E argmin 2xx yy M

Ε

0)().(.xxE 0 )(grad 2 M

M zdzpy xx xx

0xE x xxΕ

1

),dist(E argmin xx yy M

Ε

[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]

MVA 01-2018

Riemannian center of mass,Fréchet or Karcher mean?

Name conventions in geometric statistics Fréchet mean: global minima [Fréchet 1944]

Local Fréchet or Karcher mean: local minima [Karcher 1977, Kendall 1990]

Exponential barycenters: critical points [Emery 1991, Oller & Corcuera 1995]

All are sometimes called Riemannian center of mass [Groove & Karcher 1973-1976] under uniqueness assumptions

Concentration assumption: Existence and uniqueness[Karcher 77 / Buser & Karcher 1981 / Kendall 90 / Afsari 10 / Le 11]

Support of distribution in a regular geodesic ball U(r) with radius ∗ min , / ( upper bound on sect. Curv. on M with convention

that ∞ 0)

Empirical Fréchet mean (finite sample): Almost surely unique [Arnaudon & Miclo 2013]

10MVA 01-2018

4

11

A gradient descent (Gauss-Newton) algorithm

Vector space

Manifold

vHvvfxfvxf fTT

..2

1.)()(

fHvvxx ftt

. with )1(

1

),()()())((exp2

1 vvHvfxfvf fx

iin

yx2

yE 2 (y)2 xx

IdH 2 2 x

xyE with )(expx x1 vvtt

Geodesic marching

MVA 01-2018

Algorithms to compute the mean

Karcher flow (gradient descent)

exp E y ∑ log

Usual algorithm with 1 can diverge on SPD matrices [Bini & Iannazzo, Linear Algebra Appl., 438:4, 2013]

Convergence for non-negative curvature (p-means) [Afsari, Tron and Vidal, SICON 2013]

Inductive / incremental weighted means

exp log

On negatively curved spaces [Sturm 2003], BHV centroid [Billera, Holmes, Vogtmann, 2001]

On non-positive spaces[G. Cheng, J. Ho, H. Salehian, B. C. Vemuri 2016]

Stochastic algorithm [Arnaudon & Miclo, Stoch. Processes and App. 124, 2014]

12MVA 01-2018

13

First Statistical Tools: Moments

Covariance (PCA) [higher moments]

Principal component analysis

Tangent-PCA:principal modes of the covariance

Principal Geodesic Analysis (PGA) [Fletcher 2004,Sommer 2014]

M

M )().(.x.xx.xE TT

zdzpzz xxx xx

[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]

MVA 01-2018

5

PCA vs PGA

PCA find the subspace that best explains the variance

Maximize the squared distance to the mean

Generative model: Gaussian

PGA (Fletcher, Joshi, Lu, Pizer, MMBIA 2004) find a low dimensional sub-manifold generated by geodesic

subspaces that best explain measurements

Minimize the squared Riemannian distance from the measurements to that sub-manifold (no closed form)

Generative model: Implicit uniform distribution within the subspace

Gaussian distribution in the vertical space

Different models in curved spaces (no Pythagore thm)

14MVA 01-2018

15

Example with 3D rotations

Principal chart:

Distance:

Frechet mean:

nr . :ectorrotation v

2)1(

121 ),dist( rrRR

Centered chart:

mean = barycenter

i

2 ),(distmin arg 3

iSOR

RRR

MVA 01-2018

16

Distributions for parametric tests

Uniform 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 Xx

rOk n /1.)det(.2 32/12/ Σ

rO / Ric3

1)1( ΣΓ

yx..yx)y( )1(2 xxx

t

n)(E 2 xx

rOn /)()( 322 xx

[ Pennec, JMIV06, NSIP’99 ]

MVA 01-2018

6

17

Gaussian on the circle

Exponential chart:

Gaussian: truncated standard Gaussian

[. ; .] rrrx

standard Gaussian(Ricci curvature → 0)

uniform pdf with

(compact manifolds)

Dirac

:r

:

:03/).( 22 r

MVA 01-2018

19

Computing on manifolds: a summary

The Riemannian metric easily gives Intrinsic measure and probability density functions

Expectation of a function from M into R (variance, entropy)

Integral or sum in M: minimize an intrinsic functional Fréchet / Karcher mean: minimize the variance

Filtering, convolution: weighted means

Gaussian distribution: maximize the conditional entropy

The exponential chart corrects for the curvature at the

reference point Gradient descent: geodesic walking

Covariance and higher order moments

Laplace Beltrami for free

[ Pennec, NSIP’99, JMIV 2006, Pennec et al, IJCV 66(1) 2006, Arsigny, PhD 2006]

MVA 01-2018

20

Statistical Analysis of the Scoliotic Spine

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 rotation var. around anterior-posterior axis

PCA of the Covariance 4 first variation modes have clinical meaning

[ J. Boisvert, X. Pennec, N. Ayache, H. Labelle, F. Cheriet,, ISBI’06 ]

MVA 01-2018

7

21

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)

MVA 01-2018

32

Per - Operative US Image

Typical Registration Resultwith Bivariate Correlation Ratio

Pre - Operative MR Image

Registered

Acquisition of images : L. & D. Auer, M. Rudolf

axial

coronal sagittal

axial

coronal sagittal

MVA 01-2018

33

US Intensity MR Intensity and Gradient

MVA 01-2018

8

34

Accuracy Evaluation (Consistency)

222/

2 2 USMRUSMRloop

MVA 01-2018

35

Validation using Bronze Standard

Best explanation of the observations (ML) : LSQ criterion

Robust Fréchet mean

Robust initialization and Newton gradient descent

Grid scheduling for efficiency

Result

221

221

2 ),,(min),( TTTTd

transrotjiT ,,,

ij ijij TTdC )ˆ,(2

[ T. Glatard & al, MICCAI 2006,

Int. Journal of HPC Apps, 2006 ]

MVA 01-2018

36

Data (per-operative US) 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

Robustness and precision

Consistency of BCR

Results on per-operative patient images

Success var rot (deg) var trans (mm)MI 29% 0.53 0.25CR 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 2.33MR/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]

MVA 01-2018

9

40







MVA 01-2018

41

Introduction to diffusion tensor imaging (DTI)

MVA 01-2018

42

Introduction to diffusion tensor imaging (DTI)

Classical MRI: white mater, grey matter, CSF

Diffusion MRI: MR technique born in the mid-80ies (Basser, LeBihan).

in vivo imaging of the white mater architecture

set of nervous fibers (axons): information highways of the brain!

MVA 01-2018

10

43

Anatomy of a diffusion MRI

T2 image

+

6 diffusion weighted images

MVA 01-2018

44

Stejskal & Tanner equation

MVA 01-2018

45

Visualization using ellipsoids

PL

S

RA

I

MVA 01-2018

11

46

Diffusion Tensor ImagingCovariance of the Brownian motion of water

-> Architecture of axonal fibers

Symmetric positive definite matrices

Cone: Convex operations are stable mean, interpolation

Null eigenvalues are reachable in a finite time : not physical!

More complex operations are not PDEs, gradient descent…

SPD-valued image processing

Clinical images: Very noisy data

Robust estimation

Filtering, regularization

Interpolation / extrapolation Diffusion Tensor Field(slice of a 3D volume)

Intrinsic computing on Manifold-valued images?

MVA 01-2018

48







Metric and Affine Geometric Settings for Lie Groups

Analysis of Longitudinal Deformations

MVA 01-2018

50

GL(n)-Invariant Metrics on SPD matrices

Action of the Linear Group GLn

Invariant metric

Isotropy group at the identity: Rotations All rotationally invariant scalar products:

Geodesics at Id

Exponential map

Log map

Distance

-1/n)( )Tr().Tr( Tr| 212121 WWWWWW Tdef

Id

Id

def

AA

tt WWAAWAAWWWt 2

2/11

2/12121 ,||

TAAA ..

2/12/12/12/1 )..exp()( Exp

22/12/12 )..log(|),(Id

dist

2/12/12/12/1 )..log()( Log

)exp()(, tWtWId

MVA 01-2018

12

51

GL(n)-Invariant Metrics on Tensors

Space of Gaussian distributions (Information geometry) (=0) Fisher information metric [Burbea & Rao J. Multivar Anal 12 1982,

Skovgaard Scand J. Stat 11 1984, Calvo & Oller Stat & Dec. 9 1991] Tensor segmentation [Lenglet RR04 & JMIV 25(3) 2006]

Affine-invariant metrics for DTI processing (=0) [Pennec, Fillard, Ayache, IJCV 66(1), Jan 2006 / INRIA RR-5255, 2004] PGA on tensors [Fletcher & Joshi CVMIA04, SigPro 87(2) 2007]

Geometric means (=0) Covariance matrices in computer vision [Forstner TechReport 1999] Math. properties [Moakher SIAM J. Matrix Anal App 26(3) 2004] Geodesic Anisotropy [Batchelor MRM 53 2005]

Homogeneous Embedding (=-1/(n+1) ) [Lovric & Min-Oo, J. Multivar Anal 74(1), 2000]

-1/n)( Tr .Tr21112

WWWW

MVA 01-2018

52

Bi-linear, tri-linear or spline interpolation?

Tensor interpolationGeodesic walking in 1D )(exp)( 211

tt

2),( )(min)( ii distxwxRephrase as weighted mean

RiemannianEuclidean

RiemannianEuclidean

MVA 01-2018

53

Gaussian filtering: Gaussian weighted mean

n

iii distxxGx

1

2),( )(min)(

Raw Coefficients =2 Riemann =2

MVA 01-2018

13

54

PDE for filtering and diffusion

Harmonic regularization

Gradient = manifold Laplacian

Trivial intrinsic numerical schemes with exponential maps!

Geodesic marching

Anisotropic diffusion Perona-Malik 90 / Gerig 92

Robust functions

2

2)1(

,, ,,

2 )()( )( uO

u

uxxx

ui

zyxi zyxiii

dxxC

x

2

)()()(

))((exp)( )(1 xCx xt t

u uxuw xxwx )( )()(

)(

dxx

x

2

)()()(Reg

[ Pennec, Fillard, Arsigny, IJCV 66(1), 2005, ISBI 2006]

MVA 01-2018

55

Anisotropic filtering

Initial

Noisy

Recovered

MVA 01-2018

56

Anisotropic filtering

Raw Riemann Gaussian Riemann anisotropic

)/exp()( with )( )()(ˆ 22

)(ttwxxwx

uuxu

2)1(2 /)()( )( uuxxx uuuu

MVA 01-2018

14

57

Extrapolation by Diffusion

Diffusion =0.01Original tensors Diffusion =

2

)(1

2 )(2

)),(()(2

1)(

x

n

iii xdxxdistxxGC

))(()()())((1

xxxxGxCn

iii

MVA 01-2018

58







Metric and Affine Geometric Settings for Lie Groups

Analysis of Longitudinal Deformations

MVA 01-2018

Log Euclidean Metric on Tensors

Exp/Log: global diffeomorphism Tensors/sym. matrices

Vector 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

logexp

2

212

21 loglog, dist

[Arsigny, MICCAI 2005 & MRM 56(2), 2006]

59MVA 01-2018

15

60

Riemannian Frameworks on tensors

Affine-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

[ Arsigny, Pennec, Fillard, Ayache, SIAM’06, MRM’06 ]

[ Pennec, Fillard, Ayache, IJCV 66(1), 2006, Lenglet JMIV’06, etc]

2/12/12/12/1 )...exp().( ttExp

22/12/12

2

)..log(|),(distL

2

212

21 loglog,dist

))log(.)exp(log().( ttExp

IdAA

TT WVAWAAVAWVT

2/12/12/12/1 |||

IdWVWV )log(|)log(|

MVA 01-2018

Log Euclidean vs Affine invariant

Means are geometric (vs arithmetic for Euclidean)

Log Euclidean slightly more anisotropic

Speedup ratio: 7 (aniso. filtering) to >50 (interp.)

Euclidean Affine invariantLog-Euclidean

61MVA 01-2018

Log Euclidean vs Affine invariant

Real DTI images: anisotropic filtering

Difference is not significant

Speedup of a factor 7 for Log-Euclidean

Original Euclidean Log-Euclidean Diff. LE/affine (x100)

62MVA 01-2018

16

63

Some other metrics on tensors

Square root metrics Cholesky [Wang Vemuri et al, IPMI’03, TMI 23(8) 2004.]

Size and shape space [Dryden, Koloydenko & Zhou, 2008]

Power Euclidean [Dryden & Pennec, arXiv 2011]

Non Riemannian distances J-Divergence [Wang & Vemuri, TMI 24(10), 2005]

Geodesic Loxodromes [Kindlmann et al. MICCAI 2007]

4th order tensors [Gosh, Descoteau & Deriche MICCAI’08]

MVA 01-2018

64

Euclidean metric

Choleski metric

Log-Euclidean metric

Geodesic shooting in tensors spaces

Undefined (non positive)

MVA 01-2018

65

DTI Estimation from DWI

Baseline + at least 6 DWI

iTii DgbgSS exp0

+

Stejskal & Tanner diffusion equation

Diffusion Tensor Field

Noise is Gaussian in complex DW signal Rician = amplitude of complex

Gaussian LSQ on Rician noise = bias for low

SNRs [Sijbers, TMI 1998]

Estimation / Regularization on complex DWI: Anisotropic diffusion on Choleski factors

[Wang & Vemuri, TMI’04]

Estimation with a Rician noise Smoothing DWI before estimation

[Basu & Fletcher, MICCAI 2006] ML (MMSE) [Aja-Fernández et al, TMI 2008] MAP with log-Euclidean prior

[Fillard et al., ISBI 2006, TMI 2007]

17

66

Rician MAP estimation with Riemannian spatial prior

ML Rician MAP RicianStandard

Estimated tensors

FA

[ Fillard, Arsigny, Pennec, Ayache ISBI’06, TMI 26(11) 2007 ]

dxxdxSS

ISSS

MAPx

N

i

iiiii .)( .ˆ)(

2

)(ˆexp

ˆlog

2

)(1

202

22

2

MVA 01-2018

67

Clinical DTI of the spinal cord: fiber tracking

MAP RicianStandard

[ Fillard, Arsigny, Pennec, Ayache ISBI’06, TMI 26(11) 2007 ]

MVA 01-2018

T1 + Activation map + fibers

Corpus callosum + cingulum

Corticospinal tract and thalamo cortical connections

Courtesy of P. Fillardusing MedINRIA

68MVA 01-2018

18

69MVA 01-2018

Database 7 canine hearts from JHU

Anatomical MRI and DTI

Freely available at http://www-sop.inria.fr/asclepios/data/heart

•Average cardiac structure

•Variability of fibers, sheets

A Statistical Atlas of the Cardiac Fiber Structure[ J.M. Peyrat, et al., MICCAI’06, TMI 26(11), 2007]

70MVA 01-2018

Database 7 canine hearts from JHU

Anatomical MRI and DTI

Method Normalization based on aMRIs

Log-Euclidean statistics of Tensors

A Statistical Atlas of the Cardiac Fiber Structure[ J.M. Peyrat, et al., MICCAI’06, TMI 26(11), 2007]

71

Norm covariance

Eigenvalues covariance (1st, 2nd, 3rd)

Eigenvectors orientation covariance (around 1st, 2nd, 3rd)

MVA 01-2018

19

Diffusion model of the human heart10 human ex vivo hearts (CREATIS-LRMN, Lyon, France)

Classified as healthy (controlling weight, septal thickness, pathology examination)

Acquired on 1.5T MR Avento Siemens bipolar echo planar imaging, 4 repetitions, 12 gradients

Volume size: 128×128×52, 2 mm resolution

72

[ H. Lombaert Statistical Analysis of the Human Cardiac Fiber Architecture from DT-

MRI, ISMRM 2011, FIMH 2011]

Fiber tractography in the left ventricle

Helix angle highly correlated to the transmural distance

MVA 01-2018

A Statistical Atlas of the Cardiac Fiber Structure

73

[ R. Mollero, M.M Rohé, et al, to appear in FIMH 2015]

10 human ex vivo hearts (CREATIS-LRMN, France)

Classified as healthy (controlling weight, septal thickness, pathology examination)

Acquired on 1.5T MR Avento Siemens bipolar echo planar imaging, 4 repetitions, 12

gradients

Volume size: 128×128×52, 2 mm resolution

MVA 01-2018

Centrale OMA MIA -MVA 2017-2018 - · 1 1 Statistic on Riemannian manifolds and Lie groups...

Documents

Transcript of Centrale OMA MIA -MVA 2017-2018 - · 1 1 Statistic on Riemannian manifolds and Lie groups...