Probabilitic Modelling of Anatomical Shapes

8/11/2019 Probabilitic Modelling of Anatomical Shapes

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Probabilistic Modeling of AnatomicalShape

Tom Fletcher, Prasanna Muralidharan, Nikhil Singh,

Miaomiao Zhang

School of Computing

Scientific Computing and Imaging Institute

University of Utah

July 10, 2013


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Shape Statistics: Averages


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Shape Statistics: Variability

Shape priors in segmentation


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Shape Statistics: Hypothesis Testing

Testing group differences

1

1Cates, et al. IPMI 2007 and ISBI 2008.


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Shape Statistics: Regression


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What is Shape?

Shape is the geometry of an object modulo position,

orientation, and size.


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Shape Representations

Boundary models (points, curves, surfaces,

level sets) Interior models (medial, solid mesh)

Transformation models (splines, diffeomorphisms)


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Shape Analysis

A shape is a point in a high-dimensional, nonlinear

shape space.


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Shape Analysis


shape space.


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Shape Analysis


shape space.


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Shape Analysis


shape space.


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Shape Analysis

A metric space structure provides a comparison

between two shapes.


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Manifolds

Amanifoldis a smooth topological space that looks

locally like Euclidean space, via coordinate charts.


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Tangent Spaces

Infinitesimal change in shape:

Atangent vectoris the velocity of a curve on M.


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Metrics and Geodesics

A Riemannian metric is a smoothly varying inner product

on the tangent spaces, denotedv,wp forv,wTpM.

Ageodesicis a curveMthat locally minimizes

E() =

1

0

(t)2dt.

Turns out it also locally minimizes arc-length,

L() =

10

(t)dt.


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The Exponential and Log Maps

The exponential map takes tangent vectors to

points along geodesics.

The length of the tangent vector equals the lengthalong the geodesic segment.

Its inverse is the log map it gives distance

between points: d(p, q) = Logp(q).


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Kendalls Shape Space

Define object withkpoints.

Represent as a vector in R2k.

Remove translation, rotation, andscale.

End up with complex projective

space, CPk2

.


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Quotient Spaces

What do we get when we remove scaling from R2?

Notation:[x] R2/R+


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Quotient Spaces


Notation:[x] R2/R+


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Quotient Spaces


Notation:[x] R2/R+


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C i K d ll Sh S


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Constructing Kendalls Shape Space

1- Consider planar landmarks to be points in the

complex plane.

2- An object is then a point(z1,z2, . . . ,zk) Ck.

3- Removingtranslationleaves us with Ck1.

4- How to removescalingandrotation?

S li d R i i h C l Pl


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Scaling and Rotation in the Complex Plane

Im

Re0

!

r

Recall a complex number can be writ-

ten asz = rei, with modulusrandargument.

Complex Multiplication:

sei

rei

= (sr)ei(+)

Multiplication by a complex numbersei is equivalent to

scaling bys and rotation by.

R i S l d T l ti


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Removing Scale and Translation

Multiplying a centered point set, z= (z1,z2, . . . ,zk1),by a constantw C, just rotates and scales it.

Thus the shape of zis an equivalence class:

[z] ={(wz1,wz2, . . . ,wzk1) :w C}

This gives complex projective space CPk2

much like

the sphere comes from equivalence classes of scalar

multiplication in Rn.

Diff hi I R i t ti


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Diffeomorphic Image Registration

I(x) I1(x)

Model shape differences as a transformation: is smooth, bijective,1 is smooth

M t i Diff hi


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Metrics on Diffeomorphisms

Sobolev metric:

v,w

V=Lv

(x

),w

(x

)dx

L is a symmetric, positive-definite differential operator,

e.g.,L = +I.

Induces distance between two images:

d(I0,I

1)2 =min

v

1

0

v(t)2Vdt+

1

2

I01I

12dx,

where d(t)

dt =v(t)(t).

Intrinsic Means (Frechet)


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Intrinsic Means (Frechet)

The intrinsic meanof a collection of pointsx1, . . . ,xNona Riemannian manifoldM is

=arg minxM

Ni=1

d(x,xi)2,

whered(, )denotes Riemannian distance onM.

Least Squares and Maximum Likelihood


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Least Squares and Maximum Likelihood

The Frechet mean and other least squares methods can

be motivated asmaximum likelihood estimatesunder

the following Gaussian distribution on manifolds:

p(x) =

1

Cexp

d(x, )2

Need the normalizing constantCto be

independent of. Holds for nice manifolds (homogeneous spaces).

Fletcher, IJCV 2012

Computing Means


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Computing Means

Gradient Descent Algorithm:

Input: x1, . . . , xNM

0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()


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Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Computing Means


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Computing Means



0= x1

Repeat:

= 1N

N

i=1Logk(xi)

k+1=Expk()

Diffeomorphic Atlas Estimation


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Diffeomorphic Atlas Estimation

Minimize SSD overIandvk

:Ni=1

d(I,Ik)2 =

Ni=1

1

22I(k)1 Ik

2+(Lvk, vk).

Joshi et al. 2004, Vialard et al. 2011

Bayesian Atlas Estimation


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Likelihood:iid Gaussian on each of theMvoxels

p(Ik| vk,I) =

1

(2)M/2Mexp

I(k)1 Ik2

22

Prior: multivariate Gaussian on discretized velocityvk

p(vk

) = 1

(2)M

2|L1|12

exp

(Lvk, vk)

2

Zhang et al., IPMI 2013

Inference


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e e ce

Log posterior:

log

Nk=1

pvk |Ik;

N

2log |L|

1

2

Nk=1

(Lvk, vk)

MN2

log 122

Nk=1

I(k)1 Ik2.

Treatvk aslatent random variables Expectation Maximization to estimate,

E-Step approximated by Monte Carlo



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y

Input: 3D MR Images Initialization Bayesian Atlas

Whats the Best?


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=28

Atlas Deformed Atlas to Individual

Whats the Best?


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=2.8



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Whats the Best?


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=0.0028


Whats the Best?


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=0.00028


Describing Shape Change


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How does shape change over time? Changes due to growth, aging, disease, etc.

Example: 100 healthy subjects, 2080 yrs. old

We need regression of shape!

Regression on Manifolds


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Given:

Manifold data: yiMScalar data: xi R



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Given:




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Given:


Want:

Relationship f : R Mhowx explainsy

Parametric vs. Nonparametric Regression


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0.2 0.4 0.6 0.8 1.0

0.

0

0.

5

1.

0

x

y

Linear Regression

0.0 0.2 0.4 0.6 0.8 1.0

0.

5

0.

6

0.

7

0.

8

0.

9

x

y

Kernel Regression

Geodesic Regression


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Generalization of linear regression.

Least-squares fitting of geodesic to the data(xi,yi).

(p, v) =arg min(p,v)TM

Ni=1

d(Exp(p,xiv),yi)2

Fletcher, MFCA 2011; Niethammer et al., MICCAI 2011


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Corpus Callosum Data


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Age range: 20 - 90 years

R2 Statistic


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DefineR2 statistic as percentage of variance explained:

R2 =variance along geodesic

total variance of data

=

var(xi)v2

id(y,yi)2 ,

whereyis the Frechet mean:

y=arg minyM

d(y,yi)2

Hypothesis Testing ofR2


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Parametric form for sampling distribution ofR2 is

difficult

Instead use a nonparametric permutation test

Null hypothesis: no relationship betweenXandY

Permute order independent parameterxi and

computeR2k

Count percentage ofR2kthat are larger thanR

2

Hypothesis Testing: Corpus Callosum


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R2 =0.12 LowR2 indicates that age does not explain a high

percentage of the variability seen in corpus

callosum shape

Ran 10,000 permutations, computingR2k p=0.009

Lowp value indicates that the trend seen in corpus

callosum shape due to age is unlikely to be byrandom chance

Kernel Regression (Nadaraya-Watson)


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Define regression function through weighted averaging:

f(t) =N

i=1

wi(t)Yi

wi(t) = Kh(tTi)

Ni=1

Kh(tTi)

Example: Gray Matter Volume


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K (t-s)

t

h

sti

wi(t) = Kh(tTi)

Ni=1Kh(tTi)

f(t) =Ni=1

wi(t)Yi

Manifold Kernel Regression


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Using Frechet weighted average:

mh(t) =arg miny

Ni=1

wi(t)d(y,Yi)2

Davis, et al. ICCV 2007

Brain Shape Regression


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Longitudinal Shape Analysis


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OASIS data:

11 healthy subjects

12 dementia subjects

3 images over 6 years

Goal: Understand howindividualschange over time.

Why Longitudinal?


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Why Longitudinal?


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Why Longitudinal?


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Why Longitudinal?


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Why Longitudinal?


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Hierarchical Geodesic Models


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Group Level: Average geodesic trend(, )

Individual Level: Trajectory forith subject(pi, ui)

Muralidharan, CVPR 2012; Singh, IPMI 2013

Comparing Geodesics: Sasaki MetricsWhat is the distance between two geodesic trends?


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Define distance between initial conditions:

dS((p1, u1), (p2, u2))

Sasaki geodesic on tangent bundle of the sphere

Results on Longitudinal Corpus Callosum


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Non-Demented Trend

Demented Trend

Permutation Test:

Variable T2 p-valueIntercept 0.734 0.248Slope 0.887 0.027

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Probabilitic Modelling of Anatomical Shapes

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