A Comprehensive Riemannian Framework for the Analysis of ...mmani/isbi2010_presentation_online.pdf · A Comprehensive Riemannian Framework for the Analysis of White Matter Fiber Tracts

A Comprehensive Riemannian Framework for theAnalysis of White Matter Fiber Tracts

Meena Mani1 Sebastian Kurtek2

Christian Barillot1 Anuj Srivastava2

1Visages Project, INRIA/IRISA, Rennes, France

2Department of Statistics, Florida State University, Tallahassee, USA

ISBI 2010April 16, 2010

M. Mani Comprehensive Riemannian Framework April, 2010 2 / 20



DTI Fiber: Physical Features

position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape



position

orientation

scale

shape

These features eitherindividually or incombination can beused to design metricsand feature spaces


MO

TIV

AT

ION White Matter Fiber Analysis

varied applications

different goals and perspectives

need to design appropriate metrics


MO

TIV

AT


varied applications




MO

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AT


varied applications




Outline

1 Mathematical FrameworkFiber Tract RepresentationS1: Shape + orientation + scale + positionS2: Shape + orientation + scaleS3: Shape + scaleS4: Shape + orientationS5: Shape

2 Example: Clustering the Corpus Callosum

3 Statistics


Outline



3 Statistics


Outline



3 Statistics


Fiber Tract Representation

Let β : [0, 1]→ R3 be a parameterized curve


Comparing Curves

Common Technique:

L2 distance: ‖β1 − β2‖ =√∫ 1

0 ‖β1(t)− β2(t)‖2dtFor example, methods based on Fourier representation use this metric.

Problem: This metric is not invariant to reparametrization.Let γ : [0, 1]→ [0, 1] be a diffeomorphism (γ acts as a re-parameterizationfunction). A curve β is re-parameterized by γ as (β ◦ γ)(t) = β(γ(t)).It can be shown that ‖β1 ◦ γ − β2 ◦ γ‖ 6= ‖β1 − β2‖ in general.

Solution: Seek a new representation that preserves L2 distance underre-parameterization. The choice of representation depends on the featureswe want to include in the analysis.


Shape + orientation + scale + position (S1)Curve Representation: “h”-function

h(t) =

√‖β̇(t)‖β(t) , h : [0, 1]→ R3

The h-function of a re-parameterized curve is: (h, γ) ≡ h(γ(t))√γ̇(t)

Motivation: ‖(h1, γ)− (h2, γ)‖ = ‖h1 − h2‖ for all γ

Curve-Comparison/Distance function:

γ∗ = argminγ∈Γ

(‖h1 − (h2, γ)‖) , da(β1, β2) = ‖h1 − (h2, γ∗)‖

da is invariant to re-parameterizations of β1 and β2

Geodesic path or Deformation: Straight line

ψ(τ) = (1− τ)h1 + τ(h2, γ∗)


Shape + orientation + scale + position (S1)Curve Representation: “h”-function

h(t) =

√‖β̇(t)‖β(t) , h : [0, 1]→ R3

The h-function of a re-parameterized curve is: (h, γ) ≡ h(γ(t))√γ̇(t)

Motivation: ‖(h1, γ)− (h2, γ)‖ = ‖h1 − h2‖ for all γ



(‖h1 − (h2, γ)‖) , da(β1, β2) = ‖h1 − (h2, γ∗)‖

da is invariant to re-parameterizations of β1 and β2


ψ(τ) = (1− τ)h1 + τ(h2, γ∗)


Shape + orientation + scale (S2)Curve Representation: “q”-function or square-root velocity function

q(t) =β̇(t)√‖β̇(t)‖

Since q is defined using β̇, it is invariant to the translation of βThe q-function of a re-parameterized curve is: (q, γ) ≡ q(γ(t))

√γ̇(t)

Motivation: ‖(q1, γ)− (q2, γ)‖ = ‖q1 − q2‖ for all γ



(‖q1 − (q2, γ)‖) , db(β1, β2) = ‖q1 − (q2, γ∗)‖


ψ(τ) = (1− τ)q1 + τ(q2, γ∗)


Shape + orientation + scale (S2)Curve Representation: “q”-function or square-root velocity function

q(t) =β̇(t)√‖β̇(t)‖

Since q is defined using β̇, it is invariant to the translation of βThe q-function of a re-parameterized curve is: (q, γ) ≡ q(γ(t))

√γ̇(t)

Motivation: ‖(q1, γ)− (q2, γ)‖ = ‖q1 − q2‖ for all γ



(‖q1 − (q2, γ)‖) , db(β1, β2) = ‖q1 − (q2, γ∗)‖


ψ(τ) = (1− τ)q1 + τ(q2, γ∗)


Shape + scale (S3)We remove rotation from the representation as follows:

Curve Representation: q-function

q(t) =β̇(t)√‖β̇(t)‖

Curve Alignment:

(O∗, γ∗) = argminO∈SO(3),γ∈Γ

(‖q1 − O(q2, γ)‖)

Distance function:

dd(β1, β2) = ‖q1 − O∗(q2, γ∗)‖

Geodesic path: Straight line

ψ(τ) = (1− τ)q1 + τ(O∗q2, γ∗)


Shape + orientation (S4)

Keep orientation but remove scale by rescaling all curves

Curve Representation: q(t) = β̇(t)√‖β̇(t)‖

Distance function: Arc-Length on a sphere

dc(β1, β2) = minγ∈Γ

(cos−1(

∫ 1

0〈(q1, γ)(t), (q2, γ)(t)〉 dt)

)Geodesic path: Great circle!

ψ(τ) =1

sin(θ)[sin(θ − τθ)q1 + sin(τθ)(q2, γ

∗)]

where θ = dc(β1, β2)


Shape (S5)Now we remove all variables: rotation, translation and scale

Curve Representation: q(t) = β̇(t)√‖β̇(t)‖

Curve Alignment:

(O∗, γ∗) = argminO∈SO(3),γ∈Γ

(cos−1(

∫ 1

0〈(q1, γ)(t),O(q2, γ)(t)〉 dt)

)Distance function:

de(β1, β2) = cos−1(

∫ 1

0〈(q1, γ)(t),O∗(q2, γ)(t)〉 dt)

Geodesic path: Great circle!

ψ(τ) =1

sin(θ)[sin(θ − τθ)q1 + sin(τθ)(O∗q2, γ

∗)]

for θ = de(β1, β2)M. Mani Comprehensive Riemannian Framework April, 2010 12 / 20

In Summary

Manifold function pre-shape shaperepresentation space space

shape +

h(t) =√‖β̇(t)‖β(t) S1 = C1/(Γ)

orientation + C1

scale + L2 spacepositionshape +

q(t) = β̇(t)√‖β̇(t)‖

C2

L2 spaceS2 = C2/(Γ)orientation +

scaleshape +

q(t) = β̇(t)√‖β̇(t)‖

C2 S3 = C2/(Γ× SO(3))scale L2 spaceshape +

q(t) = β̇(t)√‖β̇(t)‖

C3 S4 = C3/(Γ)orientation hypersphere

shape q(t) = β̇(t)√‖β̇(t)‖

C3 S5 = C3/(Γ× SO(3))hypersphere


Geodesic PathsEvolution of one curve into another

S2: shape+orientation+scale S3: shape+scale

S4: shape+orientation S5: shape


Application: Clustering Fibers in the Corpus Callosum





(a) shape+orientation+scale (db) (b) shape (de)

Clustering the genu and splenium, the anterior and posterior sectionsof the CC. Here, shape information alone (b) is not adequate forclustering.



(a) shape+orientation (dc) (b) shape+orientation+scale (db)

Clustering of the genu, corpus and splenium, the anterior, middle andposterior sections of the CC. Including the scale information results inpoorer clustering (b).



(a) shape+orientation (dc) (b) shape+scale (dd)

Clustering the isthmus and splenium, the posterior CC


Statistics

Summary statistics

Statistical inference

I Stochastic models

I Hypothesis testing


StatisticsMean Curves of a DTI Fiber Bundle: Splenium

fib

er

bu

nd

le

shap

esh

ape+

orie

nta

tion



fib

er

bu

nd

lesh

ape

shap

e+

orie

nta

tion



fib

er

bu

nd

lesh

ape

shap

e+

orie

nta

tion


StatisticsMean Curves of a Population: Spleniums of 6 Subjects

shap

esh

ape+

orie

nta

tion


References

A. Srivastava, S. Joshi, W. Mio, and X. Liu,“Statistical shape analysis: Clustering, learning and testing,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.27, no. 4, pp. 590–602, 2005.

S. H. Joshi, E. Klassen, A. Srivastava, and I. Jermyn,“A Novel Representation for Riemannian Analysis of Elastic Curves inRn,”in CVPR, 2007.

E. Klassen, A. Srivastava, W. Mio, and S. Joshi,“Analysis of planar shapes using geodesic paths on shape spaces,”IEEE Patt. Analysis and Machine Intell., vol. 26, no. 3, pp. 372–383,March, 2004.

Search terms: Anuj Srivastava, FSU, Publications


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A Comprehensive Riemannian Framework for the Analysis of ...mmani/isbi2010_presentation_online.pdf · A Comprehensive Riemannian Framework for the Analysis of White Matter Fiber Tracts