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Multi-scale Riemann-Finsler GeometryApplications to Diffusion Tensor Imaging and

High Angular Resolution Diffusion Imaging

Laura Astola

CASA PhD-day 28. October 2009

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Introduction to Applications

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Introduction to Applications

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Introduction to Applications

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Introduction to Applications

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Introduction to Applications

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

”Multi-scale differential geometry and applications”

Image analysis at multiple scales

Use of differential geometry in image analysis

Applications to diffusion tensor imaging (DTI)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

To compute differential geometric quantities (curvatures etc.)on data one needs a metric tensor at each point.

In Diffusion Tensor ImagingDij(x) is the diffusion tensor and the corresponding metrictensor is

Gij(x) = D−1ij (x) . (1)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

In High Angular Resolution Diffusion Imaging the ”diffusiontensor” is

D(x ,y) = Di1···inyi1 · · · yin , (2)

where

y = (y1,y2,y3) = (sin θ cosϕ, sin θ sinϕ, cos θ) (3)

and the inverse

G(x ,y) =D(x ,y)y∈S2

|D(x ,y)|2D(x ,y) . (4)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

Inverse in 2D.

-5 5

-6

-4

-2

2

4

6

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

Inverse in 3D

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

Finsler-normF (x , y) = (G(x , y))

1n . (5)

Finsler-metric

Gij =12∂2F 2

∂yi∂yj . (6)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Project Outline

Finsler-normF (x , y) = (G(x , y))

1n . (5)

Finsler-metric

Gij =12∂2F 2

∂yi∂yj . (6)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Orientation distribution function (ODF) is symmetric w.r.t. theorigin.

Example: A symmetric polynomial of order two.

basis 1 orthogonal{Y 0

0 ,Y−22 ,Y−1

2 ,Y 02 ,Y

12 ,Y

22 }, where Y m

` is a sphericalharmonic of order `.

basis 2 monomials of fixed order{y1y1,y1y2,y1y3,y2y2,y2y3,y3y3} .

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Relation of the two bases. SH vs. ”tensor”

cos(2ϕ) = (cosϕ)2 − (sinϕ)2 (7)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Fixed order monomial approach:

A linear combination of monomials of fixed order is easy tohomogenize.

(Ti1···inayi1 · · · ayin )1/n = a(Ti1···inyi1 · · · yin )1/n (8)

An intuitive extension of diffusion tensor

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Spherical harmonic approach:

Eigenfunctions of Laplace-Beltrami operator

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

T n(signal)(4)−−→ T n

τ (ODF )↓ (1) ↑ (3)∑n

k=0 ckY k (2)−−→∑n

k=0 e−k(k+1)τ2πPk (0)ckY k

(9)

(1) Clebsch-projection(2) Weight components to obtain regularized (scale τ ) ODF(3) Expand iteratively lower order components to nth order(4) Multiply T n with a matrix M

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Clebsch-projection:

Dn(q) :=

[ n2 ]∑

k=1

(−1)k−1

4k

Γ(n − k + q−22 )

k !Γ(n + q−22 )|y(q)|2k4k , (10)

”divides out the unities” ((y1) + (y2) + · · ·+ (yq) = 1).

Yn := Tn − (Dn(q)Tn) , (11)

where Yn is the harmonic part of Tn and (Dn(q)Tn) thenon-harmonic part.

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Further

Dn(q)Tn := |y(q)|2Tn−2 , (12)

and we can iteratively decompose Tn to lower order harmonics.

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

From raw signal to regularized ODF:

n∑k=0

ckY k =⇒n∑

k=0

e−k(k+1)τ2πPk (0)ckY k (13)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

Back to single tensor:

Ti1...ik =1k !

∑σ∈Sk

Tσ(i1)...σ(ik−2)Iσ(ik−1)σ(ik ), (14)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Efficient Computation of Regularized ODF

T n(signal)(4)−−→ T n

τ (ODF )↓ (1) ↑ (3)∑n

k=0 ckY k (2)−−→∑n

k=0 e−k(k+1)τ2πPk (0)ckY k

(15)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

Streamline tracking in DTI: c(t) = arg max|h|=1

(Dij(c(t))hihj)

c(0) = p .(16)

Streamline tracking in HARDI with Finsler-diffusion:c(t) = arg max

|h|=1(Dij(c(t), c(t))hihj) ,

c(0) = p ,c(0) = vα ,

(17)

where vα unit vectors in the nth order tessellation of the sphere.

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

Real HARDI data of brain:

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

Connectivity strength of fibers in DTI:

m(c) =

∫ √δij c i(t)c j(t)dt∫ √

gij(c(t))c i(t)c j(t)dt. (18)

Connectivity strength of fibers in HARDI:

m(c) =

∫ √δij c i(t)c j(t)dt∫ √

gij(c(t), c(t))c i(t)c j(t)dt(19)

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Finsler Streamline Fiber Tracking

Real HARDI data of brain:

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work

/centre for analysis, scientific computing and applications

Introduction to Applications Project Outline Recent Developments Summary and Future work

Summary

Reflection. . .

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Summary

Peer reviewed (to be reviewed) activities during the last year. . .

journals:L. Astola, A. Jalba, E. Balmashnova and L. Florack”Finsler Streamline Tracking with Single Tensor Orientation Distribution Function for High Angular ResolutionDiffusion Imaging”Submitted to Journal of Mathematical Imaging and VisionL. Astola and L. Florack”Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging”Submitted to International Journal of Computer Vision: minor revisions required.

conferences:L. Astola and L. Florack”Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging”In proceedings of Scale Space and Variational Methods: Second International Conference (SSVM 2009), June,2009, Norway.A. Fuster, L. Astola, and L. Florack”A Riemannian Scalar Measure for Diffusion Tensor Images” In Proceedings of 13th International Conference onComputer Analysis of Images and Patterns (CAIP09), September, 2009, Munster, Germany

thesis:L. Astola”Lattes type Uniformly Quasiregular Mappings on Compact Manifolds”

Licentiate Thesis, Helsinki University of Technology, February, 2009, Finland.

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Introduction to Applications Project Outline Recent Developments Summary and Future work

(Near) Future Work

Submission to ISBI (International Symposium onBiomedical Imaging, Rotterdam) 2010 ?Study various Finsler curvatures and compute them insome interesting data.

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Introduction to Applications Project Outline Recent Developments Summary and Future work

Questions?

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