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Introduction to Applications Project Outline Recent Developments Summary and Future work

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

High Angular Resolution Diffusion Imaging

Laura Astola

CASA PhD-day 28. October 2009



Outline

1 Introduction to Applications

2 Project Outline

3 Recent Developments

4 Summary and Future Work



Introduction to Applications



Outline


2 Project Outline





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)



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)



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)



Project Outline

Inverse in 2D.

-5 5

-6

-4

-2

2

4

6



Project Outline

Inverse in 3D



Project Outline

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

1n . (5)

Finsler-metric

Gij =12∂2F 2

∂yi∂yj . (6)



Outline


2 Project Outline





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} .




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

cos(2ϕ) = (cosϕ)2 − (sinϕ)2 (7)




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




Spherical harmonic approach:

Eigenfunctions of Laplace-Beltrami operator




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




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.




Further

Dn(q)Tn := |y(q)|2Tn−2 , (12)

and we can iteratively decompose Tn to lower order harmonics.




From raw signal to regularized ODF:

n∑k=0

ckY k =⇒n∑

k=0

e−k(k+1)τ2πPk (0)ckY k (13)




Back to single tensor:

Ti1...ik =1k !

∑σ∈Sk

Tσ(i1)...σ(ik−2)Iσ(ik−1)σ(ik ), (14)




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)



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.




Real HARDI data of brain:




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)




Real HARDI data of brain:



Outline


2 Project Outline





Summary

Reflection. . .



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.



(Near) Future Work

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



Questions?

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