Multi-scale Differential Geometry and Applications - CASA PhD-day€¦ · 13-11-2008 · Multi-scale Differential Geometry and Applications CASA PhD-day Laura Astola 13 November

/centre for analysis, scientific computing and applications

Intro Tensor fitting Finsler Geometry Applications Future work

Multi-scale Differential Geometry andApplicationsCASA PhD-day

Laura Astola

13 November 2008



Outline

1 Introduction

2 Two ways of fitting a HOT to HARDI data

3 Finsler Geometry on a HOT field

4 Applications in the analysis of HARDI data

5 Goals in near future



Outline

1 Introduction







Introduction

Main application: The human brain white matter architecture.



Introduction

MRI: Diagnosis of brain tumor, Multiple Sclerosis,Schizophrenia(a scalar per voxel)

DTI: Promising in diagnosis of Alzheimer, pre-operativetract detectionOne dominant fiber direction, may be found(3× 3 two-tensor per voxel)

HARDI:Several fiber directions, may be found(4.− 8.th order tensor per voxel)



Introduction






Introduction






Introduction






Introduction






Introduction






Introduction






Introduction

To find the diffusion tensor D, in

S(y) = S0e−b·yT Dy ,

D =

(d11 d12d12 d22

), m(ϕ) = −1

b ln(SϕS0

),

we solve

(cosϕ sinϕ

) (d11 d12d12 d22

)(cosϕsinϕ

)= m(ϕ) ,

i.e. cos2(0) 2 cos(0) sin(0) sin2(0)

cos2(π4 ) 2 cos(π4 ) sin(π4 ) sin2(π4 )


d11

d12d22

=

m(0)m(π4 )m(π2 )



Introduction



D =

(d11 d12d12 d22

), m(ϕ) = −1

b ln(SϕS0

),

we solve

(cosϕ sinϕ

) (d11 d12d12 d22

)(cosϕsinϕ

)= m(ϕ) ,

i.e. cos2(0) 2 cos(0) sin(0) sin2(0)



d11

d12d22

=

m(0)m(π4 )m(π2 )



Introduction



D =

(d11 d12d12 d22

), m(ϕ) = −1

b ln(SϕS0

),

we solve

(cosϕ sinϕ

) (d11 d12d12 d22

)(cosϕsinϕ

)= m(ϕ) ,

i.e. cos2(0) 2 cos(0) sin(0) sin2(0)



d11

d12d22

=

m(0)m(π4 )m(π2 )



Introduction



D =

(d11 d12d12 d22

), m(ϕ) = −1

b ln(SϕS0

),

we solve

(cosϕ sinϕ

) (d11 d12d12 d22

)(cosϕsinϕ

)= m(ϕ) ,

i.e. cos2(0) 2 cos(0) sin(0) sin2(0)



d11

d12d22

=

m(0)m(π4 )m(π2 )



Introduction

More than 5 measurements ?

Solve a four-tensor dijkl , best (in L2) approximating theODF-data :

cos4 ϕ41 4 cos3 ϕ1 sinϕ1 6 cos2 ϕ1 sin2 ϕ1 4 cosϕ1 sin3 ϕ1 sin4 ϕ1


.

.

.

.

.

.cos4 ϕ4

5 4 cos3 ϕ5 sinϕ5 6 cos2 ϕ5 sin2 ϕ5 4 cosϕ5 sin3 ϕ5 sin4 ϕ5

d1111d1112d1122d1222d2222

=

ODF (ϕ1)ODF (ϕ2)ODF (ϕ3)ODF (ϕ4)ODF (ϕ5)

More than seven measurements ?

Solve a sixth order tensor dijklmn and so forth . . .



Introduction

More than 5 measurements ?

Solve a four-tensor dijkl , best (in L2) approximating theODF-data :



.

.

.

.

.

.cos4 ϕ4

5 4 cos3 ϕ5 sinϕ5 6 cos2 ϕ5 sin2 ϕ5 4 cosϕ5 sin3 ϕ5 sin4 ϕ5

d1111d1112d1122d1222d2222

=

ODF (ϕ1)ODF (ϕ2)ODF (ϕ3)ODF (ϕ4)ODF (ϕ5)

More than seven measurements ?

Solve a sixth order tensor dijklmn and so forth . . .



Introduction

Order of the tensor ≤ 2, Order of the tensor > 2,



Introduction

A sixth order totallysymmetric tensor Dijklmn,i , j , k , l ,m,n = 1,2,3 incolors.



Outline

1 Introduction







Direct fitting vs. hierarchical fitting

Previous examples were direct n.th order monomial fittings

Fitting can also be done hierarchically

As functions on the sphere, these two are equivalent.
















Hierarchical D,

D(y) =∞∑

n=0

Di1...iny i1 · · · y in n ∈ 2N .

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

Direct D,

D(y) = Di1···iny i1 · · · y in . (2)

cos(6ϕ− π7 ) + 3.4




Hierarchical D, Laplace-Beltrami smoothing is easy !

Dτ (y) = e−τ(2+1)2Dijy iy j + e−τ(4+1)4Dijkly iy jyky l + · · ·

=∞∑

k=0

e−τk(k+1)Di1...iny i1 · · · y in (3)




Direct D, assigning a Finslernorm F is easy !

F (y) = (Di1···iny i1 · · · y in)(1/n)

(4)




(y1)2 + (y2)2 + (y3)2 = 1.

Example, a fourth order tensor.

Dijkly iy jyky l = D4ijklyiy jyky l + D2ijy

iy j + D0 =⇒ (5)

Dijkl = Dijkl

+∑

i,j,k ,(i≤j)

1µ(ijkk)

(µ∑

N=1

DiσN jσN kσN kσN

)|k=l

+∑i,k

1µ(iikk)

(µ∑

N=1

DiσN iσN kσN kσN

)|k=l,i=j ,

(6)

where µ(ijkl) is the number of permutations of (i , j , k , l) and σN theN.th permutation.




(y1)2 + (y2)2 + (y3)2 = 1.

Example, a fourth order tensor.

Dijkly iy jyky l = D4ijklyiy jyky l + D2ijy

iy j + D0 =⇒ (5)

Dijkl = Dijkl

+∑

i,j,k ,(i≤j)

1µ(ijkk)

(µ∑

N=1

DiσN jσN kσN kσN

)|k=l

+∑i,k

1µ(iikk)

(µ∑

N=1

DiσN iσN kσN kσN

)|k=l,i=j ,

(6)

where µ(ijkl) is the number of permutations of (i , j , k , l) and σN theN.th permutation.






Outline

1 Introduction







Finsler geometry, roughly

Vector space Rn

Inner product 〈u,v〉, in Rn

Norm ||y || in Rn

Manifold M

Riemann metricgx = 〈u, v〉x , x ∈ M

Finsler metric F (x , y),x ∈ M, y ∈ TxM




Vector space Rn


Norm ||y || in Rn

Manifold M






Vector space Rn


Norm ||y || in Rn

Manifold M





Finsler geometry, in applications

Optics

Seismology

Mathematical ecology

Relativity theory

Medical field



Finsler structure, definition

Let M be n-dimensional C∞ manifold, TxM the tangent spaceat x ∈ M, TM = {(x , y) | x ∈ M, y ∈ TxM} .

Finsler structure on M is a function F : TM → [0,∞) satisfying:

Differentiability: F is C∞ in TM.

Homogeneity: F (λy) = λF (y), ∀λ > 0.

Strong convexity: 12∂2F 2

∂y i∂y j positive definite.



Finsler geometry, with a tensor-valued F

Differentiability, OK.

Homogeneity, OK.

Strong convexity, OK with a condition on unit level set.



Finsler geometry, with tensor-valued F

Let g(θ, ϕ) satisfy F (g) = 1. Define matrices

m =

g1 g2 g3

g1θ g2

θ g3θ

g1ϕ g2

ϕ g3ϕ

,mθ =

g1θ g2

θ g3θ

g1θ g2

θ g3θ

g1ϕ g2

ϕ g3ϕ

,m =

g1ϕ g2

ϕ g3ϕ

g1θ g2

θ g3θ

g1ϕ g2

ϕ g3ϕ

.

Then

det(mθ)

det(m)> 0 and

det(mϕ)

det(m)>

(gij y iθy

jϕ)2

gij y iθy

jθ

(7)

implies strong convexity.



Outline

1 Introduction







Whenever strongconvexity is satisfied, weobtain directional metrictensors gij(y) = ∂2F 2

∂y i∂y j .

A sixth order tensor withone of its metric tensors.Blue line is principaleigenvector. Red lineparameter y .



Bunch of metric tensors in a higher order tensor



”If 〈c(t),vg(x(t), y(t))〉 > αproceed,else stop”

Tracking in subthalamicnucleus-area of a rat brain, 20steps with step-size 0.2/voxelheight.



Outline

1 Introduction







soon Track Finsler-fibers in HARDI data

next Compute various Finsler curvatures insome interesting data

. . . later ? Study Laplace-Beltrami smoothing vs. Ricci flow.



Questions?

?

Documents

Multi-scale Differential Geometry and Applications - CASA PhD-day€¦ · 13-11-2008 · Multi-scale Differential Geometry and Applications CASA PhD-day Laura Astola 13 November