/centre for analysis, scientific computing and applications Intro Tensor fitting Finsler Geometry Applications Future work Multi-scale Differential Geometry and Applications CASA PhD-day Laura Astola 13 November 2008
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Intro Tensor fitting Finsler Geometry Applications Future work
Multi-scale Differential Geometry andApplicationsCASA PhD-day
Laura Astola
13 November 2008
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
Introduction
Main application: The human brain white matter architecture.
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Intro Tensor fitting Finsler Geometry Applications Future work
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)
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Intro Tensor fitting Finsler Geometry Applications Future work
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)
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Intro Tensor fitting Finsler Geometry Applications Future work
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)
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Intro Tensor fitting Finsler Geometry Applications Future work
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)
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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)
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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)
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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)
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Intro Tensor fitting Finsler Geometry Applications Future work
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 )
cos2(π2 ) 2 cos(π2 ) sin(π2 ) sin2(π2 )
d11
d12d22
=
m(0)m(π4 )m(π2 )
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Intro Tensor fitting Finsler Geometry Applications Future work
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 )
cos2(π2 ) 2 cos(π2 ) sin(π2 ) sin2(π2 )
d11
d12d22
=
m(0)m(π4 )m(π2 )
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Intro Tensor fitting Finsler Geometry Applications Future work
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 )
cos2(π2 ) 2 cos(π2 ) sin(π2 ) sin2(π2 )
d11
d12d22
=
m(0)m(π4 )m(π2 )
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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 )
cos2(π2 ) 2 cos(π2 ) sin(π2 ) sin2(π2 )
d11
d12d22
=
m(0)m(π4 )m(π2 )
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Intro Tensor fitting Finsler Geometry Applications Future work
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 ϕ42 4 cos3 ϕ2 sinϕ2 6 cos2 ϕ2 sin2 ϕ2 4 cosϕ2 sin3 ϕ2 sin4 ϕ2
.
.
.
.
.
.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 . . .
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Intro Tensor fitting Finsler Geometry Applications Future work
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 ϕ42 4 cos3 ϕ2 sinϕ2 6 cos2 ϕ2 sin2 ϕ2 4 cosϕ2 sin3 ϕ2 sin4 ϕ2
.
.
.
.
.
.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 . . .
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Intro Tensor fitting Finsler Geometry Applications Future work
Introduction
Order of the tensor ≤ 2, Order of the tensor > 2,
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Intro Tensor fitting Finsler Geometry Applications Future work
Introduction
A sixth order totallysymmetric tensor Dijklmn,i , j , k , l ,m,n = 1,2,3 incolors.
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
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.
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Intro Tensor fitting Finsler Geometry Applications Future work
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.
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
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.
/centre for analysis, scientific computing and applications
Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
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
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Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
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)
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Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
Direct D, assigning a Finslernorm F is easy !
F (y) = (Di1···iny i1 · · · y in)(1/n)
(4)
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Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
(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.
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Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
(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.
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Intro Tensor fitting Finsler Geometry Applications Future work
Direct fitting vs. hierarchical fitting
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
Finsler geometry, in applications
Optics
Seismology
Mathematical ecology
Relativity theory
Medical field
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Intro Tensor fitting Finsler Geometry Applications Future work
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.
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Intro Tensor fitting Finsler Geometry Applications Future work
Finsler geometry, with a tensor-valued F
Differentiability, OK.
Homogeneity, OK.
Strong convexity, OK with a condition on unit level set.
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Intro Tensor fitting Finsler Geometry Applications Future work
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.
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
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 .
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Intro Tensor fitting Finsler Geometry Applications Future work
Bunch of metric tensors in a higher order tensor
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Intro Tensor fitting Finsler Geometry Applications Future work
”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.
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Intro Tensor fitting Finsler Geometry Applications Future work
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
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Intro Tensor fitting Finsler Geometry Applications Future work
soon Track Finsler-fibers in HARDI data
next Compute various Finsler curvatures insome interesting data
. . . later ? Study Laplace-Beltrami smoothing vs. Ricci flow.
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Intro Tensor fitting Finsler Geometry Applications Future work
Questions?
?