Riemannian and Finslerian Geometry for Diﬀusion Weighted Magnetic Resonance Imaging

Riemannian and Finslerian geometry for diﬀusion weighted magnetic resonance imaging

Luc Florack

Computational Brain Connectivity Mapping Winter School Workshop 2017 - November 20-24, Jeans-les-Pins, France

A Finsler manifold is a space (M,F) of spatial base points x∈M, furnished with a notion of a ‘line’ or ‘length element’ ds ≐ F(x,dx). The ‘infinitesimal displacement vectors’ dx are ‘infinitely scalable’ into finite ‘tangent’ or ‘velocity vectors’ ẋ, viz. dx = ẋ dt. The collection of all (x, ẋ) is called the tangent bundle TM over M. Integrating the line element along a curve C produces the ‘length’ of that curve:

L (C)= ds = F (x, dx) ZC ZC

Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler Applications

Examples (anisotropic media).

Mechanics e.g. F(x,dx) := infinitesimal displacement, or infinitesimal travel time, etc. Optimal control e.g. F(x,dx) := local cost function for infinitesimal movement of Reeds-Shepp car Optics e.g. F(x,dx) := infinitesimal travel time for light propagation Seismology e.g. F(x,dx) := infinitesimal travel time for seismic ray propagation Ecology e.g. F(x,dx) := infinitesimal energy for coral reef state transition Relativity e.g. F(x,dx) := infinitesimal (pseudo-Finslerian) spacetime line element Diffusion MRI e.g. F(x,dx) := infinitesimal hydrogen spin diffusion Axiomatics

Literature.

DWMRI signal attenuation. E(x, q, ⌧)=exp[ ⌧D(x, q, ⌧)]

q = ¯ g(t)dt q-space variable  Z

2⇡iq ⇠ Propagator. P (x, ⇠, ⌧)= e · E(x, q, ⌧) dq

Einstein ∑ -convention 3 ZR diﬀusion tensor

DTI. E (x, q, ⌧)=exp[ ⌧D (x, q, ⌧)] ij DTI DTI DDTI(x, q, ⌧)=D (x)qiqj quadratic assumption…

2⇡iq ⇠ 1 1 i j PDTI(x, ⇠, ⌧)= e · EDTI(x, q, ⌧) dq = 3 exp Dij(x)⇠ ⇠ 3 2 4⌧ ZR p4⇡⌧  another quadratic form…

kj i Dik(x)D (x)=j

inverse diﬀusion tensor The Riemann-DTI paradigm

n c 2 = g cicj k k ij i,j=1 X

quadratic assumption… The Riemann-DTI paradigm

Xiang Hao

Lauren O’Donnell Christophe Lenglet Andrea Fuster

Ansatz [1,2].

kj i gij(x)=Dij(x) Dik(x)D (x)=j

Adaptations [3,4]. Riemann metric tensor inverse diﬀusion tensor

↵(x) gij(x)=e Dij(x)

kj j gij(x) = (adj D)ij(x) (adj D)ik(x)D (x)=i det D•(x)

References. References. adjugate diﬀusion tensor 1. Lauren O’Donnell et al, LNCS 2488:459-466 (2002) 2. Christophe Lenglet et al, LNCS 3024: 127-140 (2004) 3. Xiang Hao et al, LNCS 6801:13-24 (2011) 4. Andrea Fuster et al, JMIV 54: 1-14 (2016) The Riemann-DTI paradigm

Hypothesis.

Tissue microstructure imparts non-random barriers to water diﬀusion.

Conjecture.

Extrinsic diﬀusion on Euclidean space ≈ intrinsic geometry of a Riemannian space The Riemann-DTI paradigm

Conjecture.

Extrinsic diﬀusion on Euclidean space ≈ intrinsic geometry of a Riemannian space The Riemann-DTI paradigm

Conjecture.

Extrinsic diﬀusion on Euclidean space ≈ intrinsic geometry of a Riemannian space The Riemann-DTI paradigm

Terminology.

gauge figure = unit sphere = indicatrix = Riemannian metric = inner product The Riemann-DTI paradigm The Riemann-DTI paradigm The Riemann-DTI paradigm The Riemann-DTI paradigm The Riemann-DTI paradigm

length2 = 6 c 2 = g cicj k k ij The Riemann-DTI paradigm

length2 = 9 c 2 = g cicj k k ij The Riemann-DTI paradigm & geodesic tractography The Riemann-DTI paradigm & geodesic tractography

Riemannian length : Euclidean length

‘short’ geodesic 5.0 : 6.0 < 1

‘long’ geodesic 7.5 : 6.0 > 1 The Riemann-DTI paradigm & geodesic tractography

gij(x)=Dij(x) gij(x) = (adj D)ij(x)

tumour inﬁltration irregular ﬁbres cf. Pujol et al. The DTI Challenge, MICCAI 2015

ventricle inﬁltration smooth ﬁbres The Riemann-DTI paradigm & geodesic tractography

‘100 % false posives’

geodesic completeness = redundant connecons pro: pixels → geodesic congruences

→

ellipsoidal gauge ﬁgure = poor angular resoluon con: destructive interference of orientation preferences Riemannian and Finslerian geometry for diﬀusion weighted magnetic resonance imaging

DTI ⊂ generic models

Akward: Geometry built upon the limitations of a model…

⇳ ⇳

Riemann geometry ⊂ Finsler geometry Heuristics

Literature.

© J. Melonakos et al, “Finsler Tractography for White Matter Connectivity Analysis of the Cingulum Bundle”. MICCAI (2007) © J. Melonakos et al, “Finsler Active Countours”. PAMI 30:3 (2008) © De Boer et al., “Statistical Analysis of Minimum Cost Path based Structural Brain Connectivity”. NeuroImage 55:2 (2011) © Astola, “Multi-Scale Riemann-Finsler Geometry: Applications to Diffusion Tensor Imaging and High Angular Resolution Diffusion Imaging”. PhD Thesis (2010) © Astola & Florack, “Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion © Astola et al., “Finsler Streamline Tracking with Single Tensor Orientation Distribution Function for High Angular Resolution Diffusion Imaging”. JMIV 41:3 (2011) © Sepasian et al., “Riemann-Finsler Multi-Valued Geodesic Tractography for HARDI”. In: “Visualisation and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data”, Westin et al. (Eds.), Springer (2014) © Fuster & Pabst, “Finsler pp-Waves”. Phys. Rev. D 94:10 (2016) © Florack et al., “Riemann-Finsler Geometry for Diffusion Weighted Magnetics Resonance Imaging”. In: “Visualisation and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data”, Westin et al. (Eds.), Springer (2014) © Florack et al., “Direction-Controlled DTI Interpolation”. In: “Visualisation and Processing of Higher Order Descriptors for Multi-Valued Data”, Hotz et al. (Eds.), Springer (2015) © Dela Haije et al., “Structural Connectivity Analysis using Finsler Geometry” (submitted) Terminology

Tangent bundle.

TM = { (x,ẋ) | x∈M, ẋ∈TxM }

Slit tangent bundle.

TM\0 = { (x,ẋ)∈TM | ẋ≠0 }

Sphere bundle.

SM = { (x,ẋ)∈TM | F(x,ẋ)=1 }

(x,ẋ) Projectivized tangent bundle.

PTM = { (x,ẋ)∈TM | F(x,ẋ)=1 , ẋ~(-ẋ) } Finsler function

Finsler function. F (x, x˙)= F (x, x˙)((homogeneity) R) | | 2 ( R , x˙ =0, ⇠ = 0) 2 6 6 F (x, x˙) > 0(˙x = 0) (positivity) 6 @2F 2(x, x˙) ⇠i⇠j > 0(⇠ =(convexity) 0) @x˙ i@x˙ j 6

Notes.

The Finsler function ‘lives’ on the 2n-dimensional tangent bundle TM.

A Finsler function deﬁnes a (smoothly varying) local norm ||ẋ||x = F(x,ẋ) for a vector ẋ at anchor point x. The line integral (✻) is independent of curve parametrisation:

t+ t+ i j L (C)= ds = F (x, dx)= F (x(t), x˙(t))dt = gij(x(t), x˙(t(✻)))x ˙ (t)˙x (t)dt C C t t Z Z Z Z q Finsler metric

2 2 . 1 @ F (x, x˙) i j Finsler metric. gij(x, x˙) = ⟺ F (x, x˙)= gij(x, x˙)˙x x˙ 2 @x˙ i@x˙ j q

Notes.

The Finsler metric is a second order symmetric positive deﬁnite covariant tensor. The Finsler metric is homogeneous of degree 0. The Finsler metric ‘lives’ on the (2n-1)-dimensional projectivized tangent bundle PTM. Riemann metric

position only

2 2 1 @ FR (x, x˙) i j Riemann metric. g (x)= FR(x, x˙)= gij(x)˙x x˙ ij 2 @x˙ i@x˙ j ⟺ q Pythagorean rule

Notes.

A Riemann metric deﬁnes an inner product induced norm (‘Pythagorean rule’). Finsler geometry is ‘just’ Riemannian geometry without the quadratic assumption. The Finsler-DTI paradigm

DWMRI signal attenuation and propagator.

2⇡iq ⇠ E(x, q, ⌧)=exp[ ⌧D(x, q, ⌧)] P (x, ⇠, ⌧)= e · E(x, q, ⌧) dq 3 ZR

Dual Finsler function.

1 1 H2(x, q)= sup q x˙ F 2(x, x˙) 2 x˙ TM h | i2 2 x  i . ij H(x, q)=F (x, x˙) x˙ = g (x, q)qj

2 ij (i) Riemann-DTI paradigm ~ central limit theorem: H (x, q) D (x)q q / i j ij (ii) Finsler-DTI paradigm: cf. PhD thesis Tom Dela Haije X The Finsler-DTI paradigm

Finsler metric & dual Finsler metric.

2 2 2 2 ij 1 @ H (x, q) 1 @ F (x, x˙) g (x, q)= gij(x, x˙)= i j 2 @qi@qj 2 @x˙ @x˙

Figuratrix & indicatrix.

2 ij 2 i j H (x, q)=g (x, q)qiqj =1 F (x, x˙)=gij(x, x˙)˙x x˙ =1

Osculating ﬁguratrices & osculating indicatrices. # 2

ij i j g (x, #)qiqj =1 gij(x, #)˙x x˙ =1

Note.

gij(x, #)(# q)q q d# = gij(x, q)q q g (x, #)(# x˙)˙xix˙ j d# = g (x, x˙)˙xix˙ j i j i j ij ij ZT⇤Mx ZTMx The Finsler-DTI paradigm

Note.

gij(x, #)(# q)q q d# = gij(x, q)q q g (x, #)(# x˙)˙xix˙ j d# = g (x, x˙)˙xix˙ j i j i j ij ij ZT⇤Mx ZTMx

Interpretation.

The dual Finsler metric represents an orientation-parametrized family of DTI tensors of the kind considered in the Riemann-DTI paradigm. In the Riemannian limit all members of this family coincide.

# 2 Application: DTI interpolation

think out of the box…

Riemannian frame

Finslerian extension Riemann metric weighted averaging Finsler manifold

Deﬁnition. (0 ↵ 1)   2 i j (i) Fg (x, x˙)=gij(x)˙x x˙ input: two 3D-DTI tensors 2 i j (ii) Fh (x, x˙)=hij(x)˙x x˙ (iii) 2 2↵ 2(1 ↵) F (x, x˙)=Fg (x, x˙) Fh (x, x˙) 1 @2F 2(x, x˙) (iv) g (x, x˙)= (✻) output: one 5D-DTI tensor ij 2 @x˙ i@x˙ j

Claim.

(i) The tensor ( ✻ ) is a Finsler metric.

(ii) An analytical, closed-form solution exists.

Cartan tensor

. 1 @3F 2(x, x˙) Cartan tensor. C (x, x˙) = ijk 4 @x˙ i@x˙ j@x˙ k

Notes.

The Cartan tensor C is a symmetric third order covariant tensor on the slit tangent bundle (sphere bundle / projectivized tangent bundle).

The Cartan tensor is the ẋ-gradient of the metric tensor: Cijk(x, x˙)=@x˙ k gij(x, x˙) Deicke’s theorem: Space is Riemannian iﬀ the Cartan tensor vanishes identically. Cartan scalar maps

5D-DTI

ij k` mn = g (x, x˙)Cijk(x, x˙)g (x, x˙)C`mn(x, x˙)g (x, x˙)

i` jm kn = Cijk(x, x˙)(x, x˙)g (x, x˙)g (x, x˙)g (x, x˙)C`mn(x, x˙)

Notes.

These scalars live on the slit tangent bundle (sphere bundle / projectivized tangent bundle). They can be projected in various ways onto the spatial base manifold. They can be used as invariant local or tractometric features. They are (locally) nontrivial iﬀ the Riemann-DTI model (locally) fails (Deicke’s theorem). = 0

= 0

≠ 0

≠ 0 Tractography from 5D??? HV-splitting

Horizontal versus vertical transport:

V-type: Spinning without walking. H-type: Spinning-walking preserving a forward gaze. V-type

H/V-generators: @ V-type: @x˙ j

. @ j @ H-type: = N (x, x˙) H-type xi @xi i @x˙ j

Nonlinear connection.

A ‘nonlinear connection’ is needed to ensure a geometrically meaningful HV-splitting.

j j k Riemannian limit (linear connection): Ni (x, x˙)=ik(x)˙x

‘Christoﬀel symbols of the 2nd kind’:

. 1 @g`k(x) @gi`(x) @gik(x) j (x) = gj`(x) + ik 2 @xi @xk @x`  HV-splitting

Nonlinear connection.

Finslerian case: nonlinear correction terms, involving the Cartan tensor:

N i(x, x˙)=i (x, x˙)˙xk Ci (x, x˙)k (x, x˙)˙x`x˙ m j jk jk `m

formal Christoﬀel symbols of the 2nd kind

i i @G (x, x˙) i 1 i j k 1 i j N (x, x˙)= ⟺ G (x, x˙)= jk(x, x˙)x˙ x˙ = Nj (x, x˙)˙x (✻) j @x˙ j 2 2

geodesic spray coeﬃcients HV-splitting

Rate of change along a curve in TM\0.

d @ @ def d f(x(t + s),y(t + s)) =˙xi(t) f(x(t),y(t)) +y ˙i(t) f(x(t),y(t)) ˙ = ds @xi @yi s=0 dt  d @ f(x(t + s),y(t + s)) =˙xi(t) f(x(t),y(t)) + y˙i(t)+N i(x(t),y(t))x ˙ j(t) f(x(t),y(t)) ds xi j @yi s=0 ⇥ ⇤ @ @xi HV-splitting

Rate of change along a curve in TM\0.

d @ @ f(x(t + s),y(t + s)) =˙xi(t) f(x(t),y(t)) +y ˙i(t) f(x(t),y(t)) ds @xi @yi s=0 d @ f(x(t + s),y(t + s)) =˙xi(t) f(x(t),y(t)) + y˙i(t)+N i(x(t),y(t))x ˙ j(t) f(x(t),y(t)) ds xi j @yi s=0 ⇥ ⇤ H-component V-component

H/V curves.

Vertical curve: x˙ i(t)=0

i i j Horizontal curve: y˙ (t)+Nj (x(t),y(t))x ˙ (t)=0

i i j Constant Finslerian speed geodesic (y=ẋ): x¨ (t)+Nj (x(t), x˙(t))x ˙ (t)=0 x¨i(t)+2Gi(x(t), x˙(t)) = 0

Finslerian ‘pseudo-force’ Geodesics

Geodesic (global deﬁnition).

A geodesic is a ‘locally’ shortest path:

L (C)= F (x, dx) min ! ZC

d ln F (x, x˙) x¨i +2Gi(x, x˙)= x˙ i dt

Recall (local deﬁnition).

Constant Finslerian speed geodesics: x¨i +2Gi(x, x˙)=0 Always possibly by slick choice of parametrization (e.g. ‘arclength’, i.e. such that F(x,ẋ)=1). The Riemann-DTI paradigm

Conjecture. neural fiber bundles correspond to relatively short geodesics in a Riemannian ‘brain space’ the Riemannian structure can be inferred from ‘3D-DTI’ The Finsler-DTI paradigm

Conjecture. neural fiber bundles correspond to relatively short geodesics in a Finslerian ‘brain space’ the Finslerian structure can be inferred from ‘5D-DTI’ Euclidean

Riemannian

Finslerian

Diffusion Weighted Magnetic Resonance Imaging

(x,p1), (x,p2), (x,p3),... The Riemann-DTI paradigm & geodesic tractography

2 G(v, v)= v Riemann metric: lengths & angles k k

x˙ =0 Levi-Civita connection: parallel transport rx˙

i i j k Christoffel symbols: “pseudo-forces” (relative to local coordinate frames) x¨ + jk(x) x˙ x˙ =0 Xjk The Riemann-DTI paradigm & geodesic tractography

2 G(v, v)= v Riemann metric: lengths & angles k k

x˙ =0 Levi-Civita connection: parallel transport rx˙

i i j k Christoffel symbols: “pseudo-forces” (relative to local coordinate frames) x¨ + jk(x) x˙ x˙ =0 Xjk The Riemann-DTI paradigm & geodesic tractography

2 G(v, v)= v Riemann metric: lengths & angles k k

x˙ =0 Levi-Civita connection: parallel transport rx˙

i i j k Christoffel symbols: “pseudo-forces” (relative to local coordinate frames) x¨ + jk(x) x˙ x˙ =0 Xjk

Euclidean geodesic The Riemann-DTI paradigm & geodesic tractography

2 G(v, v)= v Riemann metric: lengths & angles k k

x˙ =0 Levi-Civita connection: parallel transport rx˙

i i j k Christoffel symbols: “pseudo-forces” (relative to local coordinate frames) x¨ + jk(x) x˙ x˙ =0 Xjk

Riemannian geodesic Riemann metric weighted averaging Finsler manifold

2 gij(x, x˙)=F (x, x˙)ijij((x,x,xx˙˙)=)

÷ + ÷ + ÷ + ÷ + ÷ + ÷

↵ 2↵(1 ↵) 2↵(1 ↵) 2↵(1 ↵) 2↵(1 ↵) 1 ↵

i j j hij(x)= hij(x)˙x x˙ = hij(x)˙x =

i j j gij(x)= gij(x)˙x x˙ = gij(x)˙x =