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Riemannian and Finslerian Geometry for Diffusion Weighted Magnetic Resonance Imaging

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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 © Nicolas Lavarenne, l’exposition ‘A ciel ouvert’ Heuristics Finsler manifold. 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. © David Bao et al. © Hanno Rund et al. The Riemann-DTI paradigm DWMRI signal attenuation. E(x, q, ⌧)=exp[ ⌧D(x, q, ⌧)] − q = γ¯ g(t)dt q-space variable Z 2⇡iq ⇠ Propagator. P (x, ⇠, ⌧)= e · E(x, q, ⌧) dq Einstein 3 ZR diffusion tensor ∑ DTI. E (x, q, ⌧)=exp[ ⌧D (x, q, ⌧)] -convention 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)=δj 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)=δj 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)=δi 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. © C. Beaulieu, NMR Biomed. 15:7-8, 2002 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 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 © Andrea Fuster & Neda Sepasian (unpublished) ventricle infiltration smooth fibres The Riemann-DTI paradigm & geodesic tractography ‘100 % false posi2ves’ geodesic completeness = redundant connec2ons 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, λx˙)= λ F (x, x˙)((homogeneity)λ R) | | 2 (λ 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 © Dela Haije, “Finsler Geometry and Diffusion MRI”. PhD Thesis (2017) Notes. 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, #)δ(# q)q q d# = gij(x, q)q q g (x, #)δ(# 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, #)δ(# q)q q d# = gij(x, q)q q g (x, #)δ(# 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 © Florack, Dela Haije & Fuster. in: Hotz & Schultz, Springer 2015 DTI interpolation paradox DTI interpolation paradox DTI interpolation paradox DTI interpolation paradox DTI interpolation paradox 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.
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