Left-Invariant Diffusions on the Space of Positions and Orientations and their Application to Crossing-Preserving Smoothing of HARDI images

Abstract

HARDI (High Angular Resolution Diffusion Imaging) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by R 3 S 2 -convolution with the corresponding Green's functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green's functions. In our design and analysis for appropriate (nonlinear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our leftinvariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our leftinvariant diffusion takes place. We provide experiments of R. Duits ( )

Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

“…4.2] on R 3 S 2 yield reasonable fiber extrapolations, cf. [36,65,67,68]. This can also be observed in Fig.…”

“…These diffusions are solved by shift-twist convolutions with the corresponding Green's functions, cf. [36]. This generalizes the well-known, efficient framework of tensor voting for contextual processing of tensors [57,58] in a PDE (scale space) setting on R 3 S 2 .…”

“…(3). This may be used to reduce the number of scanning directions in areas where the (hypo-elliptic) diffusion [36,Chap. 4.2] on R 3 S 2 yield reasonable fiber extrapolations, cf.…”

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

“…Despite intriguing heuristic analogies it remains an open problem how to exactly relate the present RiemannFinsler framework to the Cartan geometric approach by Duits et al [27,28,29]. A suitable construction of the n-beins of Section 2.10 might provide the necessary insight.…”

“…Clinical use of dwMRI is hampered by the fact that dwMRI analysis requires radically new approaches, based on abstract representations, a development still in its infancy. Examples are rank-2 symmetric positive-definite tensor representations in diffusion tensor imaging (DTI), pioneered by Basser, Mattiello and Le Bihan et al [1,2] and explored by many others [3,4,5,6,7,8,9,10,11,12,13,14], higher order symmetric positive-definite tensor representations [15,16,17,18,19,20,21], spherical harmonic representations in high angular resolution diffusion imaging (HARDI) [22,23,24,25,26], and SE(3) Lie group representations [27,28,29]. The latter type of representation, developed by Duits et al, appears to bear a particularly close relationship to the theory outlined below.…”