2010 # 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 spac…

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

confidence: 66%

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

confidence: 66%

“…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 .…”

confidence: 56%

“…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.…”

confidence: 97%

“…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.…”

confidence: 99%