Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images

R. Duits, E.M. Franken

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52 Citaten (Scopus)
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Samenvatting

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 \mbox{(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 $\R^3 \rtimes S^{2}$-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 left-invariant 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 left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers.
Originele taal-2Engels
Pagina's (van-tot)231-264
Aantal pagina's34
TijdschriftInternational Journal of Computer Vision
Volume92
Nummer van het tijdschrift3
DOI's
StatusGepubliceerd - 2011

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