Journal
INTERNATIONAL JOURNAL OF COMPUTER VISION
Volume 92, Issue 3, Pages 231-264Publisher
SPRINGER
DOI: 10.1007/s11263-010-0332-z
Keywords
High angular resolution diffusion imaging (HARDI); Scale spaces; Lie groups; Partial differential equations
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Funding
- Netherlands Organization for Scientific Research (NWO)
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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 ae(3) a < S 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.
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