Spatial instability of boundary layer along impedance wall

S.W. Rienstra, G.G. Vilenski

Research output: Book/ReportReportAcademic

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Abstract

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulae and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diffusions on HARDI-data we put emphasis on 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.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages13
Publication statusPublished - 2008

Publication series

NameCASA-report
Volume0818
ISSN (Print)0926-4507

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Impedance
Boundary Layer
High Angular Resolution
Invariant
Imaging
Cartan Connection
Euclidean
Convection-diffusion
Green's function
Gaussian Estimates
Covariant Derivative
Kolmogorov Equation
Motion
Magnetic Resonance Imaging
Differential Geometry
Diffusion equation
Muscle
Diffusion Process
Torsion
Brownian motion

Cite this

Rienstra, S. W., & Vilenski, G. G. (2008). Spatial instability of boundary layer along impedance wall. (CASA-report; Vol. 0818). Eindhoven: Technische Universiteit Eindhoven.
Rienstra, S.W. ; Vilenski, G.G. / Spatial instability of boundary layer along impedance wall. Eindhoven : Technische Universiteit Eindhoven, 2008. 13 p. (CASA-report).
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Rienstra, SW & Vilenski, GG 2008, Spatial instability of boundary layer along impedance wall. CASA-report, vol. 0818, Technische Universiteit Eindhoven, Eindhoven.

Spatial instability of boundary layer along impedance wall. / Rienstra, S.W.; Vilenski, G.G.

Eindhoven : Technische Universiteit Eindhoven, 2008. 13 p. (CASA-report; Vol. 0818).

Research output: Book/ReportReportAcademic

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Rienstra SW, Vilenski GG. Spatial instability of boundary layer along impedance wall. Eindhoven: Technische Universiteit Eindhoven, 2008. 13 p. (CASA-report).