Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Left-invariant parabolic evolutions on $ SE(2)$ and contour enhancement via invertible orientation scores Part I: Linear left-invariant diffusion equations on $ SE(2)$

Authors: Remco Duits and Erik Franken
Journal: Quart. Appl. Math. 68 (2010), 255-292
MSC (2000): Primary 58J65, 49Q20, 22E30; Secondary 34B30, 37L05, 47D06
DOI: https://doi.org/10.1090/S0033-569X-10-01172-0
Published electronically: February 18, 2010
MathSciNet review: 2663001
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Abstract: We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group $ SE(2)=\mathbb{R}^2 \rtimes \mathbb{T}$. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green's functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green's functions (i.e., the heat kernels on $ SE(2)$) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on $ SE(2)$ by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Green's functions. The exact Green's functions are used in so-called collision distributions on $ SE(2)$, which are the product of two left-invariant resolvent diffusions given an initial distribution on $ SE(2)$. We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.

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Additional Information

Remco Duits
Affiliation: Department of Mathematics/Computer Science and Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: R.Duits@tue.nl

Erik Franken
Affiliation: Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
Email: E.M.Franken@tue.nl

DOI: https://doi.org/10.1090/S0033-569X-10-01172-0
Keywords: Forward Kolmogorov equations, image analysis, harmonic analysis on Lie groups
Received by editor(s): March 14, 2008
Published electronically: February 18, 2010
Additional Notes: The Netherlands Organization for Scientific Research is gratefully acknowledged for financial support.
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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