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 II: Nonlinear left-invariant diffusions on invertible orientation scores

Authors: Remco Duits and Erik Franken
Journal: Quart. Appl. Math. 68 (2010), 293-331
MSC (2000): Primary 58J65, 49Q20, 22E30; Secondary 34B30, 37L05, 47D06
DOI: https://doi.org/10.1090/S0033-569X-10-01173-3
Published electronically: February 18, 2010
MathSciNet review: 2663002
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Abstract: By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group $ SE(2)$ and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard ``coherence enhancing diffusion'' equations on images as they can enhance images with crossing contours. Here we employ differential geometry on $ SE(2)$ to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions.

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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-10-01173-3
Keywords: Nonlinear diffusion equations on Lie groups, Cartan connection, fiber bundles, image analysis, coherence enhancing diffusion
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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