Abstract
In this paper we explicitly derive a level set formulation for mean curvature flow in a Riemannian metric space. This extends the traditional geodesic active contour framework which is based on conformal flows. Curve evolution for image segmentation can be posed as a Riemannian evolution process where the induced metric is related to the local structure tensor. Examples on both synthetic and real data are shown.
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References
Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. Journal of Differential Geometry 23, 69–96 (1986)
Grayson, M.: The heat equation shrinks embedded plane curves to round points. Journal of Differential Geometry 26, 285–314 (1987)
Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22(1), 61–79 (1997)
Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Archives on Rational Mechanics and Analysis 134, 275–301 (1996)
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. Journal of Differential Geometry 33, 635–681 (1991)
Giga, Y., Chen, Y.G., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Journal of Differential Geometry 33, 749–786 (1991)
Ilmanen, T.: Generalized flow of sets by mean curvature on a manifold. Indiana J. Math. 41(3), 671–705 (1992)
Sarti, A., Citti, G.: Subjective surfaces and riemannian mean curvature flow of graphs. Acta Math. Univ. Comenianae LXX, 85–103 (2001)
DoCarmo, M.: Riemannian Geometry. Birkhäuser, Basel (1992)
Osher, S.J., Sethian, J.A.: Fronts propagation with curvature dependent speed: Algorithms based on hamilton-jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)
Estépar, R.S.J.: Local Structure Tensor for Multidimensional Signal Processing. Applications to Medical Image Analysis. PhD thesis, University of Valladolid, Spain (2005)
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© 2005 Springer-Verlag Berlin Heidelberg
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Estépar, R.S.J., Haker, S., Westin, CF. (2005). Riemannian Mean Curvature Flow. In: Bebis, G., Boyle, R., Koracin, D., Parvin, B. (eds) Advances in Visual Computing. ISVC 2005. Lecture Notes in Computer Science, vol 3804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11595755_75
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DOI: https://doi.org/10.1007/11595755_75
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30750-1
Online ISBN: 978-3-540-32284-9
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