On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
Several recently introduced and studied planar curve evolutionequations turn out to be iterative smoothing procedures that areinvariant under the actions of the Euclidean and affine groups ofcontinuous transformations. This paper discusses possible ways toextend these results to the projective group of transformations.Invariant polygon evolutions are also investigated.
- L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Axioms and fundamental equations of image processing,” Archive of Rational Mechanics, Vol. 123, pp. 199–257, 1993.
- S. Angenent, “Parabolic equations for curves and surfaces Part II. Intersections, blow-up and generalized solutions,” Annals of Math., Vol. 133, pp. 171–215, 1991.
- J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duda, “Uniqueness of the Gaussian kernel for scale-space filtering,” IEEE PAMI, Vol. 8, pp. 26–33, 1986.
- W. Blaschke, Vorlesungen uber Differezialgeometrie II, Verlag von Julius Springer: Berlin, 1923.
- A.M. Bruckstein, R.J. Holt, A.N. Netravali, and T.J. Richardson, “Invariant signatures for planar shape recognition under partial occlusion,” CVGIP Image Understanding, Vol. 58, pp. 49–65, 1993.
- A.M. Bruckstein, N. Katzir, M. Lindenbaum, and M. Porat, “Similarity invariant recognition of partly occluded planar curves and shapes,” Int. J. Comput. Vision, Vol. 7, pp. 271–285, 1992.
- A.M. Bruckstein and A.N. Netravali, “On differential invariants of planar curves and the recognition of partly occluded planar shapes,” AT&T Technical Memo, July 1990; also in Int. Workshop on Visual Form, Capri, May 1992.
- A.M. Bruckstein, G. Sapiro, and D. Shaked, “Affine invariant evolutions of planar polygons,” CIS Report No. 9202, Computer Science Department, Technion, I.I.T., Haifa 32000, Israel, 1992.
- A.M. Bruckstein and D. Shaked, “On projective invariant smoothing and evolution of planar curves and polygons,” CIS Report No. 9328, Computer Science Department, Technion, Israel, Nov. 1993.
- A.M. Bruckstein and D. Shaked, “On projective invariant smoothing and evolution of planar curves,” Aspects of Visual Form Processing, C. Arcelli, L.P. Cordella, and G. Sanniti di Baja (Eds.), World Scientific, pp. 109–118, 1994.
- S. Buchin, Affine Differential Geometry, Science Press: Beijing, China, Gordon & Breach Science: New York, 1983.
- M.G. Darboux, “Sur un probleme de geometrie elementaire,” Bull. Sci. Math., Vol. 2, pp. 298–304, 1878.
- F. Dibos, “Projective invariant multiscale analysis,” CEREMADE Report 9533, Universite Paris 9 Dauphine, 1995.
- C.L. Epstein and M. Gage, “The curve shortening flow,” in Wave Motion, A.J. Chorin and A.J. Madja (Eds.), Springer Verlag, 1987.
- O. Faugeras, “On the evolution of simple curves of the real projective plane,” Comptes Rendus Acad. Sci. (Paris), Vol. 317, pp. 565–570, 1993.
- O. Faugeras, “Cartan's moving frame method and its application to the geometry and evolution of curves in the euclidean, affine, and projective planes,” INRIA TR-2053, Sept. 1993.
- M. Gage and R. Hamilton, “The shrinking of convex plane curves by the heat equation,” J. of Diff. Geometry, Vol. 23, pp. 69–96, 1986.
- M. Grayson, “The heat equation shrinks embedded plane curves to round points,” J. of Diff. Geometry, Vol. 26, pp. 285–314, 1987.
- B.B. Kimia, A.R. Tannenbaum, and S.W. Zucker, “Shapes, shocks, and deformations I: The components of shape and the reaction-diffusion space,” Int. J. of Computer Vision, Vol. 15, pp. 189–224, 1995.
- J.J. Koenderink, “The structure of images,” Biological Cybernetics, Vol. 50, pp. 363–370, 1984.
- E.P. Lane, A Treatise on Projective Differential Geometry, Univ. of Chicago Press: Chicago, 1941.
- F. Mokhtarian and A.K. Mackworth, “Scale-based description and recognition of planar curves and two dimensional shapes,” IEEE Trans. on PAMI, Vol. 8, pp. 34–43, 1986.
- F. Mokhtarian and A.K. Mackworth, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE Trans. on PAMI, Vol. 14, pp. 789–805, 1992.
- T. Moons, E.J. Pauwels, L. Van Gool, and A. Oosterlinck, “Foundations of semi differential invariants,” Int. J. of Computer Vision, Vol. 14, pp. 25–47, 1995.
- P. Olver, G. Sapiro, and A. Tannenbaum, “Differential invariant signatures and flows in computer vision,” Comptes Rendus Acad. Sci. (Paris), Vol. 319, pp. 339–344, 1994 (see also, “Classification and Uniqueness of Invariant Geometric Flows,” MIT-LIDS Technical Report, 1993).
- B.M. ter Haar Romeny, Geometry Drive Diffusion in Computer Vision, Kluwer: Netherlands, 1994.
- G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,” Int. J. of Computer Vision, Vol. 11, pp. 25–44, 1993.
- G. Sapiro and A. Tannenbaum, “Area and length preserving geometric invariant scale-spaces,” IEE PAMI, Vol. 17, pp. 67–72, 1995.
- R. Schwartz, “The pentagram map,” Experimental Mathematics, Vol. 1, pp. 71–81, 1992.
- L. Van Gool, T. Moons, E. Pauwels, and A. Oosterlink, “Semi-differential invariants,” DARPA/ESPRIT Workshop on Invariants, Reykjavik, Iceland, 1991.
- I. Weiss, “Projective invariants of shape,” Center for Automation Research Report, CAR-TR-339, 1988.
- I. Weiss, “Noise resistant invariants of curves,” IEEE Trans. on PAMI, Vol. 15, pp. 943–948, 1993.
- I. Weiss, “Geometric invariants and object recognition,” Int. J. of Computer Vision, Vol. 10, pp. 207–231, 1993.
- E.J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, 1906.
- A.P. Witkin, “Scale-space filtering,” Int. Joint Conf. Art. Intelligence, pp. 1019–1021, 1983.
- On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons
Journal of Mathematical Imaging and Vision
Volume 7, Issue 3 , pp 225-240
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- shape analysis
- projective invariants
- curve and polygon smoothing
- geometric diffusions
- invariant signatures
- Industry Sectors