On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons

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.

[1]  G. Darboux,et al.  Sur un problème de géométrie élémentaire , 1878 .

[2]  Farzin Mokhtarian,et al.  Scale-Based Description and Recognition of Planar Curves and Two-Dimensional Shapes , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Guillermo Sapiro,et al.  Area and Length Preserving Geometric Invariant Scale-Spaces , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  M. Grayson The heat equation shrinks embedded plane curves to round points , 1987 .

[5]  Isaac Weiss,et al.  Geometric invariants and object recognition , 1993, International Journal of Computer 11263on.

[6]  Alfred M. Bruckstein,et al.  Invariant signatures for planar shape recognition under partial occlusion , 1992, [1992] Proceedings. 11th IAPR International Conference on Pattern Recognition.

[7]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[8]  Farzin Mokhtarian,et al.  A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Françoise Dibos Projective invariant multiscale analysis , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[10]  V. G. Grove,et al.  A Treatise On Projective Differential Geometry , 1942 .

[11]  Luc Van Gool,et al.  Foundations of semi-differential invariants , 2005, International Journal of Computer Vision.

[12]  Olivier D. Faugeras,et al.  Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes , 1993, Applications of Invariance in Computer Vision.

[13]  J. Koenderink The structure of images , 2004, Biological Cybernetics.

[14]  L. Gool,et al.  Semi-differential invariants , 1992 .

[15]  Richard Evan Schwartz,et al.  The Pentagram Map , 1992, Exp. Math..

[16]  Isaac Weiss,et al.  Projective invariants of shapes , 1988, Proceedings CVPR '88: The Computer Society Conference on Computer Vision and Pattern Recognition.

[17]  K. Nomizu Affine Differential Geometry , 1994 .

[18]  Isaac Weiss Noise-Resistant Invariants of Curves , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[19]  M. Gage,et al.  The Curve Shortening Flow , 1987 .

[20]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[21]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[22]  Josef Grünvald,et al.  Projective differential geometry of curves and ruled surfaces , 1908 .

[23]  Benjamin B. Kimia,et al.  Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space , 1995, International Journal of Computer Vision.

[24]  Alfred M. Bruckstein,et al.  Similarity-invariant signatures for partially occluded planar shapes , 1992, International Journal of Computer Vision.

[25]  A. Bruckstein,et al.  Differential invariants of planar curves and recognizing partially occluded shapes , 1992 .

[26]  Guillermo Sapiro,et al.  Affine invariant scale-space , 1993, International Journal of Computer Vision.

[27]  Guillermo Sapiro,et al.  Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach , 1994, Geometry-Driven Diffusion in Computer Vision.

[28]  Andrew P. Witkin,et al.  Uniqueness of the Gaussian Kernel for Scale-Space Filtering , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  Bart M. ter Haar Romeny,et al.  Geometry-Driven Diffusion in Computer Vision , 1994, Computational Imaging and Vision.

[30]  Guillermo Sapiro,et al.  Evolutions of Planar Polygons , 1995, Int. J. Pattern Recognit. Artif. Intell..

[31]  S. Angenent Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions , 1991 .