Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes

This article is a general introduction to Cartan's moving frame method which is elegant, simple, and of an algorithmic nature. We have demonstrated how to use it systematically on three examples relevant to computer vision, curves in the euclidean, affine and projective planes, and derived the corresponding Frenet equations. We have then used these equations to show that the analysis of the deformation of plane curves according to an intrinsic heat equation could be done in a common framework, yielding very similar expressions for the evolution of the three curvature invariants.

[1]  Benjamin B. Kimia,et al.  On the evolution of curves via a function of curvature , 1992 .

[2]  Jan J. Koenderink,et al.  Solid shape , 1990 .

[3]  André Weil,et al.  The classical case , 1971 .

[4]  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.

[5]  Max A. Viergever,et al.  Scale and the differential structure of images , 1992, Image Vis. Comput..

[6]  E. Cartan,et al.  Lecons sur la théorie des espacea : a connexion projective , 1937 .

[7]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[8]  M. Gage.  Curve shortening makes convex curves circular , 1984 .

[9]  G. Sapiro,et al.  On affine plane curve evolution , 1994 .

[10]  M. Gage,et al.  An isoperimetric inequality with applications to curve shortening , 1983 .

[11]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[12]  Andrew Zisserman,et al.  Geometric invariance in computer vision , 1992 .

[13]  RomenyMax,et al.  A ne and projective di erential geometric invariants of space curvesAlfons , 1993 .

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

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

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

[17]  Max A. Viergever,et al.  Affine and projective differential geometric invariants of space curves , 1993, Optics & Photonics.