Foundations of semi-differential invariants

This paper elaborates the theoretical foundations of a semi-differential framework for invariance. Semi-differential invariants combine coordinates and their derivatives with respect to some contour parameter at several points of the image contour, thus allowing for an optimal trade-off between identification of points and the calculation of derivatives. A systematic way of generating complete and independent sets of such invariants is presented. It is also shown that invariance under reparametrisation can be cast in the same framework. The theory is illustrated by a complete analysis of 2D affine transformations. In a companion paper (Pauwels et al. 1995) these affine semi-differential invariants are implemented in the computer program FORM (Flat Object Recognition Method) for the recognition of planar contours under pseudo-perspective projection.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  H. Weyl.  The Classical Groups , 1940 .

[3]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[4]  C. Chester.  Techniques in partial differential equations , 1971 .

[5]  R. Walde,et al.  Introduction to Lie groups and Lie algebras , 1973 .

[6]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[7]  A. Fordy.  APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS (Graduate Texts in Mathematics) , 1987 .

[8]  Yehezkel Lamdan,et al.  On recognition of 3-D objects from 2-D images , 2011, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

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

[10]  S. Ullman.  Aligning pictorial descriptions: An approach to object recognition , 1989, Cognition.

[11]  Johan Wagemans,et al.  Similarity extraction and modeling , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[12]  Wesley E. Snyder,et al.  Application of Affine-Invariant Fourier Descriptors to Recognition of 3-D Objects , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  K. Kanatani.  Group-Theoretical Methods in Image Understanding , 1990 .

[14]  Robert M. Haralick,et al.  Optimal affine-invariant point matching , 1990, [1990] Proceedings. 10th International Conference on Pattern Recognition.

[15]  David A. Forsyth,et al.  Transformational invariance - a primer , 1990, BMVC.

[16]  Luc Van Gool,et al.  Recognition and semi-differential invariants , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[17]  D. Forsyth,et al.  Using Projective Invariants for Constant Time Library Indexing in Model Based Vision , 1991 .

[18]  P. M. Payton,et al.  Projective invariants for curves in two and three dimensions , 1992 .

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

[20]  Luc Van Gool,et al.  Semi-differential invariants for nonplanar curves , 1992 .

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

[22]  André Oosterlinck,et al.  Shape recognition under affine distortions , 1992 .

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

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

[25]  L. Gool,et al.  Foundations of semi-differential invariance with applications to recognition under affine distortion , 1994 .

[26]  Luc Van Gool,et al.  Recognition of planar shapes under affine distortion , 2005, International Journal of Computer Vision.

[27]  Alfred M. Bruckstein,et al.  On differential invariants of planar curves and recognizing partially occluded planar shapes , 1995, Annals of Mathematics and Artificial Intelligence.