Point configuration invariants under simultaneous projective and permutation transformations

Abstract The projective invariants used in computer vision today are permutation-sensitive since their value depends on the order in which the features were considered in the computation. We derive, using tools from representation theory, the projective and permutation ( p 2 ) invariants of the four collinear and the five coplanar points configurations. The p 2 -invariants are insensitive to both projective transformations and changes in the labeling of the points. When used as model database indexing functions in object recognition systems, the p 2 -invariants yield a significant speedup. Permutation invariants for affine transformations are also discussed.

[1]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[2]  Reiner Lenz,et al.  Group Theoretical Methods in Image Processing , 1990, Lecture Notes in Computer Science.

[3]  Yehezkel Lamdan,et al.  Affine invariant model-based object recognition , 1990, IEEE Trans. Robotics Autom..

[4]  Patrick Gros,et al.  Invariants of a pair of conics revisited , 1992, Image Vis. Comput..

[5]  Reiner Lenz,et al.  Group-theoretical model of feature extraction , 1989 .

[6]  Edwin Hewitt,et al.  Theory of Group Representations , 1982 .

[7]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[8]  C. E. Springer,et al.  Geometry and Analysis of Projective Spaces , 1967 .

[9]  Peter C. Wayner,et al.  Efficiently using invariant theory for model-based matching , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[10]  S. Maybank.  The projection of two non-coplanar conics , 1992 .

[11]  Eamon B. Barrett,et al.  Closed-form extension of the anharmonic ratio to N-space , 1983, Comput. Vis. Graph. Image Process..

[12]  Andrew Zisserman,et al.  Applications of Invariance in Computer Vision , 1993, Lecture Notes in Computer Science.