On the geometry and algebra of the point and line correspondences between N images

We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to use the formalism of the Grassmann-Cayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed). We have a fairly complete picture of the situation in the case of points; there are only three types of algebraic relations which are satisfied by the coordinates of the images of a 3-D point: bilinear relations arising when we consider pairs of images among the N and which are the well-known epipolar constraints, trilinear relations arising when we consider triples of images among the N, and quadrilinear relations arising when we consider four-tuples of images among the N. In the case of lines, we show how the traditional perspective projection equation can be suitably generalized and that in the case of three images there exist two independent trilinear relations between the coordinates of the images of a 3-D line.<<ETX>>

[1]  Bernard Mourrain.  Approche effective de la theorie des invariants des groupes classiques , 1991 .

[2]  Michael Werman,et al.  Fundamental Tensor: On the Geometry of Three Perspective Views , 1995 .

[3]  Y. J. Tejwani,et al.  Robot vision , 1989, IEEE International Symposium on Circuits and Systems,.

[4]  Linda G. Shapiro,et al.  Computer and Robot Vision , 1991 .

[5]  Bernard Mourrain.  New Aspects of Geometrical Calculus With Invariants , 1994 .

[6]  S. P. Mudur,et al.  Three-dimensional computer vision: a geometric viewpoint , 1993 .

[7]  Amnon Shashua,et al.  Algebraic Functions For Recognition , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Rachid Deriche,et al.  A Robust Technique for Matching two Uncalibrated Images Through the Recovery of the Unknown Epipolar Geometry , 1995, Artif. Intell..

[9]  Olivier D. Faugeras,et al.  What can be seen in three dimensions with an uncalibrated stereo rig , 1992, ECCV.

[10]  J. Dieudonné,et al.  Éléments d'analyse , 1972 .

[11]  Emanuele Trucco,et al.  Computer and Robot Vision , 1995 .

[12]  Azriel Rosenfeld,et al.  Computer Vision , 1988, Adv. Comput..

[13]  Quang-Tuan Luong.  Matrice Fondamentale et Calibration Visuelle sur l''''Environnement - Vers une plus grande autonomie , 1992 .

[14]  O. D. Faugeras,et al.  Camera Self-Calibration: Theory and Experiments , 1992, ECCV.

[15]  J. Todd.  Methods of Algebraic Geometry , 1948, Nature.

[16]  Thierry Viéville,et al.  Canonic Representations for the Geometries of Multiple Projective Views , 1994, ECCV.

[17]  Rajiv Gupta,et al.  Stereo from uncalibrated cameras , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[18]  J. G. Semple,et al.  Algebraic Projective Geometry , 1953 .

[19]  Gian-Carlo Rota,et al.  On the Exterior Calculus of Invariant Theory , 1985 .

[20]  B. Triggs.  The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image , 1995 .

[21]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[22]  O. Faugeras,et al.  On determining the fundamental matrix : analysis of different methods and experimental results , 1993 .

[23]  Olivier D. Faugeras,et al.  What can two images tell us about a third one? , 1994, ECCV.

[24]  Andrew Zisserman,et al.  A Case Against Epipolar Geometry , 1993, Applications of Invariance in Computer Vision.

[25]  Amnon Shashua,et al.  Projective Structure from Uncalibrated Images: Structure From Motion and Recognition , 1994, IEEE Trans. Pattern Anal. Mach. Intell..