Geometric relationships governing multiple images of points and lines and associated algorithms have been studied to a large extent separately in multiple view geometry. However, most of the theory carries out the geometric analysis by breaking image sequence into pairwise, triple-wise or quadraple-wise sets of view. Also, the different image features: points, lines and planes, and different incidence relations among these features: inclusion, intersection and restriction, were studied and presented separately, or case by case at best. Due to these reasons, it is difficult to incorporate all features and incidence relations for a global and consistent analysis.
In order to account for these shortcomings, We propose a unifying geometric representation of the constraints generated by multiple views of a scene. Our approach is based on a rank constraint on the so-called multiple view matrix for arbitrarily combined point and line features across multiple views. The condition gives rise to a complete set of constraints among multiple images. The rank condition in fact clearly reveals the relationship among all the previously known multilinear constraints and further implies some novel; non-linear constraints among multiple images. The formulation also clearly demonstrates that quadrilinear constraints are algebraically redundant; and that non-trivial constraints among four views are actually non-linear.
We, hence, allow for a meaningful global analysis of arbitrarily many views with arbitrarily mixed incidence relations among features, with no need to cascade pair-wise, triple-wise or quadraple-wise views.The additional appeal of our approach is the sole use of linear algebraic techniques, with no need to introduce tensorial notation or algebraic geometry.
1. Y. Ma, K. Huang, R. Vidal, J. Kosecka and S. Sastry
Rank Condition on the Multiple View Matrix
International Journal of Computer Vision, volume 59, number 2, pp. 115-137, September 2004
2. Y. Ma, R. Vidal, S. Hsu and S. Sastry
Optimal Motion Estimation from Multiple Images by Normalized Epipolar Constraint
Journal of Communications in Information and Systems, no. 1, pages 51-73, 2001
3. R. Vidal, Y. Ma, S. Hsu and S. Sastry
Optimal Motion Estimation from Multiview Normalized Epipolar Constraint
International Conference on Computer Vision, (1):34-41, Vancouver, Canada, July 2001
4. Y. Ma, R. Vidal, J. Kosecka and S. Sastry
Kruppa's Equations Revisited: its Degeneracy, Renormalization and Relations to Chirality.
In Proceedings of the European Conference on Computer Vision EECV, vol. 2, pp. 561-577, Dublin, Ireland, July 2000.