Least Square for Grassmann-Cayley Agelbra in Homogeneous Coordinates
This paper presents some tools for least square computation in Grassmann-Cayley algebra, more specifically for elements expressed in homogeneous coordinates. We show that building objects with the outer product from k-vectors of same grade presents some properties that can be expressed in term of linear algebra and can be treated as a least square problem. This paper mainly focuses on line and plane fitting and intersections computation, largely used in computer vision. We show that these least square problems written in Grassmann-Cayley algebra have a direct reformulation in linear algebra, corresponding to their standard expression in projective geometry and hence can be solved using standard least square tools.
KeywordsGrassmann-Cayley algebra least square line fitting plane fitting intersection
- 5.Gebken, C., Perwass, C., Sommer, G.: Parameter Estimation from Uncertain Data in Geometric Algebra. In: Advances in Applied Clifford Algebras, vol. 18(3-4), pp. 647–664 (2008)Google Scholar
- 6.Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2004)Google Scholar
- 8.Kanatani, K., Sugaya, Y., Niitsuma, H.: Optimization without Minimization Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation. IEICE Transactions 93-D, 2836–2845 (2010)Google Scholar
- 9.Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science: an object-oriented approach to geometry, revised 1st edn. Morgan Kaufmann series in computer graphics. Elsevier, San Francisco (2009)Google Scholar
- 10.Dorst, L., Lasenby, J.: Guide to geometric algebra in practice. Springer (2011)Google Scholar