Direct Estimation of Homogeneous Vectors: An Ill-Solved Problem in Computer Vision
Computer Vision theory is firmly rooted in Projective Geometry, whereby geometric objects can be effectively modeled by homogeneous vectors. We begin from Gauss’s 200 year old theorem of least squares to derive a generic algorithm for the direct estimation of homogeneous vectors. We uncover the common link of previous methods, showing that direct estimation is not an ill-conditioned problem as is the popular belief, but has merely been an ill-solved problem. Results show improvements in goodness-of-fit and numerical stability, and demonstrate that “data normalization” is unnecessary for a well-founded algorithm.
KeywordsCost Function Direct Estimation Geometric Object Residual Vector Generalize Eigenvector
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- 4.Harker, M., O’Leary, P., Zsombor-Murray, P.: Direct type-specific conic fitting and eigenvalue bias correction. In: Image and Vision Computing: 2004 British Machine Vision Conference Special Issue 17 (submitted, 2005)Google Scholar
- 7.Taubin, G.: Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence 13 (1991)Google Scholar
- 8.Gauss, C.F.: Méthode des moindres carrés. Mémoires sur la combinaison des observations. Mallet-Bachelier, Paris (1855)Google Scholar
- 9.Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003)Google Scholar
- 15.Taubin, G.: An improved algorithm for algebraic curve and surface fitting. In: International Conference on Computer Vision, Berlin, Germany, pp. 658–665 (1993)Google Scholar