Abstract
Several computer vision applications require estimating a rank deficient matrix from noisy observations of its entries. When the observation matrix has no missing data, the LS solution of such problem is known to be given by the SVD. However, in practice, when several entries of the matrix are not observed, the problem has no closed form solution. In this paper, we study two iterative algorithms for minimizing the non-linear LS cost function obtained when estimating rank deficient matrices from partial observations. In the first algorithm, the iterations are the well known Expectation and Maximization (EM) steps that have succeeded in several estimation problems with missing data. The second algorithm, which we call Row-Column (RC), estimates, in alternate steps, the row and column spaces of the solution matrix. Our conclusions are that RC performs better than EM in what respects to the robustness to the initialization and to the convergence speed. We also demonstrate the algorithms when inferring 3D structure from video sequences.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Basri, R., Ullman, S.: The alignment of objects with smooth surfaces. Computer Graphics, Vision, and Image Processing: Image Understanding 57 (1988)
Ullman, S., Basri, R.: Recognition by linear combinations of models. IEEE TPAMIÂ 13 (1991)
Shashua, A.: Geometry and photometry in 3D visual recognition. MIT TR 1401, Massachussets Institute of Technology, MA, USA (1992)
Moses, Y.: Face recognition: generalization to novel images. PhD thesis, Weizmann Institute of Science (1993)
Belhumeur, P., Kriegman, D.: What is the set of images of an object under all possible lighting conditions? In: IEEE CVPR (1996)
Zelnik-Manor, L., Irani, M.: Multi-frame alignement of planes. In: Hancock, E.R., Pelillo, M. (eds.) EMMCVPR 1999. LNCS, vol. 1654, pp. 99–114. Springer, Heidelberg (1999)
Zelnik-Manor, L., Irani, M.: Multi-view subspace constraints on homographies. In: IEEE ICCV, Kerkyra, Greece (1999)
Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography: a factorization method. International Journal of Computer Vision 9 (1992)
Sturm, P., Triggs, B.: A factorization based algorithm for multi-image projective structure and motion. In: Buxton, B.F., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1064. Springer, Heidelberg (1996)
Poelman, C.J., Kanade, T.: A paraperspective factorization method for shape and motion recovery. IEEE TPAMIÂ 19 (1997)
Morita, T., Kanade, T.: A sequential factorization method for recovering shape and motion from image streams. IEEE TPAMI (1997)
Aguiar, P.M.Q., Moura, J.M.F.: Factorization as a rank 1 problem. In: IEEE CVPR, Fort Collins CO, USA (1999)
Irani, M., Anandan, P.: Factorization with uncertainty. In: Vernon, D. (ed.) ECCV 2000. LNCS, vol. 1842, pp. 539–553. Springer, Heidelberg (2000)
Shapiro, L.: Affine Analysis of Image Sequences. Cambridge University Press, Cambridge (1995)
Quan, L., Kanade, T.: A factorization method for affine structure from line correspondences. In: IEEE CVPR, San Francisco CA, USA (1996)
Morris, D., Kanade, T.: A unified factorization algorithm for points, line segments and planes with uncertainty models. In: IEEE ICCV (1998)
Aguiar, P.M.Q., Moura, J.M.F.: Three-dimensional modeling from two dimensional video. IEEE Transactions on Image Processing 10 (2001)
Golub, G.H., Van-Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)
Jacobs, D.: Linear fitting with missing data: Applications to structure-from motion and to characterizing intensity images. In: IEEE CVPR, St Barbara, USA (1997)
Maruyama, M., Kurumi, S.: Bidirectional optimization for reconstructing 3D shape from an image sequence with missing data. In: IEEE ICIP, Kobe, Japan (1999)
McLachlan, G., Krishnan, T.: The EM Algorithm and Extensions. John Wiley & Sons, New York (1997)
Wiberg, T.: Computation of principal components when data are missing. In: Proc. of the Second Symposium Computational Statistics, Berlin, German (1976)
Shum, H., Ikeuchi, K., Reddy, R.: Principal component analysis with missing data and its applications to polyhedral object modeling. IEEE TPAMIÂ 17 (1995)
Guerreiro, R.F.C., Aguiar, P.M.Q.: 3D structure from video streams with partially overlapping images. In: IEEE ICIP, New York, USA (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Guerreiro, R.F.C., Aguiar, P.M.Q. (2003). Estimation of Rank Deficient Matrices from Partial Observations: Two-Step Iterative Algorithms. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2003. Lecture Notes in Computer Science, vol 2683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45063-4_29
Download citation
DOI: https://doi.org/10.1007/978-3-540-45063-4_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40498-9
Online ISBN: 978-3-540-45063-4
eBook Packages: Springer Book Archive