Abstract
Many challenging computer vision problems can be formulated as a multilinear model. Classical methods like principal component analysis use singular value decomposition to infer model parameters. Although it can solve a given problem easily if all measurements are known this prerequisite is usually violated for computer vision applications. In the current work, a standard tool to estimate singular vectors under incomplete data is reformulated as an energy minimization problem. This admits for a simple and fast gradient descent optimization with guaranteed convergence. Furthermore, the energy function is generalized by introducing an L 2-regularization on the parameter space. We show a quantitative and qualitative evaluation of the proposed approach on an application from structure-from-motion using synthetic and real image data, and compare it with other works.
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References
Ackermann, H., Rosenhahn, B.: Trajectory Reconstruction for Affine Structure-from-Motion by Global and Local Constraints. In: CVPR (June 2009)
Brand, M.: Morphable 3d models from video. In: IEEE Computer Vision and Pattern Recognition (CVPR), Kauai, Hawaii, USA, pp. 456–463 (2001)
Brand, M.: Incremental singular value decomposition of uncertain data with missing values. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 707–720. Springer, Heidelberg (2002)
Buchanan, A., Fitzgibbon, A.: Damped Newton Algorithms for Matrix Factorization with Missing Data. In: CVPR, Washington, DC, USA, pp. 316–322 (2005)
Cai, J.F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM Journal on Optimization 20(4), 1956–1982 (2010)
Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Foundations of Computational Mathematics 9(6), 717–772 (2009)
Cremers, D., Kolev, K.: Multiview stereo and silhouette consistency via convex functionals over convex domains. IEEE TPAMI (2010)
Eriksson, A., van den Hengel, A.: Efficient computation of robust low-rank matrix approximations in the presence of missing data using the l1 norm. In: CVPR, San Francisco, USA (2010)
Gabriel, K., Zamir, S.: Lower rank approximation of matrices by least squares with any choice of weights. Techonometrics 21(4), 489–498 (1979)
Gruber, A., Weiss, Y.: Factorization with uncertainty and missing data: Exploiting temporal coherence. In: NIPS, Vancouver, Canada (December 2003)
Hartley, R., Schaffalizky, F.: PowerFactorization: 3D Reconstruction with Missing or Uncertain Data. In: Australia-Japan Advanced Workshop on Computer Vision (June 2002)
Hasler, N., Ackermann, H., Rosenhahn, B., Thormählen, T., Seidel, H.P.: Multilinear pose and body shape estimation of dressed subjects from image sets. In: CVPR, San Francisco, USA (June 2010)
Turk, M.A., Pentland, A.P.: Face recognition using eigenfaces. In: CVPR (1991)
Marques, M., Costeira, J.: Estimating 3d shape from degenerate sequences with missing data. CVIU 113(2), 261–272 (2009)
Peng, Y., Ganesh, A., Wright, J., Ma, Y.: Rasl: Robust alignment by sparse and low-rank decomposition for linearly correlated images. In: CVPR, San Francisco, USA, pp. 763–770 (June 2010)
Ruhe, A.: Numerical computation of principal components when several observations are missing. Tech. rep., Dept. Information Processing, University of Umeda, Umeda, Sweden (April 1974)
Ruhe, A., Wedin, P.: Algorithms for separable nonlinear least squares problems. Society for Industrial and Applied Mathematics Review 22(3), 318–337 (1980)
Sugaya, Y., Kanatani, K.: Extending interrupted feature point tracking for 3-D affine reconstruction. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3021, pp. 310–321. Springer, Heidelberg (2004)
Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography: A factorization method. IJCV 9(2), 137–154 (1992)
Ullman, S., Basri, R.: Recognition by linear combinations of models. IEEE TPAMI 13, 992–1006 (1991)
Vu, H.H., Keriven, R., Labatut, P., Pons, J.P.: Towards high-resolution large-scale multi-view stereo. In: CVPR. Miami (June 2009)
Wold, H.: Estimation of principal components and related models by iterative least squares. In: Krishnaiah (ed.) Multivariate Analysis, pp. 391–420 (1966)
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Schmidt, F.R., Ackermann, H., Rosenhahn, B. (2011). Multilinear Model Estimation with L 2-Regularization. In: Mester, R., Felsberg, M. (eds) Pattern Recognition. DAGM 2011. Lecture Notes in Computer Science, vol 6835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23123-0_9
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DOI: https://doi.org/10.1007/978-3-642-23123-0_9
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