Abstract
We propose a new approach to the problem of robust estimation for a class of inverse problems arising in multiview geometry. Inspired by recent advances in the statistical theory of recovering sparse vectors, we define our estimator as a Bayesian maximum a posteriori with multivariate Laplace prior on the vector describing the outliers. This leads to an estimator in which the fidelity to the data is measured by the L ∞-norm while the regularization is done by the L 1-norm. The proposed procedure is fairly fast since the outlier removal is done by solving one linear program (LP). An important difference compared to existing algorithms is that for our estimator it is not necessary to specify neither the number nor the proportion of the outliers; only an upper bound on the maximal measurement error for the inliers should be specified. We present theoretical results assessing the accuracy of our procedure, as well as numerical examples illustrating its efficiency on synthetic and real data.
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Bickel, P., Ritov, Y., Tsybakov, A.B.: Simultaneous analysis of Lasso and Dantzig selector. Ann. Stat. (2008). doi:10.1214/08-AOS620
Candès, E.J., Randall, P.A.: Highly robust error correction by convex programming. IEEE Trans. Inf. Theory 54(7), 2829–2840 (2008)
Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)
Candès, E., Tao, T.: The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351 (2007)
Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)
Dalalyan, A.S., Juditsky, A., Spokoiny, V.: A new algorithm for estimating the effective dimension-reduction subspace,. J. Mach. Learn. Res. 9, 1648–1678 (2008)
Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001)
Donoho, D., Elad, M., Temlyakov, V.: Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52(1), 6–18 (2006)
Enqvist, O., Kahl, F.: Robust optimal pose estimation. In: European Conference on Computer Vision, vol. I, pp. 141–153 (2008)
Hartley, R., Kahl, F.: Global optimization through rotation space search. Int. J. Comput. Vis. 82(1), 64–79 (2009)
Hartley, R.I., Schaffalitzky, F.: L ∞ minimization in geometric reconstruction problems. In: Conference on Computer Vision and Pattern Recognition, vol. I, pp. 504–509 (2004)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2004)
Kahl, F.: Multiple view geometry and the L∞-norm. In: International Conference on Computer Vision, pp. 1002–1009. IEEE Computer Society, Los Alamitos (2005)
Kahl, F., Hartley, R.I.: Multiple-view geometry under the L ∞ norm. IEEE Trans. Pattern Anal. Mach. Intell. 30(9), 1603–1617 (2008)
Kanade, T., Ke, Q.: Quasiconvex optimization for robust geometric reconstruction. In: International Conference on Computer Vision, vol. II, pp. 986–993 (2005)
Ke, Q., Kanade, T.: Uncertainty models in quasiconvex optimization for geometric reconstruction. In: Conference on Computer Vision and Pattern Recognition, vol. I, pp. 1199–1205 (2006)
Li, H.D.: A practical algorithm for L ∞ triangulation with outliers. In: Conference on Computer Vision and Pattern Recognition (2007)
Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60(2), 91–110 (2004)
Martinec, D., Pajdla, T.: Robust rotation and translation estimation in multiview reconstruction. In: Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2007)
Nistér, D.: An efficient solution to the five-point relative pose problem. IEEE Trans. Pattern Anal. Mach. Intell. 26(6), 756–777 (2004)
Olsson, C., Eriksson, A.P., Kahl, F.: Efficient optimization for L ∞ problems using pseudoconvexity. In: International Conference on Computer Vision, pp. 1–8 (2007)
Olsson, C., Eriksson, A., Hartley, R.: Outlier removal using duality (2010)
Seo, Y.D., Hartley, R.I.: A fast method to minimize L ∞ error norm for geometric vision problems. In: International Conference on Computer Vision, pp. 1–8. (2007)
Seo, Y.D., Lee, H.J., Lee, S.W.: Sparse structures in L-infinity norm minimization for structure and motion reconstruction. In: European Conference on Computer Vision, vol. I, pp. 780–793 (2008)
Seo, Y., Lee, H., Lee, S.W.: Outlier removal by convex optimization for l-infinity approaches. In: PSIVT ’09: Proceedings of the 3rd Pacific Rim Symposium on Advances in Image and Video Technology, pp. 203–214 (2009)
Sim, K., Hartley, R.: Removing outliers using the L ∞ norm. In: Conference on Computer Vision and Pattern Recognition, vol. I, pp. 485–494 (2006)
Strecha, C., von Hansen, W., Gool, L.V., Fua, P., Thoennessen, U.: On benchmarking camera calibration and multi-view stereo for high resolution imagery. In: Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2009)
Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12(1–4), 625–653 (1999)
Vedaldi, A., Fulkerson, B.: VLFeat: An open and portable library of computer vision algorithms (2008)
Zhao, P., Yu, B.: On model selection consistency of Lasso. J. Mach. Learn. Res. 7, 2541–2563 (2006)
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Dalalyan, A., Keriven, R. Robust Estimation for an Inverse Problem Arising in Multiview Geometry. J Math Imaging Vis 43, 10–23 (2012). https://doi.org/10.1007/s10851-011-0281-3
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DOI: https://doi.org/10.1007/s10851-011-0281-3