Unscented Kalman filtering on Riemannian manifolds. (English) Zbl 1316.94028
Summary: In recent years there has been a growing interest in problems, where either the observed data or hidden state variables are confined to a known Riemannian manifold. In sequential data analysis this interest has also been growing, but rather crude algorithms have been applied: either Monte Carlo filters or brute-force discretisations. These approaches scale poorly and clearly show a missing gap: no generic analogues to Kalman filters are currently available in non-Euclidean domains. In this paper, we remedy this issue by first generalising the unscented transform and then the unscented Kalman filter to Riemannian manifolds. As the Kalman filter can be viewed as an optimisation algorithm akin to the Gauss-Newton method, our algorithm also provides a general-purpose optimisation framework on manifolds. We illustrate the suggested method on synthetic data to study robustness and convergence, on a region tracking problem using covariance features, an articulated tracking problem, a mean value optimisation and a pose optimisation problem.
MSC:
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 62M20 | Inference from stochastic processes and prediction |
| 68T45 | Machine vision and scene understanding |
| 93E11 | Filtering in stochastic control theory |
References:
| [1] | Balan, A.O., Sigal, L., Black, M.J.: A quantitative evaluation of video-based 3D person tracking. In: 2nd Joint IEEE International Workshop on Visual Surveillance and Performance Evaluation of Tracking and Surveillance, pp. 349–356 (2005) |
| [2] | Bandouch, J., Engstler, F., Beetz, M.: Accurate human motion capture using an ergonomics-based anthropometric human model. In: AMDO’08: Proceedings of the 5th International Conference on Articulated Motion and Deformable Objects. Lecture Notes in Computer Science, vol. 5098, pp. 248–258. Springer, Berlin (2008) |
| [3] | Bell, B.M., Cathey, F.W.: The iterated Kalman filter update as a Gauss-Newton method. IEEE Trans. Autom. Control 38, 294–297 (1993) · Zbl 0775.93237 · doi:10.1109/9.250476 |
| [4] | Cappé, O., Godsill, S.J., Moulines, E.: An overview of existing methods and recent advances in sequential Monte Carlo. Proc. IEEE 95(5), 899–924 (2007) · doi:10.1109/JPROC.2007.893250 |
| [5] | do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992) |
| [6] | Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22, 61–79 (1997) · Zbl 0894.68131 · doi:10.1023/A:1007979827043 |
| [7] | Engell-Nørregård, M., Erleben, K.: A projected back-tracking line-search for constrained interactive inverse kinematics. Comput. Graph. 35(2), 288–298 (2011) · doi:10.1016/j.cag.2010.12.011 |
| [8] | Erleben, K., Sporring, J., Henriksen, K., Dohlmann, H.: Physics Based Animation. Charles River Media, Newton Center (2005) |
| [9] | Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87, 250–262 (2007) · Zbl 1186.94126 · doi:10.1016/j.sigpro.2005.12.018 |
| [10] | Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004) · doi:10.1109/TMI.2004.831793 |
| [11] | Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2004) · Zbl 0994.65135 |
| [12] | Hauberg, S., Pedersen, K.S.: Stick it! Articulated tracking using spatial rigid object priors. In: Asian Conference on Computer Vision. Lecture Notes in Computer Science, vol. 6494. Springer, Berlin (2010) |
| [13] | Hauberg, S., Pedersen, K.S.: Predicting articulated human motion from spatial processes. Int. J. Comput. Vis. 94, 317–334 (2011) · Zbl 1235.68267 · doi:10.1007/s11263-011-0433-3 |
| [14] | Hauberg, S., Pedersen, K.S.: HUMIM software for articulated tracking. Tech. Rep. 01/2012, Department of Computer Science, University of Copenhagen (2012) |
| [15] | Hauberg, S., Sommer, S., Pedersen, K.S.: Gaussian-like spatial priors for articulated tracking. In: ECCV. Lecture Notes in Computer Science, vol. 6311, pp. 425–437. Springer, Berlin (2010) |
| [16] | Hauberg, S., Sommer, S., Pedersen, K.S.: Natural metrics and least-committed priors for articulated tracking. Image Vis. Comput. 30(6–7), 453–461 (2012) · doi:10.1016/j.imavis.2011.11.009 |
| [17] | Julier, S.J., Uhlmann, J.K.: A new extension of the Kalman filter to nonlinear systems. In: International Symposium Aerospace/Defense Sensing, Simulation and Controls, pp. 182–193 (1997) |
| [18] | Kalman, R.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(D), 35–45 (1960) · doi:10.1115/1.3662552 |
| [19] | Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977) · Zbl 0354.57005 · doi:10.1002/cpa.3160300502 |
| [20] | Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984) · Zbl 0579.62100 · doi:10.1112/blms/16.2.81 |
| [21] | Kjellström, H., Kragić, D., Black, M.J.: Tracking people interacting with objects. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 747–754 (2010) |
| [22] | Kraft, E.: A quaternion-based unscented Kalman filter for orientation tracking. In: Proceedings of the Sixth International Conference on Information Fusion, pp. 47–54 (2003) |
| [23] | Kwon, J., Lee, K.M.: Monocular SLAM with locally planar landmarks via geometric rao-blackwellized particle filtering on Lie groups. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1522–1529 (2010) |
| [24] | Kwon, J., Lee, K.M., Park, F.C.: Visual tracking via geometric particle filtering on the affine group with optimal importance functions. In: Computer Vision and Pattern Recognition, pp. 991–998 (2009) |
| [25] | Lewis, F.L.: Optimal Estimation: With an Introduction to Stochastic Control Theory. Wiley, New York (1986) · Zbl 0665.93065 |
| [26] | Li, R., Chellappa, R.: Aligning spatio-temporal signals on a special manifold. In: ECCV. Lecture Notes in Computer Science, vol. 6315, pp. 547–560. Springer, Berlin (2010) |
| [27] | Liu, X., Srivastava, A., Gallivan, K.: Optimal linear representations of images for object recognition. IEEE Trans. Pattern Anal. Mach. Intell. 26(5), 662–666 (2004) · doi:10.1109/TPAMI.2004.1273986 |
| [28] | van der Merwe, R., Doucet, A., Freitas, N.D., Wan, E.: The unscented particle filter. In: Advances in Neural Information Processing Systems (NIPS 2000), vol. 13, pp. 584–590. MIT Press, Cambridge (2001) |
| [29] | Misner, C., Thorne, K., Wheeler, J.: Gravitation. W.H. Freeman, New York (1973) |
| [30] | Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006) · Zbl 1478.94072 · doi:10.1007/s10851-006-6228-4 |
| [31] | Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66, 41–66 (2004) · Zbl 1287.53031 · doi:10.1007/s11263-005-3222-z |
| [32] | Poppe, R.: Vision-based human motion analysis: an overview. Comput. Vis. Image Underst. 108(1–2), 4–18 (2007) · doi:10.1016/j.cviu.2006.10.016 |
| [33] | Porikli, F., Tuzel, O., Meer, P.: Covariance tracking using model update based on Lie algebra. In: Computer Vision and Pattern Recognition, vol. 1, pp. 728–735 (2006) |
| [34] | Rabiner, L.R.: A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77(2), 257–286 (1989) · doi:10.1109/5.18626 |
| [35] | Sidenbladh, H., Black, M.J., Fleet, D.J.: Stochastic tracking of 3D human figures using 2D image motion. In: ECCV, vol. II. Lecture Notes in Computer Science, vol. 1843, pp. 702–718. Springer, Berlin (2000) |
| [36] | Sigal, L., Black, M.J.: HumanEva: synchronized video and motion capture dataset for evaluation of articulated human motion. Tech. Rep. CS-06-08, Brown University (2007) |
| [37] | Singhal, S., Wu, L.: Training multilayer perceptrons with the extended Kalman algorithm. In: Advances in Neural Information Processing Systems, vol. 1, pp. 133–140 (1989) |
| [38] | Sipos, B.J.: Application of the Manifold-Constrained unscented Kalman filter. In: Position, Location and Navigation Symposium, IEEE/ION, pp. 30–43 (2008) |
| [39] | Sommer, S., Tatu, A., Chen, C., Jørgensen, D.R., de Bruijne, M., Loog, M., Nielsen, M., Lauze, F.: Bicycle chain shape models. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 157–163. IEEE Computer Society, Los Alamitos (2009) |
| [40] | Sommer, S., Lauze, F., Nielsen, M.: The differential of the exponential map, Jacobi fields and exact principal geodesic analysis. CoRR (2010). arXiv:1008.1902 |
| [41] | Srivastava, A., Klassen, E.: Bayesian and geometric subspace tracking. Adv. Appl. Probab. 36(1), 43–56 (2004) · Zbl 1047.93044 · doi:10.1239/aap/1077134463 |
| [42] | Subbarao, R., Meer, P.: Nonlinear mean shift over Riemannian manifolds. Int. J. Comput. Vis. 84(1), 1–20 (2009) · Zbl 1477.68426 · doi:10.1007/s11263-008-0195-8 |
| [43] | Tidefelt, H., Schön, T.B.: Robust point-mass filters on manifolds. In: Proceedings of the 15th IFAC Symposium on System Identification (SYSID), pp. 540–545 (2009) |
| [44] | Tuzel, O., Porikli, F., Meer, P.: Region covariance: A fast descriptor for detection and classification. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV. Lecture Notes in Computer Science, vol. 3952, pp. 589–600. Springer, Berlin/Heidelberg (2006) |
| [45] | Tyagi, A., Davis, J.W.: A recursive filter for linear systems on Riemannian manifolds. In: Computer Vision and Pattern Recognition, pp. 1–8 (2008) |
| [46] | Wan, E.A., van der Merwe, R.: The unscented Kalman filter for nonlinear estimation. In: Adaptive Systems for Signal Processing, Communications, and Control Symposium, IEEE, pp. 153–158 (2002) |
| [47] | Ward, R.C.: Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Numer. Anal. 14, 600–610 (1977) · Zbl 0363.65031 · doi:10.1137/0714039 |
| [48] | Wu, Y., Wu, B., Liu, J., Lu, H.: Probabilistic tracking on Riemannian manifolds. In: International Conference on Pattern Recognition, pp. 1–4 (2008) |
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