Skip to main content
SpringerLink
Log in

Bounding Multiple Gaussians Uncertainty with Application to Object Tracking

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

This paper proves the uncertainty bound for the multiple Gaussian functions, termed multiple Gaussians Uncertainty (MGU), which significantly generalizes the uncertainty principle for the single Gaussian function. First, as a theoretical contribution, we prove that the momentum (velocity) and position for the sum of multiple Gaussians wave function are theoretically bounded. Second, as for a practical application, we show that the bound can be well exploited for object tracking to detect anomalies of local movement in an online learning framework. By integrating MGU with a given object tracker, we demonstrate that uncertainty principle can provide remarkable robustness in tracking. Extensive experiments are done to show that the proposed MGU can significantly help base trackers overcome the object drifting and reach state-of-the-art results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Similar considerations hold in quantum theory that momentum(mass and velocity) and position cannot be exactly measured simultaneously

  2. Gaussian pdfs are particular cases.

  3. We will consider the non-zero mean situation in the additional material, see Theorem 2.

  4. Note that the Multiple Gaussian Uncertainty is still valid when \(\mu \) is non-zero, and the proof is given in Theorem 2 in the additional material

  5. It is obtained by substituting \({\mathcal {U}}_H(.)\) with \({\mathcal {D}}(.)\)

References

  • Adam, A., Rivlin, E., & Shimshoni, I. (2006). Robust fragments-based tracking using the integral histogram. CVPR.

  • Ali, S., & Shah, M. (2008). Floor fields for tracking in high density crowd scenes. In Proceedings of ECCV.

  • Avidan, S. (2004). Support vector tracking. IEEE Transactions on PAMI, 26(8), 1064–1072.

    Article  Google Scholar 

  • Babenko, B., Yang, M., & Belongie, S. (2011). Robust object tracking with online multiple instance learning. IEEE Trans. PAMI, 6.

  • Betke, M., Hirsh, D., Bagchi, A., Hristov, N., Makris, N., & Kunz, T. (2007). Tracking large variable numbers of objects in clutter. In Proceedings of IEEE CVPR (pp. 1–8).

  • Born & Jordan. (1926). In the relation between the quantum mechanics of Heisenberg. Analen der Physik(4), 79.

  • Brostow, G., & Cipolla, R. (2006). Unsupervised bayesian detection of independent motion in crowds. In Proceedings of IEEE CVPR, (Vol. 1, pp. 594–601).

  • Canny, J. (2009). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence.

  • Cehovin, L., Kristan, M., & Leonardis, A. (2013). Robust visual tracking using an adaptive coupled-layer visual model. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(4), 941–953.

    Article  Google Scholar 

  • Chu, C.-T., Hwang, J.-N., Pai, H.-I., & Lan, K.-M. (2013). Tracking human under occlusion based on adaptive multiple kernels with projected gradients. IEEE Transactions on Multimedia, 15(7), 1602–1615.

    Article  Google Scholar 

  • de Broglie, L. (1929). The wave nature of the electron. Nobel Lecture, 12.

  • Doulamis, A. D. (2009). Adaptable neural networks for objects’ tracking re-initialization. ICANN, 2, 715–724.

    Google Scholar 

  • Duffner, S., & Garcia, C. (2013). PixelTrack: A fast adaptive algorithm for tracking non-rigid objects. In Proceedings of ICCV.

  • Felzenszwalb, P. F., Girshick, R. B., McAllester, D. A., & Ramanan, D. (2010). Object detection with discriminatively trained part-based models. IEEE Transactions on PAMI, 32(9), 1627–1645.

    Article  Google Scholar 

  • Gabor, D. (1946). Theory of communication. The Journal of the Institute of Electrical Engineers, 93(26), 429C457. pt. III.

    Google Scholar 

  • Gilbert, A., & Bowden, R. (2007). Multi person tracking within crowded scenes. In IEEE workshop on human motion (pp. 166–179).

  • Godec, M., Roth, P. M., & Bischof, H. (2013). Hough-based tracking of non-rigid objects. Computer Vision and Image Understanding, 117(10), 1245–1256.

    Article  Google Scholar 

  • Grabner, H., Grabner, M., & Bischof, H. (2008). Semi-supervised on-line boosting for robust tracking. In Proceedings of ECCV (pp. 234–247).

  • Grabner, H., Grabner, M., & Bischof, H. (2006). Real-time tracking via on-line boosting. Proceedings of BMVC, 1, 47–56.

    Google Scholar 

  • Gustafson, F., Gunnison, F., & Nicolas, B. (2002). Particle filters for position, navigation, and tracking. IEEE Transactions on Signal Processing, 50(2), 425–437.

    Article  Google Scholar 

  • Han, Z., Jiao, J., Zhang, B., Ye, Q., & Liu, J. (2011). Visual object tracking via sample-based adaptive sparse representation. Pattern Recognition Journal.

  • Hardy, G. H., Littlewood, J. E., & Pólya. (1967). Inequalities. Cambridge Mathematical Library.

  • Hare, S., Saffari, A., & Torr, P. (2011). Struck: Structured output tracking with kernels. In Proceedings of IEEE ICCV (pp. 263-270).

  • Hare, S., Saffari, A., & Torr,P. H. S. (2012). Efficient online structured output learning for keypoint-based object tracking. In Proceedings of CVPR (p. 1894)

  • Hariharakrishnan, K., & Schonfeld, D. (2005). Fast object tracking using adaptive block matching. IEEE Transactions on Multimedia, 853–859.

  • Hodge, V.J., & Austin, J. (2004). A survey of outlier detection methodologies. Artificial Intelligence Review.

  • Hu, W., Li, X., Zhang, X., Shi, X., Maybank, S., & Zhang, Z. (2011). Incremental tensor subspace learning and its applications to foreground segmentation and tracking. International Journal of Computer Vision (IJCV), 91(3), 303–327.

    Article  MATH  Google Scholar 

  • Jacobs, N., Dixon, M., & Pless, R. (2008). Location-specific transition distributions for tracking. In IEEE workshop on motion and video computing.

  • Jepson, A., Fleet, D., & El-Maraghi, T. ( 2003). Robust online appearance models for visual tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1296–1311.

  • Kalal, Z., Mikolajczyk, K., & Matas, J. (2010). Forward-backward error: automatic detection of tracking failures. In International conference on pattern recognition (pp. 23–26).

  • Kalal, Z., Mikolajczyk, K., & Matas, J. (2010). Tracking-learning-detection. IEEE Transactions on PAMI, 6.

  • Knutsson, H. (1994). Signal processing for computer vision. Springer: New York.

  • Kristan et al. (2014). The visual object tracking VOT2014 challenge results. In ECCV visual object tracking challenge workshop.

  • Kwon, J., & Lee, K. M. (2010). Visual tracking decomposition. In Proceedings of IEEE CVPR (pp. 1269–1276).

  • Kwon, J., & Lee, K. M. (2013). Minimum uncertainty gap for robust visual tracking. In Proceedings of IEEE CVPR.

  • Kwon, J., & Lee, K. Mu. (2014). Interval Tracker: Tracking by interval analysis. In Proceedings of IEEE CVPR.

  • Li, S. (1999). Discrete multi-gabor expansions. IEEE Tranactions on Information theory, 45.

  • Nestares, O., & Fleet, D. J. (2001). Probabilistic tracking of motion boundaries with spatiotemporal predictions. In Proceedings of IEEE CVPR (pp. 358–365).

  • Peng, C., et al. (2009). Contextual mixture tracking. IEEE Transactions on Multimedia, 11.2, 333–341.

    Article  Google Scholar 

  • Purdy, T. P., Peterson, R. W., & Regal, C. A. (2013). Observation of radiation pressure shot noise on a macroscopic object. Science, 339(6121), 801–804.

    Article  Google Scholar 

  • Qu, W., et al. (2007). Real-time distributed multiobject tracking using multiple interactive trackers and a magnetic-inertia potential model. IEEE Transactions on Multimedia, 9(3), 511–519.

    Article  Google Scholar 

  • Ramakrishna, V., Sheikh, Y., & Kanade, T. (2013). Tracking human pose by tracking symmetric parts. In Proceedings of IEEE CVPR.

  • Rodriguez, M., Ali, S., & Kanade, T. (2009). Tracking in unstructured crowded scenes. In Proceedings of IEEE ICCV.

  • Ross, D. A., Lim, J., Lin, R.-S., & Yang, M.-H. (2008). Incremental learning for robust visual tracking. International Journal Computer Vision, 77, 125–141.

    Article  Google Scholar 

  • Shinde, S., & Gadre, V. M. (2001). An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Transactions on Signal Processing, 49(11).

  • Stauder, J., & Ostermann, J. (1999). Detection of moving cast shadows for object segmentation. IEEE Transactions on Multimedia, 65–76.

  • Stauffer, C., & Grimson, W.E.L. (1999). Adaptive background mixture models for real-time tracking. In Proceedings of IEEE CVPR, 2, 246–252.

  • Wang, X., Hua, G., & Han, T. X. (2010). Discriminative tracking by metric learning. In Proceedings of ECCV (pp. 200–214).

  • Wilson, R., & Granlund, G. (1984). The uncertainty principle in image processing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6.

  • Wilson, R., & Goesta, H. (1984). Granlund the uncertainty principle in image processing. IEEE Transactions on PAMI, 6(6), 758–767.

    Article  Google Scholar 

  • Wu, Yi., Jongwoo L., & Ming-Hsuan Y. (2013). Online object tracking: A benchmark. In Proc. of CVPR (pp. 2411–2418).

  • Wu, B., & Nevatia, R. (2006). Tracking of multiple, partially occluded humans based on static body part detection. In Proceedings of IEEE CVPR (pp. 951–958).

  • Yao, R., Shi, Q., Shen, C., Zhang, Y., & Hengel, A. V. (2013). Part-based visual tracking with online latent structural learning. In Proceedings of IEEE CVPR (pp. 25–27).

  • Zhang, B., & Li, Z. (2016). Alessandro Perina, Alessio Del Bue, Vittorio Murino, Jianzhuang Liu. IEEE Transactions on CSVT: Adaptive Local Movement Modeling (ALMM) for Object Tracking.

  • Zhang, K., Zhang, L., Yang, M., Zhang, D. (2014). Fast tracking via spatio-temporal context learning. ECCV.

  • Zhang, B., Gao, Y., Zhao, S., & Zhong, B. (2011). Kernel Similarity modeling of texture pattern flow for motion detection in complex background. IEEE Transactions on CSVT, 21(1), 29–38.

    Google Scholar 

  • Zhang, K., Zhang, L., & Yang, M. H. (2014). Real-time compressive tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(10), 2002–2015.

    Article  Google Scholar 

  • Zhuang, B., Lu, H., Xiao, Z., & Wang, D. (2014). Visual tracking via discriminative sparse similarity map. IEEE Transactions on Image Processing, 23(4), 1872–1881.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported in the part by Natural Science Foundation of China, under Contracts 61272052 and 61473086, and by the Program for New Century Excellent Talents University of Ministry of Education of China, and the National Basic Research Program of China (2015CB352501). The work of R. Ji is supported by the Special Fund for Earthquake Research in the Public Interest No. 201508025, the Open Projects Program of National Laboratory of Pattern Recognition, and the Nature Science Foundation of China (Nos. 61422210 and 61373076). Thanks for the suggestions from Alessio Del Bue and Wanquan Liu to improve the paper. Alessandro Perina and Zhigang Li have the same contribution to the paper.

Author information

Author notes

    Authors and Affiliations

    1. School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

      Baochang Zhang & Zhigang Li

    2. Fujian Key Laboratory of Sensing and Computing for Smart City, School of Information Science and Engineering, Xiamen University, Xiamen, China

      Rongrong Ji

    3. Pattern Analysis and Computer Vision (PAVIS), Istituto Italiano di Tecnologia (IIT), Genova, Italy

      Alessandro Perina & Vittorio Murino

    4. Media Laboratory, Huawei Technologies Company Ltd., Shenzhen, 518129, China

      Jianzhuang Liu

    Authors
    1. Baochang Zhang
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Alessandro Perina
      View author publications

      You can also search for this author in PubMed Google Scholar

    3. Zhigang Li
      View author publications

      You can also search for this author in PubMed Google Scholar

    4. Vittorio Murino
      View author publications

      You can also search for this author in PubMed Google Scholar

    5. Jianzhuang Liu
      View author publications

      You can also search for this author in PubMed Google Scholar

    6. Rongrong Ji
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Rongrong Ji.

    Additional information

    Communicated by Jiri Matas.

    Alessandro Perina and Zhigang Li have the same contribution to the paper.

    Electronic supplementary material

    Below is the link to the electronic supplementary material.

    Supplementary material 1 (pdf 151 KB)

    Rights and permissions

    Reprints and permissions

    About this article

    Check for updates. Verify currency and authenticity via CrossMark

    Cite this article

    Zhang, B., Perina, A., Li, Z. et al. Bounding Multiple Gaussians Uncertainty with Application to Object Tracking. Int J Comput Vis 118, 364–379 (2016). https://doi.org/10.1007/s11263-016-0880-y

    Download citation

    • Received:

    • Accepted:

    • Published:

    • Issue Date:

    • DOI: https://doi.org/10.1007/s11263-016-0880-y

    Keywords

    Search

    Navigation