Abstract
Polarization of tri-axial signals is defined using instantaneous rotational characteristics of the three-dimensional (3D) trajectory. We propose a rotational model to parameterize the time evolution of the 3D trajectory as a sequence of scaled rotations. Using this model, the velocity-to-rotation transform is defined to estimate the eigenangle, eigenaxis and orientation quaternion that quantify the instantaneous rotational parameters of the trajectory. These rotational parameters correspond to p-dimensional directional random vectors (DRVs). We propose two approaches to discriminate between the presence and absence of an elliptically polarized trajectory generated by human footsteps. In the first approach, we fit a von Mises–Fisher probability density function to the DRVs and estimate the concentration parameter. In the second approach, we employ the Kullback–Leibler divergence between the estimated nonparametric hyperspherical probability densities. The detection performance of the proposed metrics is shown to achieve an accuracy of \(97\%\) compared to existing approaches of \(82\%\) for footstep signals.
Similar content being viewed by others
References
J. Albusac, J. Castro-Schez, L. Lopez-Lopez, D. Vallejo, L. Jimenez-Linares, A supervised learning approach to automate the acquisition of knowledge in surveillance systems. Signal Process. 89(12), 2400–2414 (2009)
R. Bahrouna, O. Michelb, F. Frassatia, M. Carmonaa, J. Lacoumea, New algorithm for footstep localization using seismic sensors in an indoor environment. J. Sound Vib. 333(3), 1046–1066 (2014)
N.L. Bihan, P.O. Amblard, Detection and estimation of Gaussian proper quaternion valued random processes. in Proceedings of IMA Conference on Mathematics (2006)
N.L. Bihan, J. Mars, Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Signal Process. 84(7), 1177–1199 (2004)
C.H. Chapman, Fundamentals of Seismic Wave Propagation (Cambridge University Press, Cambridge, 2004)
J.P. Claassen, Robust bearing estimation for three-component stations. Pure Appl. Geophys. J. 158(1–2), 349–374 (2001)
I.S. Dhillon, S. Sra, Modelling Data Using Directional Distributions. Technical Report, The University of Texas at Austin (2003)
D. Donno, A. Nehorai, U. Spagnolini, Seismic velocity and polarization estimation for wavefield separation. IEEE Trans. Signal Process. 56, 4794–4809 (2008)
T. Gandhi, R. Chang, M.M. Trivedi, Video and seismic sensor-based structural health monitoring: framework, algorithms, and implementation. IEEE Trans. Intell. Transp. Syst. 8, 169–180 (2007)
P. Ginzberg, A. Walden, Testing for quaternion propriety. IEEE Trans. Signal Process. 59(7), 3025–3034 (2011)
R. Goldman, Understanding quaternions. Gr. Models 73(2), 21–49 (2011)
S. Greenhalgh, I.M. Mason, B. Zhou, An analytical treatment of single station triaxial seismic direction finding. J. Geophys. Eng. 2, 8–15 (1995)
A.J. Hanson, Visualising Quaternions, The Morgan Kaufmann Series in Interactive 3D Technology (Morgan Kaufmann, 2006)
V. Hari, A. Premkumar, X. Zhong, A decoupled approach for near-field source localization using a single acoustic vector sensor. Circuits Syst. Signal Process. 32, 843–859 (2013)
K.M. Houston, D.P. McGaffigan, Spectrum analysis techniques for personnel detection using seismic sensors. IEEE Trans. Signal Process. 5090, 162–173 (2003)
C.J. Huang, H.Y. Yin, C.Y. Chen, C.H. Yeh, C.L. Wang, Ground vibrations produced by rock motions and debris flows. J. Geophys. Res. Earth Surf. 112 (2007)
J. Huang, S. Xiao, Q. Zhou, F.G.X. You, H. Li, B. Li, A robust feature extraction algorithm for the classification of acoustic targets in wild environments. Circuits Syst. Signal Process. 34, 2395–2406 (2015)
B. Jablonski, Anisotropic filtering of multidimensional rotational trajectories as a generalization of 2D diffusion process. Multidimens. Syst. Signal Process. 19, 379–399 (2008)
Y. Kawahara, M. Sugiyama, Change-point detection in time-series data by direct density-ratio estimation. in Proceedings of 9th SIAM International Conference on Data Mining, pp. 389–400 (2009)
B. Koszteczky, G. Vakulya, G. Simon, Forest intrusion detection system with sensor network. in Proceedings IEEE International Conference on Instrumentation and Measurement Technology, pp. 1672–1676 (2015)
S. Li, R.M. Mnatsakanov, M.E. Andrew, k-nearest neighbor based consistent entropy estimation for hyperspherical distributions. Entropy 13(3), 650–667 (2011)
J.M. Lilly, Modulated oscillations in three dimensions. IEEE Trans. Signal Process. 59, 5930–5943 (2011)
J.M. Lilly, S.C. Olhede, Bivariate instantaneous frequency and bandwidth. IEEE Trans. Signal Process. 58(2), 591–603 (2010)
K.V. Mardia, P.E. Jupp, Directional Statistics, Wiley Series in Probability and Statistics (Wiley, 1999)
G. Mazarakis, J. Avaritsiotis, A prototype sensor node for footstep detection. in Proceedings of 2nd European Workshop Wireless Sensor Networks, pp. 415–418 (2005)
A. Merrifield, An investigation of mathematical models for animal group movement, using classical and statistical approaches. Ph.D. Thesis, The University of Sydney (2006)
F. Meysel, Map-based change detection (MBCD) in urban traffic scenes. in Proceedings of Joint Urban Remote Sensing Event, pp. 1–4 (2011)
G. Miller, H. Pursey, On the partition of energy between elastic waves in a semi-infinite solid. in Proceedings of Royal Society London, Series A, vol. 233 (1955)
S. Miron, N. Le Bihan, J. Mars, Quaternion-music for vector-sensor array processing. IEEE Trans. Signal Process. 57, 1316–1327 (2006)
R. Narayanaswami, A. Gandhe, A. Tyurina, R.K. Mehra, Sensor fusion and feature-based human/animal classification for unattended ground sensors. in Proceedings of IEEE International Conference on Technology for Homeland Security, pp. 344–350 (2010)
V.D. Nguyen, M.T. Le, A.D. Do, H.H. Duong, T.D. Thai, D.H. Tran, An efficient camera-based surveillance for fall detection of elderly people. in Proceedings of IEEE 9th Conference on Industrial Electronics and Applications, pp. 994–997 (2014)
T.D. Popescu, D. Aiordǎchioaie, New procedure for change detection operating on Rényi entropy with application in seismic signals processing. Circuits Syst. Signal Process. 36, 1–21 (2017)
M.S.S. Reddy, K. Nathwani, R.M. Hegde, Probabilistic detection methods for acoustic surveillance using audio histograms. Circuits Syst. Signal Process. 34, 1977–1992 (2015)
V. Reddy, D. Venkatraman, A.W.H. Khong, B.P. Ng, Footstep detection and denoising using a single triaxial geophone. in Proceedings of IEEE Asia Pacific Conference on Circuits and Systems, pp. 1171–1174 (2010)
H. Singh, N. Misra, V. Hnizdo, A. Fedorowicz, E. Demchuk, Nearest neighbor estimates of entropy. Am. J. Math. Manag. Sci. 23, 301–321 (2003)
J.C. Souyris, C. Tison, Polarimetric analysis of bistatic sar images from polar decomposition: a quaternion approach. IEEE Trans. Geosci. Remote Sens. 45(9), 2701–2713 (2007)
J.C. Souyris, C. Tison, Quaternion neural-network-based polsar land classification in poincare-sphere-parameter space. IEEE Trans. Geosci. Remote Sens. 52(9), 5693–5703 (2014)
J.Z. Stafsudd, S. Asgari, R. Hudson, K. Yao, E. Taciroglu, Localization of short-range acoustic and seismic wideband sources: algorithms and experiments. J. Sound Vib. 312, 74–93 (2008)
G. Succi, D. Clapp, R. Gampert, G. Prado, Footstep detection and tracking. Proc. SPIE 4393, 22–29 (2001)
G. Succi, G. Prado, R. Gampert, T. Pedersen, H. Dhaliwal, Problems in seismic detection and tracking. Proc. SPIE 4040, 165–173 (2000)
D. Venkatraman, V. Reddy, A.W.H. Khong, On the use of the quaternion generalized Gaussian distribution for footstep detection. in Proceedings IEEE International Conference on Acoustics, Speech, and Signal Processing (2013)
D. Venkatraman, V. Reddy, A.W.H. Khong, B.P. Ng, Polarization-cum-energy metric for footstep detection using vector-sensor. in Proceedings of IEEE Internationl Conference on Technologies for Homeland Security (2011)
H.F. Xing, F. Li, Y.L. Liu, Wavelet denoising and feature extraction of seismic signal for footstep detection. in Proceedings of IEEE International Conference on Wavelet Analysis and Pattern Recognition, vol. 1, pp. 218–223 (2007)
W. Zhang, Q.J. Wu, H.B. Yin, Moving vehicles detection based on adaptive motion histogram. Digit. Signal Process. 20(3), 793–805 (2010)
E. Zupan, M. Saje, D. Zupan, The quaternion-based three-dimensional beam theory. Comput. Methods Appl. Mech. Eng. 198, 3944–3956 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Venkatraman, D., Khong, A.W.H. Directional Statistics Approach Based on Instantaneous Rotational Parameters of Tri-axial Trajectories for Footstep Detection. Circuits Syst Signal Process 37, 1958–1987 (2018). https://doi.org/10.1007/s00034-017-0647-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-017-0647-x