Recursive Bingham filter for directional estimation involving 180 degree symmetry

© 2014 JAIF. This work considers filtering of uncertain data defined on periodic domains, particularly the circle and the manifold of orientations in 3D space. Filters based on the Kalman filter perform poorly in this directional setting as they fail to take the structure of the underlying manifold into account. We present a recursive filter based on the Bingham distribution, which is defined on the considered domains. The proposed filter can be applied to circular filtering problems with 180 degree symmetry and to estimation of orientations in three dimensional space. It is easily implemented using standard numerical techniques and suitable for real-time applications. We evaluate our filter in a challenging scenario and compare it to a Kalman filtering approach adapted to the particular setting.

[1]  Ignacio Santa-María Megía.  Which spheres admit a topological group structure , 2007 .

[2]  Leslie Pack Kaelbling,et al.  Tracking the spin on a ping pong ball with the quaternion Bingham filter , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[3]  H. Schaeben,et al.  The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis , 2004 .

[4]  Gerhard Kurz,et al.  Recursive nonlinear filtering for angular data based on circular distributions , 2013, 2013 American Control Conference.

[5]  Suvrit Sra,et al.  A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of Is(x) , 2012, Comput. Stat..

[6]  Andrew T. A. Wood,et al.  On the derivatives of the normalising constant of the Bingham distribution , 2007 .

[7]  J. Kent.  The Fisher‐Bingham Distribution on the Sphere , 1982 .

[8]  S. R. Jammalamadaka,et al.  Topics in Circular Statistics , 2001 .

[9]  John T. Kent,et al.  Asymptotic expansions for the Bingham distribution , 1987 .

[10]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[11]  Serge Reboul,et al.  A multi-sensor circular particle filter applied to the fusion of the GPS-L2C channels , 2011, 14th International Conference on Information Fusion.

[12]  Gerhard Kurz,et al.  Constrained object tracking on compact one-dimensional manifolds based on directional statistics , 2013, International Conference on Indoor Positioning and Indoor Navigation.

[13]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[14]  Tamio Koyama,et al.  Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$d-dimensional sphere , 2012, Comput. Stat..

[15]  Udo Frese,et al.  Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds , 2011, Inf. Fusion.

[16]  Kanti V. Mardia,et al.  Table of maximum likelihood estimates for the bingham distribution , 1977 .

[17]  Gerhard Kurz,et al.  Recursive estimation of orientation based on the Bingham distribution , 2013, Proceedings of the 16th International Conference on Information Fusion.

[18]  Leslie Pack Kaelbling,et al.  Tracking 3-D Rotations with the Quaternion Bingham Filter , 2013 .

[19]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[20]  C. Herz.  BESSEL FUNCTIONS OF MATRIX ARGUMENT , 1955 .

[21]  K. Mardia.  Characterizations of Directional Distributions , 1975 .

[22]  Seth J. Teller,et al.  Scalable, absolute position recovery for omni-directional image networks , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[23]  R. Fisher.  Dispersion on a sphere , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[24]  Arak M. Mathai,et al.  Functions of Matrix Argument , 2010 .

[25]  Peter Deuflhard,et al.  Numerische Mathematik. I , 2002 .

[26]  J. Kuipers.  Quaternions and Rotation Sequences , 1998 .

[27]  Yudell L. Luke,et al.  Algorithms for the Computation of Mathematical Functions , 1977 .

[28]  Alan Edelman,et al.  The efficient evaluation of the hypergeometric function of a matrix argument , 2006, Math. Comput..

[29]  Gary R. Bradski,et al.  Monte Carlo Pose Estimation with Quaternion Kernels and the Bingham Distribution , 2011, Robotics: Science and Systems.

[30]  Keith E. Muller,et al.  Computing the confluent hypergeometric function, M(a,b,x) , 2001, Numerische Mathematik.

[31]  Christopher Bingham.  An Antipodally Symmetric Distribution on the Sphere , 1974 .

[32]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[33]  Gerhard Kurz,et al.  Unscented Orientation Estimation Based on the Bingham Distribution , 2013, IEEE Transactions on Automatic Control.

[34]  Serge Reboul,et al.  A multi-temporal multi-sensor circular fusion filter , 2014, Inf. Fusion.

[35]  S. R. Jammalamadaka,et al.  Directional Statistics, I , 2011 .

[36]  A. Shiryaev,et al.  Probability (2nd ed.) , 1995, Technometrics.

[37]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[38]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[39]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[40]  K. Mardia,et al.  Maximum Likelihood Estimators for the Matrix Von Mises-Fisher and Bingham Distributions , 1979 .

[41]  J. Kent.  The Complex Bingham Distribution and Shape Analysis , 1994 .

[42]  A. Wood,et al.  Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants , 2005 .

[43]  Gerhard Kurz,et al.  Nonlinear measurement update for estimation of angular systems based on circular distributions , 2014, 2014 American Control Conference.

[44]  Serge Reboul,et al.  A recursive fusion filter for angular data , 2009, 2009 IEEE International Conference on Robotics and Biomimetics (ROBIO).

[45]  K. Mardia.  Directional statistics in geosciences , 1981 .

[46]  Joseph J. LaViola,et al.  On Kalman Filtering With Nonlinear Equality Constraints , 2007, IEEE Transactions on Signal Processing.