Optimal quantization of circular distributions

In this work, a new method for approximating circular probability distributions by a mixture of weighted discrete samples is proposed. Particularly, the wrapped normal distribution, the von Mises distribution, and the Bingham distribution are considered. The approximation approach is based on formulating a quantizer and a global optimality measure, which can be optimized directly. Furthermore, a relationship between the Bingham distribution and the von Mises distribution are formulated showing that it is sufficient to approximate a von Mises distribution with suitably chosen parameters in order to obtain an optimal approximation of the Bingham distribution. The resulting approximation is of particular interest for filtering applications, because the involved optimality measure gives rise to a general error estimate in propagation of uncertainties through nontrivial functions in the circular domain.

[1]  Ivan Markovic,et al.  Bearing-only tracking with a mixture of von Mises distributions , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[2]  Gilles Pagès,et al.  Optimal quadratic quantization for numerics: the Gaussian case , 2003, Monte Carlo Methods Appl..

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

[4]  Augusto Aubry,et al.  Phase noise modeling and its effects on the performance of some radar signal processors , 2015, 2015 IEEE Radar Conference (RadarCon).

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

[6]  Ivan Markovic,et al.  Active speaker localization with circular likelihoods and bootstrap filtering , 2013, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[7]  G. Chirikjian.  Stochastic Models, Information Theory, and Lie Groups, Volume 1 , 2009 .

[8]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[9]  R. McKilliam.  Lattice theory, circular statistics and polynomial phase signals , 2010 .

[10]  A. Willsky,et al.  Estimation for Rotational Processes with One Degree of Freedom , 1972 .

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

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

[13]  S. Graf,et al.  Foundations of Quantization for Probability Distributions , 2000 .

[14]  Serge Reboul,et al.  GNSS dataless signal tracking with a delay semi-open loop and a phase open loop , 2013, Signal Process..

[15]  Gillis Pagés,et al.  A space quantization method for numerical integration , 1998 .

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

[17]  G. S. Watson.  Statistics on Spheres , 1983 .

[18]  Uwe D. Hanebeck,et al.  Moment-based Dirac Mixture Approximation of Circular Densities , 2014 .

[19]  Paris Smaragdis,et al.  A Wrapped Kalman Filter for Azimuthal Speaker Tracking , 2013, IEEE Signal Processing Letters.

[20]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[21]  Gerhard Kurz,et al.  Recursive Bayesian filtering in circular state spaces , 2015, IEEE Aerospace and Electronic Systems Magazine.

[22]  A. Willsky,et al.  Estimation for rotational processes with one degree of freedom--Part I: Introduction and continuous- , 1975 .

[23]  G. Chirikjian.  Stochastic models, information theory, and lie groups , 2012 .

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