Spline filter for target tracking

In this paper an efficient approach to nonlinear non-Gaussian state estimation based on spline filtering is presented. The estimation of the conditional probability density of the unknown state can be ideally achieved through Bayes rule. However, the associated computational requirements make it impossible to implement this online filter in practice. In the general particle filtering problem, estimation accuracy increases with the number of particles at the expense of increased computational load. In this paper, B-Spline interpolation is used to represent the density of the state pdf through a low order continuous polynomial. The motivation is to reduce the computational cost. The motion of spline control points and corresponding coefficients is achieved through implementation of the Fokker-Planck equation, which describes the propagation of state probability density function between measurement instants. This filter is applicable for a general state estimation problem as no assumptions are made about the underlying probability density.

[1]  Leonard A. Ferrari,et al.  Discrete splines and spline filters , 1992 .

[2]  J. Huang,et al.  Curse of dimensionality and particle filters , 2003, 2003 IEEE Aerospace Conference Proceedings (Cat. No.03TH8652).

[3]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[4]  Fred Daum,et al.  Nonlinear filters with particle flow induced by log-homotopy , 2009, Defense + Commercial Sensing.

[5]  Dong Yu,et al.  Solving Nonlinear Estimation Problems Using Splines , 2009 .

[6]  F. Daum.  Nonlinear filters: beyond the Kalman filter , 2005, IEEE Aerospace and Electronic Systems Magazine.

[7]  Hong Yue,et al.  Improved identification algorithm for B-spline modelling of output probability density functions , 2004, Proceedings of the 2004 IEEE International Symposium on Intelligent Control, 2004..

[8]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[9]  S. Haykin,et al.  Cubature Kalman Filters , 2009, IEEE Transactions on Automatic Control.

[10]  Kumaradevan Punithakumar,et al.  Spline filter for multidimensional nonlinear/non-Gaussian Bayesian tracking , 2008, SPIE Defense + Commercial Sensing.

[11]  Sung Yong Shin,et al.  Scattered Data Interpolation with Multilevel B-Splines , 1997, IEEE Trans. Vis. Comput. Graph..

[12]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[13]  Bhashyam Balaji,et al.  Feynman path integral inspired computational methods for nonlinear filtering , 2010, Defense + Commercial Sensing.

[14]  Charalambos D. Charalambous,et al.  Nonlinear Estimation for a Class of Systems , 2006, ISIT.

[15]  W. Boehm.  Inserting New Knots into B-spline Curves , 1980 .

[16]  I. J. Schoenberg,et al.  ON POLYA FREQUENCY FUNCTIONS. III. THE POSITIVITY OF TRANSLATION DETERMINANTS WITH AN APPLICATION TO THE INTERPOLATION PROBLEM BY SPLINE CURVES( , 1953 .

[17]  R. Carroll,et al.  Equivalent Kernels of Smoothing Splines in Nonparametric Regression for Clustered/Longitudinal Data , 2004 .

[18]  Fred Daum,et al.  Particle flow for nonlinear filters with log-homotopy , 2008, SPIE Defense + Commercial Sensing.

[19]  Kumaradevan Punithakumar,et al.  Spline filter for nonlinear/non-Gaussian Bayesian tracking , 2007, SPIE Optical Engineering + Applications.

[20]  Dmitrijs Docenko,et al.  Spline Histogram Method for Reconstruction of Probability Density Functions of Clusters of Galaxies , 2003 .

[21]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[22]  G. Wahba,et al.  A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines , 1970 .

[23]  P. N. Paraskevopoulos,et al.  Modern Control Engineering , 2001 .

[24]  K. P. Mohandas.  Modern Control Engineering , 2006 .

[25]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[26]  W. Boehm,et al.  Bezier and B-Spline Techniques , 2002 .

[27]  R. Klein,et al.  Implementation of non-linear estimators using monospline , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[28]  Bhashyam Balaji.  Feynman path integrals and continuous nonlinear filtering , 2010, Defense + Commercial Sensing.

[29]  Richard S. Bucy,et al.  Nonlinear Filter Representation via Spline Functions , 1974 .