On nonlinear state estimation in a Riemannian manifold

We consider state estimation for partially observed processes evolving in a Riemannian manifold and obtain some new results on un-normalized nonlinear filters. We develop both local (coordinatised) versions and for the first time, extrinsic results for embedded Riemannian manifolds; the latter have computational advantages. Our results also cover continuous discrete filters. We also obtain extrinsic conditions for a process evolving in a Riemannian manifold to be Brownian motion.

[1]  Giorgio Picci,et al.  Dynamic vision and estimation on spheres , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[2]  F. Tompkins,et al.  Bayesian Filtering on the Stiefel Manifold , 2007, 2007 2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing.

[3]  Daniel W. Stroock,et al.  On the growth of stochastic integrals , 1971 .

[4]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[5]  Taeyoung Lee,et al.  Global symplectic uncertainty propagation on SO(3) , 2008, 2008 47th IEEE Conference on Decision and Control.

[6]  J. Griffiths.  The Theory of Stochastic Processes , 1967 .

[7]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[8]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[9]  R. Brockett.  Lie Algebras and Lie Groups in Control Theory , 1973 .

[10]  Luigi Accardi,et al.  Some stationary Markov processes in discrete time for unit vectors , 1987 .

[11]  Roger W. Brockett.  Notes on Stochastic Processes on Manifolds , 1997 .

[12]  Mark A. Pinsky,et al.  Stochastic Riemannian Geometry , 1978 .

[13]  Elton P. Hsu.  Stochastic analysis on manifolds , 2002 .

[14]  James Ting-Ho Lo,et al.  Exponential Fourier densities and optimal estimation and detection on the circle , 1977, IEEE Trans. Inf. Theory.

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

[16]  Linda R. Eshleman,et al.  Exponential Fourier Densities on $SO( 3 )$ and Optimal Estimation and Detection for Rotational Processes , 1979 .

[17]  Nathaniel Coburn,et al.  Vector and Tensor Analysis , 2019, Introduction to the Variational Formulation in Mechanics.

[18]  P. Perona,et al.  Motion estimation via dynamic vision , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[19]  A. T. Bharucha-Reid.  Elements of the theory of Markov processes and their applications , 1961 .

[20]  C. E. Weatherburn,et al.  An Introduction to Riemannian Geometry and the Tensor Calculus , 1938 .

[21]  Alessandro Chiuso,et al.  A wide-sense estimation theory on the unit sphere , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[22]  P. Caines,et al.  Nonlinear Filtering in Rieznnnnifln Manifolds , 1985 .

[23]  Y. Chikuse,et al.  State space models on special manifolds , 2006 .

[24]  A. Goetz.  Introduction to differential geometry , 1970 .

[25]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[26]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[27]  G. Chirikjian,et al.  Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups , 2000 .

[28]  'Unobserved' Monte Carlo method for identification of partially observed nonlinear state space systems. Part II. Counting process observations , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[29]  Brian D. Ripley,et al.  Statistics on Spheres , 1983 .

[30]  P. Tait.  Vector Analysis , 1893, Nature.

[31]  Rama Chellappa,et al.  Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

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

[33]  B. O'neill.  Elementary Differential Geometry , 1966 .

[34]  Linda R. Eshleman,et al.  Exponential Fourier densities on SO(3) and optimal estimation and detection for rotational processes , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[35]  James Ting-Ho Lo,et al.  Exponential Fourier densities on S2 and optimal estimation and detection for directional processes , 1977, IEEE Trans. Inf. Theory.

[36]  J. Breckling.  The Analysis of Directional Time Series: Applications to Wind Speed and Direction , 1989 .

[37]  P. Caines,et al.  Nonlinear Filtering in Riemannian Manifolds , 1984 .

[38]  Tyrone E. Duncan.  Some Filtering Results in Riemann Manifolds , 1977, Inf. Control..

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

[40]  A. Doma.  Generalized Inverses of Matrices and Its Applications. , 1983 .

[41]  Y. Chikuse.  Statistics on special manifolds , 2003 .