Bearing-Only Tracking with a Mixture of von Mises Distributions - PowerPoint PPT Presentation

Faculty of Electrical Engineering University of Zagreb and Computing Bearing-Only Tracking with a Mixture of von Mises Distributions Ivan Petrović Ivan Marković University of Zagreb, Faculty of Electrical Engineering and Computing, Departement of Control and Computer Engineering October 8, 2012 Centre of Research Excellence for Advanced Cooperative Systems I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 1 / 29

Outline 1 Introduction 2 von Mises Distribution 3 Tracking with a von Mises Mixture Recursive Bayesian Tracking Key operations 4 Experiments 5 Conclusion I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 2 / 29

Outline 1 Introduction 2 von Mises Distribution 3 Tracking with a von Mises Mixture Recursive Bayesian Tracking Key operations 4 Experiments 5 Conclusion I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 3 / 29

Motivation • mixtures can smoothly represent complex distributions I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 4 / 29

Motivation • mixtures can smoothly represent complex distributions • angular random variables ⇒ von Mises distribution I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 4 / 29

Motivation • mixtures can smoothly represent complex distributions • angular random variables ⇒ von Mises distribution • captures well non-euclidean properties of angular data I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 4 / 29

Outline 1 Introduction 2 von Mises Distribution 3 Tracking with a von Mises Mixture Recursive Bayesian Tracking Key operations 4 Experiments 5 Conclusion I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 5 / 29

von Mises Distribution • the pdf has the following form [von Mises, 1918] 1 p ( x ; µ, κ ) = 2 π I 0 ( κ ) exp [ κ cos ( x − µ )] , where µ is the mean direction, κ is the concentration parameter, I 0 ( κ ) is the modified bessel function of the first kind and order zero 2 p ( x; µ, κ ) 1 0 0 50 100 150 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 6 / 29 x [ ◦ ]

Outline 1 Introduction 2 von Mises Distribution 3 Tracking with a von Mises Mixture Recursive Bayesian Tracking Key operations 4 Experiments 5 Conclusion I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 7 / 29

Recursive Bayesian Tracking • goal is to estimate p ( x k | z 1 : k ) I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 8 / 29

Recursive Bayesian Tracking • goal is to estimate p ( x k | z 1 : k ) • cyclic procedure of prediction–update steps I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 8 / 29

Recursive Bayesian Tracking • goal is to estimate p ( x k | z 1 : k ) • cyclic procedure of prediction–update steps • prediction via total probability theorem � p ( x k | z 1 : k − 1 ) = p ( x k | x k − 1 ) p ( x k − 1 | z 1 : k − 1 ) d x k − 1 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 8 / 29

Recursive Bayesian Tracking • goal is to estimate p ( x k | z 1 : k ) • cyclic procedure of prediction–update steps • prediction via total probability theorem � p ( x k | z 1 : k − 1 ) = p ( x k | x k − 1 ) p ( x k − 1 | z 1 : k − 1 ) d x k − 1 • update via Bayes’ rule p ( x k | z 1 : k ) = p ( z k | x k ) p ( x k | z 1 : k − 1 ) p ( z k | z 1 : k − 1 ) I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 8 / 29

Convolution • prediction — convolution of von Mises distribution [Mardia and Jupp, 1999] 1 �� � 1 / 2 � κ 2 i + κ 2 h ( x ) = 2 π I 0 ( κ i ) I 0 ( κ j ) · I 0 j + 2 κ i κ j + cos ( x − [ µ i + µ j ]) I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 9 / 29

Convolution • prediction — convolution of von Mises distribution [Mardia and Jupp, 1999] 1 �� � 1 / 2 � κ 2 i + κ 2 h ( x ) = 2 π I 0 ( κ i ) I 0 ( κ j ) · I 0 j + 2 κ i κ j + cos ( x − [ µ i + µ j ]) • can be well approximated by h ( x ) ≈ p ( x ; µ i + µ j , A − 1 ( A ( κ i ) A ( κ j )) , where A ( κ ) = I 1 ( κ ) I 0 ( κ ) I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 9 / 29

Convolution µ i = 120 , κ i = 70 µ i = 30 , κ i = 20 µ c = 120 , κ c = 20 . 81 3 . 5 3 2 . 5 2 1 . 5 1 0 . 5 0 0 100 200 300 400 x [ ◦ ] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 10 / 29

Product • product of von Mises distribution [Murray and Morgenstern, 2010] 1 g ( x ) = 4 π 2 I 0 ( κ i ) I 0 ( κ j ) exp [ κ ij cos ( x − µ ij )] , where µ ij = µ i + atan2 [ − sin ( µ i − µ j ) , κ i /κ j + cos ( µ i − µ j )] , � κ 2 i + κ 2 κ ij = j + 2 κ i κ j cos ( µ i − µ j ) , I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 11 / 29

Product • product of von Mises distribution [Murray and Morgenstern, 2010] 1 g ( x ) = 4 π 2 I 0 ( κ i ) I 0 ( κ j ) exp [ κ ij cos ( x − µ ij )] , where µ ij = µ i + atan2 [ − sin ( µ i − µ j ) , κ i /κ j + cos ( µ i − µ j )] , � κ 2 i + κ 2 κ ij = j + 2 κ i κ j cos ( µ i − µ j ) , • we approximate the product with g ( x ) ≈ p ( x ; µ ij , κ ij ) I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 11 / 29

Product µ i = 100 , κ i = 70 µ j = 120 , κ j = 100 µ p = 111 . 78 , κ p = 167 . 49 5 4 3 2 1 0 80 90 100 110 120 130 140 x [ ◦ ] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 12 / 29

Product µ i = 90 , κ i = 70 µ j = 270 , κ j = 70 µ P = 180 , κ P = 0 3 . 5 3 2 . 5 2 1 . 5 1 0 . 5 0 0 100 200 300 400 x [ ◦ ] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 13 / 29

von Mises Mixture • state representation is a mixture N 1 � p ( x k | z 1 : k ) = γ i 2 π I 0 ( κ i ) exp [ κ i cos ( x k − µ i )] i = 1 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 14 / 29

von Mises Mixture • state representation is a mixture N 1 � p ( x k | z 1 : k ) = γ i 2 π I 0 ( κ i ) exp [ κ i cos ( x k − µ i )] i = 1 • motion model is a single von Mises 1 p ( x k | x k − 1 ) = 2 π I 0 ( κ ) exp [ κ cos ( x k − x k − 1 )] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 14 / 29

von Mises Mixture • state representation is a mixture N 1 � p ( x k | z 1 : k ) = γ i 2 π I 0 ( κ i ) exp [ κ i cos ( x k − µ i )] i = 1 • motion model is a single von Mises 1 p ( x k | x k − 1 ) = 2 π I 0 ( κ ) exp [ κ cos ( x k − x k − 1 )] • sensor model is a mixture M 1 � p ( z k | x k ) = γ i 2 π I 0 ( κ i ) exp [ κ i cos ( x k − z k , i )] i = 1 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 14 / 29

Component Reduction • we used a variant of West’s algorithm [West, 1993] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 15 / 29

Component Reduction • we used a variant of West’s algorithm [West, 1993] • Bhatacharyya coefficient as a distance metric � 2 π � c B ( p , q ) = p ( ξ ) q ( ξ ) d ξ 0 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 15 / 29

Component Reduction • we used a variant of West’s algorithm [West, 1993] • Bhatacharyya coefficient as a distance metric � 2 π � c B ( p , q ) = p ( ξ ) q ( ξ ) d ξ 0 • for von Mises pdfs closed form result [Calderara et al., 2011] I 0 ( κ ij / 2 ) c B ( p ( x ; µ i , κ i ) , p ( x ; µ j , κ j )) = { I 0 ( κ i ) I 0 ( κ j ) } 1 / 2 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 15 / 29

Entropy • quadratic Rényi entropy � p 2 ( x ) d x H 2 ( x ) = − log I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 16 / 29

Entropy • quadratic Rényi entropy � p 2 ( x ) d x H 2 ( x ) = − log • for von Mises mixture closed form result N N I 0 ( κ ij ) � � H 2 ( x ) = − log γ ij 2 π I 0 ( κ i ) I 0 ( κ j ) i = 1 j = 1 I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 16 / 29

Outline 1 Introduction 2 von Mises Distribution 3 Tracking with a von Mises Mixture Recursive Bayesian Tracking Key operations 4 Experiments 5 Conclusion I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 17 / 29

Synthetic data • two trajectories: continuous and turn-take, two filters: mixture and particle I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 18 / 29

Synthetic data • two trajectories: continuous and turn-take, two filters: mixture and particle • simulated multimodal measurement model (von Mises mixture) [Marković and Petrović, 2010] I. Marković, I. Petrović (FER) von Mises Mixture Tracking October 8, 2012 18 / 29