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

SLIDE 1

Bearing-Only Tracking with a Mixture of von Mises Distributions

Ivan Marković Ivan Petrović

University of Zagreb, Faculty of Electrical Engineering and Computing, Departement of Control and Computer Engineering

October 8, 2012

University of Zagreb Faculty of Electrical Engineering and Computing

Centre of Research Excellence for Advanced Cooperative Systems

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 1 / 29

SLIDE 2

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

SLIDE 3

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

SLIDE 4

Motivation

mixtures can smoothly represent complex distributions
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 4 / 29

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

von Mises Distribution

the pdf has the following form [von Mises, 1918]

p(x; µ, κ) = 1 2πI0(κ) exp [κ cos(x − µ)] , where µ is the mean direction, κ is the concentration parameter, I0(κ) is the modified bessel function of the first kind and order zero

50 100 150 1 2 x [◦] p(x; µ, κ)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 6 / 29

SLIDE 9

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

SLIDE 10

Recursive Bayesian Tracking

goal is to estimate p(xk|z1:k)
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 8 / 29

SLIDE 11

Recursive Bayesian Tracking

goal is to estimate p(xk|z1:k)
cyclic procedure of prediction–update steps
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 8 / 29

SLIDE 12

Recursive Bayesian Tracking

goal is to estimate p(xk|z1:k)
cyclic procedure of prediction–update steps
prediction via total probability theorem

p(xk|z1:k−1) =

p(xk|xk−1)p(xk−1|z1:k−1) dxk−1
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 8 / 29

SLIDE 13

Recursive Bayesian Tracking

goal is to estimate p(xk|z1:k)
cyclic procedure of prediction–update steps
prediction via total probability theorem

p(xk|z1:k−1) =

p(xk|xk−1)p(xk−1|z1:k−1) dxk−1
update via Bayes’ rule

p(xk|z1:k) = p(zk|xk)p(xk|z1:k−1) p(zk|z1:k−1)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 8 / 29

SLIDE 14

Convolution

prediction — convolution of von Mises distribution

[Mardia and Jupp, 1999] h(x) = 1 2πI0(κi)I0(κj)·I0

κ2

i + κ2 j + 2κiκj + cos(x − [µi + µj])

1/2

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 9 / 29

SLIDE 15

Convolution

prediction — convolution of von Mises distribution

[Mardia and Jupp, 1999] h(x) = 1 2πI0(κi)I0(κj)·I0

κ2

i + κ2 j + 2κiκj + cos(x − [µi + µj])

1/2

can be well approximated by

h(x) ≈ p(x; µi + µj, A−1(A(κi)A(κj)), where A(κ) = I1(κ)

I0(κ)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 9 / 29

SLIDE 16

Convolution

100 200 300 400 0.5 1 1.5 2 2.5 3 3.5 x[◦] µi = 120, κi = 70 µi = 30, κi = 20 µc = 120, κc = 20.81

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 10 / 29

SLIDE 17

Product

product of von Mises distribution [Murray and Morgenstern, 2010]

g(x) = 1 4π2I0(κi)I0(κj) exp [κij cos(x − µij)] , where µij = µi + atan2 [− sin(µi − µj), κi/κj + cos(µi − µj)] , κij =

κ2

i + κ2 j + 2κiκj cos(µi − µj),

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 11 / 29

SLIDE 18

Product

product of von Mises distribution [Murray and Morgenstern, 2010]

g(x) = 1 4π2I0(κi)I0(κj) exp [κij cos(x − µij)] , where µij = µi + atan2 [− sin(µi − µj), κi/κj + cos(µi − µj)] , κij =

κ2

i + κ2 j + 2κiκj cos(µi − µj),

we approximate the product with

g(x) ≈ p(x; µij, κij)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 11 / 29

SLIDE 19

Product

80 90 100 110 120 130 140 1 2 3 4 5 x[◦] µi = 100, κi = 70 µj = 120, κj = 100 µp = 111.78, κp = 167.49

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 12 / 29

SLIDE 20

Product

100 200 300 400 0.5 1 1.5 2 2.5 3 3.5 x[◦] µi = 90, κi = 70 µj = 270, κj = 70 µP = 180, κP = 0

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 13 / 29

SLIDE 21

von Mises Mixture

state representation is a mixture

p(xk|z1:k) =

N

i=1

γi 1 2πI0(κi) exp [κi cos(xk − µi)]

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 14 / 29

SLIDE 22

von Mises Mixture

state representation is a mixture

p(xk|z1:k) =

N

i=1

γi 1 2πI0(κi) exp [κi cos(xk − µi)]

motion model is a single von Mises

p(xk|xk−1) = 1 2πI0(κ) exp [κ cos(xk − xk−1)]

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 14 / 29

SLIDE 23

von Mises Mixture

state representation is a mixture

p(xk|z1:k) =

N

i=1

γi 1 2πI0(κi) exp [κi cos(xk − µi)]

motion model is a single von Mises

p(xk|xk−1) = 1 2πI0(κ) exp [κ cos(xk − xk−1)]

sensor model is a mixture

p(zk|xk) =

M

i=1

γi 1 2πI0(κi) exp [κi cos(xk − zk,i)]

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 14 / 29

SLIDE 24

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

SLIDE 25

Component Reduction

we used a variant of West’s algorithm [West, 1993]
Bhatacharyya coefficient as a distance metric

cB(p, q) =

2π

p(ξ)q(ξ) dξ
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 15 / 29

SLIDE 26

Component Reduction

we used a variant of West’s algorithm [West, 1993]
Bhatacharyya coefficient as a distance metric

cB(p, q) =

2π

p(ξ)q(ξ) dξ
for von Mises pdfs closed form result [Calderara et al., 2011]

cB (p(x; µi, κi), p(x; µj, κj)) = I0 (κij/2) {I0(κi)I0(κj)}1/2

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 15 / 29

SLIDE 27

Entropy

quadratic Rényi entropy

H2(x) = − log

p2(x) dx
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 16 / 29

SLIDE 28

Entropy

quadratic Rényi entropy

H2(x) = − log

p2(x) dx
for von Mises mixture closed form result

H2(x) = − log

N

i=1

N

j=1

γij I0(κij) 2πI0(κi)I0(κj)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 16 / 29

SLIDE 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

SLIDE 30

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

SLIDE 31

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

SLIDE 32

Synthetic data

two trajectories: continuous and turn-take, two filters: mixture and

particle

simulated multimodal measurement model (von Mises mixture)

[Marković and Petrović, 2010]

outlier probability was 0.3, while detection probability was 0.9
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 18 / 29

SLIDE 33

Synthetic data

two trajectories: continuous and turn-take, two filters: mixture and

particle

simulated multimodal measurement model (von Mises mixture)

[Marković and Petrović, 2010]

outlier probability was 0.3, while detection probability was 0.9
measurements were corrupted with von Mises noise
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 18 / 29

SLIDE 34

Synthetic data (continuous)

2 4 6 8 10 100 200 300 time [s] bearing [◦] measurements true bearing particle filter estimation kernel mixture estimation −5.5 −5 −4.5 −4 −3.5 −3 −2.5 entropy entropy

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 19 / 29

SLIDE 35

Video (continuous)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 20 / 29

SLIDE 36

Synthetic data (turn-take)

2 4 6 8 10 100 200 300 time [s] bearing [◦] measurements true bearing particle filter estimation kernel mixture estimation −5.5 −5 −4.5 −4 −3.5 −3 −2.5 entropy entropy

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 21 / 29

SLIDE 37

Video (turn-take)

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 22 / 29

SLIDE 38

Real-world experiments

four omnidirectional microphones in a Y configuration
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 23 / 29

SLIDE 39

Real-world experiments

four omnidirectional microphones in a Y configuration
Fs = 48 kHz, L = 1024, frame rate of approx. 47 Hz
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 23 / 29

SLIDE 40

Real-world experiments

four omnidirectional microphones in a Y configuration
Fs = 48 kHz, L = 1024, frame rate of approx. 47 Hz
two experiments: continuous from 0◦ − 360◦ and turn-take
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 23 / 29

SLIDE 41

Real-world experiments (0–360)

5 10 15 20 25 30 50 100 150 200 250 300 350 time [s] bearing [◦] particle filter mixture filter

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 24 / 29

SLIDE 42

Real-world experiments (turn-take)

2 4 6 8 10 12 50 100 150 200 250 300 350 time [s] bearing [deg] particle filter mixture filter

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 25 / 29

SLIDE 43

Discussion

we used 12 components for the mixture filter, and 360 particles
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 26 / 29

SLIDE 44

Discussion

we used 12 components for the mixture filter, and 360 particles
36 vs. 360 parameters for state representation
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 26 / 29

SLIDE 45

Discussion

we used 12 components for the mixture filter, and 360 particles
36 vs. 360 parameters for state representation
mean time of an iteration was 81.2 ms and 72.5 ms for kernel and

regularized particle filter, respectively

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 26 / 29

SLIDE 46

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 27 / 29

SLIDE 47

Conclusion

theoretical steps of Bayesian estimation with von Mises mixture
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 28 / 29

SLIDE 48

Conclusion

theoretical steps of Bayesian estimation with von Mises mixture
convolution, product, component reduction and entropy
I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 28 / 29

SLIDE 49

Conclusion

theoretical steps of Bayesian estimation with von Mises mixture
convolution, product, component reduction and entropy
demonstrated on, but not limited to, the problem of speaker tracking

with a microphone array

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 28 / 29

SLIDE 50

Thank you for your attention

Questions?

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 29 / 29

SLIDE 51

Calderara, S., Prati, A., and Cucchiara, R. (2011). Mixtures of von Mises Distributions for People Trajectory Shape Analysis. IEEE Transactions on Circuits and Systems for Video Technology, 21(4):457–471. Mardia, K. V. and Jupp, P. E. (1999). Directional Statistics. Wiley, New York. Marković, I. and Petrović, I. (2010). Speaker Localization and Tracking with a Microphone Array on a Mobile Robot Using von Mises Distribution and Particle Filtering. Robotics and Autonomous Systems, 58(11):1185–1196. Murray, R. F. and Morgenstern, Y. (2010). Cue Combination on the Circle and the Sphere. Journal of Vision, 10(11):1–11. von Mises, R. (1918). Uber die ‘Ganzzahligkeit’ der Atomgewicht und Verwandte Fragen. Physikalische Zeitschrift, (19):490–500. West, M. (1993). Approximating Posterior Distributions by Mixtures. Journal of Royal Statistical Society, Series B, 55(2):409–442.

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 52

Calculating azimuth

Mic1 Mic2 Sound source x y

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 53

Calculating azimuth

Mic1 Mic2 Sound source x y

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 54

Calculating azimuth

Mic1 Mic2 Sound source x y

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 55

Calculating azimuth

Mic1 Mic2 Sound source x y

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 56

Calculating azimuth

Mic1 Mic2 Sound source x y

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 57

Calculating azimuth

Mic1 Mic2 Sound source x y ϕ+

12

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 58

Calculating azimuth

Mic1 Mic2 Sound source x y ϕ+

12

ϕ−

12

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 59

Calculating azimuth

Mic1 Mic2 Sound source x y Mic3 ϕ+

12

ϕ−

12

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 60

Calculating azimuth

Mic1 Mic2 Sound source x y Mic3 ϕ+

12

ϕ−

12

ϕ+

23

ϕ−

23

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 61

Calculating azimuth

Mic1 Mic2 Sound source x y Mic3 ϕ+

12

ϕ−

12

ϕ+

23

ϕ−

23

ϕ+

13

ϕ−

13

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 30 / 29

SLIDE 62

Sensor model

azimuth [◦] 50 100 150 200 250 300 350 ϕ+

12

ϕ−

12

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 31 / 29

SLIDE 63

Sensor model

azimuth [◦] 50 100 150 200 250 300 350 ϕ+

12

ϕ−

12

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 31 / 29

SLIDE 64

Sensor model

azimuth [◦] 50 100 150 200 250 300 350 ϕ+

12

ϕ−

12

ϕ+

23

ϕ−

23

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 31 / 29

SLIDE 65

Sensor model

azimuth [◦] 50 100 150 200 250 300 350 ϕ+

12

ϕ−

12

ϕ+

23

ϕ−

23

ϕ+

13

ϕ−

13

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 31 / 29

SLIDE 66

Sensor model

azimuth [◦] 50 100 150 200 250 300 350 ϕ+

12

ϕ−

12

ϕ+

23

ϕ−

23

ϕ+

13

ϕ−

13

Strong mode at the correct azimuth

I. Marković, I. Petrović (FER)

von Mises Mixture Tracking October 8, 2012 31 / 29