[PPT] - Bayesian Tracking: Basic Idea Iterative updating of conditional PowerPoint Presentation

SLIDE 1

Bayesian Tracking: Basic Idea

Iterative updating of conditional probability densities!

kinematic target state xk at time tk, accumulated sensor data Zk a priori knowledge: target dynamics models, sensor model, road maps

prediction:

p(xk−1|Zk−1)

dynamics model

− − − − − − − − − − →

road maps

p(xk|Zk−1)

filtering:

p(xk|Zk−1)

sensor data Zk

− − − − − − − − − − →

sensor model

p(xk|Zk)

retrodiction:

p(xl−1|Zk)

filtering output

← − − − − − − − − − −

dynamics model

p(xl|Zk) − finite mixture: inherent ambiguity (data, model, road network) − optimal estimators: e.g. minimum mean squared error (MMSE) − initiation of pdf iteration: multiple hypothesis track extraction

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 1

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p(xk|Zk−1) is a prediction of the target state at time tk

based on all measurements in the past.

p(xk|Zk−1) = dxk−1 p(xk, xk−1|Zk−1) marginal pdf = dxk−1 p(xk|xk−1, Zk−1)

bject dynamics!

p(xk−1|Zk−1)

idea: iteration!

notion of a conditional pdf

ften: p(xk|xk−1, Zk−1) = p(xk|xk−1)

(MARKOV) sometimes: p(xk|xk−1) = Nxk; Fk|k−1

deterministic

xk−1, Dk|k−1

random

(linear GAUSS-MARKOV)
p(Zk, mk|xk) ∝ ℓ(xk; Zk, mk) describes, what the current sensor output Zk, mk

can say about the current target state xk and is called likelihood function.

sometimes: ℓ(zk; xk) = N

xk; Hkxk, Rk
(1 target, 1 measurement)

iteration formula: p(xk|Zk) = ℓ(xk; Zk, mk) dxk−1 p(xk|xk−1) p(xk−1|Zk−1)

dxk ℓ(xk; Zk, mk)

dxk−1 p(xk|xk−1) p(xk−1|Zk−1)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 2

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GAUSSian transition pdf: p(xk|xk−1, Zk−1) = N(xk; Fk|k−1xk−1, Dk|k−1)

with:

Fk|k−1 (evolution matrix)

describes deterministic motion

,

Dk|k−1 (dynamics covariance matrix)

models of random maneuvers

GAUSSian posterior: p(xk−1|Zk−1) = N(xk−1; xk−1|k−1, Pk−1|k−1)

p(xk|Zk−1) =

dxk−1 N
xk; Fk|k−1xk−1, Dk|k−1
dynamics model

N

xk−1; xk−1|k−1, Pk−1|k−1
posterior at time tk−1

= Nxk; Fk|k−1xk−1|k−1

=:xk|k−1

, Fk|k−1Pk−1|k−1F⊤

k|k−1 + Dk|k−1

=:Pk|k−1
×
dxk−1 Nxk−1; . . . , . . .
=1

(normalization!)

(exploit product formula!) = N(xk; xk|k−1, Pk|k−1)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 3

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Kalman filter: linear GAUSSian likelihood/dynamics, xk = (r⊤

k , ˙

r⊤

k ,¨

r⊤

k )⊤, Zk = {zk, Zk−1}

initiation: p(x0) = N

x0; x0|0, P0|0
,

initial ignorance:

P0|0 ‘large’

prediction: N

xk−1; xk−1|k−1, Pk−1|k−1
dynamics model

− − − − − − − − − →

Fk|k−1, Dk|k−1

N

xk; xk|k−1, Pk|k−1
xk|k−1 = Fk|k−1xk−1|k−1

Pk|k−1 = Fk|k−1Pk−1|k−1Fk|k−1

⊤ + Dk|k−1

filtering: Nxk; xk|k−1, Pk|k−1

current measurement zk

− − − − − − − − − − − − − →

sensor model: Hk, Rk

N

xk; xk|k, Pk|k
xk|k

=

xk|k−1 + Wk|k−1νk|k−1, νk|k−1 = zk − Hkxk|k−1 Pk|k

=

Pk|k−1 − Wk|k−1Sk|k−1Wk|k−1⊤, Sk|k−1 = HkPk|k−1Hk⊤ + Rk Wk|k−1 = Pk|k−1Hk⊤Sk|k−1−1

‘KALMAN gain matrix’ retrodiction: N

xl; xl|k, Pl|k
filtering, prediction

← − − − − − − − − − −

dynamics model

N

xl+1; xl+1|k, Pl+1|k
xl|k

=

xl|l + Wl|l+1(xl+1|k − xl+1|l), Wl|l+1 = Pl|lF⊤

l+1|lP−1 l+1|l

Pl|k

=

Pl|l + Wl|l+1(Pl+1|k − Pl+1|l)Wl|l+1⊤

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 4

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Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 5

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in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 6

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Quite general: agent switching between different modes of over-all behavior

M1 M3 M2

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 7

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Quite general: agent switching between different modes of over-all behavior

M1 M3 M2

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 8

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Quite general: agent switching between different modes of over-all behavior

M1 M3 M2 P(2|1) P(3|1) P(1|1)

P(1|1) + P(2|1) + P(3|1) = 1

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 9

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Quite general: agent switching between different modes of over-all behavior

M1 M3 M2 P(2|1) P(1|2) P(3|2) P(2|3) P(1|3) P(3|1) P(1|1) P(2|2) P(3|3)

P(1|1) + P(2|1) + P(3|1) = 1 P(1|2) + P(2|2) + P(3|2) = 1 P(1|3) + P(2|3) + P(3|3) = 1

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 10

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A quite general mathematical structure: a graph, characterized by nodes (here: evolution models) and directed edges defining an adjacency matrix (here: transition matrix P, stochastic matrix: columns sum up to one) initial information on which model is currently being in effect: pk = (p1

k, p2 k, p3 k)⊤

Markov propagation:

pk = P pk−1 =  

p(1|1) p(1|2) p(1|3) p(2|1) p(2|2) p(2|3) p(3|1) p(3|2) p(3|3)

   

p1

k−1

p2

k−1

p3

k−1

 

Perron-Frobenius: the spectral radius of stochastic matrices is 1, 1 is also an eigenvalue and the corresponding eigenvector is positive.

Exercise: Consider the example:

 

0.5 0.3 0.2 0.2 0.4 0.4 0.3 0.3 0.4

 

and calculate the invariant state (eigenvector for eingenvalue 1). Show numerically or mathematically that each initial state converges to the invariant state.

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 11

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Excursus: Stochastic Characterization of Object Interrelations: Estimation and Tracking of Adjacency Matrices

Multiple object tracking: estimate from uncertain data Z at each time the kinematic

state vector of all relevant objects: p(x|Z).

Sometimes of interest: interrelations between tracked objects. Example: reachability

between two objects (communications, mutual help).

Interrelations completely described by the adjacency matrix X of a graph (nodes:

tracked objects, matrix elements: properties of the interrelation).

Uncertainty of sensor data (kinematics, attributes) z, Z: adjacency matrix is a random

matrix (matrix variate probability densities).

State to be estimated: kinematics x of all objects, adjacency matrix X. Based on the

sensor data, the knowledge on x, X is contained in: p(x, X|z, Z).

suitable families of matrix variate densities and likelihood functions: Bayes!

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 12

SLIDE 13

in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

p(xk, jk|xk−1, jk−1) = p(xk|jk, xk−1, jk−1) p(jk|xk−1, jk−1)

!

= p(xk|xk−1, jk) p(jk|jk−1) (MARKOV) = p(jk|jk−1)

interaction

Nxk; Fjk

k|k−1xk−1, Djk k|k−1

dynamics model jk

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 13

SLIDE 14

in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

p(xk, jk|xk−1, jk−1) = p(xk|xk−1, jk) p(jk|jk−1) (MARKOV) = p(jk|jk−1)

interaction

N

xk; Fjk

k|k−1xk−1, Djk k|k−1

dynamics model jk

previous posterior written as a GAUSSian mixture: p(xk−1|Zk−1) =

r

jk−1=1

p(xk−1, jk−1|Zk−1) =

r

jk−1=1

p(jk−1|Zk−1) N

xk−1; xjk−1

k−1|k−1, Pjk−1 k−1|k−1

,

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 14

SLIDE 15

in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

p(xk, jk|xk−1, jk−1) = p(xk|xk−1, jk) p(jk|jk−1) = p(jk|jk−1)

interaction

N

xk; Fjk

k|k−1xk−1, Djk k|k−1

dynamics model jk

previous posterior written as a GAUSSian mixture: p(xk−1|Zk−1) =

r

jk−1=1

p(xk−1, jk−1|Zk−1) =

r

jk−1=1

p(jk−1|Zk−1) N

xk−1; xjk−1

k−1|k−1, Pjk−1 k−1|k−1

,

prediction: p(xk|Zk−1) =

r

jk=1

r

jk−1=1
dxk−1 p(xk, jk|xk−1, jk−1)
IMM dynamics

p(xk−1, jk−1|Zk−1)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 15

SLIDE 16

in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

p(xk, jk|xk−1, jk−1) = p(xk|xk−1, jk) p(jk|jk−1) = p(jk|jk−1)

interaction

N

xk; Fjk

k|k−1xk−1, Djk k|k−1

dynamics model jk

previous posterior written as a GAUSSian mixture: p(xk−1|Zk−1) =

r

jk−1=1

p(xk−1, jk−1|Zk−1) =

r

jk−1=1

p(jk−1|Zk−1) N

xk−1; xjk−1

k−1|k−1, Pjk−1 k−1|k−1

,

prediction: p(xk|Zk−1) =

r

jk=1

r

jk−1=1
dxk−1 p(xk, jk|xk−1, jk−1)
IMM dynamics

p(xk−1, jk−1|Zk−1) =

r

jk=1

r

jk−1=1

p(jk|jk−1) p(jk−1|Zk−1) N

xk; xjkjk−1

k|k−1, Pjkjk−1 k|k−1

with x

(product formula)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 16

SLIDE 17

in practical applications: uncertainty on which dynamics model jk out of a set of r alternatives is in effect at tk (IMM: Interacting Multiple Models)

p(xk, jk|xk−1, jk−1) = p(xk|xk−1, jk) p(jk|jk−1) = p(jk|jk−1)

interaction

N

xk; Fjk

k|k−1xk−1, Djk k|k−1

dynamics model jk

previous posterior written as a GAUSSian mixture: p(xk−1|Zk−1) =

r

jk−1=1

p(xk−1, jk−1|Zk−1) =

r

jk−1=1

p(jk−1|Zk−1) N

xk−1; xjk−1

k−1|k−1, Pjk−1 k−1|k−1

,

prediction: p(xk|Zk−1) =

r

jk=1

r

jk−1=1
dxk−1 p(xk, jk|xk−1, jk−1)
IMM dynamics

p(xk−1, jk−1|Zk−1) =

r

jk=1

r

jk−1=1

p(jk|jk−1) p(jk−1|Zk−1) N

xk; xjkjk−1

k|k−1, Pjkjk−1 k|k−1

≈p(jk|Zk−1) N(xk; x

jk k|k−1, P jk k|k−1)

with x

(product formula)

Approximate GAUSSian mixture representation of p(xk|Zk−1) with r mixture components!

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 17

SLIDE 18

Moment Matching: Approximate an arbitrary pdf p(x) with E[x] = x, C[x] = P by p(x) ≈ N

x; x, P
!

here especially: p(x) =

i

pi N(x; xi, Pi) (GAUSSian mixtures)

Exercise 5.1 Show:

x =

i

pi xi

P =

i

pi

Pi +

spread term

(xi − x)(xi − x)⊤

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 18

SLIDE 19

BAYESian filtering update based on IMM predictions (r = 1: KALMAN filter)

p(xk|Zk) = ℓ(xk; Zk, mk) p(xk|Zk−1)

dxk ℓ(xk; Zk, mk) p(xk|Zk−1)

=

r

jk=1 ℓ(xk; Zk, mk) p(jk|Zk−1) N

xk; xjk

k|k−1, Pjk k|k−1

r

jk=1 p(jk|Zk−1)

dxk ℓ(xk; Zk, mk) N

xk; xjk

k|k−1, Pjk k|k−1

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018

— slide 19

SLIDE 20

Bayesian filtering update based on IMM predictions (r = 1: KALMAN filter)

p(xk|Zk) = ℓ(xk; Zk, mk) p(xk|Zk−1)

dxk ℓ(xk; Zk, mk) p(xk|Zk−1)

=

r

jk=1 ℓ(xk; Zk, mk) p(jk|Zk−1) N

xk; xjk

k|k−1, Pjk k|k−1

r

jk=1 p(jk|Zk−1)

dxk ℓ(xk; Zk, mk) N

xk; xjk

k|k−1, Pjk k|k−1

Consider as a simple example ℓ(xk; Zk, mk) = Nxk; Hkxk, Rk
!

=

r

jk=1

p(jk|Zk−1) N

xk; Hkxk, Rk
N(xk; xjk

k|k−1, Pjk k|k−1)

r

j′

k=1 p(j′

k|Zk−1)

dxk Nxk; Hkxk, Rk

Nxk; xj′

k

k|k−1, Pj′

k

k|k−1

=

r

jk=1

p(jk|Zk) N

xk; xjk

k|k, Pjk k|k

(due to the product formula)

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 20

SLIDE 21

Bayesian filtering update based on IMM predictions (r = 1: KALMAN filter)

p(xk|Zk) = ℓ(xk; Zk, mk) p(xk|Zk−1)

dxk ℓ(xk; Zk, mk) p(xk|Zk−1)

=

r

jk=1 ℓ(xk; Zk, mk) p(jk|Zk−1) N

xk; xjk

k|k−1, Pjk k|k−1

r

jk=1 p(jk|Zk−1)

dxk ℓ(xk; Zk, mk) N

xk; xjk

k|k−1, Pjk k|k−1

Consider as a simple example ℓ(xk; Zk, mk) = Nxk; Hkxk, Rk
!

=

r

jk=1

p(jk|Zk−1) N

xk; Hkxk, Rk
N(xk; xjk

k|k−1, Pjk k|k−1)

r

j′

k=1 p(j′

k|Zk−1)

dxk Nxk; Hkxk, Rk

Nxk; xj′

k

k|k−1, Pj′

k

k|k−1

=

r

jk=1

p(jk|Zk) N

xk; xjk

k|k, Pjk k|k

(due to the product formula)

with: p(jk|Zk) =

N

zk; Hkx

jk k|k−1, HkP jk k|k−1Hk+Rk

r

j′ k=1 p(j′ k|Zk−1) N

zk; Hkx

j′ k k|k−1, HkP j′ k k|k−1Hk+Rk

(mixture coefficients)

xjk

k|k = xjk k|k−1 + Wjk k|k(zk − Hkxjk k|k−1),

Wjk

k|k = Pjk k|k−1H⊤ k Sj−1

k

k|k

(KALMAN update)

Pjk

Sjk

k|k = HkPjk k|k−1Hk + Rk.

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 21

SLIDE 22

multiple models: p(xk, jk|xk−1, jk−1) = p(jk|jk−1) N

xk; Fjk

k|k−1xk−1, Djk k|k−1

prediction:

p(xk|Zk−1) =

r

jk=1

r

jk−1=1

p(jk|jk−1) p(jk−1|Zk−1) N

xk; xjkjk−1

k|k−1, Pjkjk−1 k|k−1

x

jkjk−1 k|k−1 = Fjk k|k+1x jk−1 k−1|k−1,

P

mixing step: p(xk|Zk−1) ≈

r

jk=1

p(jk|Zk−1) N(xk; xjk

k|k−1, Pjk k|k−1) p(jk|Zk−1)=r

jk−1=1 p(jk|jk−1) p(jk−1|Zk−1)

x

jk k|k−1= 1 p(jk|Zk−1)

r

jk−1=1 p(jk|jk−1) p(jk−1|Zk−1)x jkjk−1 k|k−1 ,

P

jk k|k−1= 1 p(jk|Zk−1)

r

filtering: p(xk|Zk) =

r

jk=1

p(jk|Zk) N

xk; xjk

k|k, Pjk k|k

with:

p(jk|Zk) =

N zk; Hkx

jk k|k−1, HkP jk k|k−1Hk+Rk

r

j′ k=1 p(j′ k|Zk−1) N

zk; Hkx

j′ k k|k−1, HkP j′ k k|k−1Hk+Rk

(mixture coefficients)

x

jk k|k=x jk k|k−1+W jk k|k(zk−Hkx jk k|k−1),

P

W

jk k|k=P jk k|k−1H⊤ k S j−1 k k|k

S

jk k|k=HkP jk k|k−1Hk+Rk

Sensor Data Fusion - Methods and Applications, 5th Lecture on November 21, 2018 — slide 22

SLIDE 23

Radar Function Control: Information Flow

✂✁☎✄✝✆✞✁✟✁✟✠☛✡✌☞✟✍✎✍☛✠✞✁✟✏☎✑ ✒ ✆✟✁☎✑✓☞✟✔✖✕✌✆✟✗✟☞✟✔ ✄ ✑ ✘ ✁✟✘ ✄✙✘ ✠✟✚ ✛✙✠✟✚ ✆✟✔ ✄ ✜✌✆☎✄✝✆☎✢✣✄✙✘ ☞✟✁☎✑ ✤☛✠✞✁✟✆✟✥☎✦✓✆✟✔ ✧✓☞✟✔★✍✎✠☎✄✝✘ ☞✞✁ ✩✓✪✓✫✟✬✟✭✟✮✰✯✟✬✟✱✙✫✞✲ ✳✴✔★✠☎✢✓✵ ✒ ✆✟✠✟✔ ✢✓✶☛✕✌✆✟✷✟✥✟✆✸✑✹✄ ✳✴✠✞✔✻✺✟✆✸✄✼✡✌✶✟✠✟✔★✠☎✢✹✄✙✆✟✔★✘ ✑✣✄✝✘ ✢✣✑ ✜✌✠☎✄✙✠☛✕✌✠☎✄✙✆✎✕✴✆✞✏✟✥☎✢✹✄✙✘ ☞✟✁ ✽ ✳✴✶✟✔★✆✟✠☎✄✟✰✑✣✑✾✆✸✑✹✑✓✍✎✆✟✁☎✄ ✽❀✿ ✔★✠☎✢✓✆☎❁✝✥✟✚✣✜✌✆✟✺✟✔★✠✟✏✟✠☎✄✙✘ ☞✟✁ ✽❀❂ ✦✓✆✟✔★✚ ☞✟✠✟✏✎❃✌✠✟✁✟✏✟✚ ✘ ✁✟✺ ✽ ✳✌✔✻✠☎✢✓✵ ✽★❄ ✜✌❅✟✕✌✆☎✦✓✘ ✑✓✘ ✄✟✳✌✘ ✍☛✆ ✽ ✕✌✠✟✏✟✠✟✔❀❆✼✠✟✔★✠✟✍✎✆☎✄✙✆✟✔ ✑ ✽ ✕✌✠✟✁✟✺✟✆☎✛✙✜✌☞✟✗✟✗✟✚ ✆✟✔ ✿ ✠☎✄✙✆☎✑ ✽✓❇ ✆☎❈❊❉✼✆✟✠✞✍❊❆✼☞☎✑✓✘ ✄✙✘ ☞✟✁ ✽ ✕✌✆☎✦✓✘ ✑✓✘ ✄ ❄ ✁☎✄✙✆✟✔ ✦✓✠✟✚ ✑ ✽ ✕✌✠✟✏✞✠✟✔✓❆✰✠✟✔★✠✟✍☛✆☎✄✙✆✟✔ ✑ ✽ ✒ ✆✟✠✟✔ ✢✓✶ ✒ ✆☎✢✣✄✙☞✟✔ ✑ ✽ ✒ ✆✟✠✟✔ ✢✓✶☛❆✰✠☎✄ ✄✝✆✟✔★✁☎✑ ✮✰✯✟✬✟✱✙✫✟✲❋✮✰✯☎●★●■❍ ❏✓❑❀▲✟▼ ◆✌❖✓P✓✬✸●■✯✎✮✰✯☎●★●■❍ ❏✓❑❀▲✟▼ ◗ ✪✓❏❙❘❚❍ ✱✙❯❋✬☎●❚❍ ✪✓❏

Tracking System

❱✰✱✙❍ ✪✓✱✙❍

★❲❨❳✎✬✟❏✓✬✟❑❀✯✟❯❋✯✟❏✣●❚▼