Summary: B AYES ian (Multi-) Sensor Tracking • Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. • Objective: Learn as much as possible about the individual target states at each time by analyzing the ‘time series’ which is constituted by the sensor data. Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 1
Summary: B AYES ian (Multi-) Sensor Tracking • Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. • Objective: Learn as much as possible about the individual target states at each time by analyzing the ‘time series’ which is constituted by the sensor data. • Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets’ temporal evolution is usually not well-known. Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 2
Summary: B AYES ian (Multi-) Sensor Tracking • Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. • Objective: Learn as much as possible about the individual target states at each time by analyzing the ‘time series’ which is constituted by the sensor data. • Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets’ temporal evolution is usually not well-known. • Approach: Interpret measurements and target vectors as random variables (RVs). Describe by probability density functions (pdf) what is known about them. Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 3
Summary: B AYES ian (Multi-) Sensor Tracking • Basis: In the course of time one or several sensors produce measurements of targets of interest. Each target is characterized by its current state vector, being expected to change with time. • Objective: Learn as much as possible about the individual target states at each time by analyzing the ‘time series’ which is constituted by the sensor data. • Problem: imperfect sensor information: inaccurate, incomplete, and eventually ambiguous. Moreover, the targets’ temporal evolution is usually not well-known. • Approach: Interpret measurements and target vectors as random variables (RVs). Describe by probability density functions (pdf) what is known about them. • Solution: Derive iteration formulae for calculating the pdfs! Develop a mech- anism for initiation! By doing so, exploit all background information available! Derive state estimates from the pdfs along with appropriate quality measures! Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 4
Elements for multisensor situation pictures: tracks of temporally evolving objects Which object properties are of interest? → state X k at time t k • road-moving vehicle: odometer count x k X k = ( x k , ˙ x k ) • position, speed, acceleration: X k = ( r k , ˙ r k , ¨ r k ) X k = ( x 1 k , x 2 • joint state of several objects: k , . . . ) • attributes, e.g. radar cross section x k ∈ R + : X k = ( x k , x k ) • maneuvering phase, object class i k ∈ N : X k = ( x k , i k ) Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 5
Elements for multisensor situation pictures: tracks of temporally evolving objects → state X k at time t k Which object properties are of interest? → from sensor data Z k = { Z k , Z k − 1 } , context How to lean states X k ? → e.g. by conditional pdfs p ( X k | Z k ) How to imprecise information? → iterative calculation of p ( X k | Z k ) What means “learning” in this context? Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 6
The general tracking equations � p ( X k − 1 | Z k − 1 ) p ( X k | Z k − 1 ) = Prediction dX k − 1 p ( X k | X k − 1 ) � �� � � �� � evolution filtering t k − 1 p ( Z k | X k ) p ( X k | Z k − 1 ) filtering p ( X k | Z k ) = � p ( X k | Z k − 1 ) dX k p ( Z k | X k ) � �� � � �� � sensor model prediction filtering t l evolution � �� � � �� � p ( X l | Z l ) � p ( X l +1 | X l ) p ( X l +1 | Z k ) retrodiction p ( X l | Z k ) = dX l +1 p ( X l +1 | Z l ) � �� � retrodiction t l +1 � �� � prediction t l +1 Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 7
Elements for situation pictures: tracks of temporally evolving objects → state X k at time t k Which object properties are of interest? → from sensor data Z k = { Z k , Z k − 1 } , context How to lean states X k ? → e.g. by conditional pdfs p ( X k | Z k ) How to imprecise information? → iterative calculation of p ( X k | Z k ) What means “learning” in this context? → evolution / sensor models p ( X k | X k − 1 ) , p ( Z k | X k ) What is needed for this? → sequential decisions How to initiate / terminate tracking processes? Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 8
Why is ‘target tracking’ a key function? infer secondary quantities from incomplete measurement data. Eliminate fluctuating false return background (clutter). Create a time basis for classification from attribute data. . . . ‘Shape’ of objects / object groups relevant in many applications. Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 9
Track-based inference of object properties • Velocity history: vehicle, helicopter, plane • Acceleration history: threat: no under-wing weapons • Rare events: truck by night on dirt road near a border • Object interrelations: resulting from formation, convoy • Object sources / sinks: classification by origin / designation • Classification: road-moving vehicle, ‘on-road’ → ‘off-road’ Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 10
Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 11
How to deal with probability density functions? • pdf p ( x ) : Extract probability statements about the RV x by integration! � dx p ( x ) = 1 ) • na¨ ıvely: positive and normalized functions ( p ( x ) ≥ 0 , • conditional pdf p ( x | y ) = p ( x,y ) p ( y ) : Impact of information on y on RV x ? � dy p ( x, y ) = � dy p ( x | y ) p ( y ) : • marginal density p ( x ) = Enter y ! • Bayes: p ( x | y )= p ( y | x ) p ( x ) p ( y | x ) p ( x ) p ( x | y ) ← p ( y | x ) , p ( x ) ! = dx p ( y | x ) p ( x ) : � p ( y ) Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 12
Recapitulation: The Multivariate G AUSS ian Pdf – wanted: probabilities ‘concentrated’ around a center x q ( x ) = 1 2 ( x − x ) P − 1 ( x − x ) ⊤ – quadratic distance: q ( x ) defines an ellipsoid around x , its volume and orien- tation being determined by a matrix P (symmetric: P ⊤ = P , positively definite: all eigenvalues > 0 ). � dx e − q ( x ) (normalized!) p ( x ) = e − q ( x ) / – first attempt: e − 1 1 2( x − x ) ⊤ P − 1 ( x − x ) p ( x ) = N ( x ; x , P ) = � | 2 π P | p ( x ) = � – G AUSS ian Mixtures: i p i N ( x ; x i , P i ) (weighted sums) Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 13
pdf: t k−1 Prädiktion: t k Exploit imprecise knowledge on the dynamical behavior of the object. = � � � � � p ( x k |Z k − 1 ) d x k − 1 N x k ; Fx k − 1 , D N x k − 1 ; x k − 1 | k − 1 , P k − 1 | k − 1 . � �� � � �� � � �� � prediction dynamics old knowledge Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 14
A Useful Product Formula for G AUSS ians � � � � � � � x ; y + W ν , P − WSW ⊤ � N z ; Fx , D N x ; y , P = N z ; Fy , S N � �� � independent of x S = FPF ⊤ + D , W = PF ⊤ S − 1 . ν = z − Fy , Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 15
k ) ⊤ , Z k = { z k , Z k − 1 } Kalman filter: x k = ( r ⊤ r ⊤ k , ˙ � � initiation: p ( x 0 ) = N x 0 ; x 0 | 0 , P 0 | 0 , initial ignorance: P 0 | 0 ‘large’ � � dynamics model � � N x k − 1 ; x k − 1 | k − 1 , P k − 1 | k − 1 − − − − − − − − − → N x k ; x k | k − 1 , P k | k − 1 prediction: F k | k − 1 , D k | k − 1 x k | k − 1 = F k | k − 1 x k − 1 | k − 1 ⊤ + D k | k − 1 P k | k − 1 = F k | k − 1 P k − 1 | k − 1 F k | k − 1 Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 16
pdf: t k−1 t k : kein plot missing sensor detection: ‘data processing’ = prediction (not always: exploitation of ‘negative’ sensor evidence) Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 17
pdf: t k−1 pdf: t k Prädiktion: t k+1 missing sensor information: increasing knowledge dissipation Sensor Data Fusion - Methods and Applications, 2nd Lecture on October 23, 2019 — slide 18
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