Information Fusion – an Omnipresent Phenomenon! · – ﬁnite mixture: inherent ambiguity (data,...

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All living creatures by nature perform sensor data and information fusion. Each in their own way, they “fuse” sensations provided by multiple and mutually

complementary sense organs with knowledge learned from previous experiences and communications from other creatures. As a result, they produce a “mental picture” of their individual environment, the basis of behaving appropriately in

their struggle to avoid harm or reach a particular goal in a given situation.

Prior to its technical realization or the scientific reflection on it:

Information Fusion – an Omnipresent Phenomenon!

Branch of Applied Informatics: „Cognitive Tools“

1. Understanding, Automation, Enhancement

2. Integration of New Sources of Information

• networking, mobile sensors of high sensitivity, range • new dimensions of apprehension otherwise hidden • data base systems containing vast context information • interaction with humans: exploit natural intelligence!

Sensor Data and Information Fusion for Producing Situation Pictures

Information to be fused: imprecise, incomplete, ambiguous, unresolved, false, deceptive, hard-to-be-formalized, contradictory, ...

Context Knowledge

Sensor Data & Information Fusion

Mapping of a dynamic over-all scenario onto near real-time “situation pictures“

Multiple Sensor Data

Basis for Decision Support

Array-Sensor Signals

Condensed Information on Objects of Interest

Existence, number, geolocation, time, behavior,

properties, class, identity, interrelations history,

sources/sinks, traffic analysis, anomaly detection, …

A Generic Tracking and Sensor Data Fusion System

Track AssociationSensor Data to Track File

Storage

Track Maintenance:

Retrodiction Prediction, Filtering

Sensing Hardware:

Signal Processing:

Parameter Estimation

Received Waveforms

Detection Process:

Data Rate Reduction

Track Initiation:

Multiple Frame

- Object Environment- Object Characteristics

A Priori Knowledge:

- Sensor Performance

- Track Cancellation- Object Classification / ID- Track-to-Track Fusion

Track Processing:

- Interaction Facilities

Man-Machine Interface:

- Displaying Functions- Object Representation

Tracking & Fusion System

Sensor System Sensor System

SensorData

SensorControl

Sensor System

Track Extraction

W. Koch, Walking Through the JDL Model

On Characterizing Tracking / Fusion Performance

a well-understood paradigm: air surveillance with multiple radarsMany results can be transfered to other sensors (IR, E/O, sonar, acoustics).

Sensor Data Fusion: ‘tracks’ represent the available information on the targetsassociated to them with appropriate quality measures, thus providing answers to:

When? Where? How many? To which direction? How fast, accelerating? What?

By sensor data fusion we wish to establish one-to-one associations between:

targets in the field of view $ identified tracks in the tracking computer

Strictly speaking, this is only possible under ideal conditions regarding the sensorperformance and underlying target situation. The tracking/fusion performance can thus

be measured by its deficiencies when compared with this ideal goal.

1. Let a target be detected at first by a sensor at time ta. Usually, a timedelay is involved until a confirmed track has finally been establishedat time te (track extraction). A ‘measure of deficiency’ is thus:

• extraction delay te � ta.

2. Unavoidably, false tracks will be extracted in case of a high falsereturn density (e.g. clutter, jamming/detection), i.e. tracks related tounreal or unwanted targets. Corresponding ‘deficiencies’ are:

• mean number of falsely extracted targets per time,• mean life time of a false track before its deletion.

3. A target should be represented by one and the same track until leav-ing the field of view. Related performance measures/deficiencies:

• mean life time of tracks related to true targets,• probability of an ‘identity switch’ between targets,• probability of a target not being represented by a track.

4. The track inaccuracy (error covariance of a state estimate) should beas small as possible. The deviations between estimated and actualtarget states should at least correspond with the error covariances pro-duced (consistency). If this is not the case, we speak of a ‘track loss’.

• A track must really represent a target!

Challenges:

• low detection probability • high clutter density • low update rate• agile targets • dense target situations • formations, convoys• target-split events (formation, weapons) • jamming, deception

Basic Tasks:

• models: sensor, target, environment ! physics• data association problems ! combinatorics• estimation problems ! probability, statistics• process control, realization ! computer science

Target Tracking: Basic Idea, Demonstration

Problem-inherent uncertainties and ambiguities!BAYES: processing scheme for ‘soft’, ‘delayed’ decision

sensor performance: • resolution conflicts • DOPPLER blindness

environment: • dense situations • clutter • jamming/deception

target characteristics: • qualitatively distinct maneuvering phases

background knowledge • vehicles on road networks • tactics

pdf: tk−1

‘Probability densities functions (pdf)’ p(xk�1|Zk�1) represent imprecise

knowledge on the ‘state’ xk�1 based on imprecise measurements Zk�1.

pdf: tk−1

Prädiktion: tk

Exploit imprecise knowledge on the dynamical behavior of the object.

p(xk|Zk�1)

| {z }prediction

Rdxk�1 p(xk|xk�1)| {z }

dynamics

p(xk�1|Zk�1)

| {z }old knowledge

pdf: tk−1

tk: kein plot

missing sensor detection: ‘data processing’ = prediction(not always: exploitation of ‘negative’ sensor evidence)

pdf: tk−1

pdf: tk

Prädiktion: tk+1

missing sensor information: increasing knowledge dissipation

pdf: tk−1

pdf: tk

tk+1: ein plot

sensor information on the kinematical object state

pdf: tk−1

pdf: tk

Prädiktion: tk+1

likelihood(Sensormodell)

BAYES’ formula: p(xk+1

| {z }new knowledge

p(zk+1

) p(xk+1

Rdxk+1

p(zk+1|{z}plot

) p(xk+1

| {z }prediction

pdf: tk−1

pdf: tk

pdf: tk+1(Bayes)

filtering = sensor data processing

pdf: tk−1

pdf: tk

tk+1: drei plots

ambiguities by false plots: 1 + 3 data interpretation hypotheses(‘detection probability’, false alarm statistics)

pdf: tk−1

pdf: tk

pdf: tk+1

Multimodal pdfs reflect ambiguities inherent in the data.

pdf: tk−1

pdf: tk

pdf: tk+1

Prädiktion: tk+2

temporal propagation: dissipation of the probability densities

pdf: tk−1

pdf: tk

pdf: tk+1

tk+2: ein plot

association tasks: sensor data$ interpretation hypotheses

pdf: tk−1

pdf: tk

pdf: tk+1

Prädiktion: tk+2

likelihood

BAYES: p(xk+2

p(zk+2

) p(xk+2

)Rdxk+2

p(zk+2

) p(xk+2

pdf: tk−1

pdf: tk

pdf: tk+1

pdf: tk+2

in particular: re-calculation of the hypothesis weights

pdf: tk−1

pdf: tk

pdf: tk+1

pdf: tk+2

How does new knowledge affect the knowledge in the past of a past state?

pdf: tk−1

pdf: tk

Retrodiktion: tk+1

pdf: tk+2

‘retrodiction’: a retrospective analysis of the past

tk−1tk

optimal information processing at present and for the past

Multiple Hypothesis Tracking: Basic IdeaIterative 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)– track initiation, termination: sequential likelihood ratio testing

Summary: BAYESian (Multi-) Sensor Tracking

• Basis: In the course of time one or several sensors produce measurements oftargets of interest. Each target is characterized by its current state vector, beingexpected to change with time.

• Objective: Learn as much as possible about the individual target states at eachtime by analyzing the ‘time series’ which is constituted by the sensor data.

• Problem: imperfect sensor information: inaccurate, incomplete, and eventuallyambiguous. 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!

How to deal with probability density functions?

• pdf p(x): Extract probability statements about the RV x by integration!

• naı̈vely: positive and normalized functions (p(x) � 0,Rdx p(x) = 1)

• conditional pdf p(x|y) =

p(x,y)p(y)

: Impact of information on y on RV x?

• marginal density p(x) =

Rdy p(x, y) =

Rdy p(x|y) p(y): Enter y!

• Bayes: p(x|y)= p(y|x)p(x)p(y)

p(y|x)p(x)Rdx p(y|x)p(x): p(x|y) p(y|x), p(x)!

• certain knowledge on x: p(x) = �(x� y) ‘=’ lim�!0

1p2⇡�

(x�y)2�2

• transformed RV y = t[x]: p(y) =

Rdxp(y, x) =

Rdxp(y|x)px(x) =

Rdx �(y � t[x]) px(x) =: [T px](y) (T : px 7! p, “transfer operator”)

The Multivariate GAUSSian Pdf

– wanted: probabilities ‘concentrated’ around a center x

– quadratic distance: q(x) =

(x� x)P�1(x� x)>

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).

– first attempt: p(x) = e

�q(x)/Rdx e

�q(x) (normalized!)

p(x) = N (x; x, P) =

q|2⇡P|

(x�x)>P�1(x�x)

E[x] = x, E[(x� x)(x� x)>] = P (covariance matrix)

– GAUSSian Mixtures: p(x) =

Pi pi N (x; xi, Pi) (weighted sums)

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) =

piN (x;xi,Pi) (GAUSSian mixtures)

pi�Pi +

spread termz }| {(xi � x)(xi � x)>

A Useful Product Formula for GAUSSians

N�z; Hx, R

�N�x; y, P

�= N

�z; Hy, S

�| {z }

independent of x

(N�x; y+W⌫, P�WSW>�

N�x; Q�1(P�1x+H>R�1z), Q

⌫ = z�Hy, S = HPH>+R, W = PH>S�1, Q�1 = P�1 +H>R�1H.

Sketch of the proof:

• Interpret N�z; Hx, R

�N�x; y, P

�as a joint pdf p(z|x)p(x) = p(z,x).

• Show that p(z,x) is a GAUSSian: p(z,x) = N��

�Hyy

�,�

S HPPH> P

��.

• Calculate from p(z,x) the marginal and conditional pdfs p(z) and p(x|z).

• From p(z,x) = p(z|x)p(x) = p(x|z)p(z) = p(x, z) we obtain the result.

More Precise Formulation of the BAYESian Approach

Consider a set of measurements Zl = {zjl }mlj=1

of a single or a multipletarget state xl at time instants tl, l = 1, . . . , k and the time series:

Zk= {Zk,mk, Zk�1,mk�1, . . . , Z1

} = {Zk,mk,Zk�1}!

Based on Zk, what can be learned about the object states xlat t

, . . . , tk, tk+1

, . . ., i.e. for the past, present, and future?

Evidently the answer is given be calculating the pdfp(xl|Zk

multiple sensor measurement fusion: Calculate p(x|Zk1

, . . . ,ZkN)!

• communication lines • common coordinate system: sensor registration

How to calculate the pdf p(xl|Zk)?

Consider at first the present time: l = k.

an observation:

BAYES’ rule: p(xk|Zk) = p(xk|Zk,mk,Zk�1

p(Zk,mk|xk,Zk�1) p(xk|Zk�1

Rdxk p(Zk,mk|xk,Zk�1

)| {z }likelihood function

p(xk|Zk�1)| {z }

prediction

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

based on all measurements in the past.

• p(Zk,mk|xk) / `(xk;Zk,mk) describes, what the current sensor output Zk,mk

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

p(xk|Zk�1) =

Rdxk�1 p(xk,xk�1|Zk�1

) marginal pdf

Rdxk�1 p(xk|xk�1,Zk�1

)| {z }object dynamics!

p(xk�1|Zk�1)| {z }

idea: iteration!

notion of a conditional pdf

often: p(xk|xk�1,Zk�1) = p(xk|xk�1) (MARKOV)

sometimes: p(xk, ik|xk�1, ik�1) = pik,ik�1| {z }phase transition

N�xk; F

ikk|k�1| {z }

deterministic

xk�1, Dikk|k�1| {z }random

�(IMM)

p(xk|Zk�1) =

Rdxk�1 p(xk,xk�1|Zk�1

) marginal pdf

Rdxk�1 p(xk|xk�1,Zk�1

)| {z }object dynamics!

p(xk�1|Zk�1)| {z }

idea: iteration!

notion of a conditional pdf

often: p(xk|xk�1,Zk�1) = p(xk|xk�1) (MARKOV)

sometimes: p(xk, ik|xk�1, ik�1) = pik,ik�1| {z }phase transition

N�xk; F

ikk|k�1| {z }

deterministic

xk�1, Dikk|k�1| {z }random

�(IMM)

• p(Zk,mk|xk) / `(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: `(Zk,mk;xk) = (1� PD)⇢c + PD

N�zjk; H

jkxk, R

�(single target in clutter)

iteration formula: p(xk|Zk) =

`(xk;Zk,mk)Rdxk�1 p(xk|xk�1) p(xk�1|Zk�1

)Rdxk `(xk;Zk,mk)

Rdxk�1 p(xk|xk�1) p(xk�1|Zk�1

How to calculate the pdf p(xl|Zk)?

Consider the past: l < k!

an observation:

p�xl| Zk

Zdxl+1

p�xl,xl+1

| Zk�=

Zdxl+1

p�xl+1

�p�xl| Z l

�Rdxl p

�xl+1

�| {z }dynamics model

p�xl| Z l

�| {z }

filtering tl

p�xl+1

| Zk�

| {z }retrodiction: tl+1

• p(xl+1

|Zk) retrodiction: known from last iteration step

• p(xk|xk�1) dynamic object behavior (possibly multiple models)

• p(xl|Z l) filtering at the time where retrodiction is considered

• GAUSSians or GAUSSian mixtures: Exploit product formula!

• linear GAUSSian likelihood/dynamics: Rauch-Tung-Striebel smoothing

A side Result: Expected Measurements

innovation statistics, expectation gates, gating

p�zk| Zk�1�

Zdxk p

�zk|xk

�p�xk| Zk�1�

Zdxk N

�zk; Hkxk, Rk

�| {z }likelihood: sensor model

N�xk; xk|k�1, Pk|k�1

�| {z }

prediction at time tk

= N�zk; Hkxk|k�1, Sk|k�1

�(product formula)

innovation: ⌫k|k�1 = zk �Hkxk|k�1,

innovation covariance: Sk|k�1 = HkPk|k�1H>k +Rk

expectation gate: ⌫>k|k�1S�1k|k�1⌫k|k�1 �(Pc)

MAHALANOBIS: ellipsoid containing zk with probability Pc

�(Pc): gating parameter (! �2-table)

Sensor data of uncertain origin

• prediction: xk|k�1, Pk|k�1 (dynamics)

• innovation: ⌫k = zk �Hxk|k�1 , white

• Mahalanobis norm: ||⌫k||2 = ⌫>k S�1k ⌫k

• expected plot: zk ⇠ N(Hxk|k�1,Sk)

• ⌫k ⇠ N(0,Sk), Sk = HPk|k�1H>+R

• gating: ||⌫k|| < �, Pc(�) correlation prob.

missing/false plots, measurement errors, scan rate, agile targets: large gates

more conventional tracking methods:• KALMAN filter + gating• Nearest-Neighbor filter (NN)• Probabilistic Data Association Filter• Joint PDAF (JPDAF, multiple targets)

problem: limited applicability

nearly optimal solution (BAYES):

multiple targetmultiple hypothesismultiple model

>;tracking

predictionfilteringretrodiction

ad-hoc tracking methods as limiting cases!

ambiguous sensor data (PD < 1, ⇢f > 0)

mk +1 possible interpretations of the sensor data Zk = {zjk}mkj=1

• E0

: target was not detected; mk false returns in the field of view (FoV)

• Ej, j = 1, . . . ,mk: detected; zjk originates from the target; mk � 1 false returns

• E0

Incorporate the interpretations Ej into the likelihood function p(Zk,mk|xk)!

p(Zk,mk|xk) =

p(Zk,mk,Ej|xk) =

p(Zk,mk|Ej,xk) p(Ej|xk)

• E0

p(Zk,mk|xk) =

p(Zk,mk,Ej|xk) =

p(Ej|xk) =

(1� PD j = 0

mkPD j 6= 0

p(Zk,mk|Ej,xk) = p(Zk|mk,Ej,xk) p(mk|Ej,xk)

⇢pF(mk) |FoV|�mk j = 0

pF(mk � 1) |FoV|�(mk�1) N�zjk; Hxk, R

�j 6= 0

• E0

p(Zk,mk|xk) =

p(Zk,mk,Ej|xk) =

/ (1� PD)⇢F + PD

N⇣zjk; Hxk, R

⌘=: `(Zk,mk|xk)

p(Ej|xk) =

(1� PD j = 0

mkPD j 6= 0

pF(mk) =

(⇢F |FoV|)mk

�⇢F |FoV| (Poisson)

p(Zk,mk|Ej,xk) = p(Zk|mk,Ej,xk) p(mk|Ej,xk)

⇢pF(mk) |FoV|�mk j = 0

pF(mk � 1) |FoV|�(mk�1) N�zjk; Hxk, R

�j 6= 0

Overview: MHT for Well-separated Targets

ambiguous data Zk ! interpretation hypotheses Ek

the target was not detected, 1� PD

zk 2 Zk originated by the target, PD

�mk +1 interpretations

hypothetical interpretation histories for Zk

• tree structure: Hk = (EHk,Hk�1)

• current: EHk, pre-histories: Hk�i

normal mixtures $ MHT (total probability)

p(xk|Zk) =

⇣xk; xHk

xHk, PHk

: update by KALMAN iteration!

p⇤HkPHk

p⇤Hk

, p⇤Hk= pHk�1

⇢(1� PD) ⇢FPD N (⌫Hk

Problem: Growing Memory Disaster:m data, N hypotheses! Nm+1 continuations

radical solution: mono-hypothesis approximation• gating: Exclude competing data with ||⌫i

k|k�1|| > �!

!KALMAN filter (KF)

+ very simple, � � too small: loss of target measurement

• Force a unique interpretation in case of a conflict!

look for smallest statistical distance: mini ||⌫ik|k�1||

!Nearest-Neighbor filter (NN)

+ one hypothesis, � hard decision, � not adaptive

• global combining: Merge all hypotheses!

!PDAF, JPDAF filter

+ all data, + adaptive, � reduced applicability

The qualitative shape of p(xk|Zk) is often much simpler

than its correct representation: a few pronounced modes

adaptive solution: nearly optimal approximation

• individual gating: Exclude irrelevant data!before continuing existing track hypotheses Hk�1

! limiting case: KALMAN filter (KF)

• pruning: Kill hypotheses of very small weight !after calculating the weights pHk

, before filtering

! limiting case: Nearest Neighbor filter (NN)

• local combining: Merge similar hypotheses!after the complete calculation of the pdfs

! limiting case: PDAF (global combining)

Retrodiction for GAUSSian Mixtures

wanted: p(xl|Zk) � p(xl+1

|Zk) for l < k

p(xl|Zk) =

p(xl, Hk|Zk) =

p(xl|Hk,Zk)| {z }

no ambiguities!

p(Hk|Zk)| {z }

filtering!

Calculation of p(xl|Hk,Zk) as in case of PD = 1, ⇢F = 0!

p(xl|Hk,Zk) = N

⇣xl; xHk

(l|k), PHk(l|k)

with parameters given by RAUCH-TUNG-STRIEBEL formulae:

xHk(l|k) = xHk

(l|l) +WHk(l|k) (xHk

(l+1|k)� xHk(l+1|l))

PHk(l|k) = PHk

(l|l) +WHk(l|k) (PHk

(l+1|k)�PHk(l+1|l))WHk

(l|k)>

gain matrix: WHk(l|k) = PHk

(l|l)F>l+1|lPHk(l+1|l)�1

Retrodiction of Hypotheses’ Weights

Consider approximation: neglect RTS step!

p(xl|Hk,Zk) = N

�xl; xHk

(l|k), PHk(l|k)

�⇡ N

�xl; xHk

(l|l), PHk(l|l)

p(xl|Hk,Zk) ⇡

p⇤HlN�xl; xHk

(l|l), PHk(l|l)

with recursively defined weights:p⇤Hk

= pHk, p⇤Hl

Pp⇤Hl+1

summation over all histories Hl+1

with equal pre-histories!

• Strong sons strengthen weak fathers.• Weak sons weaken even strong fathers.• If all sons die, also the father must die.

IMM Modeling: Suboptimal Realization

• Conventional KALMAN filteringOnly one component: worst-case assumtion

• standard IMM filter (as discussed!)Approximate after prediction, before update byr components! Effort: ⇠ r KALMAN filter

• GPB: Generalized Pseudo-BAYESianApproximate after measurement processing byr components! Effort: ⇠ r2 KALMAN filter

• IMM-MHT filter (nearly optimal)Accept longer dynamics histories! variablenumber of components!

Extendable to ambiguity with respect to sensor models!

Track Extraction: Initiation of the PDF Iteration

extraction of target tracks: detection on a higher level of abstraction

start: data sets Zk = {zjk}mkj=1

(sensor performance: PD, ⇢F , R)

goal: Detect a target trajectory in a time series: Zk= {Zi}ki=1

at first simplifying assumptions:

• The targets in the sensors’ field of view (FoV) are well-separated.• The sensor data in the FoV in scan i are produced simultaneously.

decision between two competing hypotheses:

: Besides false returns Zk contains also target measurements.h0

: There is no target existing in the FoV; all data in Zk are false.

statistical decision errors:

= Prob(accept h1

) analogous to the sensors’ PD

= Prob(accept h1

) analogous to the sensors’ PF

Practical Approach: Sequential Likelihood Ratio Test

Goal: Decide as fast as possible for given decision errors P0

Consider the ratio of the conditional probabilities p(h1

|Zk), p(h

|Zk) and the

likelihood ratio LR(k) = p(Zk|h1

)/p(Zk|h0

) as an intuitive decision function:

p(Zk|h1

p(Zk|h0

a priori: p(h1

) = p(h0

Starting from a time window with length k = 1, calculate the test functionLR(k) successively and compare it with two thresholds A, B:

If LR(k) < A, accept hypothesis h0

(i.e. no target is existing)!

If LR(k) > B, accept hypothesis h1

(i.e. target exists in FoV)!

If A < LR(k) < B, wait for new data Zk+1

, repeat with LR(k +1)!

Sequential LR Test: Some Useful Properties

1. Thresholds and decision errors are approximately related to each other by:

A ⇡1� P

1� P0

and B ⇡P1

2. The actual decision length (number of scans required) is a random variable.

3. On average, the test has a minimal decision length for given errors P0

4. The quantity P0

) affects the mean decision length given h1

) holds.

5. Choose the probability P1

close to 1 for actually detecting real target tracks.

should be small for not overloading the tracking system with false tracks.

Iterative Calculation of the Likelihood Ratio

LR(k) =

p(Zk|h1

p(Zk|h0

Rdxk p(Zk,mk,xk,Zk�1|h

p(Zk,mk,Zk�1, h0

Rdxk p(Zk,mk|xk) p(xk|Zk�1, h

) p(Zk�1|h1

|FoV|�mk pF(mk) p(Zk�1|h0

LR(k) =

p(Zk|h1

p(Zk|h0

p(Zk,mk,Zk�1, h0

) p(Zk�1|h1

Rdxk p(Zk,mk|xk, h1

) p(xk|Zk�1, h1

|FoV|�mk pF(mk)LR(k � 1)

basic idea: iterative calculation!

LR(k) =

p(Zk|h1

p(Zk|h0

p(Zk,mk,Zk�1, h0

) p(Zk�1|h1

Rdxk p(Zk,mk|xk, h1

) p(xk|Zk�1, h1

|FoV|�mk pF(mk)LR(k � 1)

basic idea: iterative calculation!

Let Hk = {Ek,Hk�1} be an interpretation history of the time series Zk= {Zk,Zk�1}.

Ek = E0

k : target was not detected, Ek = Ejk: zjk 2 Zk is a target measurement.

p(xk|Zk�1, h1

Hk�1

p(xk|Hk�1Zk�1, h1

) p(Hk�1|Zk�1, h1

) The standard MHT prediction!

p(Zk,mk|xk, h1

p(Zk,Ek|xk, h1

) The standard MHT likelihood function!

The calculation of the likelihood ratio is just a by-product of Bayesian MHT tracking.

Sequential Track Extraction: Discussion• LR(k) is given by a growing number of summands, each related to a par-

ticular interpretation history. The tuple {�jk,xjkPjk} is called a sub-track.

• For mitigating growing memory problems all approximations discussed fortrack maintenance can be used if they do not significantly affect LR(k):

– individual gating: Exclude data not likely to be associated.

– pruning: Kill sub-tacks contributing marginally to the test function.

– local combining: Merge similar sub tracks:

{�i,xi,Pi}i ! {�,x,P} with: � =

Pi �i,

Pi �ixi, P =

Pi �i[Pi + (xi � x)(. . .)>].

• The LR test ends with a decision in favor of or against the hypotheses: h0

(no target) or h1

(target existing). Intuitive interpretation of the thresholds!

track extraction at tk: Decide in favor of h1

initiation of pdf iteration (track maintenance):

Normalize coefficients �jk: pjk =

�jkPjk �jk

(�jk,xjk,Pjk) ! p(xk|Zk) =

pjk N⇣xk; xjk, Pjk

Continue track extraction with the remaining sensor data!

sequential LR test for track monitoring:

After deciding in favor of h1

reset LR(0) = 1! Calculate LR(k) from p(xk|Zk)!

track confirmation: LR(k) > P1

: reset LR(0) = 1!

track deletion: LR(k) < 1�P1

1�P0

; ev. track re-initiation

Information Fusion – an Omnipresent Phenomenon! · – ﬁnite mixture: inherent ambiguity (data,...

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