t opic 6: l evel 1 c orrelation david l. hall. t opic o bjectives introduce the concept of level 1...

TOPIC 6: LEVEL 1 CORRELATION

David L. Hall

TOPIC OBJECTIVES

Introduce the concept of Level 1 processing Focus on the problem of association and

correlation Discuss techniques for association and

correlation

LEVEL 1 (CORRELATION)

NOTE ABOUT LEVEL-1 FUSION

During this and the next two topics, we will focus on level 1 fusion. Classically this refers to locating, tracking, characterizing and identifying objects. In military systems, these objects are primarily targets or entities such as vehicles, sensors, installations, etc.

However, the mathematics and techniques of level 1 fusion apply to any type of object, entity, activity which can be characterized by a dynamic state vector. Examples of “entities” that can be located, characterized, tracked and identified include; Fault conditions in a complex machine Individuals or groups of people Viruses or bugs in a computer network system The evolution of a communicable disease Environmental conditions such as an oil spill, plume of emissions, etc.

The key concept is the use of multiple observations (e.g. angles, symptoms, vibrations, etc.) which can be linked to an underlying vector of parameters which, if known, would allow us to predict future conditions and observables

JDL LEVEL ONE PROCESSINGOBJECT REFINEMENT

JDL Level One Processing Object Refinement

Data Alignment

•Spatial Reference Adjustment

•Temporal Reference Adjustment

•Units Adjustment

Data/Object Correlation

Object

Positional Estimation

• System Models• Optimization Criteria• Optimization Approach• Processing Approach

Object Identity

Estimation

• Physical Models• Feature-based Inference Techniques

• Cognitive-based Models

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NP

RO

CES

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• Gating• Association Measures• Assignment Strategies

Sources HumanComputerInteraction

DATA FUSION DOMAIN

Level OSignal

Refinement

Level OneObject

Refinement

Level TwoSituation

Refinement

Level ThreeThreat

Refinement

Level FourProcess

Refinement

Database Management System

SupportDatabase

FusionDatabase

WHY DO WE TRACK OBJECTS?WHY DO WE TRACK OBJECTS?

• Position, direction of movement, and history of movement can imply:- Purpose or function of object- Intent of object- Indirectly, type of object (kinematics as discriminator)

• Support need to react, kinematics data provides general framework for response -- for example

- Future position and time (where and when)- Available reaction time (T)- Relative position (own-position, object)

• Kinematic data provides specific framework for a targeting/shooting response -- for example,

- Aimpoint selection- Kinematic engagement parameters (thrust, guidance)

• Data guides sensors to permit closed loop, economic sensor employment

TRACKING OBJECTS/EVENTS/ACTIVITIES

SINGLE OR MULTIPLE SENSORS

WHAT DO SINGLE OR MULTIPLE SENSORSWHAT DO SINGLE OR MULTIPLE SENSORSUSUALLY GIVE US TO WORK WITH?USUALLY GIVE US TO WORK WITH?

• Point Measurements- Ranges (single values or cell distributions)- Angles (single values or cell distributions)- Images (frames)- Latitude/longitude

• Kinematics- Relative radial velocity (Doppler)- Velocity via delta position- No direct acceleration -- only via delta Doppler or position- Frame/frame change analysis imagery

In most these are In most these are gross pointgross point measurements in that they do measurements in that they donot reveal the local/body kinematics such as pitch and roll, etc.not reveal the local/body kinematics such as pitch and roll, etc.In most these are In most these are gross pointgross point measurements in that they do measurements in that they do

not reveal the local/body kinematics such as pitch and roll, etc.not reveal the local/body kinematics such as pitch and roll, etc.

COMBINATORICS

Complicated situations can involve --

Multiple sensors for detecting Multiple position measurements/predictions

Includes false alarms Includes Electronic Counter-Measure effects

Multiple targets, i.e., Real targets Deliberate false targets

This leads to ambiguities in allocating, (i.e., associating measurements to hypothesized targets)

Manage problem by defining gates around the measurements or predictions

THE FIRST PROBLEM -- COMBINATORICSTHE FIRST PROBLEM -- COMBINATORICS

In principle, if we have N observations and M tracks, it is necessary to systematically consider every N x M pair of observations to tracks or N x (N-1) observations to observations to determine which belongs together

CONCEPTUAL PROCESSING FLOW FOR LEVEL 1 FUSION

BulkGating

DataAssociation

& Correlation

Position/Kinematic/Attribute

Estimation

IdentityEstimation

• Observation File• Track File• Sensor Information

Sensor#1

PreprocessingData

Alignment

Sensor#2

PreprocessingData

Alignment

SensorN

PreprocessingData

Alignment

•

•

•

•

•

•

•

•

•

Note: This is a simplified partitioning of functions for level-1 processing; it is used here to help explain key functions. In an actual system, these functions are often interleaved.

DATA ALIGNMENT

DATAALIGNMENT

Units Conversion

Reference PointAdjustments

Bias Corrections

UNITSADJUSTMENT

Time Synchronization

Transformation to Reference Time

UTC UT1, UT2

Ephemeris Time

Interpolation

Extrapolation

TEMPORALREFERENCE

ADJUSTMENT

Coordinate Transformations earth fixed geocentric

Platform/Sensor Displacement

Motion Corrections

Sensor Model Transforms

SPATIALREFERENCE

ADJUSTMENT

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NTEC

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I QU

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TRACKING COORDINATE SYSTEMS

ISSUESISSUES:• Computational resources available for filtering versus association/correlation

leads to issue of coupling• Transformation of coordinates can lead to computing bias errors• Choice of coordinate system is application dependent

SOME METHODS:SOME METHODS:• North-East-Down

– Useful for airborne equations– Approximately inertial

• Cartesian– Simple extrapolation equations– Measurement errors coupled

• Polar Coordinates (various forms)– Usually same system as radar– Pseudo accelerations complicate extrapolation

COORDINATE SYSTEM ISSUES

• R and have to accelerate and jerk, to continue to represent the constant velocity, V

• If the state vector is truncated at the rate terms, then something is missing in the predictions for both the state vector and the error covariance terms,

• •

Ri+1 = Ri + Rt+

i+1 = i + t +

•

•

These required terms are missing.

Constant X or Y velocitywith r, filter

AirborneEarly Warning(AEW)

R1 •

1 •

TGT(t1); x1- CONST• TGT(t2); x2 - x1

• • R2 •

2 •

x

y


BulkGating

DataAssociation

& Correlation


Estimation

IdentityEstimation


Sensor#1

PreprocessingData

Alignment

Sensor#2

PreprocessingData

Alignment

SensorN

PreprocessingData

Alignment

•

•

•

•

•

•

•

•

•

BULK GATING

Goal: to eliminate unlikely observation to observation, observation to track, or track to track pairs that could be associated

Methods: utilize physical or identity knowledge to reduce the candidate pairs of observations to observations, observations to tracks or tracks to tracks

EXAMPLE USE OF KINEMATICS FOR BULK GATING


BulkGating

DataAssociation

& Correlation


Estimation

IdentityEstimation


Sensor#1

PreprocessingData

Alignment

Sensor#2

PreprocessingData

Alignment

SensorN

PreprocessingData

Alignment

•

•

•

•

•

•

•

•

•

THE DATA ASSOCIATION/CORRELATION

PROBLEM

• Partition ObservationsPartition ObservationsGiven N observations, zi, from one or more sensors, how do we determine which observation pairs belong together, representing observations of the same entity?

• Data Association DifficultiesData Association Difficulties– Limited resolution sensors– Dense object environment– Low SNR results in false alarms, etc.– Countermeasures– Dynamic objects– Parametric overlap in feature space– Out-of-sequence/time delayed reports

EXAMPLE OF ASSIGNMENT/CORRELATION

PROBLEMSensor collects

and forwards data on multiple targets

DATA/OBJECT CORRELATION PROCESS

Computer associates data

with its target

Target 1 Target 2 Target 2Data

Target 1Data

ASSIGNMENT CHALLENGES

A

B

C

S1

S2

Sn

S3

TARGETS SENSORS OBSERVATIONS

y1, y2, … yn

Associati

on/Correlatio

n

Evolving Situation Display

Track 1

Track 2

Track N

False alarms

MULTI-TARGET ASSIGNMENT

THE BASIC ASSOCIATION/CORRELATION

QUESTION

Entity A – track A

Entity B – track B

ti

ti+1

ti+2

Does the new observation at time ti+2 “belong “ to track A, track B or neither?

ONE APPROACH: SINGLE SCAN NEAREST NEIGHBOR

Entity A – track A

Entity B – track B

ti

ti+1

ti+2

• Update the predicted position of each entity A & B

• Using the current estimated position of each entity, predict the location of each entity at the time of the new observation (viz., “move entity A from time ti to ti+2, and entity B from time ti+1 to ti+2

• Predict the observations for entity A & B at time, ti+2

• Compare the “distance” between the predicted observation of entity A at ti+2 with the actual observation (do the same for entity B

• Determine which is most likely, the observation belongs to track A, to track B or to neither.

Predicted Observation of entity A at ti+2

Predicted Observation of entity B at time ti+2

• Basic concept is to determine the likelihood that a measurement, z,

could have been the result of a known entity and observation process

• Define the observation residual: the difference between observed and

predicted measurement, z ;^

-Z• Because of observation and prediction errors, has properties of a

random variable with covariance

= Z - Z-Z

-cov (Z) = S = HPHT + R

GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)


Where P = prediction covariance

R = measurement covariance

H = measurement matrix (i.e., Z = HX + )

• Under certain assumptions, the probability density function (PDF) for

-f(Z) =

e-d /2

(2)m/2S1/2

Where m = dimension of the observation

2

Note: these slides are for the mathematically minded, and not required for those not possessing the requisite math background


• The probability that an observation residual, (Z), is within a specified volume (about a known entity) is

~

~Pr = ∫∫ • • • f(Z)dZ1dZ2 • • • dZM

~ ~ ~

• Under assumptions of independence of Z1, Z2, • • • , and ellipsoidal volumes, we can use the chi-square test:

~ d2 = ZT S-1 Z 2M

• This provides a basis for gating:

Pr(2M > G) = PG



Sequential ScanSequential Scan

Optimality based on maximum likelihood in effect -- want to

MAX [f(Zij)] over all i, j~

i.e., overall measurement track assignments of the measurements

This is equivalent to

MIN [d2ij + Ln si]

i.e., minimizing the sum of all these distances

Important:Important: We have an adjunct quality condition which is to maximize the number of assignments -- using possible measurements.



Multi-Scan

- Use a maximum likelihood approach toward building a probabilistic model of the various association possibilities across the multi-scan process

- Overall probability, S, maximized as the basis for the assignments

- The modeled processes are:

QK = PO(nKnFK) PTL(Di)PDT(NUiDi) PER(yil)nK

i=j i=1

NUi



where, in summary, PO(nKnFK) = probability that nk true targets and nFK false targets arise in

the scan volume during the K scans Di = track length for track, iPTL(Di) = probability of track length DPDT(NUiDi) = probability that track, i, produces NU1 detections (used for

track update) given that the track length is Di

PER(yil) = probability of residual error, yil , for the lth observation included in the ith track

Track length is assumed to last as long as it is in the scan volume, but Track length is assumed to last as long as it is in the scan volume, but this probability of distribution function (PDF) is governed by track this probability of distribution function (PDF) is governed by track birth/death process within the scan volume.birth/death process within the scan volume.

This leads to the expression: This leads to the expression: NT

FT

nK

LK = LK - r lnFT =i=1

nK ln + (ln [PTL (Di)]

d2il

2

PD

FT (2)M2 Si + (ln [ ] - ))

i=1

NUi

+ (D1 - NU1) ln (1 - PD)

ln[Pn (Di)] + (Di - NUi) ln(1-PO)



BAYESIAN APPROACHBAYESIAN APPROACH

Bayesian Rule: P(H1D) = P(DH1) P(H1)

P(D)

where H is equal to a feasible location of the measurements.

The maximization of the a posteriori probability given by Bayes can be used as another approach for assigning the measurements.

When Bayes is used in multi-scan, it leads to an approach to tracking called multiple hypothesis tracking (MHT).

Note: The hypothesized models of maximum likelihood estimation (MLE) are still required for the Bayesian approach.


THE ASSIGNMENT PROBLEMTHE ASSIGNMENT PROBLEMIn a dense target environment:

• Multiple measurements will occur in one track gate

• Single measurements will occur in multiple gates, as shown in the diagram

•

•

• P1

03

02

01

P3

P2

GATE

GATE

GATE

01, 02, 03 = Observation PositionsP1, P2, P3 = Predicted Target Positions

© Blackman, S.S., Multiple-Target Tracking with Radar Applications, Artech House, p. 92, 1986.

How to (optimally) resolve conflicts?How to (optimally) resolve conflicts?

• For sequential scans -- decision every scan/data set

• For multi-scan -- multiple sets of assignment matrices

BENEFIT OF OPTIMAL SOLUTION TO THE ASSIGNMENT PROBLEM

1.0

0.8

0.6

0.4

0.2

0.0

SUBOPTIMAL ASSIGNMENT

OPTIMAL ASSIGNMENT

9 18 27 36 45 54 63 72 81

EX

PE

CT

ED

NU

MB

ER

OF

EX

PE

CT

ED

NU

MB

ER

OF

MIS

CO

RR

EL

AT

ION

SM

ISC

OR

RE

LA

TIO

NS/S

CA

N/S

CA

N

TIME(s)

COMPARATIVE MISCORRELATION PROBABILITY --OPTIMALCOMPARATIVE MISCORRELATION PROBABILITY --OPTIMALVERSUS SUBOPTIMAL ASSIGNMENTVERSUS SUBOPTIMAL ASSIGNMENT

© Blackman, S.S., Multiple-Target Tracking with Radar Applications, Artech House, p. 96, 1986.

MULTIPLE ASSOCIATION HYPOTHESIS AND TRACKING

NEWTRACK

FALSE ALARM

EXISTINGTRACKS

0

2

5

0

5

0

2

5

0

2

5

0

5

0

2

5

0

5

0

5

0

2

5

0

5

0

2

5

0

2

4

0

2

4

0

4

0

2

4

0

1

2

3

HYPOTHESISNUMBER

AFTER 1 MEASUREMENT

AFTER 2 MEASUREMENTS

AFTER 3 MEASUREMENTS

1 2 3 4

5 6 7 8 91011

1213141516171819202122232425262728

0123

0130123

01301301230130123

0000

2224444

00044400002224444

0000

0000000

22222255555555555

TRACK

MEASUREMENT

CHI-SQUAREERROR ELLIPSE

1

2

11

1312

Adapted from Blackman, S.S., Multiple-Target Tracking with Radar Applications, Artech House, p. 287 and 289, 1986.

PROBABILISTIC DATA ASSOCIATION: THE CONCEPT

TRACK #1

TRACK #2

CHI-SQUARE ELLIPSE

CORRELATING SENSOR REPORTS

PREDICTED POSITION

Does each observation have to “belong” to one (and only one) track?

A SINGLE SCAN ASSOCIATION PROCESS

Database of known entities x1(t1) or prior observations, y1(t1)

Database of a priori entity behavior and characteristics

Retrieve candidate entities from database

Update entities to observation time,(t1)

Compute predicted observation Zpredicted(t1)

Perform gating

Form (ixj) association matrix

Assignment logic

Assigned (observation-observation) or (observation-track) pairs

- Boolean Query - Solve equations ofmotion to predict x(t)

- Solve observation equations zpredicted(t1)

x1(t1)

Candidate entities x1

or y1

CandidateCandidateObservationObservation

Z1(t1)

- Establish feasible (i,j) pairs

- Compute similarity measure for each (i,j) pair

- Utilize hypothesis testing to assign Zj(t1) to x1(t1)

Database of sensor characteristics

DESIGN OPTIONS FOR MULTI-TARGET TRACKING

Assignment of observations to tracks Hard (unique) assignment Soft (non-unique) assignment

Allowable explanations for observations Single hypothesis Multiple hypothesis

When to make a final decision about observations After each observation (single scan) After N observations (multiple scan) After all observations are received (batch processing)

Processing approach Sequential estimation Batch estimation Covariance error analysis

TOPIC 6 ASSIGNMENTS

Preview the on-line topic 6 materials Read chapter 3 of Hall and McMullen Read Uhlmann (1992) paper Visit the web sites provided in the on-line materials Writing assignment 5: Write a brief description of how

correlation and association is involved (or not involved) with your selected application; what causes the need for data association and correlation; under what circumstances are correlation/association challenging for your application?

DATA FUSION TIP OF THE WEEK

The problem of association and correlation fundamentally involves sorting observations into piles or bins, each bin representing a group of observations that “belong” together, representing observations of the same object, entity or activity. It is important to perform this function, since all subsequent processing assumes that the observations being processed belong to a unique object or entity.

t opic 6: l evel 1 c orrelation david l. hall. t opic o bjectives introduce the concept of level 1...

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