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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Transcript of 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
CA
TEG
OR
YFU
NC
TIO
NP
RO
CES
S
• 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
CA
TEG
OR
YFU
NC
TIO
NTEC
HN
I QU
E
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
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
•
•
•
•
•
•
•
•
•
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
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
•
•
•
•
•
•
•
•
•
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)
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
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
• 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
Note: these slides are for the mathematically minded, and not required for those not possessing the requisite math background
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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.
Note: these slides are for the mathematically minded, and not required for those not possessing the requisite math background
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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
Note: these slides are for the mathematically minded, and not required for those not possessing the requisite math background
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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)
Note: these slides are for the mathematically minded, and not required for those not possessing the requisite math background
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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.
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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
GATING, DATA ASSOCIATION AND ASSIGNMENT (CONTINUED)
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.