Engineering Statistics for Multi-Object Tracking

1
Engineering Statistics forMulti-Object Tracking
Short Tutorial
Ronald Mahler Lockheed Martin NESS Tactical
Systems Eagan, Minnesota, USA 651-456-4819 /
ronald.p.mahler_at_lmco.com IEEE Workshop on
Multi-Object Tracking Madison WI June 21, 2003
This work was supported by the U.S. Army Research
Office under contracts DAAH04-94-C-0011 and
DAAG55-98-C-0039, and by the U.S. Air Force
Office of Scientific Research under contract
FF49620-01-C-0031. The content does not
necessarily reflect the position or the policy of
the Government. No official endorsement should
be inferred.
2
Top-Down vs. Bottom-Up Data Fusion
usual bottom-up approach to data fusion
heuristically integrate a patchwork of optimal
or heuristic algorithms that address specific
functions (mostly done by operators)
data fusion
detection
sensor mgmt
?
tracking
diverse data
I.D.
ill-characterized data
data association
fuzzy logic
3
Our Topic Finite-Set Statistics (FISST)

• Systematic probabilistic foundation for
  multisensor, multitarget problems
• Preserves the Statistics 101 formalism that
  engineers prefer
• Corrects unexpected difficulties in multitarget
  information fusion
• Potentially powerful new computational techniques
  (1st-order moment filtering)
• Unifies much of multisource, multitarget,
  multiplatform data fusion

sensor manage- ment
performance estima- tion
detection
fuzzy logic
unified Bayesian data fusion
Dempster- Shafer
tracking
information theory
group targets
FISST
control theory
identification
Bayes
rules
4
FISST-Related Books and Monographs
hardcover proceedings
book (out of print)
monograph
book chapter 14
Theory and Decision Library
Volune 97
SPIE Milestone Series Volume MS 124
HANDBOOK OF MULTISENSOR DATA FUSION
Lockheed Martin
John Goutsias, Ronald. P.S. Mahler Hung T.
Nguyen Editors
I.R. Goodman, Ronald. P.S. Mahler and Hung T.
Nguyen
MATHEMATICS OF DATA FUSION
An Introduction to Multisource-Multitarget
Statistics and its Applications Ronald
Mahler March 15, 2000
Random Sets Theory and Applications
Edited by DAVID L. HALL JAMES LLINAS
Kluwer Academic Publishers
Springer
Invited Presentations
Scientific Workshops
1995 ONR/ARO/LM Workshop Invited Session SPIE
AeroSense'99
ATR Working Group USAF Correlation Symp. SPIE
AeroSense Conf. Nat'l Symp. on Data Fusion IEEE
Conf. Dec. Contr. Optical Discr. Alg's
Conf. ONR Workshop on Tracking U. Wisconsin
MDA/POET AFIT NRaD Harvard Johns Hopkins U.
Massachusetts New Mexico State IDC2002
5
Current FISST Technology Transition
basic research project
applied research project
completed research project
scientific performance estimation for Levels
2,3,4 data fusion
robust joint tracking and NCTI using HRRR and
track radar
basic research in data fusion, tracking, NCTI
AFRL
ARO
robust INTELL fusion
AFRL
scientific performance estimation for Level 1
fusion
LMTS
finite-set statistics
AFRL
anti-air multisource radar fusion
MRDEC
SSC/ONR
MDA
BMD technology
robust ASW acoustic target ID
LMTS
AFOSR
MDA
AFRL
BMD technology
unified collection control for UAVs
AFOSR Air Force Office of Scientific
Research AFRL Air Force Research Laboratory ARO
Army Research Office LMTS Lockheed Martin
Tactical Systems MDA Missile Defense
Agency MRDEC Army Missile Research, Dev Eval
Center ONR Office of Naval Research SSC
SPAWAR Systems Center
robust SAR ATR against ground targets
robust multitarget ASW /ASUW fusion NCTI
6
Topics

• Introducing Dr. Dogbert, Critic of FISST
• What is engineering statistics?
• Finite-set statistics (FISST)
• Multitarget Bayes Filtering
• Brief introduction to applications

7
Dr. Dogbert, Critic of FISST
The multisource-multitarget engineering problems
addressed by FISST actually require nothing more
complicated than Bayes rule
which means that FISST is of mere theoretical
interest at best. At worst, it is nothing more
than pointless mathematical obfuscation.
multitarget calculus multitarget
statistics multitarget modeling
What is extraordinary about this assertion is not
the ignorance it displays regarding FISST but,
rather, the ignorance it displays regarding
BayesRule!
8
The Bayesian Iceberg
Nothing deep here!
Bayes rule
non-standard application
The optimality and
simplicity of the
Bayesian framework can be
taken for granted only within
the confines of standard
applications addressed by standard
textbooks. When one ventures out of
these confines one must exercise proper
engineering prudencewhich includes verifying
that standard textbook assumptions still apply.
This is especially true in multitarget
problems!
9
How Multi- and Single-Object Statistics Differ
I sit astride a mountaintop!
Bayes rule
multitarget maximum likelihood estimator (MLE)
is defined but multitarget maximum a
posteriori (MAP) estimator is not!
need explicit methods for constructing provably
true multisensor-multitarget likelihood functions
from explicit sensor models!
need explicit methods for constructing provably
true multitarget Markov transition densities
from explicit motion models!
new multitarget state estimators must be
defined and proved optimal!
multitarget Shannon entropy cannot be defined!
multitarget L2 (i.e., RMS distance) metric cannot
be defined!
multitarget expected values are not defined!
10
Dr. Dogbert (Ctd.)
I, the great Dr. Dogbert , have discovered a
powerful new research strategy! Strip off the
mathematics that makes FISST systematic,
general, and rigorous, copy its main ideas,
change notation, and rename it Dogbert Theory!
a mere change of name and notation does not
constitute a great advance.
It is easy to claim invention of a simple and
yet elastically all-subsuming theory if one does
so by neglecting, avoiding, or glossing over the
specifics required to actually substantiate
puffed-up boasts of generality and first
principles.
11
What is Engineering Statistics?

• Engineering statistics, like all engineering
  mathematics, is a tool and not an end in itself.
  It must have two (inherently conflicting)
  characteristics
• Trustworthiness Constructed upon systematic,
  reliable mathematical foundations, to which we
  can appeal when the going gets rough.
• Fire and forget These foundations can be safely
  neglected in most applications, leaving a
  parsimonious and serviceable mathematical
  machinery in their place.

12
Single-Sensor / Target Engineering Statistics

• Progress in single-sensor, single-target
  applications has been greatly facilitated by the
  existence of a systematic, rigorous, and yet
  practical engineering statistics that supports
  the development of new concepts in this field.

observations
z x
states
formal Bayes modeling
13
What is Single-Target Bayes Statistics?
Classical statistics unknown state-vector x
of the target is a fixed parameter
14
Single-Target Bayes Statistics Distributions
specify single-target state space and its measure
structure
specify single-sensor/target measurement space
and its measure structure
X
Z
15
The Recursive Bayes Filter
1. time-update step via prediction integral
random target state
? Xkkx
target
interim target state-change
predicted random state
? Xk1ky
k
16
The Recursive Bayes Filter
2. data-update step via Bayes rule
predicted random state
? Xk1k
sensor
? zk1
observation
data- updated random state
? Xk1k1
k

17
The Recursive Bayes Filter
3. estimation step via Bayes-optimal state
estimator
? Xk1k

xk1k1
current best estimate of xk1
? Xk1k1

18
Basic Issues in Single-Target Bayes Filtering
over-tuned likelihoods may be non-robust
w/r/t deviations between model reality
without true likelihood for both target
background, Bayes- optimal claim is hollow
sensor models
f(zx)
19
Bayes-Optimal State Estimators
pre-specified cost function
state estimator family of state-valued
func- tions of the obser- vations z1,, zm, mgt0
C(x,y)

x(z1,, zm)
20
True Likelihood Functions Markov Densities
Bayesian modeling
Step 1 construct model
Step 2 use probability- mass function to
describe model statistics
Step 3 construct true density function that
faithfully describes model
21
Finite-Set Statistics (FISST)

• One would expect that multisensor, multitarget
  applications rest upon a similar engineering
  statistics
• Not so, despite long existence of point process
  theory (PPT)
• FISST is in part an engineering friendly
  version of PPT

multisensor- multitarget observations
Z X
multitarget states
formal Bayes modeling?
22
FISST, I
systematic, probabilistic framework for modeling
uncertainties in data
likelihood function, f(zx)
common probabilistic framework random closed
sets, ?
random set model
statistical
radar report
23
Dealing With Ill-Characterized Data
Observation Gustav is NEAR the tower
p1
p2
p3
p4
interpretations of NEAR
probabilities for each interpretation
(random set model Q of natural language
statement)
24
FISST, II
reformulate multi-object problem as generalized
single-object problem
sensors
diverse observations
targets

25
FISST, III
multisource-multitarget Statistics 101
single-sensor/target
sensor target vector observation, z vector
state, x derivative, dpZ /dz integral, ???f(x)
dx prob.-mass func., pZ(S) likelihood,
fZ(zx) prior PDF, f0(x) information
theory filtering theory
26
FISST, IV
systematic, rigorous foundation for -
ill-characterized data likelihoods -
multitarget filtering and estimation -
multigroup filtering sensor management -
computational approximation
sensor manage- ment
performance estima- tion
detection
fuzzy logic
unified Bayesian data fusion
Dempster- Shafer
tracking
information theory
group targets
FISST
control theory
identification
Bayes
rules
27
What is Bayes Multitarget Statistics?
Classical statistics unknown state-set X of
the target is a fixed parameter
28
Multitarget Bayes Statistics Distributions
multitarget likelihoods and distributions must
have a specific form
specify multitarget state space and its measure
structure (e.g. finite subsets of single-target
state space). Subtle issues are involved here!
specify multisensor-multitarget measurement space
and its measure structuree.g., finite subsets of
Z Z1 ? ??? ?Zs union of the measurement
spaces of the individual sensors)
X
Z
29
Point Processes / Random Finite Sets
a point process is a random finite multi-set
finite multi-set ( set with repetition of
elements)
- random finite sets are best for engineering
- are geometric and preserve "Statistics 101"
formalism
- other notations add complexity, no new
information, and lose simple/useful set-theoretic
tools
30
Set Integral (Multi-Object Integral)
Set integral
31
Set Derivative (Multi-Object Derivative)
32
Probability Law of a Random Finite Set
probability generating functional (p.g.fl.)
h bounded real-valued test function
33
Multitarget Posterior Density Functions
fkk(X?Z(k))
multitarget posterior
34
The Multitarget Bayes Filter
observation space
random observation- sets Z produced by targets
state space
random state-set
random state-set
multi-target motion
Xkk
Xk1k1
Xk1k
three targets
five targets
multitarget Bayes filter
fkk(XZ(k))
fk1k1(XZ(k1))
fk1k(XZ(k))
???
???
Zk1
usually computationally intractable!
35
Issues in Multitarget Bayes Filtering
what does true multisensor- multitarget likeliho
od even mean?
how do we construct multisensor-
multitarget measurement models
and multisensor- multitarget likelihood functions?
sensor models
f(ZX)
FISST explicitly addresses these problems
36
True Multitarget Likelihoods Markov Densities
Bayesian modeling
Step 1 construct model
Step 2 use belief- mass function to describe
model statistics
Step 3 construct true density function that
faithfully describes model
37
Failure of Classical Bayes Estimators
simple posterior distribution
0 1 2
?
state
no-target state
position states (km)
posterior probability
1/2 f(x) 1/4 km -1
Some problems go away if continuous states are
discretized into cells, BUT - approach is
no longer comprehensive - power of
continuous mathematics is lost New multitarget
state estimators must be defined and proved
optimal, consistent, etc.!
38
Application 1 Poorly Characterized Data
SAR features
alphanumeric format messages corrupted by
unknowable variations (fat-fingering, fatigue,
overloading, differences in training ability)
LRO features
interpretation by operators
?
MTI features
sent out on coded datalink
ELINT features
SIGINT features
systematic Bayes-rule methodology for modeling
poorly characterized data
39
Modeling Ill-Characterized Data
model ambiguous data as random closed subsets
of observation space
Q
SA(f ) Q SF(X?S) SA(W) vague
imprecise contingent general (fuzzy)
(Dempster- (rules)
Shafer)
choose specific modeling technique
40
Generalized Likelihood of Datalink Message Line
SAR line from formatted datalink message
IMA/SAR/22.0F/29.0F/14.9F/ Y / - / Y / - / - / T
/ 6 / - //
41
Application 2 Poorly Characterized Likelihoods
dents
statistically uncharacterizable extended
operating conditions (EOCs)
wet mud
turret articulations
non-standard equipment (e.g. hibachi welded on
motor)
irregular placement of standard equipment (e.g.
grenade launchers)
systematic Bayes-rule methodology for hedging
against unknowable statistics
42
Modeling Ill-Characterized Likelihoods
likelihood value
nominal likelihood function
intensity value in pixel
43
Application 3 Scientific Performance Estimation
miss distance
I.D. miss
ordinary metrics
multisensor- multitarget algorithm
track purity
mathematical information produced by
algorithm (multitarget Kullback-Leibler)
multitarget miss distance
44
Multitarget Miss Distance
Oliver Drummond et. al. have proposed optimal
truth-to-track association for evaluating
multitarget tracker algorithms
multi-target tracker
data
find best association xpi ? gi between truths gi
and tracks xi
g ground truth targets, G
n tracks, X
Wasserstein distance rigorously and tractably
generalizes intuitive approach
45
Application 4 Group-Target Filtering
- systematic Bayes foundation for group target
filtering
- systematic, principled approximation strategies
46
Approximate 1st-Order Multi-Group Bayes Filter
observation space
random observation- sets Z produced by targets
in group targets
state space
random group-set, Gk1k1
random group- set, Gkk
multi-group motion
three groups
five groups
multi- group Bayes filter
fk1k(Xk1Z(k))
fk1k1(Xk1Z(k1))
fkk(XkZ(k))
usually computationally hopeless!
47
Application 5 Sensor Management
multitarget system
all targets regarded as single system
(possibly undetected low-observable)
- data - attributes - language - rules
multisensor-multitarget observation
all observations regarded as single observation
multiple sensors on multiple platforms
all sensors on all platforms regarded as single
system
- systematic Bayesian control-theoretic
foundation for sensor management
- systematic, principled approximation strategies
48
Multi-Sensor / Target Mgmt Based on MHCs
use multi-hypothesis correlator (MHC) as filter
part of sensor mgmt
observation space
single-target state space
Xkk
Xk1k1
Xk1k
multitarget motion
multitarget Bayes filter
fkk(XZ(k))
fk1k1(XZ(k1))
fk1k(XZ(k))
???
???
optimize Csiszár information objective function
49
Summary / Conclusions

• Finite-set statistics (FISST) provides a
  systematic, Bayes-probabilistic foundation and
  unification for multisource, multitarget data
  fusion
• Is an engineering friendly formulation of point
  process theory, the recognized theoretical
  foundation for stochastic multi-object systems
• Has led to systematic foundations for previously
  murky problems e.g. group-target fusion
• Is leading to new computational approaches, e.g.
  PHD filter multistep look-ahead sensor
  management
• Currently being investigated by other
  researchers
• Defense Sciences Technology Office (Australia),
  U. Melbourne (Australia), Swedish Defense
  Research Agency, Georgia Tech Research Institute,
  SAIC Corp.

50
(No Transcript)