Continuum Fusion:A New Approach to
Composite Hypothesis Testing
A. SchaumNaval Research Laboratory
Washington, [email protected]
Quantitative Methods in Defense and National Security 2010
George Mason University May 25-26, 2010
CONTINUUM FUSION:A NEW THEORY of INFERENCE
A FRAMEWORK FOR GENERATING DETECTION ALGORITHMS
WHEN USING AMBIGUOUS MODELS
PROBLEM CLASS• FOR MAKING DECISIONS BASED ON MODELS CONTAINING PARAMETERS WHOSE VALUES
ARE FIXED BUT UNKNOWN (CALLED THE “COMPOSITE HYPOTHESIS” TESTING PROBLEM)• CAN BE SUBSTITUTED FOR ANY GENERALIZED LIKELIHOOD RATIO (GLR) TEST
SYNOPSIS• FUSES A CONTINUUM OF OPTIMAL METHODS WHEN YOU DON’T KNOW WHICH ONE IS
REALLY OPTIMAL
ADVANTAGES• GROWS THE GLR RECIPE INTO A FULL MENU OF DETECTION ALGORITHM “FLAVORS” • CAN PRODUCE DETECTORS FOR MODELS WHERE GLR IS UNSOLVABLE• FLEXIBILITY ALLOWS SIMULTANEOUS TREATMENT OF STATISTICAL AND NON-
STATISTICAL MODELS• ALLOWS OPTIMIZATION OF NEW DESIGN METRICS
OUTLINE
• CONTEXT/BACKGROUND
• MOTIVATING EXAMPLE: ANOMALY DETECTION
• CREATING A SYSTEMATIC METHODOLOGY
• RESULTS
CONTEXT:DETECTION & DISCRIMINATION ALGORITHMS
• DATA DRIVEN/AGNOSTIC (“MACHINE LEARNING”)– ARTIFICIAL NEURAL NETWORKS– GENETIC– SVMs...
• MODEL-BASED– UBIQUITOUS IN MANY SENSING MODALITIES– KNOWN PHYSICS, UNKNOWN PARAMETERS
• COMMONEST: SIGNAL AMPLITUDE– VARIABLE RANGE– UNKNOWN SIGNAL DRIVER
– ALLOW GENERALIZATION TO UNTRAINED SITUATIONS
NAVY CONTEXTMANNED & UNMANNED LONG STANDOFF RANGE
RECCE/SURVEILLANCE
ANALYST DISPLAY STATION
APPROACH HSI autonomous detection system cues image analyst to region of interest on high resolution imager
NRL CONOPS is theINDUSTRY STANDARD
BACKGROUND:REPRESENTATIONS OF HYPERSPECTRAL DATA
Algorithms operate in an N-dimensional spectral space (N=64 for WAR HORSE)• Similar objects in HSI imagery occupy similar regions in the spectral space.• Multivariate detection algorithms generate a “decision surface” that identifies where
targets lie in the vector space.• Number of dimensions should exceed number of different constituents.
Each pixel is an N-dim. vector.
Hyperspectral Imagery
Hyperspectral Scatter Plot
RedG
reen
Target of interest
A point represents a single HSI pixel.
Decision boundary
Likelihood ratio decision boundary
“Linear matched filter”
ADDITIVE TARGET MODELS
pC x 1
2 Ndet M
1
2
exp 1
2x C t
M 1 x C
TARGET DISTRIBUTION
pT x 1
2 Ndet M
1
2
exp 1
2x T t
M 1 x T
Target mean T can depend on parameters with unknown values.
Clutter mean and covariance can usually be estimated from field data.
CLUTTER DISTRIBUTION
Whitening
x M
1
2 x
EUCLIDEAN SPACE
OUTLINE
• CONTEXT/BACKGROUND
• MOTIVATING EXAMPLE: ANOMALY DETECTION
• CREATING A SYSTEMATIC METHODOLOGY
• RESULTS
INITIAL MOTIVATION: SUPPRESSING ONE MECHANISM
OF FALSE ALARMS IN ANOMALY DETECTION
WHITENED SPACE
GLR SOLUTION DOES NOT
KNOW THE PHENOMENOLOGY
THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER
PRIMARY CLUTTER
CLUTTER IN SHADOW GLR SURFACES
(CFAR) FLR SURFACES
PRIMARY
CLUTTER
REDUCED-SCALE VERSIONS OF
PRIMARY DETECTOR,
MATCHED TO CLUTTER SCALE (CFAR)
UNION OF ALL “CLUTTER”
DECISION REGIONS
STANDARD ANOMALY DETECTOR
DECISION BOUNDARY
pC x 1
2 N exp 1
2x C t
x C
OUTLINE
• CONTEXT
• MOTIVATING EXAMPLE
• CREATING A SYSTEMATIC METHODOLOGY
• RESULTS
CREATING A SYSTEMATIC METHODOLOGY
1. FUSION LOGIC
2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”
3. HANDLING THE GENERAL CASE
CREATING A SYSTEMATIC METHODOLOGY
1. FUSION LOGIC– Form UNION of “decide clutter” regions (if fusing over clutter parameters)
– Form UNION of “decide target” regions (if fusing over target parameters)
– Fusion flavors
2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”– Generating the optimal answer, when it exists
3. HANDLING THE GENERAL CASE– Deriving the “Fusion Relations”
– A surprise: Unification
CREATING A SYSTEMATIC METHODOLOGY
1. FUSION LOGIC– Form UNION of “decide clutter” regions (if fusing over clutter parameters)
– Form UNION of “decide target” regions (if fusing over target parameters)
– Fusion flavors
2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”– Generating the optimal answer, when it exists
3. HANDLING THE GENERAL CASE– Deriving the “Fusion Relations”
– A surprise: Unification
Summary of “Getting the Right Results”
• CFAR and CPD flavors both give the matched filter answer to the Gaussian additive target problem (unknown target amplitude)
• GLR solution does not always give the right answer to the Gaussian additive target problem
• CFAR, CPD, and GLR flavors all give the correct Gaussian anomaly detector
Some “unknown parameter” problems have optimal solutions (UMP: “uniformly most powerful”) that do no depend on those parameters.
CREATING A SYSTEMATIC METHODOLOGY
1. FUSION LOGIC– Form UNION of “decide clutter” regions (if fusing over clutter parameters)
– Form UNION of “decide target” regions (if fusing over target parameters)
– Fusion flavors
2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”– Generating the optimal answer, when it exists
3. HANDLING THE GENERAL CASE– Introductory example
– The “Fusion Relations”
– A surprise: Unification
In-scene target radiance prediction
• 1st order: reflectance to radiance– (Solar spectrum) x (reflectivity)
• 2nd order: column densities– aerosols– water vapor– CO2
– BRDF effects– contamination
• Other unknowns– Downwelling radiances
• solar• sky
– Background interactions• reflections• adjacency effects
– Upwelling effects
UNKNOWN PARAMETER VALUES IN TARGET SIGNATURE PREDICTION
1
2
3
eet
Mean target spectrum
Covariancematrix
VRC: Virtual Relative Calibration
Most Short Wave IR mineral reflectance spectra are flat (“graybody”)
• Model the mean reflectance of an image as gray body• A non-flat mean background radiance spectrum seen by a remote systems reflects the spectral content of illumination/attenuation effects
Mean spectrum can serve as relative calibration source
Application: use of laboratory reflectance signature to detect material in remote sensing system
Issue: Sensor measures radiance, not reflectance
Clutter
DIMENSION 1
DIM
EN
SIO
N 2
AN AFFINE TARGET SUBSPACE MODEL FOLLOWS FROM VRC
Target
Subsp
ace
TARGET DISTRIBUTIONHAS UNKNOWN MEAN
LAB REFLECTANCESIGNATURE
.
Clutter mean
X
Target
Subsp
ace
Clutter
DIMENSION 1
DIM
EN
SIO
N 2
AN AFFINE SUBSPACE TARGET MODEL
TARGET HAS KNOWN VARIANCE,KNOWN MEAN DIRECTION,
BUT UNKNOWN MEAN AMPLITUDE
GENERALIZED LIKELIHOODRATIO TEST
Target
Subsp
ace
Clutter
AMF DecisionBoundaries
DIMENSION 1
DIM
EN
SIO
N 2
THE AFFINE MATCHED FILTERSOLVES A GLR PROBLEM
DIMENSION 1
DIM
EN
SIO
N 2
CFAR FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL
TARGET HAS KNOWN VARIANCEBUT UNKNOWN MEAN
“DECLARE TARGET” REGION
Target
Subsp
ace
DIMENSION 1
DIM
EN
SIO
N 2
FUSED CFAR DECISION SURFACE FOR THE
AFFINE SUBSPACE PROBLEM
LR FUSIONDecision Boundary(Comet shape) is a
combination ofasymptotes and envelopes of the constituent curves
Target
Subsp
ace
DIMENSION 1
FLR DECISION SURFACES FOR THE
AFFINE SUBSPACE PROBLEM
CFAR FUSIONDecision
Boundaries
DIM
EN
SIO
N 2
Target
Subsp
ace
GLR DecisionBoundaries
DIMENSION 1
GLR vs FLR DECISION SURFACES FOR THE
AFFINE SUBSPACE PROBLEM
CFAR FUSIONDecision
Boundaries
DIM
EN
SIO
N 2
DIMENSION 1
DIM
EN
SIO
N 2
CPD FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL
TARGET HAS KNOWN VARIANCEBUT UNKNOWN MEAN
“DECLARE TARGET” REGION
ENVELOPE CANNOT BE DEDUCED FROM GEOMETRICAL ARGUMENTS
OPTIMAL DETECTORS BASED ON THE LIKELIHOOD RATIO TEST
The FUNDAMENTAL THEOREM of statistical binary testing.
LR pT x : t pC x : c threshold x represents target
d x : t, c pT x : t pC x : c discriminant function
IF the values of all parameters t, c are knownIF the values of all parameters t, c are known
DECISION BOUNDARY IS DEFINED BY NULLS IN THE DISCRIMINANT FUNCTION: d(x:t,c) = 0
(CLR) FUSION IN ACTION
CLUTTER AT 1
1-D TARGET SUBSPACE
FUSED DECISION SURFACE
The FUNDAMENTAL THEOREM OFCONTINUUM FUSION
d x; t, c pT (x; t)
pC x; c t, c 0
d x; t, c ti
0;d x; t, c
ci
0
2d x; t, c 2ti
0;2d x; t, c
2ci
0.
LAPLACIAN ECD ADDITIVE TARGET MODEL
CLUTTER
BALLTARGET DIRECTION
Each curve is characterized by a number:
the likelihood ratio
d x : t pT x : t 8 pC x 0
Clutter (& target) modeled as Laplacian-distributed-more realistic-matched filter alarms falsely on half of all outliers
MEAN TARGET = 8
“Seed” algorithm for fusion
CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODEL
CLR FUSION SOLUTION
CONSTRAINT ON CONSTITUENT DETECTORSALL MUST HAVE THE SAME LR VALUES FOR THE
CORRESPONDING TARGET DISTRIBUTION MODELS
MEAN TARGET VALUES: 9 8 7 6 5 4.5
MEAN TARGET = 8EXAMPLE
MEAN TARGET = 9EXAMPLE
CLUTTER
BALL TARGET DIRECTION
Each curve is characterized by a number:
the likelihood ratio
d x : t pT x : t pC x 0
DETECTORS BASED ON THE GENERALIZED LIKELIHOOD RATIO TEST
GLR x t
Max pT x; t
c
Max pC x; c threshold x represents target
t = target parameters, c = clutter parameters
Standard recipe for composite hypothesis problem:
GLR test
RELATION OF GLR TO FUSION
GLR recipe:
GLR x t
Max pT x; t
c
Max pC x; c
pT x; t ti
0;pC x; c
ci
0
2 pT x; t ti
2 0;2 pC x; c
ci2 0
General Fusion Equations:
d x; t, c pT (x; t)
pC x; c t, c 0
d x; t, c ti
0;d x; t, c
ci
0
2d x; t, c 2ti
0;2d x; t, c
2ci
0
1. GLR is equivalent to a fusion method!2. independent of t,c means GLR can be derived from a“Constant Likelihood Ratio” (CLR) Fusion Method.Therefore:3. The fusion formalism always includes the GLR as a
special case!
CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODEL
IS THE GLR TEST
CLR FUSION/GLR TEST
MEAN TARGET VALUES: 9 8 7 6 5 4.5
CLUTTER
BALL TARGET DIRECTION
Each curve produces the same likelihood ratio for its own target distribution
d x : t pT x : t pC x 0
that is, is independent of t
GLR (= CLR FUSION FLAVOR) FOR THE LAPLACIAN ECD ADDITIVE TARGET MODEL
CLUTTER
BALL TARGET DIRECTION
CLR FUSION SOLUTION
1. LR was constant in the fusion process.2. Constituents were hyperboloids.3. Asymptotes grow linearly with target
mean because LR kept constant.
Surface is a paraboloidResembles matched filter asymptoticallyOutlier rejection is lost in the fusion process
Can prevent growth of asymptotic slopes by allowing log(LR) to vary (linearly) with target meanLLLR FUSION (Log Linear Likelihood Ratio)
• Recoups outlier rejection• Captures bulk statistical rejection
CONTINUUM OF LLLR FUSION FLAVORS FOR THE LAPLACIAN ECD
ADDITIVE TARGET MODEL
CLUTTER
BALL TARGET DIRECTION
CLR FUSION SOLUTION
MINIMUM TARGET CONTRAST SET TO 4 STANDARD DEVIATIONS FOR ALL 4 DETECTORS
THREE OTHER DETECTOR FLAVORS(LINEAR LOG LIKELIHOOD RATIO)
FOR THE ADDITIVE TARGET MODEL
DESIGNED TO MINIMIZE OUTLIER DETECTIONS
CPD
CFAR
GLR
DECISION BOUNDARIES FOR THREE VERSIONS OF THE AFFINE MATCHED FILTER
3 FUSION FLAVORSfor
GAUSSIAN DISTRIBUTIONS
*
*
*
* *
OUTLIERS PRESENT?
GLR FOR THE AFFINE LAPLACIAN MODEL
CLUTTER AT 1
1-D TARGET SUBSPACE
FUSED DECISION SURFACE
**
* *
OUTLIERS?
LLLR FUSION FLAVOR FORAFFINE LAPLACIAN MODEL
**
* *
OUTLIERS?
FUSION FLAVOR TAILORED TO1. REJECT BULK CLUTTER2. REJECT OUTLIERS
SUMMARY: CONTINUUM FUSION
• Provides a new framework for designing detection algorithms in model-based problem sets
• Reduces to the desired results in the appropriate limits– Matched filter, RX– Does so more naturally and generally than GLR
• Comes in many flavors– Includes “vanilla,” the GLR, the only prior solution to the general CH problem– New metrics of performance can be optimized (min/max)– Any CF method can be customized by manipulating the fusion process
• (including GLR)• Non-statistical criteria can be accommodated
• Constitutes a new branch of Statistical Detection Theory• Future
– Any model that has used a GLR can be revisited• Thousands of published results
– Some CH problems unsolvable with GLR can be solved with other CF methods– Theoretical issues
• Relationship to specialized apps (“parameter testing holy trinity”)• UMPs and Invariance• Studying “fusion characteristics”
References-A. Schaum, Continuum Fusion, a theory of inference, with applications to hyperspectral detection, 12 April 2010 / Vol. 18, No. 8 / OPTICS EXPRESS 8171-8188. -A. Schaum, Continuum Fusion Detectors for Affine and Oblique Spectral Subspace Models, Special Issue of IEEE Transactions on Geoscience and Remote Sensing on Hyperspectral Image and Signal Processing, in review
Matrix of CF Problems
(System model) x (Statistical model) x (Flavor)
• System Model– Physical: Sensing mode/environment– Structural
• Statistical models– Gaussian, Laplacian, t-score, ...– Homoskedastic, Ampliskedastic, Heteroskedastic
• Some flavors– CFAR: constant false alarm rate – CPD: constant probability of detection– CRL: constant likelihood ratio (= GLR)– FIF: Fixed Intercept Fraction– FI: Fixed Intercept– LLLR: Linear log likelihood ratio– Geometrical
MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION
PROTOTYPE PROBLEMTWO SENSORSFUSE SIGNALS FROM BOTH
SENSOR 1
SENSOR 2 PROCESSING
1-D SIGNALS
FUSION
DETECTIONS
MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION
MODE 1
MODE 2
TARGET DISTRIBUTION
TARGET DISTRIBUTION
CLUTTER DISTRIBUTION
MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION
DIMENSION 1
DIMENSION 2
CLUTTER DISTRIBUTION
BIVARIATETARGET DISTRIBUTION
MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION
DIMENSION 2
DIMENSION 1CLUTTER
TARGET
1. Clutter signals are uncorrelated, due to whitening transformation.
2. Target and clutter distributions have different means, but identical variances.
3. Target signals have unknown level of correlation .
= 3/4
= 0 = -3/4
The correlation is a target parameter with unknown value, which defines a composite hypothesis testing problem.
GLR method is nearly unsolvable!
= 1/2
BIMODAL FUSION
Detectors corresponding to3 different threshold values
BIMODAL FUSION
= -1/2
f = .8
Picking a seed algorithm
Detectors corresponding to3 different threshold values
= 1/2
BIMODAL FUSION
f = .8
FI Fusion Flavor
Expectation for the sensor fusion problem:1. FI should be approximately CFAR and approximately CPD2. Therefore it should also be approximately CLR (i.e. GLR)
Definition of Fixed Intercept fusion:
For different parameter values, fuse optimal algorithms whose decision boundaries have the same intercept with line from target-to-clutter means
FI Fusion is solvable in closed formGLR method is virtually unsolvable
BIMODAL FUSION f = .8
Selected values of > 0
BIMODAL FUSION
f = .8Selected values of < 0
BIMODAL FUSION f = .8All
d x; pT (x;)
pC x 0
d x;
0
BIMODAL FUSION f = .8
Bounding surfaces correspondsto extreme allowed values of parameters
= -1
= 1
BIMODAL FUSION f = .8
Removing the spurious boundaries
BIMODAL FUSION f = .8
FI fuseesvs
GLR fusees
= 1/4
= -.95
= -.95 = 1/2
= 1/4
= 1/2
SUMMARY: CONTINUUM FUSION
• Provides a new framework for designing detection algorithms in model-based problem sets
• Reduces to the desired results in the appropriate limits– Matched filter, RX– Does so more naturally and generally than GLR
• Comes in many flavors– Includes “vanilla,” the GLR, the only prior solution to the general CH problem– New metrics of performance can be optimized (min/max)– Any CF method can be customized by manipulating the fusion process
• (including GLR)• Non-statistical criteria can be accommodated
• Constitutes a new branch of Statistical Detection Theory• Future
– Any model that has used a GLR can be revisited• Thousands of published results
– Some CH problems unsolvable with GLR can be solved with other CF methods– Theoretical issues
• Relationship to specialized apps (“parameter testing holy trinity”)• UMPs and Invariance• Studying “fusion characteristics”
References-A. Schaum, Continuum Fusion, a theory of inference, with applications to hyperspectral detection, 12 April 2010 / Vol. 18, No. 8 / OPTICS EXPRESS 8171-8188. -A. Schaum, Continuum Fusion Detectors for Affine and Oblique Spectral Subspace Models, Special Issue of IEEE Transactions on Geoscience and Remote Sensing on Hyperspectral Image and Signal Processing, in review
Matrix of CF Problems
(System model) x (Statistical model) x (Flavor)
• System Model– Physical: Sensing mode/environment– Structural
• Statistical models– Gaussian, Laplacian, t-score, ...– Homoskedastic, Ampliskedastic, Heteroskedastic
• Some flavors– CFAR: constant false alarm rate – CPD: constant probability of detection– CRL: constant likelihood ratio (= GLR)– FIF: Fixed Intercept Fraction– FI: Fixed Intercept– LLLR: Linear log likelihood ratio– Geometrical
CPD FUSION COMPARED
SPACE HAS BEEN WHITENED USING
MEASURED 2ND-ORDER STATISTICS
HYPERSPECTRAL SPACE
GLR SOLUTION DOES NOT
KNOW THE PHENOMENOLOGY
THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER
PRIMARY CLUTTER
CLUTTER IN SHADOW GLR SURFACES
(CFAR) FLR SURFACES
PRIMARY
CLUTTER
SAME-SCALE VERSIONS OF
PRIMARY DETECTOR,
FOR MAINTAINING DETECTION PROBABILITY
UNION OF ALL “CLUTTER”
DECISION REGIONS
STANDARD ANOMALY DETECTOR
DECISION BOUNDARY
VRC: Virtual Relative Calibration
Most SWIR reflectance spectra of natural materials are nearly flat• Green vegetation is the exception
Remove with NDVI• Model the mean reflectance of the rest of an image as gray body• Non-flat mean background radiance spectrum serves as relative calibration source
Tr i iSi T
i,
r i iSi B
i iSi B
i iSi ,
Tr i r
iT
i.
i = band numberTr = target radiance = transmissivityT = target reflectivityB = clutter reflectivity
Procedure1. Measure clutter mean (after chlorophyll removal)2. Use target reflectivity to create 1-D subspace through
shade point
VRC: Virtual Relative Calibration
Most Short Wave IR reflectance spectra of natural materials are nearly flat
• Green vegetation is the exception Remove vegetation pixels with standard algorithm: NDVI
• Model the mean reflectance of the rest of an image as gray body• A non-flat mean background radiance spectrum reflects the spectral content of illumination/attenuation effects
Mean spectrum can serve as relative calibration source
Procedure
1. Measure clutter mean (after chlorophyll removal)2. Use target reflectivity to create 1-D subspace through
shade point
Application: use of laboratory reflectance signature to detect material in remote sensing system
