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Continuum Fusion: A New Approach to Composite Hypothesis Testing

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A. Schaum

Naval Research Laboratory

Washington, D.C

schaum@nrl.navy.mil

Quantitative Methods in

Defense and National Security 2010

George Mason University May 25-26, 2010

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

- CONTEXT/BACKGROUND
- MOTIVATING EXAMPLE: ANOMALY DETECTION
- CREATING A SYSTEMATIC METHODOLOGY
- RESULTS

- 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

- COMMONEST: SIGNAL AMPLITUDE
- ALLOW GENERALIZATION TO UNTRAINED SITUATIONS

NRL CONOPS is the

INDUSTRY STANDARD

APPROACH

HSI autonomous detection system cues image analyst to region of interest on high resolution imager

ANALYST DISPLAY STATION

Hyperspectral Scatter Plot

A point represents a single HSI pixel.

Target of interest

Green

Each pixel is an N-dim. vector.

Red

Hyperspectral Imagery

- 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.

Target mean T can depend on

parameters with unknown values.

Likelihood ratio

decision boundary

“Linear matched filter”

TARGET

DISTRIBUTION

Decision boundary

Whitening

EUCLIDEAN SPACE

CLUTTER DISTRIBUTION

Clutter mean and covariance can usually

be estimated from field data.

- CONTEXT/BACKGROUND
- MOTIVATING EXAMPLE: ANOMALY DETECTION
- CREATING A SYSTEMATIC METHODOLOGY
- RESULTS

UNION OF ALL “CLUTTER”

DECISION REGIONS

(CFAR) FLR SURFACES

STANDARD ANOMALY DETECTOR

DECISION BOUNDARY

PRIMARY

CLUTTER

CLUTTER IN SHADOW

GLR SURFACES

REDUCED-SCALE VERSIONS OF

PRIMARY DETECTOR,

MATCHED TO CLUTTER SCALE (CFAR)

WHITENED SPACE

PRIMARY CLUTTER

GLR SOLUTION DOES NOT

KNOW THE PHENOMENOLOGY

THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER

- CONTEXT
- MOTIVATING EXAMPLE
- CREATING A SYSTEMATIC METHODOLOGY
- RESULTS

- FUSION LOGIC
- GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”
- HANDLING THE GENERAL CASE

- 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

- GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”
- Generating the optimal answer, when it exists

- HANDLING THE GENERAL CASE
- Deriving the “Fusion Relations”
- A surprise: Unification

- 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

- GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”
- Generating the optimal answer, when it exists

- HANDLING THE GENERAL CASE
- Deriving the “Fusion Relations”
- A surprise: Unification

Some “unknown parameter” problems have optimal solutions (UMP: “uniformly most powerful”) that do no depend on those parameters.

- 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

- 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

- GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”
- Generating the optimal answer, when it exists

- HANDLING THE GENERAL CASE
- Introductory example
- The “Fusion Relations”
- A surprise: Unification

Mean target spectrum

1

3

2

Covariance

matrix

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

- Downwelling radiances

VRC: Virtual Relative Calibration

Application: use of laboratory reflectance signature

to detect material in remote sensing system

Issue: Sensor measures radiance, not reflectance

- 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

Target Subspace

Clutter mean

.

X

AN AFFINE TARGET SUBSPACE MODEL FOLLOWS FROM VRC

TARGET DISTRIBUTION

HAS UNKNOWN MEAN

Clutter

DIMENSION 2

LAB REFLECTANCE

SIGNATURE

DIMENSION 1

TARGET HAS KNOWN VARIANCE,

KNOWN MEAN DIRECTION,

BUT UNKNOWN MEAN AMPLITUDE

GENERALIZED LIKELIHOOD

RATIO TEST

AN AFFINE SUBSPACE

TARGET MODEL

Clutter

Target Subspace

DIMENSION 2

DIMENSION 1

THE AFFINE MATCHED FILTER

SOLVES A GLR PROBLEM

AMF Decision

Boundaries

Clutter

Target Subspace

DIMENSION 2

DIMENSION 1

CFAR FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL

“DECLARE TARGET” REGION

DIMENSION 2

TARGET HAS KNOWN VARIANCE

BUT UNKNOWN MEAN

DIMENSION 1

FUSED CFAR DECISION SURFACE

FOR THE

AFFINE SUBSPACE PROBLEM

LR FUSION

Decision Boundary

(Comet shape) is a combination of

asymptotes and envelopes of the constituent curves

Target Subspace

DIMENSION 2

DIMENSION 1

FLR DECISION SURFACES

FOR THE

AFFINE SUBSPACE PROBLEM

CFAR FUSION

Decision Boundaries

Target Subspace

DIMENSION 2

DIMENSION 1

GLR vs FLR DECISION SURFACES

FOR THE

AFFINE SUBSPACE PROBLEM

CFAR FUSION

Decision Boundaries

GLR Decision

Boundaries

Target Subspace

DIMENSION 2

DIMENSION 1

CPD FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL

“DECLARE TARGET” REGION

ENVELOPE CANNOT BE DEDUCED

FROM GEOMETRICAL ARGUMENTS

DIMENSION 2

TARGET HAS KNOWN VARIANCE

BUT UNKNOWN MEAN

DIMENSION 1

The FUNDAMENTAL THEOREM

of statistical binary testing.

IF the values of all parameters t, care 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

“Seed” algorithm for fusion

MEAN TARGET = 8

TARGET DIRECTION

CLUTTER

BALL

- Clutter (& target) modeled as Laplacian-distributed
- more realistic
- matched filter alarms falsely on half of all outliers

CLR FUSION SOLUTION

MEAN TARGET = 8

EXAMPLE

MEAN TARGET = 9

EXAMPLE

TARGET DIRECTION

MEAN TARGET VALUES:

9 87654.5

CLUTTER

BALL

CONSTRAINT ON CONSTITUENT DETECTORS

ALL MUST HAVE THE SAME LR VALUES FOR THE CORRESPONDING

TARGET DISTRIBUTION MODELS

Standard recipe for composite hypothesis problem:

GLR test

GLR is equivalent to a fusion method!

independent of t,c means GLR can be derived from a

“Constant Likelihood Ratio” (CLR) Fusion Method.

Therefore:

The fusion formalism always includes the GLR as a special case!

CLR FUSION/GLR TEST

TARGET DIRECTION

MEAN TARGET VALUES:

9 87654.5

CLUTTER

BALL

CLR FUSION

SOLUTION

CLUTTER

BALL

Surface is a paraboloid

Resembles matched filter asymptotically

Outlier rejection is lost in the fusion process

TARGET DIRECTION

LR was constant in the fusion process.

Constituents were hyperboloids.

Asymptotes grow linearly with target mean because LR kept constant.

- Can prevent growth of asymptotic slopes by allowing log(LR) to vary (linearly) with target mean
LLLR FUSION (Log Linear Likelihood Ratio)

- Recoups outlier rejection
- Captures bulk statistical rejection

DESIGNED TO MINIMIZE OUTLIER DETECTIONS

CLR FUSION

SOLUTION

CLUTTER

BALL

TARGET DIRECTION

THREE OTHER DETECTOR FLAVORS

(LINEAR LOG LIKELIHOOD RATIO)

FOR THE ADDITIVE TARGET MODEL

MINIMUM TARGET CONTRAST SET TO 4 STANDARD DEVIATIONS FOR ALL 4 DETECTORS

*

*

*

*

OUTLIERS PRESENT?

*

DECISION BOUNDARIES FOR THREE VERSIONS

OF THE AFFINE MATCHED FILTER

GLR

CFAR

CPD

3 FUSION FLAVORS

for

GAUSSIAN DISTRIBUTIONS

*

*

*

*

OUTLIERS?

CLUTTER AT 1

FUSED

DECISION SURFACE

1-D TARGET

SUBSPACE

*

*

*

*

OUTLIERS?

FUSION FLAVOR TAILORED TO

REJECT BULK CLUTTER

REJECT OUTLIERS

- 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”

- Any model that has used a GLR can be revisited

- 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

(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

PROTOTYPE PROBLEM

TWO SENSORS

FUSE SIGNALS FROM BOTH

SENSOR 1

FUSION

SENSOR 2

PROCESSING

DETECTIONS

1-D

SIGNALS

TARGET DISTRIBUTION

TARGET DISTRIBUTION

MODE 2

MODE 1

CLUTTER

DISTRIBUTION

BIVARIATE

TARGET DISTRIBUTION

DIMENSION 2

DIMENSION 1

CLUTTER

DISTRIBUTION

= 3/4

= 0

= -3/4

DIMENSION 2

Clutter signals are uncorrelated, due to whitening transformation.

Target and clutter distributions have different means, but identical variances.

Target signals have unknown level of correlation .

TARGET

DIMENSION 1

CLUTTER

The correlation is a target parameter with unknown value, which defines a composite hypothesis testing problem.

GLR method is nearly unsolvable!

Detectors corresponding to

3 different threshold values

= 1/2

f = .8

Detectors corresponding to

3 different threshold values

= -1/2

Picking a seed algorithm

= 1/2

f = .8

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

Expectation for the sensor fusion problem:

- FI should be approximately CFAR and approximately CPD
- Therefore it should also be approximately CLR (i.e. GLR)

FI Fusion is solvable in closed form

GLR method is virtually unsolvable

f = .8

Selected values of > 0

f = .8

Selected values of < 0

f = .8

All

f = .8

= -1

= 1

Bounding surfaces corresponds

to extreme allowed values of parameters

f = .8

Removing the spurious boundaries

f = .8

FI fusees

vs

GLRfusees

= -.95

= 1/2

= 1/4

= 1/2

= -.95

= 1/4

- 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”

- Any model that has used a GLR can be revisited

- 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

(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

UNION OF ALL “CLUTTER”

DECISION REGIONS

(CFAR) FLR SURFACES

STANDARD ANOMALY DETECTOR

DECISION BOUNDARY

PRIMARY

CLUTTER

CLUTTER IN SHADOW

GLR SURFACES

SAME-SCALE VERSIONS OF

PRIMARY DETECTOR,

FOR MAINTAINING DETECTION PROBABILITY

HYPERSPECTRAL SPACE

PRIMARY CLUTTER

GLR SOLUTION DOES NOT

KNOW THE PHENOMENOLOGY

THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER

SPACE HAS BEEN WHITENED USING

MEASURED 2ND-ORDER STATISTICS

- i = band number
- Tr = target radiance
- = transmissivity
T= target reflectivity

B= clutter reflectivity

Procedure

Measure clutter mean (after chlorophyll removal)

Use target reflectivity to create 1-D subspace through shade point

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

- Green vegetation is the exception

VRC: Virtual Relative Calibration

Application: use of laboratory reflectance signature

to detect material in remote sensing system

- 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

- Green vegetation is the exception

Procedure

Measure clutter mean (after chlorophyll removal)

Use target reflectivity to create 1-D subspace through shade point