Abstract

A new theoretical framework is created for the class of detection problems traditionally addressed by the generalized likelihood ratio test. Absent prior knowledge that would permit implementation of the optimal detector, a family of optimal detectors is fused according to any one of a group of criteria. Geometrical solutions are presented to several specific problems motivated by hyperspectral signal processing. For the general case, a set of partial differential relations is derived. The generalized likelihood ratio test is shown to be equivalent to one of several flavors of continuum fusion detector.

© 2010 OSA

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References

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  1. Louis Scharf, Statistical Signal Processing (USA, Addison-Wesley, 1990).
  2. K. Pearson, “Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia,” Phil. Trans. 187, 253–318, (1895), http://www.jstor.org/stable/90707 .
  3. A. Schaum, Hyperspectral Target Detection using a Bayesian Likelihood Ratio Test, Proceedings 2002 IEEE Aerospace Conference, Vol. 3, Pages 3–1537 to 3–1540, 9–16 March 2002.
  4. Eric Weisstein, “Envelope,” http://mathworld.wolfram.com/Envelope.html .

Other

Louis Scharf, Statistical Signal Processing (USA, Addison-Wesley, 1990).

K. Pearson, “Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia,” Phil. Trans. 187, 253–318, (1895), http://www.jstor.org/stable/90707 .

A. Schaum, Hyperspectral Target Detection using a Bayesian Likelihood Ratio Test, Proceedings 2002 IEEE Aerospace Conference, Vol. 3, Pages 3–1537 to 3–1540, 9–16 March 2002.

Eric Weisstein, “Envelope,” http://mathworld.wolfram.com/Envelope.html .

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Figures (4)

Fig. 1
Fig. 1

A clutter distribution in whitened feature space with anomaly detector decision boundary.

Fig. 2
Fig. 2

CFAR decision boundaries for different values of a clutter parameter.

Fig. 3
Fig. 3

SNO-CONE decision boundaries from CFAR fusion of anomaly detectors, compared to standard results from GLR test.

Fig. 4
Fig. 4

Envelopes for fused detectors are either mergers or bounding surfaces.

Equations (23)

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d ( x ) > λ " d e c l a r e t arg e t " ,
d ( x ) = p T ( x : t ) p C ( x : c )
d ( x ) = M a x t [ p T ( x : t ) ] M a x c [ p C ( x : c ) ] .
p C ( x ) = ( 1 2 π ) N exp ( 1 2 ( x μ C ) 2 ) ,
p C ( x : c ) = ( 1 2 π c ) N exp ( ( x c μ C ) 2 2 c 2 ) ,
d ( x ) c N exp ( ( x c μ C ) 2 2 c 2 ) , with c = x μ C 2 N ( 1 + 4 N x 2 ( x μ C ) 2 1 ) .
p C ( x ) = ( 1 2 π ) N exp ( x 2 2 ) ; p T ( x ) = ( 1 2 π ) N exp ( ( x T ) 2 2 ) .
d ( x ) = T T     x > λ .
f ( x : t , c ) = d ( x : t , c ) λ ( t , c ) = 0
g ( x : t , c ) = ln [ d ( x : t , c ) ] k ( t , c ) ( k ( t , c ) ln [ λ ( t , c ) ] ) .
g ( x : c ) = ( x c μ C ) 2 r 2 c 2 with r a constant,
r c p C ( x + c μ C : c ) ​ ​   d N x = r p C ( y + μ C : 1 ) ​ ​   d N y ,
f ( x 12 : t 2 ) f ( x 12 : t 1 ) = f ( x 12 : t 1 + d t ) f ( x 12 : t 1 ) = d f ( x 12 : t 1 ) = 0 , i .e . f ( x : t ) t = 0
[ f ( x 13 : t 3 ) f ( x 13 : t 2 ) ] [ f ( x 13 : t 2 ) f ( x 13 : t 1 ) ] .
2 f ( x : t ) t 2 < 0.
f ( x : t , c ) = d ( x : t , c ) λ ( t , c ) = 0 f ( x : t , c ) t = 0 ; 2 f ( x : t , c ) t 2 < 0 f ( x : t , c ) c = 0 ; 2 f ( x : t , c ) c 2 > 0 ,
F ( x : c ) f ( x : t ( x : c ) , c ) = 0.
g c = 2 ( μ C ( x c μ C ) + c r 2 ) = 0.
2 g c 2 = 2 ( μ C 2 r 2 ) ,
c = μ C x μ C 2 r 2
( μ C x x ) 2 = μ C 2 r 2 ,
p T ( x : t ) t = 0 ; 2 p T ( x : t ) t 2 < 0 p C ( x : c ) c = 0 ; 2 p C ( x : c ) c 2 < 0.
f ( x : t , c ) c = d ( x : t , c ) c = ( p T ( x : t ) p C ( x : c ) ) c = p T ( x : t ) [ p C ( x : c ) ] 2 p C ( x : c ) c = 0 a n d 2 f c 2 = p T p C 3 [ 2 ( p C c ) 2 p C 2 p C c 2 ] = p T p C 2 2 p C c 2 > 0.

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