Abstract

The additive target model is used routinely in the statistical detection of opaque targets, despite its phenomenological inaccuracy. The more appropriate replacement target model is seldom used, because the standard method for producing a detection algorithm from it proves to be intractable, unless narrow restrictions are imposed. Now, the recently developed continuum fusion (CF) methodology allows an expanded solution set to the general replacement target problem. It also provides a mechanism for producing approximate solutions for the standard approach. We illustrate the principles of CF by using them to generate both types of answers for the correct detection model.

© 2014 Optical Society of America

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  1. D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19(1), 29–43 (2002).
    [CrossRef]
  2. S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
    [CrossRef]
  3. C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
    [CrossRef]
  4. D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
    [CrossRef]
  5. D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
    [CrossRef]
  6. A. Stocker and A. Schaum, “Spectrally-selective target detection,” in Proceedings of International Symposium on Spectral Sensing Research, International Society for Photogrammetry and Remote Sensing, B. A. Mandel, ed. (1997), p. 23.
  7. J. W. Boardman, “Automating spectral unmixing of AVIRIS data using convex geometry concepts,” in Fourth Annual JPL Airborne Geoscience Workshop (Jet Propulsion Lab, 1993), Vol. 1, pp. 11–14.
  8. D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
    [CrossRef]
  9. A. Schaum and A. D. Stocker, “Spectral detection methods: spectral unmixing, correlation processing, and when they are appropriate,” in Proceedings of the Second Annual Symposium on Spectral Sensing Research, July, 1994.
  10. H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
    [CrossRef]
  11. L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
    [CrossRef]
  12. L. Scharf, Statistical Signal Processing (Wesley, 1991).
  13. A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012).
    [CrossRef]
  14. A. Schaum, “The continuum fusion theory of signal detection applied to a bi-modal fusion problem,” Proc. SPIE 8064, 806403 (2011).
    [CrossRef]
  15. B. J. Daniel and A. Schaum, “Linear log-likelihood ratio (L3R) algorithm for spectral detection,” Proc. SPIE 8048, 804804 (2011).
    [CrossRef]
  16. S. Kay, “Fundamentals of statistical signal processing,” in Detection Theory (Prentice-Hall, 1998), Vol. 2.

2014

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

2012

A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012).
[CrossRef]

2011

A. Schaum, “The continuum fusion theory of signal detection applied to a bi-modal fusion problem,” Proc. SPIE 8064, 806403 (2011).
[CrossRef]

B. J. Daniel and A. Schaum, “Linear log-likelihood ratio (L3R) algorithm for spectral detection,” Proc. SPIE 8048, 804804 (2011).
[CrossRef]

2009

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

2005

S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
[CrossRef]

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

2002

D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19(1), 29–43 (2002).
[CrossRef]

2001

D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
[CrossRef]

2000

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Boardman, J. W.

J. W. Boardman, “Automating spectral unmixing of AVIRIS data using convex geometry concepts,” in Fourth Annual JPL Airborne Geoscience Workshop (Jet Propulsion Lab, 1993), Vol. 1, pp. 11–14.

Brueggeman, M.

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

Bucholtz, F.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Butler, R. W.

S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
[CrossRef]

Cooley, T.

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

Daniel, B. J.

A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012).
[CrossRef]

B. J. Daniel and A. Schaum, “Linear log-likelihood ratio (L3R) algorithm for spectral detection,” Proc. SPIE 8048, 804804 (2011).
[CrossRef]

Hazel, G. G.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Huang, X.

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

Jacobson, J.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

Kay, S.

S. Kay, “Fundamentals of statistical signal processing,” in Detection Theory (Prentice-Hall, 1998), Vol. 2.

Kraut, S.

S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
[CrossRef]

Kwon, H.

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

Lockwood, R.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

Manolakis, D.

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19(1), 29–43 (2002).
[CrossRef]

D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
[CrossRef]

Michalowicz, J. V.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Nasrabadi, N. M.

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

Pieper, M.

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

Schaaf, W.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Scharf, L.

L. Scharf, Statistical Signal Processing (Wesley, 1991).

Scharf, L. L.

S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
[CrossRef]

Schaum, A.

A. Schaum, “The continuum fusion theory of signal detection applied to a bi-modal fusion problem,” Proc. SPIE 8064, 806403 (2011).
[CrossRef]

B. J. Daniel and A. Schaum, “Linear log-likelihood ratio (L3R) algorithm for spectral detection,” Proc. SPIE 8048, 804804 (2011).
[CrossRef]

A. Schaum and A. D. Stocker, “Spectral detection methods: spectral unmixing, correlation processing, and when they are appropriate,” in Proceedings of the Second Annual Symposium on Spectral Sensing Research, July, 1994.

A. Stocker and A. Schaum, “Spectrally-selective target detection,” in Proceedings of International Symposium on Spectral Sensing Research, International Society for Photogrammetry and Remote Sensing, B. A. Mandel, ed. (1997), p. 23.

Schaum, A. P.

A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012).
[CrossRef]

Shaw, G.

D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19(1), 29–43 (2002).
[CrossRef]

D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
[CrossRef]

Siracusa, C.

D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
[CrossRef]

Stellman, C. M.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

Stocker, A.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

A. Stocker and A. Schaum, “Spectrally-selective target detection,” in Proceedings of International Symposium on Spectral Sensing Research, International Society for Photogrammetry and Remote Sensing, B. A. Mandel, ed. (1997), p. 23.

Stocker, A. D.

A. Schaum and A. D. Stocker, “Spectral detection methods: spectral unmixing, correlation processing, and when they are appropriate,” in Proceedings of the Second Annual Symposium on Spectral Sensing Research, July, 1994.

Tao, D.

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

Truslow, E.

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

Zhang, L.

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

IEEE Signal Process. Mag.

D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19(1), 29–43 (2002).
[CrossRef]

D. Manolakis, E. Truslow, M. Pieper, T. Cooley, and M. Brueggeman, “Detection algorithms in hyperspectral imaging systems,” IEEE Signal Process. Mag. 31(1), 24–33 (2014).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geosci. Remote Sens. 39, 1392–1409 (2001).
[CrossRef]

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

L. Zhang, L. Zhang, D. Tao, and X. Huang, “Sparse transfer manifold embedding for hyperspectral target detection,” IEEE Trans. Geosci. Remote Sens. 52, 1030–1043 (2014).
[CrossRef]

IEEE Trans. Signal Process.

S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator,” IEEE Trans. Signal Process. 53, 427–438 (2005).
[CrossRef]

Opt. Eng.

C. M. Stellman, G. G. Hazel, F. Bucholtz, J. V. Michalowicz, A. Stocker, and W. Schaaf, “Real-time hyperspectral detection and cuing,” Opt. Eng. 39, 1928–1935 (2000).
[CrossRef]

A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012).
[CrossRef]

Proc. SPIE

A. Schaum, “The continuum fusion theory of signal detection applied to a bi-modal fusion problem,” Proc. SPIE 8064, 806403 (2011).
[CrossRef]

B. J. Daniel and A. Schaum, “Linear log-likelihood ratio (L3R) algorithm for spectral detection,” Proc. SPIE 8048, 804804 (2011).
[CrossRef]

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402 (2009).
[CrossRef]

Other

A. Schaum and A. D. Stocker, “Spectral detection methods: spectral unmixing, correlation processing, and when they are appropriate,” in Proceedings of the Second Annual Symposium on Spectral Sensing Research, July, 1994.

A. Stocker and A. Schaum, “Spectrally-selective target detection,” in Proceedings of International Symposium on Spectral Sensing Research, International Society for Photogrammetry and Remote Sensing, B. A. Mandel, ed. (1997), p. 23.

J. W. Boardman, “Automating spectral unmixing of AVIRIS data using convex geometry concepts,” in Fourth Annual JPL Airborne Geoscience Workshop (Jet Propulsion Lab, 1993), Vol. 1, pp. 11–14.

S. Kay, “Fundamentals of statistical signal processing,” in Detection Theory (Prentice-Hall, 1998), Vol. 2.

L. Scharf, Statistical Signal Processing (Wesley, 1991).

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Figures (6)

Fig. 1.
Fig. 1.

Some possible distributions for the replacement target model, in whitened spectral space.

Fig. 2.
Fig. 2.

Uniformly most powerful (UMP) solutions for the ZCL model are the spherical clairvoyants.

Fig. 3.
Fig. 3.

CF decision boundaries for the ICL model with flavor defined by Eq. (19).

Fig. 4.
Fig. 4.

Clairvoyant radius dependence on center of decision sphere in the ICL model.

Fig. 5.
Fig. 5.

Approximating the GLRT for lnλ=6 in the ICL model.

Fig. 6.
Fig. 6.

Approximation to the GLRT using a CF interpretation.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

H0:XN(μB,I)H1:XN(μf,σf2I),
p(x:f)=1(2πσf2)M2exp(12σf2xμf2),
μf=fμT+(1f)μB
σf2=(1f)+fσT2,
GLRT(x)=MaxθT[pT(x:θT)]MaxθB[pB(x:θB)]>λDeclaretarget.
GLRT(x)=Maxf[p(x:f)]p(x:0)>λ,
GLRT(x)=p(x:fM)p(x:0),
fM={0q>0q0<q<11q>0q=(bb24ac)/2aa=μT2(1σT2),b=2μT2M(1σT2)c=2μTx(1σT2)(x2+y2).
d(x:θT,θB,λ)pT(x:θT)pB(x:θB)λ,
dGLRT(x)=MaxθT[MinθB(d(x:θT,θB,λ))].
D(x:θT,θB,λ)=kln(pT(x:θT)pB(x:θB))klnλ,
DGLRT(x)=MaxθT[MinθB(D(x:θT,θB,λ))].
k=2(σf21)1
D(x:f,λ)=Rλ2(f)(xCf)2y2,
Rλ2(f)=σf2(1σf2)2[μf2(1σf2)(2lnλ+Mln(σf2))]
Cf=μf1σf2.
CfCZCL=μT1σT2.
σf2=(1f)2+f2σT2.
CICL=μT2f(1+σT2),
Rα2(C)=R02+αC
Dα(x:C,R0)=Rα2(C)(xC)2y2.
C={μT2forx<μT+α2xα2forμT+α2<x<μT1σT2+α2μT1σT2forx>μT1σT2+α2.
T=i=1MtiandB=i=1Mbi,
μT=E(i=1Mti)=MμtandμB=E(i=1Mbi)=Mμb.
X=i=1fMti+i=1(1f)Mbi,
var(i=1fMti)=fMvar(ti)=fMσt2.
σT2=Mσt2.
var(X)=fMσt2+(1f)Mσb2=fσT2+(1f)σB2,
var(i=1fMti)=var(fMti)=(fM)2σt2
σT2=M2σt2.
var(X)=f2M2σt2+(1f)2M2σb2=f2σT2+(1f)2σB2,

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