Abstract

A new upper bound is derived on the area under the receiver operating characteristic curve for the ideal observer in a signal-detection task. This upper bound is determined by the values of the likelihood-generating function and its second derivative at the origin. This bound is compared with other bounds on ideal-observer performance that have been derived recently, and it is also shown how this bound leads to some asymptotic results for approximations to ideal-observer performance.

© 2002 Optical Society of America

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References

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  1. J. Shapiro, “Bounds on the area under the ROC curve,” J. Opt. Soc. Am. A 16, 53–57 (1999).
    [CrossRef]
  2. M. V. Burnashev, “On one useful inequality in the testing of hypotheses,” IEEE Trans. Inf. Theory 44, 1668–1670 (1998).
    [CrossRef]
  3. D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
    [CrossRef]
  4. U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
    [CrossRef]
  5. H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
    [CrossRef]
  6. E. Clarkson, H. H. Barrett, “Approximations to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
    [CrossRef]

2002 (1)

D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
[CrossRef]

2000 (2)

U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

E. Clarkson, H. H. Barrett, “Approximations to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
[CrossRef]

1999 (1)

1998 (2)

Abbey, C. K.

Barrett, H. H.

Burnashev, M. V.

M. V. Burnashev, “On one useful inequality in the testing of hypotheses,” IEEE Trans. Inf. Theory 44, 1668–1670 (1998).
[CrossRef]

Clarkson, E.

Gay, R.

D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
[CrossRef]

Grenauder, U.

U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

Gronenboom, A.

D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
[CrossRef]

Miller, M. I.

U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

Pastor, D.

D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
[CrossRef]

Shapiro, J.

Srivastava, A.

U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Inf. Theory (3)

M. V. Burnashev, “On one useful inequality in the testing of hypotheses,” IEEE Trans. Inf. Theory 44, 1668–1670 (1998).
[CrossRef]

D. Pastor, R. Gay, A. Gronenboom, “A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise,” IEEE Trans. Inf. Theory 48, 228–238 (2002).
[CrossRef]

U. Grenauder, A. Srivastava, M. I. Miller, “Asymptotic performance analysis of Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

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Figures (3)

Fig. 1
Fig. 1

Exact DΛ, approximate DΛ from relation (9), upper bound from relation (16), and both bounds from relation (26) versus the contrast parameter δ for independent exponential noise with a flat background (M=5).

Fig. 2
Fig. 2

Exact DΛ, approximate DΛ from relation (9), upper bound from relation (16), and both bounds from relation (26) versus the contrast parameter κ for independent Poisson noise with a flat background (Mb=1.5).

Fig. 3
Fig. 3

Exact DΛ, approximate DΛ from relation (9), upper bound from relation (16), and both bounds from relation (26) versus total mean photon count Mb for independent Poisson noise with a flat background (κ=1.2).

Equations (66)

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Λ(g)=pr(g|H1)pr(g|H0).
TPF(Λ0)=Λ0pr(Λ|H1)dΛ
FPF(Λ0)=Λ0pr(Λ|H0)dΛ
AUCΛ=01TPFd(FPF)=0TPF(Λ)pr(Λ|H0)dΛ=0Λpr(Λ|H1)pr(Λ|H0)dΛdΛ=1-00Λpr(Λ|H1)pr(Λ|H0)dΛdΛ.
M0(β)=0Λβpr(Λ|H0)dΛ=Λβ0.
Λ=pr(Λ|H1)pr(Λ|H0),
M0(β)=exp[β(β-1)G(β-12)].
G(0)=-4 lnM012=-4 ln0[pr(Λ|H1)pr(Λ|H0)]1/2dΛ=-4 ln0[pr(g|H1)pr(g|H0)]1/2dg=-4dB,
2(1-AUCΛ)1-erf{[12G(0)]1/2},
2(1-AUCT)1-erf[12SNRT],
DΛ=-ln[2(1-AUCΛ)].
12G(0)DΛ.
minΛ0{FPF(Λ0)+FNF(Λ0)}=1+FPF(1)-TPF(1)=2Pe
1-{1-exp[-12G(0)]}1/22Peexp[-14G(0)]
-ln(2Pe)DΛ-2 ln(2Pe)
14G(0)ΔΛG(0)+2 ln(1+{1-exp[-12G(0)]}1/2).
μ(β)=log M0(β)=β(β-1)G(β-12),
μ(β)+(12-β)μ(β)-12[(μ(β))2+μ(β)]1/2
ln(2Pe)μ(β)
14G(0)DΛ12G(0)+{2G(0)+116[G(0)]2-14G(0)}1/2
2(1-AUCΛ)=00 min(Λ1,Λ2)pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2
2(1-AUCΛ)=00 exp-12lnΛ2Λ1(Λ1Λ2)1/2pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2.
exp-12lnΛ2Λ1(Λ1Λ2)1/2=min(Λ1,Λ2).
00min(Λ1,Λ2)pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2
=00Λ2Λ1pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2+0Λ2Λ2pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2=00Λ2Λ1pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2+00Λ1Λ2pr(Λ1|H0)pr(Λ2|H0)dΛ2dΛ1=200Λ1Λ2pr(Λ1|H0)pr(Λ2|H0)dΛ2dΛ1=200Λpr(Λ1|H0)pr(Λ2|H1)dΛ2dΛ1
=2(1-AUCΛ).
2(1-AUCΛ)=0[FPF(Λ)]2dΛ,
-μ(β)-μ(1-β)DΛ-2μ(β)-(1-2β)μ(β)+12[2μ(β)]1/2.
12G(0)DΛ12G(0)+[G(0)-18G(0)]1/2.
pβ(Λ)=Λβpr(Λ|H0)M0(β),
2(1-AUCΛ)=M0(α)M0(β)00exp-12lnΛ2Λ1×Λ11/2-αΛ21/2-βpα(Λ1)pβ(Λ2)dΛ1dΛ2=M0(α)M0(β)exp-12lnΛ2Λ1Λ11/2-αΛ21/2-βα,β.
ln[2(1-AUCΛ)]μ(α)+μ(β)-12lnΛ2Λ1α,β+12-αln Λ1α,β+12-βln Λ2α,βμ(α)+μ(β)-12ln2Λ2Λ1α,β1/2+12-αln Λ1α,β+12-βln Λ2α,β.
(12-α)ln Λ1α,β=(12-α)μ(α),
(12-β)ln Λ2α,β=(12-β)μ(β),
(ln Λ-ln Λ)2α,β=μ(α)+[μ(α)]2-2μ(α)μ(β)+μ(β)+[μ(β)]2.
ln[2(1-AUCΛ)]2μ(β)+(1-2β)μ(β)-12[2μ(β)]1/2.
ln[2(1-AUCΛ)]2μ(12)-12[2μ(12)]1/2.
ln[2(1-AUCΛ)]-12G(0)-12[4G(0)-12G(0)]1/2.
2(1-AUCΛ)=00minΛ1Λ2,1Λ2pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ200Λ1Λ2βΛ2pr(Λ1|H0)pr(Λ2|H0)dΛ1dΛ2=M0(β)M0(1-β).
ln[2(1-AUCΛ)]μ(β)+μ(1-β).
ln[2(1-AUCΛ)]2μ(12)=-12G(0).
pr(g|H1)=m=1M1bmexp-gmbm,
pr(g|H2)=m=1M1bm+smexp-gmbm+sm.
M0(β)=m=1Mαmβ1-γmβ.
AUCΛ=1+δ2+δMk=0M-1(M+k-1)!k!(M-1)!12+δk.
2(1-AUCΛ)1-erf M2ln(2+δ)24(1+δ)1/2.
M2ln(2+δ)24(1+δ)DΛM2ln(2+δ)24(1+δ)+δ(2M)1/22+δ,
DΛM ln(2+δ)24(1+δ)+2 ln1+1-4(1+δ)(2+δ)2M/21/2.
β˜=121+12M1/2.
pr(g|H1)=m=1Mexp[gm log(bm)-bm]gm!,
pr(g|H2)=m=1Mexp[gm log(bm+sm)-bm-sm]gm!.
M0(β)=exp-βm=1Msmexpm=1Mbm(κmβ-1),
AUCΛ=1-exp[-Mb(1+κ)]k=0l=0k(Mb)k+lκl(1+δkl)k!l!.
2(1-AUCΛ)1-erf[(Mb)1/b(κ-1)].
Mb(κ-1)2DΛMb(κ-1)2+ln κ2(2Mbκ)1/2,
DΛ2Mb(κ-1)2+2 ln(1+{1-exp[-Mb(κ-1)2]}1/2).
(1-2β˜)κβ˜/2=12(2Mb)1/2.
β˜121-12(2Mbκ)1/2.
(1-2β˜)[2μ(β˜)]3/2=μ(β˜).
(1-2β˜)[2μ1(β˜)]3/2=1Mμ1(β˜).
1DΛ12MG1(0)1+2G1(0)MG1(0)-18G1(0)1/2.
limMDΛ12MG1(0)=1
DˆΛ=-ln(1-erf {[12G(0)]1/2}).
DˆΛ12MG1(0)-12MG1(0)lnπ212MG1(0)1/2+12MG1(0)+21/2,
limMDˆΛ12MG1(0)=1.
limMDΛDˆΛ=1.

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