Abstract

The asymptotic form for the likelihood ratio is derived for list-mode data generated by an imaging system viewing a possible signal in a randomly generated background. This calculation provides an approximation to the likelihood ratio that is valid in the limit of large number of list entries, i.e., a large number of photons. These results are then used to derive surrogate figures of merit, quantities that are correlated with ideal-observer performance on detection tasks, as measured by the area under the receiver operating characteristic curve, but are easier to compute. A key component of these derivations is the determination of asymptotic forms for the Fisher information for the signal amplitude in the limit of a large number of counts or a long exposure time. This quantity is useful in its own right as a figure of merit (FOM) for the task of estimating the signal amplitude. The use of the Fisher information in detection tasks is based on the fact that it provides an approximation for ideal-observer detectability when the signal is weak. For both the fixed-count and fixed-time cases, four surrogate figures of merit are derived. Two are based on maximum likelihood reconstructions; one uses the characteristic functional of the random backgrounds. The fourth surrogate FOM is identical in the two cases and involves an integral over attribute space for each of a randomly generated sequence of backgrounds.

© 2012 Optical Society of America

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References

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  1. H. H. Barrett, C. K. Abbey, and 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]
  2. E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A 27, 2313–2326 (2010).
    [CrossRef]
  3. L. Caucci and H. H. Barrett, “Objective assessment of image quality. V. Photon counting detectors and list-mode data,” J. Opt. Soc. Am. A 29, 1003–1016 (2012).
    [CrossRef]
  4. H. H. Barrett, T. White, and L. C. Parra, “List-mode likelihood,” J. Opt. Soc. Am. A 14, 2914–2923 (1997).
    [CrossRef]
  5. J. Shao, Mathematical Statistics (Springer, 1999).
  6. S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).
  7. F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A 23, 2406–2414 (2006).
    [CrossRef]
  8. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
  9. M. A. Kupinski, J. W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
    [CrossRef]
  10. E. Clarkson, M. A. Kupinski, H. H. Barrett, and L. Furenlid, “A task-based approach to adaptive and multimodality imaging,” Proc. IEEE 96, 500–511 (2008).
    [CrossRef]

2012 (1)

2010 (1)

2008 (1)

E. Clarkson, M. A. Kupinski, H. H. Barrett, and L. Furenlid, “A task-based approach to adaptive and multimodality imaging,” Proc. IEEE 96, 500–511 (2008).
[CrossRef]

2006 (1)

2003 (1)

1998 (1)

1997 (1)

Abbey, C. K.

Barrett, H. H.

Caucci, L.

Clarkson, E.

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).

Furenlid, L.

E. Clarkson, M. A. Kupinski, H. H. Barrett, and L. Furenlid, “A task-based approach to adaptive and multimodality imaging,” Proc. IEEE 96, 500–511 (2008).
[CrossRef]

Hoppin, J. W.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

Kupinski, M. A.

E. Clarkson, M. A. Kupinski, H. H. Barrett, and L. Furenlid, “A task-based approach to adaptive and multimodality imaging,” Proc. IEEE 96, 500–511 (2008).
[CrossRef]

M. A. Kupinski, J. W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

Parra, L. C.

Shao, J.

J. Shao, Mathematical Statistics (Springer, 1999).

Shen, F.

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).

White, T.

J. Opt. Soc. Am. A (6)

Proc. IEEE (1)

E. Clarkson, M. A. Kupinski, H. H. Barrett, and L. Furenlid, “A task-based approach to adaptive and multimodality imaging,” Proc. IEEE 96, 500–511 (2008).
[CrossRef]

Other (3)

J. Shao, Mathematical Statistics (Springer, 1999).

S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).

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Equations (121)

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pr(A^|f)=Spr(A^|r)s(r)f(r)drSs(r)f(r)dr=p(A^)fsf,
Lp(A^)dA^=s,
s(r)f(r)Ss(r)f(r)dr
Ss(r)f(r)dr>0.
pr(A^|f)=j=1Jpr(A^j|f),
pr(A^|fa+fn)=pr(A^|fa)
A=Ss(r)f(r)dr>0,
Aa=Ss(r)fa(r)dr>0.
Ss(r)f(r)dr=Ss(r)fa(r)dr=1,
Spr(A^|r)s(r)f(r)dr=Spr(A^|r)s(r)fa(r)dr.
Ss(r)fn(r)dr=0,
Spr(A^|r)s(r)fn(r)dr=0.
f=(AAa)fa+Afn=fa+[(AAa)1]fa+Afn.
Spr(A^|r)s(r)f˜(r)dr=0
pr(A^|f)=j=1Jpr(A^j|f).
pr(A^)=Upr(A^|f)pr(f)df.
pr(A^)=pr(A^|f)f,
pr(A^|f)=exp{j=1Jln[pr(A^j|f)]}.
pr(A^|f)exp{JLpr(A^|fa)ln[pr(A^|f)]dA^}.
|1Jln[pr(A^|f)]Lpr(A^|fa)ln[pr(A^|f)]dA^|<ϵ,
L(ϵ)pr(A^|fa)dA^>1ϵ
D[pr(A^|fa),pr(A^|f)]=Lpr(A^|fa)ln[pr(A^|fa)pr(A^|f)]dA^,
H[pr(A^|fa)]=Lpr(A^|fa)ln[pr(A^|fa)]dA^.
pr(A^|f)exp[JDL(fa,f)]exp[JHL(fa)].
Λ(A^)=Upr(A^|f+Δf)pr(f)dfUpr(A^|f)pr(f)df.
Λ(A^)=Upr(A^|f)pr(fΔf)dfUpr(A^|f)pr(f)df.
Λ(A^)Uexp[JDL(fa,f)]pr(fΔf)dfUexp[JDL(fa,f)]pr(f)df.
Λ(A^)Uexp{J2(ffa)F(fa)(ffa)}pr(fΔf)dfUexp{J2(ffa)F(fa)(ffa)}pr(f)df.
sc(A^|f)=p(A^)p(A^)fssf.
F(f)=p(A^)p(A^)[p(A^)f]2A^|fss(sf)2.
fdF(fa)fd=[p(A^)fdp(A^)fa]2A^|fa(sfdsfa)2.
fdF(fa)fd=[p(A^)fdp(A^)fasfdsfa]2A^|f.
p(A^)fdp(A^)fa=sfdsfa.
p(A^)fdsfd=p(A^)fasfa.
Uexp[J2fdF(fa)fd]pr(fd+fa)dfd.
fd=n=1Nfdnun.
fdF(fa)fd=n=1Rλn|fdn|2=fd1F(fa)fd1,
U1exp[J2fd1F(fa)fd1]U0pr(fd0+fd1+fa)dfd0dfd1.
Λ(A^)U0pr(fd0+faΔf)dfd0U0pr(fd0+fa)dfd0.
Λ(A^)U0pr(fd0Δf)dfd0U0pr(fd0)dfd0.
Λ(A^)U(fa)pr(fΔf)dfU(fa)pr(f)df,
pr(A^|α)=Upr(A^|f+αΔf)pr(f)df.
sc(A^)=ddαln[pr(A^|α)]|α=0.
sc(A^)=UΔfsc(A^|f)pr(f|A^)df,
sc(A^)=UΔf[j=1Jsc(A^j|f)]pr(f|A^)df,
d2(α)=α2[sc(A^)]2A^+.
d2(α)α2[sc(A^)]2A^|fafa.
sc(A^)JUΔfsc(A^|f)A^|fapr(f|A^)df.
d2(α)α2J2[UΔfsc(A^|f)A^|fapr(f|A^)df]2A^|fafa.
d2(α)α2J2[UΔfsc(A^|f)pr(f|A^)dfA^|fa]2A^|fafa.
d2(α)α2J2[Δfsc[A^|f^ML(A)]A^|fa]2A^|fafa.
Δfsc(A^|f)A^|fa=ΔffLln[pr(A^|f)]pr(A^|fa)dA^.
Δfsc(A^|f)A^|fa12Δff[(ffa)F(fa)(ffa)].
Δfsc(A^|f)A^|faΔfF(fa)(ffa).
d2(α)α2J2ΔfF(fa)[f^PM(A^)fa][f^PM(A^)fa]A^|faF(fa)Δffa.
d2(α)α2J2ΔfF(fa)[f^ML(A^)fa][f^ML(A^)fa]A^|faF(fa)Δffa.
F(fa)[f^PM(A^)fa]F(fa)U(ffa)exp[J2(ffa)F(fa)(ffa)]pr(f)dfUexp[J2(ffa)F(fa)(ffa)]pr(f)df.
d2(α)α2J2{UΔfF(fa)fdexp[J2fdF(fa)fd]pr(fd+fa)dfdUexp[J2fd4F(fa)fd]pr(fd+fa)dfd}2fa.
F(fa)[f^PM(A^)fa]F(fa)U1fd1exp{J2fd1F(fa)fd1}U0pr(fd0+fd1+fa)dfd0dfd1U1exp{J2fd1F(fa)fd1}U0pr(fd0+fd1+fa)dfd0dfd1.
f^PM(A^)faU1exp{J2fd1F(fa)fd1}U0(fd0+fd1)pr(fd0+fd1+fa)dfd0dfd1U1exp{J2fd1F(fa)fd1}U0pr(fd0+fd1+fa)dfd0dfd1.
f^PM(A^)faU0fd0pr(fd0+fa)dfd0U0pr(fd0+fa)dfd0.
f^PM(A^)U(fa)fpr(f)dfU(fa)pr(f)df.
pr(A^|f)exp{J2(ffa)F(fa)(ffa)},
j=1Jfln[pr(A^j|f^ML)]=0.
j=1Jfln[pr(A^j|fa)]+j=1Jffln[pr(A^j|fβ)](f^MLfa)=0,
y=j=1Jfln[pr(A^j|fa)].
1Jj=1Jffln[pr(A^j|fβ)]F(fβ)F(fa)
d2(α)α2JΔfF(fa)Δffa.
d2(α)α2J{[p(A^)Δfp(A^)fa]2A^|fafa(sΔfsfa)2fa}.
d2(α)α2J{1sfaL[p(A^)Δf]2p(A^)fadA^fa(sΔfsfa)2fa}.
J¯(f)=τSs(r)f(r)dr=τsf.
pr(G|f)=[J¯(f)]JJ!exp[J¯(f)]j=1Jpr(A^j|f)=Pr(J|f)pr(A^|f,J).
pr(G)=Upr(G|f)pr(f)df.
Λ(G)UPr(J|f)exp{JDKL[pr(A^|fa),pr(A^|f)]}pr(fΔf)dfUPr(J|f)exp{JDKL[pr(A^|fa),pr(A^|f)]}pr(f)df.
Λ(G)UPr(J|f)exp{J2(ffa)F(fa)(ffa)}pr(fΔf)dfUPr(J|f)exp{J2(ffa)F(fa)(ffa)}pr(f)df.
UPr(J|fd+fa)exp[J2fdF(fa)fd]pr(fd+fa)dfd.
U1exp[J2fd1F(fa)fd1]U0Pr(J|fd0+fd1+fa)pr(fd0+fd1+fa)dfd0dfd1.
Λ(A^)U0Pr(J|fd0+faΔf)pr(fd0+faΔf)dfd0U0Pr(J|fd0+fa)pr(fd0+fa)dfd0.
Λ(A^)U0Pr(J|fd0Δf)pr(fd0Δf)dfd0U0Pr(J|fd0)pr(fd0)dfd0.
Λ(A^)U(fa)Pr(J|fΔf)pr(fΔf)dfU(fa)Pr(J|f)pr(f)df.
Pr(J|f)1J!exp{J¯(fa)+Jln[J¯(fa)]+[JJ¯(fa)J¯(fa)]J¯(ffa)J2J¯2(fa)[J¯(ffa)]2}.
Fs(fa)=F(fa)+1(sfa)2ss=p(A^)p(A^)[p(A^)fa]2A^|fa.
Λ(G)Uexp{J2fdFs(fa)fd+[JJ¯(fa)J¯(fa)]J¯(fd)}pr(fd+faΔf)dfdUexp[J2fdFs(fa)fd+[JJ¯(fa)J¯(fa)]J¯(fd)]pr(fd+fa)dfd.
[p(A^)fnp(A^)fa2]2A^|fa=0,
U˜1exp{J2f˜d1Fs(fa)f˜d1+[JJ¯(fa)J¯(fa)]J¯(f˜d1)}U˜0pr(f˜d0+f˜d1+fa)df˜d0df˜d1.
Λ(A^)U˜0pr(f˜d0+faΔf)df˜d0U˜0pr(f˜d0+fa)df˜d0
Λ(A^)U˜(fa)pr(fΔf)dfU˜(fa)pr(f)df.
u¯(A^)=Spr(A^|r)s(r)fa(r)dr
pr(G|α)=Upr(G|f+αΔf)pr(f)df,
sc(G)=ddtln[pr(G|α)]|α=0,
sc(G)=UΔfsc(G|f)pr(f|G)df.
sc(G)=UΔf[j=1Jsc(A^j|f)+JτJ¯(f)sτs]pr(f|G)df.
sc(G)=UΔf[j=1Jp(A^j)p(A^j)f]pr(f|G)dfτΔfs.
d2(α)α2[sc(G)]2G|fafa.
sc(G)JUp(A^)Δfp(A^)fA^|fapr(f|G)dfτΔfs.
sc(G)JU[p(A^)Δfp(A^)f]pr(f|G)dfA^|faτΔfs.
sc(G)Jp(A^)Δfp(A^)fML(G)A^|faτΔfs
sc(G)JUΔfsc(A^|f)A^|fapr(f|G)df+[JUpr(f|G)sfdfτ]sΔf.
sc(G)JΔfF(fa)[f^PM(G)fa]+[τ^PM(G)τ]Δfs.
τ^PM(G)=UJsfpr(f|G)df.
τ=J¯(fa)sfa,
τ^PM(G)JsfPM(G).
sc(G)JΔfF(fa)[f^ML(G)fa]+[τ^ML(G)τ]Δfs,
τ^ML(G)=JsfML(G).
[τ^PM(G)τ]JsfPM(G)τJsfaτJs(sfa)2[f^PM(G)fa].
sc(G)JΔfFs(fa)[f^PM(G)fa]+(Jsfaτ)Δfs.
Fs(fa)[f^PM(G)fa]Fs(fa)U(ffa)exp{J2(ffa)Fs(fa)(ffa)+[JJ¯(fa)J¯(fa)]J¯(ffa)}pr(f)dfUexp[J2(ffa)Fs(fa)(ffa)+[JJ¯(fa)J¯(fa)]J¯(ffa)]pr(f)df.
Fs(fa)[f^PM(G)fa]Fs(fa)Ufdexp{J2fdFs(fa)fd+[JJ¯(fa)J¯(fa)]J¯(fd)}pr(fd+fa)dfdUexp[J2fd4Fs(fa)fd+[JJ¯(fa)J¯(fa)]J¯(fd)]pr(fd+fa)dfd.
Fs(fa)[f^PM(G)fa]Fs(fa)U˜1fd1exp{J2fd1Fs(fa)fd1+[JJ¯(fa)J¯(fa)]J¯(fd1)}U˜0pr(fd0+fd1+fa)dfd0dfd1U˜1exp{J2fd1Fs(fa)fd1+[JJ¯(fa)J¯(fa)]J¯(fd1)}U˜0pr(fd0+fd1+fa)dfd0dfd1.
f^PM(G)faU˜0fd0pr(fd0+fa)dfd0U˜0pr(fd0+fa)dfd0.
f^PM(A^)U˜(fa)fpr(f)dfU˜(fa)pr(f)df.
pr(G|f)1J!exp{J¯(fa)+Jln[J¯(fa)]J2(ffa)Fs(fa)(ffa)+[JJ¯(fa)J¯(fa)]J¯(ffa)}.
(Jsf^MLτ)s+j=1Jfln[pr(A^j|f^ML)]=0.
(Jsfaτ)s+j=1Jfln[pr(A^j|fa)]+{J(sfβ)2ss+j=1Jffln[pr(A^j|fβ)]}(f^MLfa)0,
y=j=1Jfln[pr(A^j|fa)].
1J{J(sfa)2ssj=1Jffln[pr(A^j|fβ)]}Fs(fβ)Fs(fa).
yJFs(fa)(f^MLfa)(Jsfaτ),
d2(α)α2τ(sfa)(ΔfF(fa)Δf)fa.
d2(α)α2τ(sfa){[p(A^)Δfp(A^)fa]2A^|fa(sΔfsfa)2}fa.
d2(α)α2τL[p(A^)Δf]2p(A^)fadA^(sΔf)2sfafa.
SFOM4=L[p(A^)Δf]2p(A^)fadA^(sΔf)2sfafa.

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