Abstract

Characteristic functionals are one of the main analytical tools used to quantify the statistical properties of random fields and generalized random fields. The viewpoint taken here is that a random field is the correct model for the ensemble of objects being imaged by a given imaging system. In modern digital imaging systems, random fields are not used to model the reconstructed images themselves since these are necessarily finite dimensional. After a brief introduction to the general theory of characteristic functionals, many examples relevant to imaging applications are presented. The propagation of characteristic functionals through both a binned and list-mode imaging system is also discussed. Methods for using characteristic functionals and image data to estimate population parameters and classify populations of objects are given. These methods are based on maximum likelihood and maximum a posteriori techniques in spaces generated by sampling the relevant characteristic functionals through the imaging operator. It is also shown how to calculate a Fisher information matrix in this space. These estimators and classifiers, and the Fisher information matrix, can then be used for image quality assessment of imaging systems.

© 2016 Optical Society of America

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References

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  1. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  2. E. Parzen, Stochastic Processes (SIAM, 1999).
  3. A. N. Shiryaev, Probability (Springer, 2000).
  4. A. Amini and M. Unser, “Sparsity and infinite divisibility,” IEEE Trans. Inf. Theory 60, 2346–2358 (2014).
    [Crossref]
  5. J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
    [Crossref]
  6. W. Rudin, Functional Analysis (McGraw-Hill, 1973).
  7. J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
    [Crossref]
  8. P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
    [Crossref]
  9. J. W. Goodman, Statistical Optics (Wiley, 2015).
  10. P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
    [Crossref]
  11. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).
  12. 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]
  13. M. A. Kupinski, E. Clarkson, J. Hoppin, and H. H. Barrett, “Experimental determination of object statistics from noisy images,” J. Opt. Soc. Am. A 20, 421–429 (2003).
    [Crossref]
  14. 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]
  15. 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]

2014 (2)

A. Amini and M. Unser, “Sparsity and infinite divisibility,” IEEE Trans. Inf. Theory 60, 2346–2358 (2014).
[Crossref]

J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
[Crossref]

2012 (1)

2010 (1)

2007 (1)

P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
[Crossref]

2006 (2)

P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
[Crossref]

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]

2003 (1)

1998 (1)

J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
[Crossref]

Aguilaera, A. M.

P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
[Crossref]

Amini, A.

A. Amini and M. Unser, “Sparsity and infinite divisibility,” IEEE Trans. Inf. Theory 60, 2346–2358 (2014).
[Crossref]

J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
[Crossref]

Barrett, H. H.

Bouzas, P. R.

P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
[Crossref]

P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
[Crossref]

Caucci, L.

Clarkson, E.

Fageot, J.

J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2015).

Hirsch, M. W.

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

Hoppin, J.

Kupinski, M. A.

Moller, J.

J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
[Crossref]

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Ocana, F. M.

P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
[Crossref]

Parzen, E.

E. Parzen, Stochastic Processes (SIAM, 1999).

Rudin, W.

W. Rudin, Functional Analysis (McGraw-Hill, 1973).

Ruiz-Fuentes, N.

P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
[Crossref]

Shen, F.

Shiryaev, A. N.

A. N. Shiryaev, Probability (Springer, 2000).

Smale, S.

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

Syversveen, A. R.

J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
[Crossref]

Unser, M.

J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
[Crossref]

A. Amini and M. Unser, “Sparsity and infinite divisibility,” IEEE Trans. Inf. Theory 60, 2346–2358 (2014).
[Crossref]

Valderamma, M. J.

P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
[Crossref]

Waagepetersen, R. P.

J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
[Crossref]

Appl. Math. Model. (1)

P. R. Bouzas, M. J. Valderamma, and A. M. Aguilaera, “On the characteristic functional of a doubly stochastic Poisson process: application to a narrow-band process,” Appl. Math. Model. 30, 1021–1032 (2006).
[Crossref]

Comput. Statist. (1)

P. R. Bouzas, N. Ruiz-Fuentes, and F. M. Ocana, “Functional approach to the random mean of a compound Cox process,” Comput. Statist. 22, 467–479 (2007).
[Crossref]

IEEE Trans. Inf. Theory (1)

A. Amini and M. Unser, “Sparsity and infinite divisibility,” IEEE Trans. Inf. Theory 60, 2346–2358 (2014).
[Crossref]

J. Fourier Anal. Appl. (1)

J. Fageot, A. Amini, and M. Unser, “On the continuity of characteristic functionals and sparse stochastic modeling,” J. Fourier Anal. Appl. 20, 1179–1211 (2014).
[Crossref]

J. Opt. Soc. Am. A (4)

Scand. J. Stat. (1)

J. Moller, A. R. Syversveen, and R. P. Waagepetersen, “Log Gaussian Cox processes,” Scand. J. Stat. 25, 451–482 (1998).
[Crossref]

Other (6)

W. Rudin, Functional Analysis (McGraw-Hill, 1973).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

E. Parzen, Stochastic Processes (SIAM, 1999).

A. N. Shiryaev, Probability (Springer, 2000).

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

J. W. Goodman, Statistical Optics (Wiley, 2015).

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Equations (80)

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A=n=1An,
Pr(A)=n=1Pr(An).
Pr(E)=nEPn.
Rpr(x)dx=1,
Pr(E)=Epr(x)dx,
aprX(x)dx=Pr[EX(a)]
F(X)=F(x)prX(x)dx.
(ϕ,fω)=Sϕ*(r)fω(r)dqr.
(ϕ,fω)=limΔV0k=1Kϕ*(rk)f(rk,ω)ΔVk,
Xϕ=(ϕ,f)=Sϕ*(r)f(r)dqr.
RNf(x)prX(x)dNx
ψX(ξ)=exp[2πiRe(ξx)],
Ψf(ϕ)=exp[2πiRe(ϕ,f)].
j,k=1JΨf(ϕjϕk)ajak*=|j=1Jajexp[2πiRe(ϕj,f)]|20,
ddτΨf(τϕ)|τ=0=2πi(ϕ,f¯).
d2dτ2Ψf(τϕ)|τ=0=4π2SSϕ(r)R(r,r)ϕ(r)dqrdqr.
d2dτ2Ψf(τϕ)|τ=0=4π2(ϕ,Rϕ),
(2τR22τI2)Ψf(τϕ)|τ=0=4π2(ϕ,Rϕ).
(2τR2+2τI2)Ψf(τϕ)|τ=0=4π2Re(ϕ,R˜ϕ*)
22τRτIΨf(τϕ)|τ=0=4π2Im(ϕ,R˜ϕ*).
Xϕ=Sϕ*(r)f(r,ω)dqr.
g(r)=Sh(r,r)f(r)dqr=Hf(r).
Ψf(ϕ)=n=1NPrnΨfn(ϕ)
Mp(ϕ)=exp[2πi(ϕ,lnp)]=exp[2πi(ϕ,f¯)]exp[2π2(ϕ,Kϕ)]
(ϕ,f)=n=1Nϕ(rn),
Pr(N)=N¯NN!exp(N¯).
N¯=Sf¯(r)dqr.
f(r)=n=1Nh(rrn).
S=j=1JSj,
(ϕ,f)=j=1Jaj(ϕj,fj).
ψI(ξ)=Rn[(2π)NdetK]exp(12fK1f)exp[2πifD(ξ)f]dNf.
ψI(ξ)={det[I+2πiKD(ξ)]}12.
ΨI(ϕ)={det[I+2πiKDϕ]}12.
ϕ(r)=n=1Nξnδ(rrn).
(ϕ,f)=n=1Nξnf(rn),
KDϕβ(r)=n=1NξnK(r,rn)β(rn),
β(r)=n=1NvnK(r,rn).
n=1NξnK(rn,rn)vn=λvn.
ϕ˜(r)=n=1Nϕ(rn)δ(rrn),
ΨI(ϕ)={det[I+2πiKDϕ]}1.
ψI(ξ)={det[I+2πiKD(ξ)]}1.
(ϕ,f)=n=1Nanexp(iφn)ϕ(rn),
ϕ˜(r)=i2π0{J0[2πa|ϕ(r)|]1}pra(a)da,
Ψw(ϕ)=exp{RqΛ[ϕ(r)]dqr},
Λ(ω)=2πiθω2π2σ2ω2+R*[exp(2πiaω)1+2πiaωstep(1|a|)]dV(a).
R*min(1,a2)dV(a)<.
f(r)=n=1NAnexp(2πikn·r).
Ψf(ϕ)=exp{N¯exp[π2σ2|Φ(k)|2]kN¯}.
ψX(ξ)=exp[N¯exp(π2σ2|ξ|2)N¯].
f(r,ω)=nF(n,ω)μn(r),
f(r)=nF(n)μn(r),
(ϕ,fω)=nFω(n)(ϕ,μn),
(ϕ,f)=nF(n)(ϕ,μn).
Af(r,t)=SA(r,r)f(r,t)dqr.
tf(r,t)=Af(r,t)+I(r,t),
f(r,t)=exp(tA)a(r)+0texp[(tt)A]I(r,t)dt,
Af(r,t)=SA(r,r)f(r,t)dqr,
tf(r,t)=Af(r,t)+I(r,t),
f(r,t)=exp(tA)a(r)+0texp[(tt)A]I(r,t)dt,
(ϕ,ft)=Sϕ(r)f(r)dqr,
(ϕ,f)=n=1Nk=1Knϕ(rn+Δrnk).
[Hf]m=Shm(r)f(r)dqr=(hm*,f),
ξ˜m=12πi[exp(2πiξm)1],
g¯(v)=SL(v,r)f(r)dqr,
N¯=Vg¯(v)dpv,
ψ^g(ξ|θ)=1Jj=1Jexp[2πiRe(ξ,gj)].
Zl*Zl=J1Jψg*(ξl|θ)ψg(ξl|θ)+1Jψg(ξlξl|θ)
ZlZl=J1Jψg(ξl|θ)ψg(ξl|θ)+1Jψg(ξl+ξl|θ).
[KZ(θ)]ll=ΔZl*ΔZl=1J[ψg(ξlξl|θ)ψg*(ξl|θ)ψg(ξl|θ)]
[CZ(θ)]ll=ΔZlΔZl=1J[ψg(ξl+ξl|θ)ψg(ξl|θ)ψg(ξl|θ)].
WZ=[ZZ*].
W¯Z(θ)=[Z¯(θ)Z¯*(θ)]
QZ(θ)=[KZ(θ)CZ(θ)CZ(θ)KZ*(θ)].
prZ(Z|θ)=1πLdetQZ(θ)exp{12[WZW¯Z(θ)]QZ1(θ)[WZW¯Z(θ)]}.
θMAP(Z)=argmaxθ[prZ(Z|θ)prθ(θ)].
argminθ{[WZW¯Z(θ)]QZ1(θ)[WZW¯Z(θ)]+lndetQZ(θ)2ln[prθ(θ)]}.
n^MAP(Z)=argmaxn[prZ(Z|n)Prn],
argminn{[WZW¯Z(n)]QZ1(n)[WZW¯Z(n)]+lndetQZ(n)2lnPrn}.
F(θ)=θln[prZ(Z|θ)]θln[prZ(Z|θ)]Z|θ.
[F(θ)]pp=[θpW¯Z(θ)]QZ1(θ)[θpW¯Z(θ)]+12tr{Q1(θ)[θpQ(θ)]Q1(θ)[θpQ(θ)]}.

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