Abstract

Matched filtering followed by a minimum Euclidean distance projection onto realizable filter values was previously shown to optimize the signal-to-noise ratio for single training images in optical correlation pattern recognition. The algorithm is now shown to solve the combination of (1) standard statistical pattern-recognition metrics with multiple training images, (2) additive input noise of known power spectral density and also additive detection noise that is irreducible by the filter, (3) the building of the filter on arbitrary subsets of the complex unit disk, and (4) the use of observable correlator outputs only. The criteria include the Fisher ratio, the Bayes error and Bayes cost, the Chernoff and Bhattacharyya bounds, the population entropy and expected information, versions of signal-to-noise ratio that use other than second power in their norm, and the area under the receiver operating characteristic curve. Different criteria are optimized by different complex scalar weights.

© 2001 Optical Society of America

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References

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  1. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  2. B. V. K. Vijaya Kumar, R. D. Juday, P. K. Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
    [CrossRef]
  3. B. F. Draayer, G. W. Carhart, M. K. Giles, “Optimum classification of correlation-plane data by Bayesian decision theory,” Appl. Opt. 33, 3034–3049 (1994).
    [CrossRef] [PubMed]
  4. See any statistical pattern recognition text. The example followed in this paper is from Keinosuke Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).
  5. R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
    [CrossRef]
  6. J. W. Goodman, Statistical Optics (Wiley, New York, 2000).
  7. B. Javidi, F. Parchekani, G. Zhang, “Minimum-mean-square-error filters for detecting a noisy target in background noise,” Appl. Opt. 35, 6964–6975 (1996).
    [CrossRef] [PubMed]
  8. C. Soutar, S. E. Monroe, J. Knopp, “Complex characterization of the Epson liquid crystal television,” Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 269–277 (1993).
    [CrossRef]
  9. L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
    [CrossRef] [PubMed]
  10. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  11. E.g., the function BESSELI(0,x,1) in Matlab Version 5 (The Mathworks, Inc., Natick MA); the function bessi0(x) from Numerical Recipes in C, 2nd ed. [Cambridge U. Press, New York (1992)]; Section 9.8 of Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., National Bureau of Standards publication AMS 55, U.S. Government Printing Office, Washington, D.C. (1964).
  12. R. D. Juday, “Generalized Rayleigh quotient approach to filter optimization,” J. Opt. Soc. Am. A 15, 777–790 (1998).
    [CrossRef]
  13. R. D. Juday, “A philosophy for optical filter optimization,” in 1999 Euro-American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1999), pp. 227–240.
  14. N. Towghi, B. Javidi, “lp-norm optimum filters for image recognition. Part I. Algorithms,” J. Opt. Soc. Am. A 16, 1928–1935 (1999).
    [CrossRef]
  15. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  16. M. Montes-Usátegui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant function filters,” Appl. Opt. 34, 3904–3914 (1995).
    [CrossRef]
  17. R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 30, 520–522 (1991).
    [CrossRef] [PubMed]
  18. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  19. B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
    [CrossRef]
  20. P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
    [CrossRef]
  21. J. M. Rollins, R. D. Juday, S. E. Monroe, “Laboratory results for the optimized Fisher ratio,” in Optical Pattern Recognition XI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE4043, 182–191 (2000).
    [CrossRef]

1999 (3)

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

N. Towghi, B. Javidi, “lp-norm optimum filters for image recognition. Part I. Algorithms,” J. Opt. Soc. Am. A 16, 1928–1935 (1999).
[CrossRef]

1998 (1)

1996 (1)

1995 (2)

1994 (2)

1993 (1)

1992 (2)

1991 (1)

1989 (1)

1980 (1)

Barton, R. S.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

Campos, J.

Carhart, G. W.

Carlson, D. W.

Casasent, D.

Draayer, B. F.

Ennis, D. J.

Fiebig, M.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Fukunaga, Keinosuke

See any statistical pattern recognition text. The example followed in this paper is from Keinosuke Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).

Giles, M. K.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

Hamlyn, G.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Hester, C. F.

Jared, D. A.

Javidi, B.

Juday, R. D.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

R. D. Juday, “Generalized Rayleigh quotient approach to filter optimization,” J. Opt. Soc. Am. A 15, 777–790 (1998).
[CrossRef]

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, R. D. Juday, P. K. Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
[CrossRef]

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 30, 520–522 (1991).
[CrossRef] [PubMed]

R. D. Juday, “A philosophy for optical filter optimization,” in 1999 Euro-American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1999), pp. 227–240.

J. M. Rollins, R. D. Juday, S. E. Monroe, “Laboratory results for the optimized Fisher ratio,” in Optical Pattern Recognition XI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE4043, 182–191 (2000).
[CrossRef]

Juvells, I.

Knopp, J.

C. Soutar, S. E. Monroe, J. Knopp, “Complex characterization of the Epson liquid crystal television,” Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 269–277 (1993).
[CrossRef]

Mahalanobis, A.

Miller, P. C.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Monroe, S. E.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

C. Soutar, S. E. Monroe, J. Knopp, “Complex characterization of the Epson liquid crystal television,” Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 269–277 (1993).
[CrossRef]

J. M. Rollins, R. D. Juday, S. E. Monroe, “Laboratory results for the optimized Fisher ratio,” in Optical Pattern Recognition XI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE4043, 182–191 (2000).
[CrossRef]

Montes-Usátegui, M.

Neto, L. G.

Parchekani, F.

Rajan, P. K.

Roberge, D.

Rollins, J. M.

J. M. Rollins, R. D. Juday, S. E. Monroe, “Laboratory results for the optimized Fisher ratio,” in Optical Pattern Recognition XI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE4043, 182–191 (2000).
[CrossRef]

Royce, M.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Sheng, Y.

Soutar, C.

C. Soutar, S. E. Monroe, J. Knopp, “Complex characterization of the Epson liquid crystal television,” Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 269–277 (1993).
[CrossRef]

Towghi, N.

Vijaya Kumar, B. V. K.

Virgo, P.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Zhang, G.

Appl. Opt. (9)

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with MEDOF, coupled modulators, and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, G. Hamlyn, “Evaluation of an optical correlator automatic target recognition system for acquisition and tracking in densely cluttered natural scenes,” Opt. Eng. 38, 1814–1825 (1999).
[CrossRef]

Opt. Lett. (1)

Other (6)

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

C. Soutar, S. E. Monroe, J. Knopp, “Complex characterization of the Epson liquid crystal television,” Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 269–277 (1993).
[CrossRef]

E.g., the function BESSELI(0,x,1) in Matlab Version 5 (The Mathworks, Inc., Natick MA); the function bessi0(x) from Numerical Recipes in C, 2nd ed. [Cambridge U. Press, New York (1992)]; Section 9.8 of Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., National Bureau of Standards publication AMS 55, U.S. Government Printing Office, Washington, D.C. (1964).

R. D. Juday, “A philosophy for optical filter optimization,” in 1999 Euro-American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1999), pp. 227–240.

J. M. Rollins, R. D. Juday, S. E. Monroe, “Laboratory results for the optimized Fisher ratio,” in Optical Pattern Recognition XI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE4043, 182–191 (2000).
[CrossRef]

See any statistical pattern recognition text. The example followed in this paper is from Keinosuke Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).

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Figures (6)

Fig. 1
Fig. 1

The influence of the filtered input clutter noise is modeled as a Gaussian, distributed in the complex plane of central correlation value C0 about B2, the squared expectation value of C0. When intensity-detected, the complex values between r and r+dr fall between intensity values I and I+dI, where I=r2. The resulting Bessel function description of the intensity distribution is derived in the text. If the annuli have equal areas (like those of a Fresnel zone plate) the intensity bins are equally spaced.

Fig. 2
Fig. 2

Physical measurement for variance of correlation-plane intensity. The plotted trace is from an area well away from the correlation peak of a high-energy object. The mean intensity does not affect the Fisher ratio—only the intensity variance does.

Fig. 3
Fig. 3

For the filter to be optimal, simultaneously at every frequency m the realizable filter value (solid dot) is chosen to be at the best possible location in the contour plot. There the partial derivative of metric with respect to allowable change in filter is zero. The contours are shown in text to be circular, so the filter point is chosen closest to the ideal value at the center. (The isocriterion circles are drawn with constant difference in radii, but a section through the pattern would show their criterion values to fall off quadratically from the center.) The realizable filter locus is fixed, but the metric contours move as the search parameters are varied.

Fig. 4
Fig. 4

Two-class Bayes decision regions. A measurement u is assigned to the class having the greater a posteriori likelihood. In the two-class case the a posteriori likelihoods sum to unity. Bayes error is the probability of making an incorrect assignment; it is the shaded area. One of the regions is examined in greater detail in Fig. 5.

Fig. 5
Fig. 5

Detail of Fig. 4 when the mth frequency’s filter magnitude changes by dM. For this segment, assigned to class Λ since it has the higher a posteriori likelihood, the Bayes error (the area under the lesser likelihood’s curve) increases by the vertically hatched area. The stippled area is simply exchanged between neighboring segments with no net change in Bayes error. To first order in dM, then, the change in Bayes error is the integral over this segment of PΨ(pΨ/Mm)dMmdu. Equation (28) is the result.

Fig. 6
Fig. 6

Receiver operating characteristic (ROC) curve and its behavior as a function of how the measurement densities overlap. A decision threshold on the u axis determines the detection and false alarm rates and parametrically traces out the ROC curve. As the densities separate more and more, the curve moves toward the upper left corner, increasing the area underneath the curve. Maximizing the area helps in the trade-off between missed targets and false alarms.

Tables (1)

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Table 1 Behavior of J(Hm ) and Optimal Selection of Hm

Equations (86)

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D=B exp(jβ)=k HkSk,
ηmag2=k|Hk|2Nk=kMk2Nk.
un=RkHkSnk2nC,
R=ubwbBbwb2.
σCT2=σC2+σn2C+σId2.
ρD(r exp iξ)=1πηmag2exp-r2-2rB cos ξ+B2ηmag2,
ρI(I)dI=dμ=ξ=02πρD(r exp jξ)rdrdξ.
ρI(I)=12πηmag2exp-I+B2ηmag2 × ξ=02πexp+2BIηmag2cos ξdξ.
ρI(u/R)=1ηmag2exp-(B-u/R)2ηmag2×exp-2Bu/Rηmag2I02Bu/Rηmag2.
I0(x)=12π02πexp(x cos θ)dθ.
ρu(u)=ρIuR exp-u22δ2/2πδ2.
ρu(u)=ρIuRfdet(u),
J=(IΛ-IΨ)2σΛT2+σΨT2  NumDen,
σC2=1NC-1iC(Ii-IC)2=1NC-1iCIi-1NClCIl2,
σn2C=(ηmag2)2+2ηmag2B2C.
σId2=σoptical2R2,
Den2JMm=DenNumMm-NumDenMm= 2Den(IΛ-IΨ) Mm (IΛ-IΨ)
-NumMm (σΛT2+σΨT2),
ηmag2Mm=2MmNm.
IMm=2BAmcos[θm-(β-ϕm)],
σCT2Mm=σC2Mm+σn2CMm,
σC2Mm=2NC-1iC (Ii-IC)IiMm-IMmC
=4NC-1iC(Ii-IC)-{BiAim×cos[θm-(βi-ϕim)]-1NCkC BkAkmcos[θk-(βk-ϕkm)]}.
σn2CMm=4MmNm,C(ηmag2+IC)+4ηmag2BAmcos[θm-(β-ϕm)]C.
Den24mJ=iΛ Sim*+ Di(IΛ-IΨ)DenNΛ - NumNΛ-1+ DikΛNum(Ik-IΛ)NΛ(NΛ-1)- Diηmag2NumNΛ+iΨ Sim*-Di(IΛ-IΨ)DenNΨ - NumNΨ-1+ DikΨNum(Ik-IΨ)NΨ(NΨ-1)-Diηmag2NumNΨ- MmNm,Λ[Num(2ηmag2+IΛ)]- MmNm,Ψ[Num(2ηmag2+IΨ)].
mJ=i(ΛΨ)TiSim*-Mm{Tnoise,ΛNmΛ+Tnoise,ΨNmΨ}.
HmI=iΛΨ TiSim*iΛΨ Tnoise,iNmi
EBayes=allminuC{Λ,Ψ}{PCpC(u)}du=PΛRΨpΛ(u)du+PΨRΛpΨ(u)du
IC=BC2=kHkSCk2,
EBayesMm=PΛRΨpΛ(u)Mmdu+PΨRΛpΨ(u)Mmdu.
ρI(I)=12πηmag2exp-I+B2ηmag2×ξ=02πexp2BIηmag2cos ξdξ.
ρI(I)=F1(I)F2(I)F3(I),
F1(I)=12πηmag2,
F2(I)=exp-I+B2ηmag2,
F3(I)=ξ=02πexp2BIηmag2cos ξdξ.
ρu(u)=ρIuRfdet(u).
ρu(u)Mm=ρIuRMm fdet(u).
ρI(u/R)Mm=F1F2F3Mm+F2F3F1Mm+F3F1F2Mm,
F1Mm=-2MmNm(2πηmag2)2,
F2Mm=MmNm2(ηmag2)2uR+B2exp-u/R+B2ηmag2-Amcos[θm-(β-ϕm)]×2Bηmag2exp-u/R+B2ηmag2,
F3Mm=Amcos[θm-(β-ϕm)] 2u/Rηmag2×ξ=02πcos ξ exp2Bu/Rηmag2cos ξdξ-MmNm4Bu/R(ηmag2)2×ξ=02πcos ξ exp2Bu/Rηmag2cos ξdξ,
F1Mm-MmNmg1,
F2Mm MmNmg2-Amcos[θm-(β-ϕm)]g3,
F3Mm -MmNmg4+Amcos[θm-(β-ϕm)]g5
ρI(u/R)Mm=Amcos[θm-(β-ϕm)](-F1F3g3+F1F2g5)-MmNm(F1F2g4+F2F3g1),
ρIuRMm=Amcos[θm-(β-ϕm)]F4(u)-MmNmF5(u).
F6(u)  F4(u)fdet(u),
F7(u)  F5(u)fdet(u),
ρu(u)Mm=Amcos[θm-(β-ϕm)]F6(u)-MmNmF7(u).
FBayesMm=Amcos[θm-(β-ϕm)]×PΛRΨF6,Λ(u)du+PΨRΛF6,Ψ(u)du- MmNΛmPΛRΨF7,Λ(u)du+NΨmPΨ×RΛF7,Ψ(u)du.
mEBayes=TΨSΨm*+TΛSΛm*-Mm(TΨ,noiseNΨm+TΛ,noiseNΛm).
min(a, b)asb1-s,0s1.
EBayesEub(s)=PΛsPΨ1-sall upΛs(u)pΨ1-s(u)du.
Eub(s)Mm=Amcos[θm-(β-ϕm)]PΛsPΨ1-s×all u[spΨ1-spΛs-1F6,Λ+(1-s)pΛspΨs-2F6,Ψ]du-MmNmPΛsPΨ1-sall u[spΨ1-spΛs-1F7,Λ+(1-s)pΛspΨs-2F7,Ψ]du,
AROC=ν=-u=νpΛ(u)dupΨ(ν)dν,
AROCMm
=ν=0u=νpΛ(u) pΨ(ν)Mm+pΨ(u) pΛ(ν)Mmdudν.
AROCMm=ν=0u=νpΛ(u){Amcos[θm-(β-ϕm)]×F6,Ψ(ν)-MmNmF7,Ψ(ν)}dudν+ν=0u=νpΨ(u){Amcos[θm-(β-ϕm)]×F6,Λ(ν)-MmNmF7,Λ(ν)}dudν
=Amcos[θm-(β-ϕm)]×ν=0u=ν[pΛ(u)F6,Ψ(ν)+pΨ(ν)F6,Λ(u)]dudν-MmNmν=0u=ν[pΛ(u)F7,Ψ(ν)+pΨ(ν)F7,Λ(u)]dudν,
D(pΛpΨ)=allupΛ(u)logpΛ(u)pΨ(u)du,
DΛ=all upΛ(u)logPΛpΛ(u)PΨpΨ(u)du.
DΛMm=all upΛ(u)MmlogPΛpΛ(u)PΨpΨ(u)+pΛ(u) PΨpΨ(u)PΛpΛ(u)MmPΛpΛ(u)PΨpΨ(u)du,
DΛMm=Amcos[θm-(β-ϕm)]all uF6,Λ(u)×1+logPΛpΛ(u)PΨpΨ(u)-F6,Ψ(u) pΛ(u)pΨ(u)du-MmNmall uF7,Λ(u)×1+logPΛpΛ(u)PΨpΨ(u)-F7,Ψ (u) pΛ(u)pΨ(u)du,
|z|p=i=0J-1|zi|p1/p.
σΛT2=σΛ2+σn2Λ+σId2
σΛTp=(σΛT2)p/2=(σΛ2+σn2Λ+σId2)p/2.
Jlp=RB2Λ(RσΛT2)p/2.
Jlp=RB2[R(σn2+σId2)]p/2,
σn2=(ηmag2)2+2ηmag2B2.
Den2JlpMm=Amcos[θm-(β-ϕm)]{[R(σn2+σId2)]p/2×2RB+2R2B3η2[R(σn2+σId2)](p-2)/2}+MmNm{2R2B2(η2+B2)p×[R(σn2+σId2)](p-2)/2},
ρCart(z)=12πt2exp-x2+y22t2,
ρpolar(z)=12πt2exp-r22t2,
Mn  0xnexp-x22t2dx
M1=t2,M3=2t4,M5=8t6.
ηmag2=r2=12πt202πdθ0r2exp-r22t2rdr
=1t2 M3=2t2,or
t2=ηmag22.
I(z)=z*z=x2+y2.
σn2=I2-I2.
ρ(z)=1πηmag2exp-(x-B)2+y2ηmag2.
I=1πηmag2--[(x+B)2+y2]×exp-x2+y2ηmag2dxdy.
πηmag2I=02π0(r cos θ+B)2exp-r2ηmag2dθrdr+02π0(r sin θ)2exp-r2ηmag2dθrdr.
I=B2+ηmag2.
I2=2(ηmag2)2+4ηmag2B2+B4.
σn2=I2-I2=(ηmag2)2+2B2ηmag2
σn2C=(ηmag2)2+2ηmag2B2C.

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