Abstract

In standard intensity imaging, apart from the availability of prior information about an image, the resolution is limited by the width of the aperture. The region of support of the autocorrelation of the pupil function is the measured spatial-frequency bandwidth of the imaging system, and thus the size of the pupil function determines the resolution limit. This relationship between the size of the pupil function and the resolution limit is generally taken to describe a fundamental limit for resolution. Contrary to conventional wisdom, the resolution is actually limited only for fields with zero higher-order cumulants. For fields with nonvanishing higher-order cumulants, higher resolution can be obtained by integrating higher powers of instantaneous intensity in the image plane and combining these images appropriately. The result is that resolution is limited only by the time required for the integral of the higher power of intensity to approximate the expected value. These claims are demonstrated, and the variance of the integrated intensity-squared image as a function of the temporal spectrum and integration time is analyzed. Simulations are provided that show imaging of spatial-frequency information outside the support of the pupil function autocorrelation for non-Gaussian fields.

© 1999 Optical Society of America

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References

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  1. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  2. M. J. Beran, J. G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  3. M. C. Dogan, J. M. Mendel, “Applications of cumulants to array processing—Part I: aperture extension and array calibration,” IEEE Trans. Signal Process. 43, 1200–1216 (1995).
    [CrossRef]
  4. S. J. Reeves, “Superresolution imaging of non-Gaussian emitters,” Signal Process. (to be published).
  5. J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
    [CrossRef]
  6. K. F. Cheung, “A multidimensional extension of Papoulis’ generalized sampling expansion with the application in minimum density sampling,” in Advanced Topics in Shannon Samping and Interpolation Theory, R.J. M., ed. (Springer-Verlag, New York, 1993), pp. 85–119.
  7. B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, Mass., 1954).

1995

M. C. Dogan, J. M. Mendel, “Applications of cumulants to array processing—Part I: aperture extension and array calibration,” IEEE Trans. Signal Process. 43, 1200–1216 (1995).
[CrossRef]

1991

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

Beran, M. J.

M. J. Beran, J. G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Cheung, K. F.

K. F. Cheung, “A multidimensional extension of Papoulis’ generalized sampling expansion with the application in minimum density sampling,” in Advanced Topics in Shannon Samping and Interpolation Theory, R.J. M., ed. (Springer-Verlag, New York, 1993), pp. 85–119.

Dogan, M. C.

M. C. Dogan, J. M. Mendel, “Applications of cumulants to array processing—Part I: aperture extension and array calibration,” IEEE Trans. Signal Process. 43, 1200–1216 (1995).
[CrossRef]

Gnedenko, B. V.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, Mass., 1954).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Kolmogorov, A. N.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, Mass., 1954).

Mendel, J. M.

M. C. Dogan, J. M. Mendel, “Applications of cumulants to array processing—Part I: aperture extension and array calibration,” IEEE Trans. Signal Process. 43, 1200–1216 (1995).
[CrossRef]

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

Parrent, J. G. B.

M. J. Beran, J. G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Reeves, S. J.

S. J. Reeves, “Superresolution imaging of non-Gaussian emitters,” Signal Process. (to be published).

IEEE Trans. Signal Process.

M. C. Dogan, J. M. Mendel, “Applications of cumulants to array processing—Part I: aperture extension and array calibration,” IEEE Trans. Signal Process. 43, 1200–1216 (1995).
[CrossRef]

Proc. IEEE

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

Other

K. F. Cheung, “A multidimensional extension of Papoulis’ generalized sampling expansion with the application in minimum density sampling,” in Advanced Topics in Shannon Samping and Interpolation Theory, R.J. M., ed. (Springer-Verlag, New York, 1993), pp. 85–119.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, Mass., 1954).

S. J. Reeves, “Superresolution imaging of non-Gaussian emitters,” Signal Process. (to be published).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. J. Beran, J. G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

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Figures (6)

Fig. 1
Fig. 1

Magnitude response (in dB) of 2nd-order (intensity) imaging with circular aperture.

Fig. 2
Fig. 2

Magnitude response (in dB) of 4th-order imaging with circular aperture.

Fig. 3
Fig. 3

Magnitude response (in dB) of 6th-order imaging with circular aperture.

Fig. 4
Fig. 4

Partitioning of image in the frequency domain. Dotted line indicates outer boundary of 4th order frequency response.

Fig. 5
Fig. 5

Building images: (a) original intensity image, (b) integrated intensity, (c) integrated squared intensity, (d) expected value of intensity image, (e) expected value of intensity-squared image.

Fig. 6
Fig. 6

(a) restored image, (b) original image spectrum, (c) restored image spectrum, (d) 2nd-order spectral region of support, (e) 4th-order spectral region of support.

Tables (1)

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Table 1 Number of Indistinguishable Occurrences of Each Case

Equations (163)

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y(p, t)=h(τ)x˜(p, t-τ)dr,
x˜(p, t)=x(p, t)exp{j[(2π/λ)(ct+px)]}.
z(p, t)=g(q)y(p-q, t)dq,
Ii(p)=E[|z(p, t)|i]=limT 1T0T|z(p, t)|idt
I2(p)=E[|z(p, t)|2]=limT 1T0T|z(p, t)|2dt=γ2|g(p)|2*c2x(p)=γ2|g(p)|2*m2x(p),
mix(p)=E{|x(p, t)|i},
cix(p)=cum[x(p, t), x*(p, t), , x(p, t), x*(p, t)n elements].
c2x(p)=m2x(p),
c4x(p)=m4x(p)-2[m2x(p)]2,
c6x(p)=m6x(p)-9m4x(p)m2x(p)+12[m2x(p)]3,
c8x(p)=m8x(p)-16m6x(p)m2x(p)-18[m4x(p)]2+144m4x(p)[m2x(p)]2-144[m2x(p)]4.
I4(p)=E[|z(p, t)|4]=limT 1T0T|z(p, t)|4dt=2[γ2|g(p)|2*c2x(p)]2+γ4|g(p)|4*c4x(p)=2[I2(p)]2+γ4|g(p)|4*c4x(p),
I6(p)=E[|z(p, t)|6]=limT 1T0T|z(p, t)|6dt=6[γ2|g(p)|2*c2x(p)]3+[γ2|g(p)|2*c2x(p)][γ4|g(p)|4*c4x(p)]+γ6|g(p)|6*c6x(p)=6[I2(p)]3+9I2(p)[I4(p)-2I2(p)]+γ6|g(p)|6*c6x(p).
J2(p)=I2(p)
J4(p)=I4(p)-2[I2(p)]2
J6(p)=I6(p)-9I4(p)I2(p)+12[I2(p)]3.
cnx(p)=αn[m2x(p)]n/2
c4x(p)=m4x(p)-3[m2x(p)]2
=-2A4(p).
f(x)x0k+kx0k-1(x-x0).
Jn(p)=γn|g(p)|n*cnx(p)=γnαn|g(p)|n*[m2x(p)]n/2γnαn|g(p)|n*{(m2x¯)n/2+(n/2)(m2x¯)(n/2)-1[m2x(p)-(m2x¯)n/2]}=βn|g(p)|n*m2x(p)-Kn,
[m2x(p)]2=[m2ix(p)]2+2m2ix(p)m2ox(p)+[m2ox(p)]2[m2ix(p)]2+2m2ix(p)m2ox(p)
J4(p)-γ4α4|g(p)|4*[m2ix(p)]2=2γ4α4|g(p)|4*[m2ix(p)m2ox(p)].
ζ1, ζ2,, ζn,
σnfn(σnx)12πexp-x22(n)
σ2=x2f(x)dx.
F{|g(p)|n}λ exp-8ω25nωr2
exp-8ωmax25nωr2=0.1,
ωmax=1.2ωrn.
var1T0T|z(p, t)|2dt
=E1T0T|z(p, t)|2dt2-E1T0T|z(p, t)|2dt2
=1T20T0Th(τ1)h*(τ2)g(q1)g*(q2)×h(α1)h*(α2)g(r1)g*(r2)
×E{x(p-q1, t1-τ1)x*(p-q2, t1-τ2)
×x(p-q1, t2-α1)x*(p-q2, t2-α2)}
×dτ1dτ2dα1dα2d1d2dr1dr2dt1dt2
-[γ2|g(p)|2*c2x(p)]2.
1ϵpϵt2c2x(p-q1)c2x(p-r1)δ(q1-q2)δ(τ1-τ2)
×δ(r1-r2)δ(α1-α2),
1ϵpϵt2c2x(p-q1)c2x(p-r1)δ(q1-q2)
×δ(τ1-α2+Δt)δ(r1-r2)δ(τ2-α1+Δt),
1ϵpϵtc4x(p-q1)δ(q1-q2)δ(τ1-τ2)δ(r1-r2)
×δ(α1-α2)δ(q1-r1)δ(τ1-α1+Δt),
var1T0T|z(p, t)|2dt
=1ϵpϵt2 1T20T0Th(τ)h*(τ+Δt)dτ2dt1dt2
×[|g(p)|2*c2x(p)]2+1ϵpϵt 1T20T0T|h(τ)|2|
×h(τ+Δt)|2dτdt1dt2|g(p)|4*c4x(p).
var1T0T|zp(t)|4dt
=E1T0T|zp(t)|4dt2-E1T0T|zp(t)|4dt2=1T241ϵpϵt40T0Th(τ)h*(τ+Δt)dτ4dt1dt2[|g(p)|2*c2x(p)]4+161ϵpϵt40T0T|h(τ)|2dτ2h(τ)h*(τ+Δt)dτ2dt1dt2[|g(p)|2*c2x(p)]4+321ϵpϵt30T0T|h(τ)|2h(τ)h*(τ+Δt)dτ|h(τ)|2dτh(τ)h*(τ+Δt)dτdt1dt2×[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+161ϵpϵt30T0T|h(τ)|2dτ2|h(τ)|2|h(τ+Δt)|2×dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+161ϵpϵt30T0Th(τ)h*(τ+Δt)dτ2|h(τ)|2×|h(τ+Δt)|2dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+41ϵpϵt30T0Th(τ)h*(τ+Δt)dτ2×h(τ)2h(τ+Δt)2dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+81ϵpϵt20T0Th(τ)dτ|h(τ)|4|×h(τ+Δt)|2dτdt1dt2[|g(p)|6*c6x(p)][|g(p)|2*c2x(p)]+81ϵpϵt20T0T|h(τ)|2h(τ)|h(τ+Δt)|2×h*(τ+Δt)dτh*(τ)h(τ+Δt)dτdt1dt2[|g(p)|6*c6x(p)][|g(p)|2*c2x(p)]+81ϵpϵt20T0T|h(τ)|2h(τ)h*(τ+Δt)dτ|h(τ)|2h*(τ)h(τ+Δt)dτdt1dt2[|g(p)|4*c4x(p)]2+1ϵpϵt20T0Th2(τ)h*2(τ+Δt)dτ2dt1dt2[|g(p)|4*c4x(p)]2+81ϵpϵt20T0T|h(τ)|2|h(τ+Δt)|2dτ2×dt1dt2[|g(p)|4*c4x(p)]2+1ϵpϵt0T0T|h(τ)|4|h(τ+Δt)|4dτdt1dt2[|g(p)|8*c8x(p)].
h(t)=12πσ2exp-t22σ2expj 2πλct
mnx(p)=An(p).
var1T0T|zp(t)|4dt
=1T1(2πσ2)7381.28ϵp4[|g(p)|2*A2(p)]4-492.12ϵp3[|g(p)|4*A4(p)][|g(p)|2*A2(p)]2+130.79ϵp2[|g(p)|6*A6(p)][|g(p)|2*A2(p)]+29.02ϵp2[|g(p)|4*A4(p)]2-24.75ϵp[|g(p)|8*A8(p)].
{[b2(p)-|g(p)|2*A2(p)]2+α[b4(p)-|g(p)|4
*A22(p)]2+β[|l(p)|2*A2(p)]2}dp,
1T20T0Th(τ1)h*(τ2)h(τ3)h*(τ4)g(q1)g*(q2)
×g(q3)g*(q4)h(α1)h*(α2)h(α3)h*(α4)g(r1)
×g*(r2)g(r3)g*(r4)E{x(p-q1, t1-τ1)
×x*(p-q2, t1-τ2)x(p-q3, t1-τ3)
×x*(p-q4, t1-τ4)x(p-q1, t2-τ1)
×x*(p-q2, t2-τ2)x(p-q3, t2-τ3)
×x*(p-q4, t2-τ4)}dτ1dτ2dτ3dτ4dα1dα2dα3
×dα4dq1dq2dq3dq4dr1dr2dr3dr4dt1dt2.
1ϵpϵt4c2x(p-q1)c2x(p-q3)c2x(p-r1)c2x(p-r3)
×δ(q1-q2)δ(q3-q4)δ(r1-r2)δ(r3-r4)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4),
1ϵpϵt4c2x(p-q1)c2x(p-q3)c2x(p-r1)c2x(p-r3)
×δ(q1-r2)δ(r1-q2)δ(q3-r4)δ(r3-q4)
×δ(α1-τ2-Δt)δ(α2-τ1-Δt)
×δ(α3-τ4-Δt)δ(α4-τ3-Δt),
1ϵpϵt4c2x(p-q1)c2x(p-q3)c2x(p-r1)c2x(p-r3)
×δ(q1-r2)δ(q2-r1)δ(q3-q4)δ(r3-r4)
×δ(α1-τ2-Δt)δ(α2-τ1-Δt)
×δ(τ3-τ4)δ(α3-α4).
1ϵpϵt3c4x(p-q1)c2x(p-r1)c2x(p-r3)δ(q1-q2)
×δ(q3-q4)δ(r1-r2)δ(r3-r4)δ(q1-q3)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4)
×δ(τ1-τ3),
1ϵpϵt3c4x(p-q1)c2x(p-r1)c2x(p-r3)δ(q1-r4)
×δ(q2-q3)δ(q1-q2)δ(r1-r2)δ(r3-q4)
×δ(α4-τ1-Δt)δ(τ2-τ3)δ(τ1-τ2)
×δ(α1-α2)δ(α3-τ4-Δt),
1ϵpϵt3c4x(p-q1)c2x(p-q3)c2x(p-r3)δ(q1-q2)
×δ(q3-q4)δ(r1-r2)δ(r3-r4)δ(q1-r1)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4)
×δ(α1-τ1-Δt),
1ϵpϵt3c4x(p-q1)c2x(p-q3)c2x(p-r3)δ(q1-q2)
×δ(q3-r4)δ(r1-r2)δ(q4-r3)δ(q1-r1)
×δ(τ1-τ2)δ(α4-τ3-Δt)δ(α1-α2)
×δ(α3-τ4-Δt)δ(α1-τ1-Δt),
1ϵpϵt3c4x(p-q1)c2x(p-r1)c2x(p-r3)δ(q1-r2)
×δ(q2-r1)δ(q3-r4)δ(q4-r3)δ(q1-q2)
×δ(α1-τ2-Δt)δ(α2-τ1-Δt)
×δ(α4-τ3-Δt)
×δ(α3-τ4-Δt)δ(τ1-τ3),
1ϵpϵt2c6x(p-q1)c2x(p-r3)δ(q1-q2)δ(q3-q4)
×δ(r1-r2)δ(r3-r4)δ(q1-r1)δ(q1-q3)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4)
×δ(α2-τ1-Δt)δ(τ1-τ3),
1ϵpϵt2c6x(p-q1)c2x(p-r3)δ(q1-q2)δ(q3-r4)
×δ(r1-r2)δ(q4-r3)δ(q1-q3)δ(q1-r1)
×δ(τ1-τ2)δ(α4-τ3-Δt)δ(α1-α2)
×δ(α3-τ4-Δt)δ(τ1-τ3)δ(α1-τ1-Δt),
1ϵpϵt2c4x(p-q1)c4x(p-r1)δ(q1-q2)δ(q3-q4)
×δ(r1-r2)δ(r3-r4)δ(q1-q3)δ(r1-r3)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4)
×δ(τ1-τ3)δ(α1-α3),
1ϵpϵt2c4x(p-q1)c4x(p-r1)δ(q1-q2)δ(q3-r4)
×δ(r1-r2)δ(r3-q4)δ(q1-q3)δ(r1-r3)
×δ(τ1-τ2)δ(α4-τ3-Δt)δ(α1-α2)
×δ(α3-τ4-Δt)δ(τ1-τ3)δ(α1-α3),
1ϵpϵt2c4x(p-q1)c4x(p-r1)δ(q1-r2)δ(q2-r1)
×δ(q3-r4)δ(q4-r3)δ(q1-q3)δ(r1-r3)
×δ(α2-τ1-Δt)δ(α1-τ2-Δt)
×δ(α4-τ3-Δt)δ(α3-τ4-Δt)
×δ(τ1-τ3)δ(α1-α3),
1ϵpϵt2c4x(p-q1)c4x(p-r1)δ(q1-q2)δ(q3-q4)
×δ(r1-r2)δ(r3-r4)δ(q1-r1)δ(q3-r3)
×δ(τ1-τ2)δ(τ3-τ4)δ(α1-α2)δ(α3-α4)
×δ(α1-τ2-Δt)δ(α3-τ3-Δt),
1ϵpϵtc8x(p-q1)
×δ(q1-q2)δ(q3-q4)δ(r1-r2)δ(r3-r4)
×δ(q1-r1)δ(q1-q3)δ(τ1-τ2)δ(τ3-τ4)
×δ(α1-α2)δ(α3-α4)δ(α1-τ1-Δt)
×δ(τ1-τ3)δ(α1-α3).
1ϵpϵt4 1T20T0T|h(q)|2dq4
×dt1dt2[|g(p)|2*c2x(p)]4,
1ϵpϵt4 1T20T0Th(τ)h*(τ+Δt)dτ4
×dt1dt2[|g(p)|2*c2x(p)]4,
1ϵpϵt4 1T20T0T|h(τ)|2dτ2h(τ)h*(τ+Δt)dτ2
×dt1dt2[|g(p)|2*c2x(p)]4,
1ϵpϵt3 1T20T0T|h(τ)|4dτ
×|h(τ)|2dτ2dt1dt2[|g(p)|4*c4x(p)]
×[|g(p)|2*c2x(p)]2,
1ϵpϵt3 1T20T0T|h(τ)|2h(τ)h*(τ+Δt)dτ
×|h(τ)|2dτh(τ)h*(τ+Δt)
×dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2,
1ϵpϵt3 1T20T0T|h(τ)|2dτ2|h(τ)|2|
×h(τ+Δt)|2dτdt1dt2[|g(p)|4*c4x(p)]
×[|g(p)|2*c2x(p)]2,
1ϵpϵt3 1T20T0Th(τ)h*(τ+Δt)dτ2
×|h(τ)|2|h(τ+Δt)|2dτdt1dt2[|g(p)|4*c4x(p)]
×[|g(p)|2*c2x(p)]2,
1ϵpϵt3 1T20T0Th(τ)h*(τ+Δt)dτ2
×h(τ)2h(τ+Δt)2dτdt1dt2[|g(p)|4*c4x(p)]
×[|g(p)|2*c2x(p)]2,
1ϵpϵt2 1T20T0T|h(τ)|2dτ|h(τ)|4|h
×(τ+Δt)|2dτdt1dt2[|g(p)|6*c6x(p)]
×[|g(p)|2*c2x(p)],
1ϵpϵt2 1T20T0T|h(τ)|2h(τ)|h(τ+Δt)|2
×h*(τ+Δt)dτ
×h*(τ)h(τ+Δt)dτdt1dt2[|g(p)|6*c6x(p)]
×[|g(p)|2*c2x(p)],
1ϵpϵt2 1T20T0T|h(τ)|4dτ2
×dt1dt2[|g(p)|4*c4x(p)]2,
1ϵpϵt2 1T20T0T|h(τ)|2h(τ)h*(τ+Δt)dτ2
×dt1dt2[|g(p)|4*c4x(p)]2,
1ϵpϵt2 1T20T0Th2(τ)h*2(τ+Δt)dτ2
×dt1dt2[|g(p)|4*c4x(p)]2,
1ϵpϵt2 1T20T0T|h(τ)|2|h(τ+Δt)|2dτ2
×dt1dt2[|g(p)|4*c4x(p)]2,
1ϵpϵt 1T20T0T|h(τ)|4|h(τ+Δt)|4dτdt1dt2[|g(p)|8*c8x(p)].
var1T0T|zp(t)|4dt
=E1T0T|zp(t)|4dt2-E1T0T|zp(t)|4dt2=1T241ϵpϵt40T0Th(τ)h*(τ+Δt)dτ4dt1dt2[|g(p)|2*c2x(p)]4+161ϵpϵt40T0T|h(τ)|2dτ2h(τ)h*(τ+Δt)dτ2dt1dt2[|g(p)|2*c2x(p)]4+321ϵpϵt30T0T|h(τ)|2h(τ)h*(τ+Δt)dτ|h(τ)|2dτh(τ)h*(τ+Δt)dτdt1dt2×[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+161ϵpϵt30T0T|h(τ)|2dτ2|h(τ)|2|h(τ+Δt)|2×dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+161ϵpϵt30T0Th(τ)h*(τ+Δt)dτ2×|h(τ)|2|h(τ+Δt)|2dτdt1dt2[|g(p)|4*c4x(p)][|g(p|2*c2x(p)]2+41ϵpϵt30T0Th(τ)h*(τ+Δt)dτ2×h(τ)2h(τ+Δt)2dτdt1dt2[|g(p)|4*c4x(p)][|g(p)|2*c2x(p)]2+81ϵpϵt20T0T|>h(τ)dτ|h(τ)|4|×h(τ+Δt)|2dτdt1dt2[|g(p)|6*c6x(p)][|g(p)2*c2x(p)]+81ϵpϵt20T0T|h(τ)|2h(τ)|h(τ+Δt)|2×h*(τ+Δt)dτh*(τ)h(τ+Δt)dτdt1dt2[|g(p)|6*c6x(p)]+81ϵpϵt20T0T|h(τ)|2h(τ)h*(τ+Δt)dτ×|h(τ)|2h*(τ)h(τ+Δt)dτdt1dt2[|g(p)|4*c4x(p)]2+1ϵpϵt20T0Th2(τ)h*2(τ+Δt)dτ2×dt1dt2[|g(p)|4*c4x(p)]2+81ϵpϵt20T0T|h(τ)|2|h(τ+Δt)|2dτ2dt1dt2[|g(p)|4*c4x(p)]2+1ϵpϵt0T0T|h(τ)|4|h(τ+Δt)|4dτdt1dt2[|g(p)|8*c8x(p)].

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