Abstract

In phase-diverse speckle imaging one collects a time series of phase-diversity image sets that are used to jointly estimate the object and each of the phase-aberration functions. Current approaches model the total phase aberration in some deterministic parametric fashion. For many imaging schemes, however, additional information can be exploited. Specifically, the total aberration function consists of the fixed aberrations combined with dynamic (time-varying), turbulence-induced aberrations, about whose stochastic behavior we often have some knowledge. One important example is that in which the wave-front phase error corresponds to Kolmogorov turbulence. In this context using the extra statistical information available may be a powerful aid in the joint aberration/object estimation. In addition, such a framework provides an attractive method for calibrating fixed aberrations in an imaging system. The discipline of Bayesian statistical inference provides a natural framework for using the stochastic information regarding the wave fronts. Here one imposes an a priori probability distribution on the turbulence-induced wave fronts. We present the general Bayesian approach for the joint-estimation problem of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data that leads to a maximum a posteriori estimator. We also present results based on simulated data, which show that the Bayesian approach provides an increase in accuracy and robustness for this joint estimation.

© 1999 Optical Society of America

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References

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  1. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Phase-diverse speckle interferometry,” in Signal Recovery and Synthesis IV, Vol. 11 of 1992 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1992), pp. 5–7.
  2. J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
    [CrossRef]
  3. R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. SPIE207, 32–39 (1979).
    [CrossRef]
  4. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  5. R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
    [CrossRef]
  6. J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, “Post-detection correction of compensated imagery using phase-diverse speckle,” in Adaptive Optics, Vol. 23 of 1995 Tech. Dig. Ser.-Opt. Soc. Am., M. Cullum, ed. (Optical Society of America, Washington, D.C., 1995) pp. 471–476.
  7. J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
    [CrossRef]
  8. J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.
  9. R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” in Real Time and Post-Facto Solar Image Correction, R. R. Raddick, ed., National Solar Observatory/Sacramento Peak Summer Workshop13 (Sunspot, N.M., 1992), pp. 112–118.
  10. T. J. Schulz, “Multi-frame blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  11. D. L. Snyder, C. W. Helstrom, A. D. Lanterman, M. Faisal, R. L. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
    [CrossRef]
  12. H. L. Van-Trees, Detection, Estimation, and Modulation Theory: Part I (Wiley, New York, 1968).
  13. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).
  14. D. C. Liu, J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
    [CrossRef]
  15. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  16. R. G. Paxman, J. H. Seldin, P. P. Sanchez, “Applied phase diversity,” [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1992].
  17. J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

1996 (1)

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

1995 (1)

1993 (1)

1992 (1)

1989 (1)

D. C. Liu, J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

1966 (1)

Chidlaw, R.

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. SPIE207, 32–39 (1979).
[CrossRef]

Dennis, J. E.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Ellerbroek, B. L.

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, “Post-detection correction of compensated imagery using phase-diverse speckle,” in Adaptive Optics, Vol. 23 of 1995 Tech. Dig. Ser.-Opt. Soc. Am., M. Cullum, ed. (Optical Society of America, Washington, D.C., 1995) pp. 471–476.

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

Faisal, M.

Fienup, J. R.

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Phase-diverse speckle interferometry,” in Signal Recovery and Synthesis IV, Vol. 11 of 1992 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1992), pp. 5–7.

Fried, D. L.

Gonsalves, R. A.

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. SPIE207, 32–39 (1979).
[CrossRef]

Helstrom, C. W.

Johnston, D. C.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.

Keller, C. U.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

Lanterman, A. D.

Liu, D. C.

D. C. Liu, J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Löfdahl, M. G.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

Nocedal, J.

D. C. Liu, J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Paxman, R. G.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Phase-diverse speckle interferometry,” in Signal Recovery and Synthesis IV, Vol. 11 of 1992 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1992), pp. 5–7.

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” in Real Time and Post-Facto Solar Image Correction, R. R. Raddick, ed., National Solar Observatory/Sacramento Peak Summer Workshop13 (Sunspot, N.M., 1992), pp. 112–118.

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, “Post-detection correction of compensated imagery using phase-diverse speckle,” in Adaptive Optics, Vol. 23 of 1995 Tech. Dig. Ser.-Opt. Soc. Am., M. Cullum, ed. (Optical Society of America, Washington, D.C., 1995) pp. 471–476.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.

R. G. Paxman, J. H. Seldin, P. P. Sanchez, “Applied phase diversity,” [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1992].

Reiley, M. F.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

Sanchez, P. P.

R. G. Paxman, J. H. Seldin, P. P. Sanchez, “Applied phase diversity,” [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1992].

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

Scharmer, G. B.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

Schnabel, R. B.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Schulz, T. J.

T. J. Schulz, “Multi-frame blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Phase-diverse speckle interferometry,” in Signal Recovery and Synthesis IV, Vol. 11 of 1992 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1992), pp. 5–7.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, “Post-detection correction of compensated imagery using phase-diverse speckle,” in Adaptive Optics, Vol. 23 of 1995 Tech. Dig. Ser.-Opt. Soc. Am., M. Cullum, ed. (Optical Society of America, Washington, D.C., 1995) pp. 471–476.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” in Real Time and Post-Facto Solar Image Correction, R. R. Raddick, ed., National Solar Observatory/Sacramento Peak Summer Workshop13 (Sunspot, N.M., 1992), pp. 112–118.

R. G. Paxman, J. H. Seldin, P. P. Sanchez, “Applied phase diversity,” [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1992].

Snyder, D. L.

Stribling, B. E.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

Van-Trees, H. L.

H. L. Van-Trees, Detection, Estimation, and Modulation Theory: Part I (Wiley, New York, 1968).

White, R. L.

Astrophys. J. (1)

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 467, 1087–1099 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Math. Program. (1)

D. C. Liu, J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Other (11)

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Phase-diverse speckle interferometry,” in Signal Recovery and Synthesis IV, Vol. 11 of 1992 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1992), pp. 5–7.

J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. SPIE207, 32–39 (1979).
[CrossRef]

R. G. Paxman, J. H. Seldin, P. P. Sanchez, “Applied phase diversity,” [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1992].

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

H. L. Van-Trees, Detection, Estimation, and Modulation Theory: Part I (Wiley, New York, 1968).

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, “Post-detection correction of compensated imagery using phase-diverse speckle,” in Adaptive Optics, Vol. 23 of 1995 Tech. Dig. Ser.-Opt. Soc. Am., M. Cullum, ed. (Optical Society of America, Washington, D.C., 1995) pp. 471–476.

J. H. Seldin, M. F. Reiley, R. G. Paxman, B. E. Stribling, B. L. Ellerbroek, D. C. Johnston, “Space-object identification using phase-diverse speckle,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 2–15 (1997).
[CrossRef]

J. H. Seldin, R. G. Paxman, B. L. Ellerbroek, D. C. Johnston, “Phase-diverse speckle restorations of artificial satellites imaged with adaptive-optics compensation,” in Adaptive Optics, Vol. 13 of 1996 Tech. Dig. Ser.-Opt. Soc. Am. (Optical Society of America, Washington, D.C., 1996), addendum, pp. 341–343.

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” in Real Time and Post-Facto Solar Image Correction, R. R. Raddick, ed., National Solar Observatory/Sacramento Peak Summer Workshop13 (Sunspot, N.M., 1992), pp. 112–118.

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Figures (10)

Fig. 1
Fig. 1

Optical layout for phase-diverse speckle imaging.

Fig. 2
Fig. 2

(a) Discrete jet object used in simulations; (b) diffraction-limited image; (c) conventional image for first aberration realization j=1, k=1; (d) diversity image for j=1, k=2; (e) conventional image for second aberration realization j=2, k=1; (f) diversity image for j=2, k=2.

Fig. 3
Fig. 3

Normalized RMSE as a function of iteration for MAP and ML object estimates from PDS data.

Fig. 4
Fig. 4

PDS object estimates: (a) MAP object estimate (from iteration 90), (b) ML object estimate (from iteration 60).

Fig. 5
Fig. 5

RMSE of MAP and ML PDS phase-aberration function estimates. The minimum value for MAP is 0.089 waves, whereas the minimum value for ML is 0.152 waves.

Fig. 6
Fig. 6

PDS aberration estimates: (a) True phase-aberration function for realization j=1, (b) MAP aberration estimate (iteration 1200), (c) ML aberration estimate (iteration 70), (d) image of absolute error for the MAP aberration estimate, (e) image of absolute error for the ML aberration estimate, displayed on same scale as (d).

Fig. 7
Fig. 7

RMSE of MAP and ML PDS PSF estimates for Kolmogorov aberrations.

Fig. 8
Fig. 8

Phase-aberration function for segmented aperture with fixed piston errors only.

Fig. 9
Fig. 9

Plots of MAP PDS piston estimates as a function of iteration. The true piston value is denoted with a dashed line; the estimates are solid curves. (a) Segment 4, (b) segment 7.

Fig. 10
Fig. 10

Plots of error for each of the MAP PDS piston estimates as a function of iteration. Note that these plots all converge to values within a fairly compact neighborhood of zero.

Tables (2)

Tables Icon

Table 1 Simulation Details for Model of Image Formation with Dynamic Aberrations Only

Tables Icon

Table 2 Fixed Piston Misalignment Values for Each of the Segments That Were Used in the Simulationsa

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

gjk(x)=xf (x)sjk(x-x),j=1,, Jk=1,, K,
sjk(x)=|hjk(x)|2x|hjk(x)|2,
hjk(x)=1N2uHjk(u)exp(i2πu, x/N)
Hjk(u)=|Hk(u)|exp{i[ϕj(u)+θk(u)]}.
ϕj(u)=ϕ(f)(u)+ϕj(d)(u),
ϕj(u)=l=1Lalαl(u)+l=1Lbjlβl(u),
a=a1a2aL,
bj=bj1bj2bjL,j=1,, J.
p(d1,, dJ, b1,, bJ; f, a)
=p(d1,, dJ|b1,, bJ; f, a)p(b1,, bJ; f, a)
=j=1Jp(dj|bj ; f, a)j=1Jp(bj)
=j=1Jp(dj|bj ; f, a)p(bj),
p(dj|bj; f, a)=k=1Kx exp[-gjk(x)] [gjk(x)]djk(x)djk(x)!,
p(bj)=1(2π)L||exp(-12bj-1bj),
LL(f, b1,, bJ, a)j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1Jbj-1bj.
αl=l=1Lρllψl,
ϕj(u)=l=1Lcjlψl(u),
ϕj(d)=ϕj-ϕj(f)=l=1Lcjlψl-l=1Lalαl=l=1Lcjlψl-l=1Lall=1Lρllψl=l=1Lcjlψl-l=1Ll=1Lalρllψl=l=1Lcjl-l=1Lρllalψl=l=1L[cjl-(ρa)l]ψl=l=1Lbjlψl,
L=j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1J(cj-ρa)-1(cj-ρa),
a*(c¯)(ρ-1ρ)-1ρ-1c¯,
c¯=1Jj=1Jcj.
L*=j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1J[cj-ρa*(c¯)]-1[cj-ρa*(c¯)]
=j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1Jcj-1cj-Jc¯-1ρ(ρ-1ρ)-1ρ-1c¯.
=j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1Jcj-1cj-Jc¯Bρ c¯
L1+L2,
Bρ-1ρ(ρ-1ρ)-1ρ-1.
aˆa*(cˆ¯)=(ρ-1ρ)-1ρ-1cˆ¯,
L2cjl=--1cjl+(Bρc¯)l.
Lm=j=1Jk=1Kx{djk(x)log[gjk(x)]-gjk(x)}-12j=1Jcj-1cjL1+L2,
L2cjl=-(-1cj)l.
NRMSE=x[fˆ (x-xo)-f (x)]2xf (x)21/2,
RMSE=j=120 u[ϕˆj(u)-ϕj(u)-mj]220 M1/2,
uψl(u)ψl1(u)=0for1lL,L+1l1L1.
αl=Poαl+P1αl=l=1Lρll(o)ψl+l=1L1-Lρll(1)ψL+l,
ϕj(u)=l=1Lcjl(o)ψl(u)+l=1L1-Lcl(1)ψL+l(u).
cj(o)=cj1cj2cjL,
c(1)=cL+1cL+2cL1,
μc(o)=ρ(o)a.
aˆ=[(ρ(o))-1ρ(o)]-1(ρ(o))-1cˆ¯(o)+[(ρ(1))ρ(1)]-1(ρ(1))cˆ(1).
-2L2=j=1J(cj-ρa)-1(cj-ρa),
a*(c¯)=(ρ-1ρ)-1ρ-1c¯.
c=c1c2cJ,
ρ=ρρρ,
Σ=Σ0000Σ00000Σ.
-2L2=(c-ρ a)Σ-1(c-ρ a).
a*=(ρΣ-1ρ)-1ρΣ-1c.
ρΣ-1=[ρ-1ρ-1],
ρΣ-1c=j=1JρΣ-1cj=JρΣ-1c¯,
(ρΣ-1ρ)-1=(JρΣ-1ρ)-1=1J(ρΣ-1ρ)-1.
[c-ρ a*(c¯)]Σ-1[c-ρ a*(c¯)]
=cΣ-1c-cΣ-1ρ (ρΣ-1ρ)-1ρΣ-1c=j=1JcjΣ-1cj-Jc¯Σ-1ρ(ρΣ-1ρ)-1ρΣ-1c¯,

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