Abstract

The phase diversity technique is studied as a wave-front sensor to be implemented with widely extended sources. The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images (in focus and out of focus) by use of a maximum-likelihood approach. The propagation of the photon noise in the images on the estimated phase is derived from a theoretical analysis. The covariance matrix of the phase estimator is calculated, and the optimal distance between the observation planes that minimizes the noise propagation is determined. The phase error is inversely proportional to the number of photons in the images. The noise variance on the Zernike polynomials increases with the order of the polynomial. These results are confirmed with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations.

© 1999 Optical Society of America

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    [CrossRef]
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1996

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

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. 466, 1087–1099 (1996).
[CrossRef]

J. A. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
[CrossRef] [PubMed]

1995

1994

1993

1992

1990

1989

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

1988

O. Von der Lühe, “Wavefront error measurement technique using extended, incoherent light sources,” Opt. Eng. 278, 1078–1087 (1988).

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

1987

1986

J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. 3, 1897–1907 (1986).
[CrossRef]

1982

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1976

1969

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

Acton, D. S.

Baba, N.

Biraud, Y.

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd revised ed. (McGraw-Hill, New York, 1986).

Carrara, D. A.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase-diversity and curvature wavefront sensing,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Carreras, R. A.

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

Conan, J. M.

Conan, J.-M.

L. M. Mugnier, J.-M. Conan, V. Michau, G. Rousset, “Imagerie à travers la turbulence par déconvolution myope multi-trame,” in Seizième Colloque sur le Traitement du Signal et des Images GRETSI (Groupe de Recherche et d’Études sur le Traitement du Signal et des Images, Grenoble, France, 1997), pp. 567–570.

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Duncan, A. L.

Ellerbroek, B. L.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Fender, J. S.

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

Fessler, J. A.

J. A. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
[CrossRef] [PubMed]

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Fontanella, J.-C.

V. Michau, G. Rousset, J.-C. Fontanella, “Wavefront sensing from extended sources,” in Real Time and Post Facto Solar Image Correction (National Solar Observatory, Sunspot, N.M., 1992), pp. 124–128.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A 4, 166–170 (1987).
[CrossRef]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

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. 466, 1087–1099 (1996).
[CrossRef]

Kendrick, R. L.

Kupke, R.

R. Kupke, F. Roddier, D. L. Mickey, “Curvature-based wavefront sensor for use on extended patterns,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 519–527 (1994).
[CrossRef]

Lane, R. G.

Lee, D. J.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

D. J. Lee, B. Welsh, M. Roggemann, “Cramer–Rao evaluation of phase diversity aberration sensing,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 161–172 (1997).
[CrossRef]

Liang, J.

D. R. Williams, J. Liang, D. T. Miller, “Adaptive optics for the human eye,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 145–147.

Ling, A.

N. Miller, A. Ling, “Imaging with phase diversity: experimental results,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 227–238 (1993).
[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. 466, 1087–1099 (1996).
[CrossRef]

M. G. Löfdahl, G. B. Scharmer, “Application of phase-diversity to solar images,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 254–267 (1994).
[CrossRef]

Love, G. D.

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

Marron, J. C.

Meynadier, L.

L. Meynadier, “Analyse de surface d’onde pour le contrôle actif d’un télescope spatial,” Ph.D. dissertation (Université de Nice-Sophia Antipolis, Nice, France, 1997).

L. Meynadier, V. Michau, M. T. Velluet, “Wavefront sensing by phase diversity,” in THEMIS Forum (Observatoire de Paris, Paris, 1996), pp. 205–210.

Michau, V.

L. Meynadier, V. Michau, M. T. Velluet, “Wavefront sensing by phase diversity,” in THEMIS Forum (Observatoire de Paris, Paris, 1996), pp. 205–210.

L. M. Mugnier, J.-M. Conan, V. Michau, G. Rousset, “Imagerie à travers la turbulence par déconvolution myope multi-trame,” in Seizième Colloque sur le Traitement du Signal et des Images GRETSI (Groupe de Recherche et d’Études sur le Traitement du Signal et des Images, Grenoble, France, 1997), pp. 567–570.

V. Michau, G. Rousset, J.-C. Fontanella, “Wavefront sensing from extended sources,” in Real Time and Post Facto Solar Image Correction (National Solar Observatory, Sunspot, N.M., 1992), pp. 124–128.

Mickey, D. L.

R. Kupke, F. Roddier, D. L. Mickey, “Curvature-based wavefront sensor for use on extended patterns,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 519–527 (1994).
[CrossRef]

Miller, D. T.

D. R. Williams, J. Liang, D. T. Miller, “Adaptive optics for the human eye,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 145–147.

Miller, N.

N. Miller, A. Ling, “Imaging with phase diversity: experimental results,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 227–238 (1993).
[CrossRef]

Miura, N.

Mugnier, L. M.

L. M. Mugnier, J.-M. Conan, V. Michau, G. Rousset, “Imagerie à travers la turbulence par déconvolution myope multi-trame,” in Seizième Colloque sur le Traitement du Signal et des Images GRETSI (Groupe de Recherche et d’Études sur le Traitement du Signal et des Images, Grenoble, France, 1997), pp. 567–570.

Noll, R. J.

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. 466, 1087–1099 (1996).
[CrossRef]

R. G. Paxman, B. J. Thelen, J. H. Seldin, “Phase-diversity correction of turbulence-induced space-variant blur,” Opt. Lett. 19, 1231–1233 (1994).
[CrossRef] [PubMed]

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, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[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. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase-diversity and curvature wavefront sensing,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

B. J. Thelen, R. G. Paxman, J. H. Seldin, D. R. Rice, “Bayes estimation of dynamic and fixed aberrations, and object from phase-diverse speckle data,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 32–42 (1996).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Restaino, S. R.

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

Rice, D. R.

B. J. Thelen, R. G. Paxman, J. H. Seldin, D. R. Rice, “Bayes estimation of dynamic and fixed aberrations, and object from phase-diverse speckle data,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 32–42 (1996).
[CrossRef]

Roddier, C.

C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wave-front distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope mirror figure from out-of-focus stellar images,” in Space Astronomical Telescopes and Instruments, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1494, 78–84 (1991).
[CrossRef]

Roddier, F.

C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wave-front distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope mirror figure from out-of-focus stellar images,” in Space Astronomical Telescopes and Instruments, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1494, 78–84 (1991).
[CrossRef]

R. Kupke, F. Roddier, D. L. Mickey, “Curvature-based wavefront sensor for use on extended patterns,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 519–527 (1994).
[CrossRef]

Roggemann, M.

D. J. Lee, B. Welsh, M. Roggemann, “Cramer–Rao evaluation of phase diversity aberration sensing,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 161–172 (1997).
[CrossRef]

Rousset, G.

V. Michau, G. Rousset, J.-C. Fontanella, “Wavefront sensing from extended sources,” in Real Time and Post Facto Solar Image Correction (National Solar Observatory, Sunspot, N.M., 1992), pp. 124–128.

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., Vol. 423 of NATO Asi Series C: Mathematical and Physical Sciences (Kluwer Academic, Dordrecht, The Netherlands, 1994).
[CrossRef]

L. M. Mugnier, J.-M. Conan, V. Michau, G. Rousset, “Imagerie à travers la turbulence par déconvolution myope multi-trame,” in Seizième Colloque sur le Traitement du Signal et des Images GRETSI (Groupe de Recherche et d’Études sur le Traitement du Signal et des Images, Grenoble, France, 1997), pp. 567–570.

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. 466, 1087–1099 (1996).
[CrossRef]

M. G. Löfdahl, G. B. Scharmer, “Application of phase-diversity to solar images,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 254–267 (1994).
[CrossRef]

Schulz, T. J.

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. 466, 1087–1099 (1996).
[CrossRef]

R. G. Paxman, B. J. Thelen, J. H. Seldin, “Phase-diversity correction of turbulence-induced space-variant blur,” Opt. Lett. 19, 1231–1233 (1994).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
[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]

B. J. Thelen, R. G. Paxman, J. H. Seldin, D. R. Rice, “Bayes estimation of dynamic and fixed aberrations, and object from phase-diverse speckle data,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 32–42 (1996).
[CrossRef]

Tarr, G. L.

R. A. Carreras, S. R. Restaino, G. D. Love, G. L. Tarr, J. S. Fender, “Phase diversity experimental results: deconvolution of ν Scorpii,” Opt. Commun. 130, 13–19 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Thelen, B. J.

R. G. Paxman, B. J. Thelen, J. H. Seldin, “Phase-diversity correction of turbulence-induced space-variant blur,” Opt. Lett. 19, 1231–1233 (1994).
[CrossRef] [PubMed]

B. J. Thelen, R. G. Paxman, J. H. Seldin, D. R. Rice, “Bayes estimation of dynamic and fixed aberrations, and object from phase-diverse speckle data,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 32–42 (1996).
[CrossRef]

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[CrossRef]

D. J. Lee, B. Welsh, M. Roggemann, “Cramer–Rao evaluation of phase diversity aberration sensing,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 161–172 (1997).
[CrossRef]

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase-diversity and curvature wavefront sensing,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Phase diversity principle.

Fig. 2
Fig. 2

Spiral galaxy object.

Fig. 3
Fig. 3

Wave front used for the numerical approach of the noise propagation theory.

Fig. 4
Fig. 4

Images of the spiral galaxy in the focal plane. (a) Noiseless diffraction-limited image. (b) Aberrated image with 107 total photon number.

Fig. 5
Fig. 5

Noise variance (in radians squared) of the first 21 (solid curve) and 36 (dotted curve) estimated Zernike coefficients for 107 photons and 64 × 64 pixels/image.

Fig. 6
Fig. 6

The total noise variance (in radians squared) of the first 21 estimated Zernike coefficients versus the defocus wave-front amplitude for 107 photons and 64 × 64 pixels/image.

Fig. 7
Fig. 7

Urban scene used for computing the images in the simulations.

Fig. 8
Fig. 8

Full-extent noisy blurred scenes in the focal plane with a total of 107 photons and with the amplitude of coefficients a 4a 21 equal to 0.05 rad (a) in the focal plane, (b) in the out-of-focus plane. The other aberrated images are similar, because they are of low amplitude.

Fig. 9
Fig. 9

Standard deviation of estimated Zernike coefficients on 50 image pairs of 32 × 32 pixels for different values of Zernike coefficients a 4a 21 used to simulate the distorted phase, dotted curve, a l = 0 rad; dashed curve, a l = 0.037 rad; solid curve, a l = 0.05 rad; and 107 photons/image.

Fig. 10
Fig. 10

Bias of estimated Zernike coefficients on 50 image pairs of 32 × 32 pixels for different values of Zernike coefficients a 4a 21 used to simulate the distorted phase, dotted curve, a l = 0 rad; dashed curve, a l = 0.037 rad; solid curve a l = 0.05 rad; and 107 photons per image. For this final case the ±3σ error bars are plotted.

Fig. 11
Fig. 11

Images in the focal plane for different fluxes and a 4-a 21 = 0.05 rad. The square indicates the actual recorded image taken into account for the algorithm. (a) Total flux level is 106 photons; (b) total flux level is 109 photons.

Fig. 12
Fig. 12

Noise variance of estimated Zernike coefficients on 50 image pairs of 64 × 64 pixels for different total photon numbers/image. From the top, N ph = 106, 107, 108, and 109 photons, and a 4a 21 = 0.05 rad.

Fig. 13
Fig. 13

Average noise variance of estimated Zernike coefficients versus the detected flux level/image.

Fig. 14
Fig. 14

Bias of estimated Zernike coefficients on 50 image pairs of 64 × 64 pixels for different numbers of photons/image. Solid line, N ph = 109; long-dashed curve, N ph = 108; dashed curve, N ph = 107; dotted curve, N ph = 106; and a 4-a 21 = 0.05 rad. For this final case, the ±3σ error bars are plotted.

Fig. 15
Fig. 15

Optical setup of phase diversity experiment.

Fig. 16
Fig. 16

Experimentally recorded images (a) in the focal plane, (b) in the out-of-focus plane.

Fig. 17
Fig. 17

Experimental noise variance of estimated Zernike coefficients on 50 image pairs of 64 × 64 pixels for different numbers of photons/image.

Tables (3)

Tables Icon

Table 1 Values in Radians of the Coefficients Used for the Simulation

Tables Icon

Table 2 Phase Residual Error in Function of the Spectral Bandwidth

Tables Icon

Table 3 Theoretical and Estimated Astigmatism Zernike Coefficients in the Function of the Angular Position θ of the Slide Plate with Respect to the Optical Axis

Equations (43)

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Ikr=Or * Skr,
Skr=-+ Akxexpi 2πλFr·xdx2,
A1x=αxexpiφx,A2x=A1xexpiϕdx,φx=2πΔxλ,
E=k,i |Ikri-Ori*Skri|2.
E  k=12i=1N2 |Ĩkfi-ÕfiS˜kfi|2,
pe=arg minp Ep, m.
Covpep,p2E-1m,p2 E×Covmtm,p2 Etp,p2 E-1,
Coval,eNphN2al,al2 E-1Ik,al2 E×tIk,al2Etal,al2 E-1,
Coval,e2 NphN4al,al2 E-1.
Coval,e  1KNph,
E=i |I˜1fi-O˜fiS˜1fi|2+|I˜2fi-O˜fiS˜2fi|2.
X˜k,l=1N2n=1Nm=1N Xn,m exp-2iπnk+mlN,
Xn,m=k=1Nl=1N X˜k,l exp2iπnk+mlN,
EX=1N2 DFT-1EX˜,
EX˜=EX˜R+i EX˜I,
EO˜f=2S˜1*O˜S˜1-I˜1+S˜2*O˜S˜2-I˜2f.
ES˜kf=2O˜*O˜S˜k-I˜kf,
EOr=1N2 DFT-1EO˜f,ESkr=1N2 DFT-1ES˜kf,
EΩr=EOOΩr=2ΩrEOr.
Eal=k=12jsupportiESkriSkφxjφal,
Skrial=-2 ImÃk*riA˜kri * Z˜lri,
Ir=IrΠr,
Πr=1if r  FOV0otherwise.
E=i |I˜1fi-O˜fiS˜1fi * Π˜fi|2+|I˜2fi-O˜fiS˜2fi * Π˜fi|2.
Ckfi=j O˜fjS˜kfjΠ˜fi-fj.
EO˜fi=O˜Rfi+i O˜Ifi E,
O˜Rf0 E=kiCkO˜R Ck*+Ck*O˜R Ck-I˜ O˜R Ck*-I˜* O˜R Ckfi,O˜If0 E=kiCkO˜I Ck*+Ck*O˜I Ck-I˜ O˜I Ck*-I˜* O˜I Ckfi.
O˜Rf0 Ckfi=O˜Rf0j O˜fjS˜kfjΠ˜fi-fj=S˜kf0Π˜fi-f0,
O˜Rf0 Ck*fi=S˜k*f0Π˜fi-f0,
O˜If0 Ckfi=iS˜kf0Π˜fi-f0,
O˜If0 Ck*fi=-iS˜k*f0Π˜fi-f0.
EÕf0=2 k S˜k*f0iCkfi-I˜fiΠ˜*fi-f0,  EO˜f0=2(S˜1*f0O˜f0S˜1f0 * Π˜f0-I˜f0 * Π˜*-f0+S˜2*f0×O˜f0S˜2f0 * Π˜f0-I˜f0 * Π˜*-f0).
EO˜f0=2(S˜1*f0O˜f0S˜1f0 * Π˜f0-I˜f0 * Π˜f0+S˜2*f0O˜f0S˜2f0 * Π˜f0-I˜f0 * Π˜f0).
ES˜kf0=2O˜*f0O˜f0S˜kf0 *Π˜f0-I˜f0 * Π˜f0,
EOr0=4N2 DFT-1EO˜f0,  ESkr0=4N2 DFT-1ES˜kf0,
I˜k-O˜S˜k,e=0.
al,al2E=2Ealal=2 Rek=12h=1N2|O˜fh|2S˜k,e*alS˜k,eal,
2Ealal=8 k=12h=1N2 |O˜fh|2(DFTImA˜k,e*×A˜k,e * Z˜l)*DFTImA˜k,e*A˜k,e * Z˜l.
I˜kfhIkri=1N2 exp-i 2πNfhri.
Ik,al2E=2EalIkri=-2N2Ori * Sk,erial,
2EalIkri=4N2k=12 ImZ˜lri * A˜k,eri×A˜k,e*ri * Ori.
Ik,al2EtIk,al2 E=4N4k=12i ReOri * Sk,erial×Ori * Sk,erial.
Ik,al2EtIk,al2E=2N2al,al2 E.

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