Abstract

In reconstructing images from coherently illuminated objects, the far-field two-dimensional phase function in the entrance pupil plane (measurement plane) of a coherent imaging system has to contend with the presence of point discontinuities or branch points. The general class of least-squares phase reconstructors that use phase gradients (or phase differences) on a grid of points in the entrance pupil measurement plane fails to correctly determine the two-dimensional pupil phase when branch points are present. The phase estimation error results from the fact that the phase gradient or phase difference inputs to the least-squares reconstructor are being wrapped by the measurement system into the principal-value range [-π, π]. Recently, the existence and the determination of a hidden phase term was presented that accounts for branch-point effects [D. L. Fried, J. Opt. Soc. Am. A 15, 2759 (1998)]. Fried’s hidden phase term is used in this study to present a branch-point-tolerant least-squares phase reconstructor for estimation of the two-dimensional measurement-plane phase function. Simulations of three different types of coherently illuminated object demonstrate the utility of this approach. The sensitivity of the reconstruction method to additive Gaussian noise is also presented.

© 1999 Optical Society of America

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References

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1998 (2)

1996 (1)

1994 (2)

1993 (1)

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

1992 (2)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Computers Electr. Eng. 18, 451–466 (1992).
[CrossRef]

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

1985 (1)

1979 (1)

1977 (2)

1970 (1)

Arrasmith, W. W.

Arsenault, H.

Fiddy, M. A.

Fornaro, G.

Franceschetti, G.

Freund, I.

Fried, D. L.

Gardner, C. S.

Ghiglia, D. C.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9 of Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

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

Hudgin, R. H.

Hunt, B. R.

Hutchin, R. A.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

Lanari, R.

Lowenthal, S.

Panofsky, W. K. H.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eq. (1-1).

Phillips, M.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eq. (1-1).

Roggemann, M.

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Computers Electr. Eng. 18, 451–466 (1992).
[CrossRef]

Romero, L. A.

Sansosti, E.

Scivier, M. S.

Takahashi, T.

Takajo, H.

Vaughn, J. L.

Welsh, B.

Welsh, B. M.

Appl. Opt. (2)

Computers Electr. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Computers Electr. Eng. 18, 451–466 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (7)

Proc. IEEE (1)

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

Other (3)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9 of Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

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

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eq. (1-1).

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Figures (10)

Fig. 1
Fig. 1

Measurement-plane phase: simulated 2D phase cuts from coherently illuminated object in the shape of a doughnut.

Fig. 2
Fig. 2

Sampled phase comparison: (a) reference phase, (b) Ghiglia–Romero least-squares reconstructed phase, (c) general least-squares reconstructed phase.

Fig. 3
Fig. 3

Comparison of measurement-plane phase maps: (a) Ghiglia–Romero least-squares phase reconstruction when the measurement-plane phase differences are known accurately within the range [-2π, 2π], (b) Ghiglia–Romero least-squares phase reconstruction when the measurement-plane phase differences are wrapped into the range [-π, π].

Fig. 4
Fig. 4

Profile corresponding to an object with a Gaussian brightness profile. The object has a square hole at the center. The Gaussian object is centered on a 64×64 grid with 0.22 m between sample points. The decay of the Gaussian envelope is set so that the e-1 width for the brightness profile is at 0.87 m. The radius of the square hole at the center of the Gaussian reflectance profile is 0.65 m.

Fig. 5
Fig. 5

(a) Measurement-plane reference phase; (b) measurement-plane branch points and their associated residues (-2π, 2π, or 0).

Fig. 6
Fig. 6

Comparison between (a) the measurement-plane reference phase and (b) the Ghiglia–Romero least-squares reconstructed phase, including the hidden phase term.

Fig. 7
Fig. 7

(a) Diffraction-limited image of a 1.733-μdeg symmetric object in the shape of a doughnut, (b) reconstructed image obtained by least-squares reconstructor with the hidden phase correction.

Fig. 8
Fig. 8

(a) Diffraction-limited image of a 1.733-μdeg asymmetric object, (b) reconstructed image obtained by least-squares reconstructor with the hidden phase correction.

Fig. 9
Fig. 9

(a) Diffraction-limited image of a 1.733-μdeg symmetric object in the shape of five spatially separated point sources, (b) reconstructed image obtained by least-squares reconstructor with the hidden phase correction.

Fig. 10
Fig. 10

Noise comparison: (a) noise-free reconstructed doughnut-shaped object, (b) reconstruction with additive Gaussian noise with a standard deviation of π/3 rad, (c) reconstruction with a noise standard deviation of π/2 rad, (d) reconstruction with a noise standard deviation of 2π/3 rad.

Equations (16)

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g(rp,q)=P[ϕ(rp+1,q)-ϕ(rp,q)]d xˆ+P{[ϕ(rp,q+1)-ϕ(rp,q)]}d yˆ.
ϕ(rp,q)=tan-1Im[u(rp,q)]Re[u(rp,q)],
ct(ξ)g(ξ)dξ
=±2πifbranchpointisenclosed0ifbranchpointisnotenclosed.
cdrzˆ×g(r)
=±2πifbranchpointispresent0ifbranchpointisnotpresent.
zˆ×g(r)=±2πδ(r-rbp).
g(r)=s(r)+×H(r).
2h(r)=±2πδ(r-rbp).
h(r)=±log(|r-rbp|).
ϕh(r)=Imlogk=1K(x-xk)+i(y-yk)k=1K(x-xk)+i(y-yk).
ϕ(r)=ϕls(r)+ϕh(r)+ϕn(r),
ϕls=MLSg.
g=d-1P[ϕ(r2,1)-ϕ(r1,1)]P[ϕ(r1,2)-ϕ(r1,1)]P[ϕ(rN,N-1)-ϕ(rN-1,N-1)]P[ϕ(rN-1,N)-ϕ(rN-1,N-1)].
MLS=(ΓTΓ)-1ΓT,
g=Γϕls.

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