Abstract

Restoring images blurred by an unknown optical system is a problem of interest in image processing and, in particular, in terrestrial astronomy, in which the incoming radiation from celestial objects is contaminated in an unforeseeable way by passing through the atmosphere in turbulent motion. Here a blind deconvolution method of image restoration of the phase-diversity class is presented. Numerical analysis clearly shows that the algorithm is capable of finding both the unknown incoherent object and the point-spread function, which is considered a function of a phase-aberration term only, from multiple images and in the absence of a reference source. Noise and measurement errors, such as calibration errors of the detecting optical system, are explicitly taken into account. Exploiting both the physical constraint on the optical path disturbance and a regularizing functional yields a rigorously stable problem.

© 1999 Optical Society of America

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1997 (2)

1996 (4)

N. F. Law, R. G. Lane, “Blind deconvolution using least squares minimization,” Opt. Commun. 128, 341–352 (1996).
[CrossRef]

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13(5), 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13(11), 61–63 (1996).
[CrossRef]

1995 (2)

1994 (4)

1993 (5)

1992 (5)

1990 (1)

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

1988 (2)

1987 (1)

1982 (1)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–295 (1982).
[CrossRef]

1976 (2)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 58–63 (1976).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

1975 (1)

T. G. Stockman, T. M. Cannon, R. G. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in stars images,” Astron. Astrophys. 6, 85–87 (1970).

1968 (1)

A. V. Oppenheim, R. W. Schafer, T. G. Stockman, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Akilov, G.

L. Kantorovitch, G. Akilov, Analyse Fonctionelle (Mir, Moscow, 1981).

Arsenine, V.

A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posées (Mir, Moscow, 1976).

Ayers, G. R.

Baba, N.

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–295 (1982).
[CrossRef]

Bertero, M.

M. Bertero, Linear Inverse and Ill-Posed Problems (Academic, San Diego, Calif., 1989).

Boas, R. P.

R. P. Boas, Entire Functions (Academic, San Diego, Calif., 1954).

Bones, P. J.

Bucci, O. M.

O. M. Bucci, A. Capozzoli, G. D’Elia, “New technique for wavefront reconstruction in optical telescopes,” J. Opt. Soc. Am. A 14, 3394–3401 (1997).
[CrossRef]

O. M. Bucci, A. Capozzoli, G. D’Elia, “A method for image restoration and wavefront sensing by using phase diversity,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 28–30.

O. M. Bucci, A. Capozzoli, G. D’Elia, “A hybrid evolutionary algorithm in the diagnosis of reflector distortions,” presented at the Journées Internationales de Nice sur les Antennes, November 17–19, 1998, Nice, France.

Cannon, M.

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 58–63 (1976).
[CrossRef]

Cannon, T. M.

T. G. Stockman, T. M. Cannon, R. G. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Capozzoli, A.

O. M. Bucci, A. Capozzoli, G. D’Elia, “New technique for wavefront reconstruction in optical telescopes,” J. Opt. Soc. Am. A 14, 3394–3401 (1997).
[CrossRef]

O. M. Bucci, A. Capozzoli, G. D’Elia, “A method for image restoration and wavefront sensing by using phase diversity,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 28–30.

O. M. Bucci, A. Capozzoli, G. D’Elia, “A hybrid evolutionary algorithm in the diagnosis of reflector distortions,” presented at the Journées Internationales de Nice sur les Antennes, November 17–19, 1998, Nice, France.

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, 32–37 (1979).
[CrossRef]

Christou, J. C.

S. M. Jeffries, J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862–874 (1993).
[CrossRef]

Conan, J.-M.

Connoly, T. J.

R. G. Lane, R. A. Johnston, R. Irwan, T. J. Connoly, “Regularized blind deconvolution,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 5–7.

D’Elia, G.

O. M. Bucci, A. Capozzoli, G. D’Elia, “New technique for wavefront reconstruction in optical telescopes,” J. Opt. Soc. Am. A 14, 3394–3401 (1997).
[CrossRef]

O. M. Bucci, A. Capozzoli, G. D’Elia, “A method for image restoration and wavefront sensing by using phase diversity,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 28–30.

O. M. Bucci, A. Capozzoli, G. D’Elia, “A hybrid evolutionary algorithm in the diagnosis of reflector distortions,” presented at the Journées Internationales de Nice sur les Antennes, November 17–19, 1998, Nice, France.

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Fienup, J. R.

Fischer, T. L.

J. A. Scales, M. L. Smith, T. L. Fischer, “Global optimization methods for multimodal inverse problems,” J. Comput. Phys. 103, 258–268 (1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Galatsanos, N. P.

Ghiglia, D. C.

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, 32–37 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hammoud, A. M.

Hatzinakos, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13(11), 61–63 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13(5), 43–64 (1996).
[CrossRef]

Holmes, T. J.

Ingebretsen, R. G.

T. G. Stockman, T. M. Cannon, R. G. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Irwan, R.

R. G. Lane, R. A. Johnston, R. Irwan, T. J. Connoly, “Regularized blind deconvolution,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 5–7.

Isernia, T.

T. Isernia, G. Leone, R. Pierri, “The phase retrieval in near zone as a nonlinear inverse problem: the planar scanning,” in Italian Recent Advances in Applied Electromagnetics, G. Franceschetti, R. Pierri, eds. (Liguori, Naples, 1991), pp. 117–134.

Isobe, S.

Jeffries, S. M.

S. M. Jeffries, J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862–874 (1993).
[CrossRef]

Johnston, R. A.

R. G. Lane, R. A. Johnston, R. Irwan, T. J. Connoly, “Regularized blind deconvolution,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 5–7.

Jones, P. J.

Kantorovitch, L.

L. Kantorovitch, G. Akilov, Analyse Fonctionelle (Mir, Moscow, 1981).

Kawamura, S.

Keller, C. U.

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

Kundur, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13(5), 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13(11), 61–63 (1996).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in stars images,” Astron. Astrophys. 6, 85–87 (1970).

Lane, R. G.

N. F. Law, R. G. Lane, “Blind deconvolution using least squares minimization,” Opt. Commun. 128, 341–352 (1996).
[CrossRef]

R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
[CrossRef]

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. G. Lane, R. A. Johnston, R. Irwan, T. J. Connoly, “Regularized blind deconvolution,” in Signal Recovery and Synthesis, Vol. 11 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 5–7.

Law, N. F.

N. F. Law, R. G. Lane, “Blind deconvolution using least squares minimization,” Opt. Commun. 128, 341–352 (1996).
[CrossRef]

Leone, G.

T. Isernia, G. Leone, R. Pierri, “The phase retrieval in near zone as a nonlinear inverse problem: the planar scanning,” in Italian Recent Advances in Applied Electromagnetics, G. Franceschetti, R. Pierri, eds. (Liguori, Naples, 1991), pp. 117–134.

Lofdahl, M. G.

M. G. Lofdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Lofdahl, M. H.

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

Mastin, G. A.

McCallum, B. C.

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

Mikhailov, V. P.

V. P. Mikhailov, Partial Differential Equations (Mir, Moscow, 1978).

Miura, N.

Nagy, J. C.

Noguchi, M.

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Ohsawa, K.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, T. G. Stockman, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Parker, C. R.

Pauca, V. P.

Paxman, R. G.

Pierri, R.

T. Isernia, G. Leone, R. Pierri, “The phase retrieval in near zone as a nonlinear inverse problem: the planar scanning,” in Italian Recent Advances in Applied Electromagnetics, G. Franceschetti, R. Pierri, eds. (Liguori, Naples, 1991), pp. 117–134.

Plemmons, R. J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Romero, L. A.

Satherley, B. L.

Scales, J. A.

J. A. Scales, M. L. Smith, T. L. Fischer, “Global optimization methods for multimodal inverse problems,” J. Comput. Phys. 103, 258–268 (1992).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, T. G. Stockman, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Scharmer, G. B.

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

M. G. Lofdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Schulz, T. J.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

Smith, M. L.

J. A. Scales, M. L. Smith, T. L. Fischer, “Global optimization methods for multimodal inverse problems,” J. Comput. Phys. 103, 258–268 (1992).
[CrossRef]

Snyder, D. L.

Stark, H.

Stockman, T. G.

T. G. Stockman, T. M. Cannon, R. G. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, T. G. Stockman, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Thiébaut, E.

Tikhonov, A.

A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posées (Mir, Moscow, 1976).

Torgersen, T. C.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Watson, R. W.

White, R. L.

Yang, Y.

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in stars images,” Astron. Astrophys. 6, 85–87 (1970).

Astron. Astrophys. Suppl. Ser. (1)

M. G. Lofdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Astrophys. J. (2)

R. G. Paxman, J. H. Seldin, M. H. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

S. M. Jeffries, J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862–874 (1993).
[CrossRef]

IEEE Signal Process. Mag. (2)

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13(5), 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13(11), 61–63 (1996).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 58–63 (1976).
[CrossRef]

J. Comput. Phys. (1)

J. A. Scales, M. L. Smith, T. L. Fischer, “Global optimization methods for multimodal inverse problems,” J. Comput. Phys. 103, 258–268 (1992).
[CrossRef]

J. Opt. Soc. Am. A (14)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Object, normalized to its maximum, versus the scene pixels (128×128 upper right-hand samples of image NGC4024 in the Matlab Toolbox on Image Processing).

Fig. 3
Fig. 3

Blurred image, normalized to its maximum, versus the scene pixels corresponding to the object in Fig. 2.

Fig. 4
Fig. 4

Simulated OPD.

Fig. 5
Fig. 5

Reconstructed object, normalized to its maximum, versus the scene pixels.

Fig. 6
Fig. 6

Difference between the simulated optical path and the retrieved path.

Fig. 7
Fig. 7

Object, normalized to its maximum, versus the scene pixels.

Fig. 8
Fig. 8

Blurred image, normalized to its maximum, versus the scene pixels corresponding to the object in Fig. 7.

Fig. 9
Fig. 9

Simulated OPD.

Fig. 10
Fig. 10

Object and diffraction-limited image, normalized to the maximum: a section versus the scene pixels.

Fig. 11
Fig. 11

Object retrieved by the method presented in the paper and retrieved object restored by an iterative transform algorithm (ITA) normalized to their maxima: a section.

Fig. 12
Fig. 12

Difference between the simulated optical path and the retrieved path.

Fig. 13
Fig. 13

Object, normalized to its maximum, versus the scene pixels.

Fig. 14
Fig. 14

Blurred image, normalized to its maximum, versus the scene pixels corresponding to the object in Fig. 13.

Fig. 15
Fig. 15

Simulated OPD.

Fig. 16
Fig. 16

Retrieved object, normalized to its maximum, versus the scene pixels: a section.

Fig. 17
Fig. 17

Difference between the simulated optical path and the retrieved path.

Tables (1)

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Table 1 Values of Strehl Ratio and Similarity at Various Noise Levels

Equations (25)

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zk=u*sk,
sk=|hk|2,
Hk=pΣ exp[j(2πv+wk)],
vV=[vC1(Σ)||v|2(f/λ)2].
A|(u, v)L2(Ω)×Vz=(z1,, zK),
zkL2(Ω)C(Ω),k=1,, K,
A(uˆ, vˆ)A(L2, V)Z,
A(L2, V)Z,
F|(u, v)L2×VA(u, v)-Pz(A(u, v))L2K2,
L 2K=L2××L2K times
F|(u, v)L2×VA(u, v)-z¯L2K2,
Fα|(u, v)W21(Ω)×VA(u, v)-Pz[A(u, v)]L2K2+αuW21(Ω)2,
u(x, y)W212=u(x, y)L22+x u(x, y)L22+y u(x, y)L22.
v=ce=1Ncnnen.
Kα|(u, c)W21(Ω)×VcA(u, ce)-Pz[A(u, ce)]L2K(Ω)2+αuW21(Ω)2,
Vc={cRN|ceV}.
Kα,β|(u, c)W21(Ω)×RNA(u, ce)-Pz[A(u, ce)]L2K(Ω)2+αuW21(Ω)2+β|(ce)|2-P(|(ce)|2)L2(Σ)2.
Mα,β(U, c, Q, r)=1KkSkU-QkL22+α(UL22+ξUL22+ηUL22)+β|(ce)|2-rL22,
q(n)=Pz{A[u(n), c(n)e]},
r(n)=P{|[c(n)e]|2},
U(n+1)=1K kSk*[c(n), ξ,η]Qk(n)(ξ, η)α(1+ξ2+η2)+1K|mSm[c(n), ξ,η]|2,
c(n+1)=infcR MNα,β[U(n+1),c,Q(n),r(n)],
z˜=(z˜1,,z˜K)L 2K=L2××L2K times
d2(z˜, Z)=infzZz˜-zL2K2=infzZ 1Kkz˜k-zkL22,
d(z˜, Z)=d[z˜, Pz(z˜)].

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