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Volumn 19, Issue 7, 2002, Pages 1334-1345

Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization

Author keywords

[No Author keywords available]

Indexed keywords

ERROR ANALYSIS; FUNCTIONS; INTERFEROMETRY; OPTIMIZATION;

EID: 0038266150     PISSN: 10847529     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAA.19.001334     Document Type: Article
Times cited : (522)

References (112)
  • 2
    • 0035359284 scopus 로고    scopus 로고
    • Dispersion relations and phase retrieval in infrared reflection spectra analysis
    • J.-F. Brun, D. de Sousa Meneses, B. Rousseau, and P. Echegut, "Dispersion relations and phase retrieval in infrared reflection spectra analysis," Appl. Spectrosc. 55, 774-780 (2001).
    • (2001) Appl. Spectrosc. , vol.55 , pp. 774-780
    • Brun, J.-F.1    De Sousa Meneses, D.2    Rousseau, B.3    Echegut, P.4
  • 3
    • 0002614571 scopus 로고
    • Phase retrieval and image reconstruction for astronomy
    • H. Stark, ed. (Academic, Orlando, Fla.)
    • J. C. Dainty and J. R. Fienup, "Phase retrieval and image reconstruction for astronomy," in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 231-275.
    • (1987) Image Recovery: Theory and Application , pp. 231-275
    • Dainty, J.C.1    Fienup, J.R.2
  • 4
    • 0002940560 scopus 로고    scopus 로고
    • A feasible set approach to the crystallographic phase problem
    • L. D. Marks, W. Sinkler, and E. Landree, "A feasible set approach to the crystallographic phase problem," Acta Crystallogr. Sect. A 55, 601-612 (1999).
    • (1999) Acta Crystallogr. Sect. A , vol.55 , pp. 601-612
    • Marks, L.D.1    Sinkler, W.2    Landree, E.3
  • 5
    • 84975597896 scopus 로고
    • Phase retrieval in crystallography and optics
    • R. P. Millane, "Phase retrieval in crystallography and optics, " J. Opt. Soc. Am. A 7, 394-411 (1990).
    • (1990) J. Opt. Soc. Am. A , vol.7 , pp. 394-411
    • Millane, R.P.1
  • 6
    • 0347564344 scopus 로고    scopus 로고
    • Phase retrieval techniques for radar ambiguity problems
    • P. Jaming, "Phase retrieval techniques for radar ambiguity problems," J. Fourier Anal. Appl. 5, 309-329 (1999).
    • (1999) J. Fourier Anal. Appl. , vol.5 , pp. 309-329
    • Jaming, P.1
  • 8
    • 0019539710 scopus 로고
    • The phase retrieval problem
    • L. S. Taylor, "The phase retrieval problem," IEEE Trans. Antennas Propag. AP-29, 386-391 (1981).
    • (1981) IEEE Trans. Antennas Propag. , vol.AP-29 , pp. 386-391
    • Taylor, L.S.1
  • 9
    • 51249193129 scopus 로고
    • Analyticity and phase retrieval
    • P. Roman and A. S. Marathay, "Analyticity and phase retrieval, " Nuovo Cimento 30, 1452-1464 (1963).
    • (1963) Nuovo Cimento , vol.30 , pp. 1452-1464
    • Roman, P.1    Marathay, A.S.2
  • 10
    • 84867549090 scopus 로고
    • The question of phase retrieval in optics
    • A. Walther, "The question of phase retrieval in optics," Opt. Acta 10, 41-49 (1963).
    • (1963) Opt. Acta , vol.10 , pp. 41-49
    • Walther, A.1
  • 11
    • 0000146113 scopus 로고
    • Is a complete determination of the energy spectrum of light possible from measurements of degree of coherence?
    • E. Wolf, "Is a complete determination of the energy spectrum of light possible from measurements of degree of coherence?" Proc. Phys. Soc. London 80, 1269-1272 (1962).
    • (1962) Proc. Phys. Soc. London , vol.80 , pp. 1269-1272
    • Wolf, E.1
  • 12
    • 0005863744 scopus 로고
    • On the interference bands of approximately homogeneous light; In a letter to Prof. A. Michelson
    • Lord Rayleigh (J. W. Strutt), "On the interference bands of approximately homogeneous light; in a letter to Prof. A. Michelson," Philos. Mag. 34, 407-411 (1892).
    • (1892) Philos. Mag. , vol.34 , pp. 407-411
    • Rayleigh, L.1
  • 13
    • 0015327061 scopus 로고
    • A practical algorithm for the determination of phase from image and diffraction plane pictures
    • R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).
    • (1972) Optik (Stuttgart) , vol.35 , pp. 237-246
    • Gerchberg, R.W.1    Saxton, W.O.2
  • 14
    • 0020173780 scopus 로고
    • Phase retrieval algorithms: A comparison
    • J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    • (1982) Appl. Opt. , vol.21 , pp. 2758-2769
    • Fienup, J.R.1
  • 15
    • 0020191832 scopus 로고
    • Image restoration by the method of convex projections: Part I - Theory
    • D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part I - theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
    • (1982) IEEE Trans. Med. Imaging , vol.MI-1 , pp. 81-94
    • Youla, D.C.1    Webb, H.2
  • 16
    • 0019625454 scopus 로고
    • An iterative method for the extrapolation of band-limited functions
    • A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554-565 (1981).
    • (1981) J. Math. Anal. Appl. , vol.83 , pp. 554-565
    • Lent, A.1    Tuy, H.2
  • 17
    • 0020766276 scopus 로고
    • Signal reconstruction from phase by projection onto convex sets
    • A. Levi and H. Stark, "Signal reconstruction from phase by projection onto convex sets," J. Opt. Soc. Am. 73, 810-822 (1983).
    • (1983) J. Opt. Soc. Am. , vol.73 , pp. 810-822
    • Levi, A.1    Stark, H.2
  • 19
    • 0001336448 scopus 로고
    • The method of successive projection for finding a common point of convex sets
    • L. M. Brègman, "The method of successive projection for finding a common point of convex sets," Sov. Math. Dokl. 6, 688-692 (1965).
    • (1965) Sov. Math. Dokl. , vol.6 , pp. 688-692
    • Brègman, L.M.1
  • 20
    • 0027541192 scopus 로고
    • The foundations of set theoretic estimation
    • P. L. Combettes, "The foundations of set theoretic estimation, " Proc. IEEE 81, 182-208 (1993).
    • (1993) Proc. IEEE , vol.81 , pp. 182-208
    • Combettes, P.L.1
  • 21
    • 77954607855 scopus 로고    scopus 로고
    • The convex feasibility problem in image recovery
    • P. W. Hawkes, ed. (Academic, Orlando, Fla.)
    • P. L. Combettes, "The convex feasibility problem in image recovery," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1996), Vol. 95, pp. 155-270.
    • (1996) Advances in Imaging and Electron Physics , vol.95 , pp. 155-270
    • Combettes, P.L.1
  • 25
    • 0026117505 scopus 로고
    • Motion artifact correction in MRI using generalized projections
    • M. Hedley, H. Yan, and D. Rosenfeld, "Motion artifact correction in MRI using generalized projections," IEEE Trans. Med. Imaging 10, 40-46 (1991).
    • (1991) IEEE Trans. Med. Imaging , vol.10 , pp. 40-46
    • Hedley, M.1    Yan, H.2    Rosenfeld, D.3
  • 27
    • 84975659910 scopus 로고
    • Image construction: Optimum amplitude and phase masks in photolithography
    • B. E. A. Saleh and K. M. Nashold, "Image construction: optimum amplitude and phase masks in photolithography," Appl. Opt. 24, 1432-1437 (1985).
    • (1985) Appl. Opt. , vol.24 , pp. 1432-1437
    • Saleh, B.E.A.1    Nashold, K.M.2
  • 28
    • 0021493772 scopus 로고
    • Image restoration by the method of generalized projections with application to restoration from magnitude
    • A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
    • (1984) J. Opt. Soc. Am. A , vol.1 , pp. 932-943
    • Levi, A.1    Stark, H.2
  • 29
    • 0002027588 scopus 로고
    • Restoration from phase and magnitude by generalized projections
    • H. Stark, ed. (Academic, Orlando, Fla.)
    • A. Levi and H. Stark, "Restoration from phase and magnitude by generalized projections," in Image Recovery; Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp 277-320.
    • (1987) Image Recovery; Theory and Applications , pp. 277-320
    • Levi, A.1    Stark, H.2
  • 30
    • 0023827797 scopus 로고
    • Signal enhancement - A composite property mapping algorithm
    • J. A. Cadzow, "Signal enhancement - a composite property mapping algorithm," IEEE Trans. Acoust. Speech Signal Process. 36, 49-62 (1988).
    • (1988) IEEE Trans. Acoust. Speech Signal Process. , vol.36 , pp. 49-62
    • Cadzow, J.A.1
  • 31
    • 0025535641 scopus 로고
    • Method of successive projections for finding a common point of sets in metric spaces
    • P. L. Combettes and H. J. Trussell, "Method of successive projections for finding a common point of sets in metric spaces," J. Optim. Theory Appl. 67, 487-507 (1990).
    • (1990) J. Optim. Theory Appl. , vol.67 , pp. 487-507
    • Combettes, P.L.1    Trussell, H.J.2
  • 32
    • 0042098472 scopus 로고
    • Signal enhancement and the method of successive projections
    • N. E. Hurt, "Signal enhancement and the method of successive projections," Acta Appl. Math. 23, 145-162 (1991).
    • (1991) Acta Appl. Math. , vol.23 , pp. 145-162
    • Hurt, N.E.1
  • 33
    • 0005791455 scopus 로고    scopus 로고
    • Cyclic projection methods on a class of nonconvex sets
    • S. Chrétien and P. Bondon, "Cyclic projection methods on a class of nonconvex sets," Numer. Funct. Anal. Optim. 17, 37-56 (1996).
    • (1996) Numer. Funct. Anal. Optim. , vol.17 , pp. 37-56
    • Chrétien, S.1    Bondon, P.2
  • 34
    • 0005881344 scopus 로고
    • Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data
    • R. Barakat and G. Newsam, "Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data," J. Opt. Soc. Am. A 2, 2027-2039 (1985).
    • (1985) J. Opt. Soc. Am. A , vol.2 , pp. 2027-2039
    • Barakat, R.1    Newsam, G.2
  • 35
    • 0022092459 scopus 로고
    • Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data. II. The nonlinear problem of phase retrieval
    • R. Barakat and G. Newsam, "Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data. II. The nonlinear problem of phase retrieval," J. Integral Eq. 9, 77-125 (1985).
    • (1985) J. Integral Eq. , vol.9 , pp. 77-125
    • Barakat, R.1    Newsam, G.2
  • 36
    • 84894009772 scopus 로고    scopus 로고
    • note
    • The issue of nonuniqueness of the projection is not to be confused with the uniqueness of solutions to the phase problem. The results surveyed, for instance, in Ref. 37, are not affected by the multivaluedness of the projection operators.
  • 37
    • 0001412939 scopus 로고
    • The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase
    • H. Stark, ed. (Academic, Orlando, Fla.)
    • M. H. Hayes, "The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase," in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 195-230.
    • (1987) Image Recovery: Theory and Application , pp. 195-230
    • Hayes, M.H.1
  • 39
    • 0036602309 scopus 로고    scopus 로고
    • Optical wavefront reconstruction: Theory and numerical methods
    • D. R. Luke, J. V. Burke, and R. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Rev. 44(2) (2002), ftp://amath.washington.edu/pub/russell/Luke_Burke_Lyon_01.ps.gz.
    • (2002) SIAM Rev. , vol.44 , Issue.2
    • Luke, D.R.1    Burke, J.V.2    Lyon, R.3
  • 40
    • 0017985106 scopus 로고
    • Reconstruction of an object from the modulus of its Fourier transform
    • J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27-29 (1978).
    • (1978) Opt. Lett. , vol.3 , pp. 27-29
    • Fienup, J.R.1
  • 41
    • 0018518696 scopus 로고
    • Space object imaging through the turbulent atmosphere
    • J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529-534 (1979).
    • (1979) Opt. Eng. , vol.18 , pp. 529-534
    • Fienup, J.R.1
  • 42
    • 0019016018 scopus 로고
    • Iterative method applied to image reconstruction and to computer-generated holograms
    • J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).
    • (1980) Opt. Eng. , vol.19 , pp. 297-305
    • Fienup, J.R.1
  • 44
    • 84894004589 scopus 로고    scopus 로고
    • note
    • "a.e." stands for "almost everywhere" in the sense of measure theory, since, strictly speaking, the elements of ℒ are classes of equivalence of signals that may differ on a set of zero measure. For technical details on, see, for instance, Ref. 45.
  • 47
    • 0028543165 scopus 로고
    • Inconsistent signal feasibility problems: Least-squares solutions in a product space
    • P. L. Combettes, "Inconsistent signal feasibility problems: least-squares solutions in a product space," IEEE Trans. Signal Process. 42, 2955-2966 (1994).
    • (1994) IEEE Trans. Signal Process. , vol.42 , pp. 2955-2966
    • Combettes, P.L.1
  • 49
    • 84902143152 scopus 로고
    • Numerical investigation of the uniqueness of phase retrieval
    • J. H. Seldin and J. R. Fienup, "Numerical investigation of the uniqueness of phase retrieval," J. Opt. Soc. Am. A 7, 412-427 (1990).
    • (1990) J. Opt. Soc. Am. A , vol.7 , pp. 412-427
    • Seldin, J.H.1    Fienup, J.R.2
  • 50
    • 0005828958 scopus 로고    scopus 로고
    • The composition of finitely many projections onto closed convex sets in Hubert space is asymptotically regular
    • to be published
    • H. H. Bauschke, "The composition of finitely many projections onto closed convex sets in Hubert space is asymptotically regular," Proc. Am. Math. Soc. (to be published).
    • Proc. Am. Math. Soc.
    • Bauschke, H.H.1
  • 51
    • 0001448913 scopus 로고
    • On the convergence of von Neumann's alternating projection algorithm for two sets
    • H. H. Bauschke and J. M. Borwein, "On the convergence of von Neumann's alternating projection algorithm for two sets," Set-Valued Anal. 1, 185-212 (1993).
    • (1993) Set-Valued Anal. , vol.1 , pp. 185-212
    • Bauschke, H.H.1    Borwein, J.M.2
  • 53
    • 0032653272 scopus 로고    scopus 로고
    • Hard-constrained inconsistent signal feasibility problems
    • P. L. Combettes and P. Bondon, "Hard-constrained inconsistent signal feasibility problems," IEEE Trans. Signal Process. 47, 2460-2468 (1999).
    • (1999) IEEE Trans. Signal Process. , vol.47 , pp. 2460-2468
    • Combettes, P.L.1    Bondon, P.2
  • 54
    • 33845708830 scopus 로고
    • The method of projections for finding the common point of convex sets
    • L. G. Gubin, B. T. Polyak, and E. V. Raik, "The method of projections for finding the common point of convex sets," USSR Comput. Math. Math. Phys. 7, 1-24 (1967).
    • (1967) USSR Comput. Math. Math. Phys. , vol.7 , pp. 1-24
    • Gubin, L.G.1    Polyak, B.T.2    Raik, E.V.3
  • 55
    • 0022705171 scopus 로고
    • Extensions of a result on the synthesis of signals in the presence of inconsistent constraints
    • D. C. Youla and V. Velasco, "Extensions of a result on the synthesis of signals in the presence of inconsistent constraints," IEEE Trans, Circuits Syst. CAS-33, 465-468 (1986).
    • (1986) IEEE Trans, Circuits Syst. , vol.CAS-33 , pp. 465-468
    • Youla, D.C.1    Velasco, V.2
  • 56
    • 84893985982 scopus 로고    scopus 로고
    • note
    • Iterative signal recovery projection algorithms have also been implemented optically without sampling the continuous waveforms (e.g., Ref. 57). In such instances, the underlying signal space is ℒ itself.
  • 57
    • 0019010463 scopus 로고
    • Coherent optical extrapolation of 2-D band-limited signals: Processor theory
    • R. J. Marks II, "Coherent optical extrapolation of 2-D band-limited signals: processor theory," Appl. Opt. 19, 1670-1672 (1980).
    • (1980) Appl. Opt. , vol.19 , pp. 1670-1672
    • Marks R.J. II1
  • 58
    • 84894011274 scopus 로고    scopus 로고
    • note
    • Let A be a set of real numbers. If R ≠ Ø, then inf(R) stands for the infimum of R, i.e., the largest number in [-∞, +x] that is smaller than or equal to all elements of R. By convention, inf(Ø) = +∞.
  • 60
    • 84893997261 scopus 로고    scopus 로고
    • note
    • For theoretical reasons, the sets (and functions) that we deal with must be "measurable" - this is not the same as being "physically measurable" or "observable"! For our purposes, measurable sets and functions constitute a sufficiently large class to work with; thus all closed and open subsets (and all continuous functions) are measurable, as well as various combinations of those.
  • 61
    • 84893993224 scopus 로고    scopus 로고
    • note
    • Mathematically, this set is assumed to have nonzero measure.
  • 62
    • 84893988601 scopus 로고    scopus 로고
    • note
    • The complex Hubert space ℒ from the phase retrieval problem is also a real Hubert space, provided that we use the real part of the inner product as the new inner product.
  • 64
    • 84894009148 scopus 로고    scopus 로고
    • Recall the notation from Remark 3.4
    • Recall the notation from Remark 3.4.
  • 66
    • 84893998190 scopus 로고    scopus 로고
    • note
    • N: A(t) ∩ Z ≠ Ø} is measurable for every closed (or, equivalently, open) set Z in C; see Section 8.1 in Ref. 59 and Section 14.A in Ref. 67.
  • 68
    • 84893994946 scopus 로고    scopus 로고
    • note
    • +: t → max{0, Re[x(t)]}.
  • 69
    • 84949069862 scopus 로고
    • Super-resolution through error energy reduction
    • R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709-720 (1974).
    • (1974) Opt. Acta , vol.21 , pp. 709-720
    • Gerchberg, R.W.1
  • 70
    • 0018008609 scopus 로고
    • Generalized image restoration by the method of alternating orthogonal projections
    • D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 694-702 (1978).
    • (1978) IEEE Trans. Circuits Syst. , vol.CAS-25 , pp. 694-702
    • Youla, D.C.1
  • 71
    • 0016552759 scopus 로고
    • A new algorithm in spectral analysis and band-limited extrapolation
    • A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
    • (1975) IEEE Trans. Circuits Syst. , vol.CAS-22 , pp. 735-742
    • Papoulis, A.1
  • 72
    • 84975560695 scopus 로고
    • Convolution-based framework for signal recovery and applications
    • A. E. Çetin and R. Ansari, "Convolution-based framework for signal recovery and applications," J. Opt. Soc. Am. A 5, 1193-1200 (1988).
    • (1988) J. Opt. Soc. Am. A , vol.5 , pp. 1193-1200
    • Çetin, A.E.1    Ansari, R.2
  • 74
    • 84968481460 scopus 로고
    • Weak convergence of the sequence of successive approximations for nonexpansive mappings
    • Z. Opial, "Weak convergence of the sequence of successive approximations for nonexpansive mappings," Bull. Am. Math. Soc. 78, 591-597 (1967).
    • (1967) Bull. Am. Math. Soc. , vol.78 , pp. 591-597
    • Opial, Z.1
  • 75
    • 51249189005 scopus 로고
    • An example concerning fixed points
    • A. Genel and J. Lindenstrauss, "An example concerning fixed points," Isr. J. Math. 22, 81-86 (1975).
    • (1975) Isr. J. Math. , vol.22 , pp. 81-86
    • Genel, A.1    Lindenstrauss, J.2
  • 76
    • 84894002082 scopus 로고    scopus 로고
    • note
    • Alternative descriptions of these algorithms have been proposed; see, for instance, Ref. 77.
  • 77
    • 0025430092 scopus 로고
    • Image recovery using iterative data refinement with relaxation
    • G. T. Herman and D.-W. Ro, "Image recovery using iterative data refinement with relaxation," Opt. Eng. 29, 513-523 (1990).
    • (1990) Opt. Eng. , vol.29 , pp. 513-523
    • Herman, G.T.1    Ro, D.-W.2
  • 78
    • 84894012228 scopus 로고    scopus 로고
    • note
    • 14
  • 79
    • 84893997058 scopus 로고    scopus 로고
    • note
    • n). In practical experiments for problem (5), however, this ambiguity has hardly an impact, as the sets γ and C-D almost always coincide.
  • 80
    • 0032620199 scopus 로고    scopus 로고
    • Reconstruction of an object from its noisy Fourier modulus: Ideal estimate of the object to be constructed and a method that attempts to find that object
    • H. Takajo, T. Shizuma, T. Takahashi, and S. Takahata, "Reconstruction of an object from its noisy Fourier modulus: ideal estimate of the object to be constructed and a method that attempts to find that object," Appl. Opt. 38, 5568-5576 (1999).
    • (1999) Appl. Opt. , vol.38 , pp. 5568-5576
    • Takajo, H.1    Shizuma, T.2    Takahashi, T.3    Takahata, S.4
  • 81
    • 0000635375 scopus 로고    scopus 로고
    • Further study on the convergence property of the hybrid input-output algorithm used for phase retrieval
    • H. Takajo, T. Takahashi, and T. Shizuma, "Further study on the convergence property of the hybrid input-output algorithm used for phase retrieval," J. Opt. Soc. Am. A 16, 2163-2168 (1999).
    • (1999) J. Opt. Soc. Am. A , vol.16 , pp. 2163-2168
    • Takajo, H.1    Takahashi, T.2    Shizuma, T.3
  • 82
    • 0032209426 scopus 로고    scopus 로고
    • Study on the convergence property of the hybrid input-output algorithm used for phase retrieval
    • H. Takajo, T. Takahashi, R. Ueda, and M. Taninaka, "Study on the convergence property of the hybrid input-output algorithm used for phase retrieval," J. Opt. Soc. Am. A 15, 2849-2861 (1998).
    • (1998) J. Opt. Soc. Am. A , vol.15 , pp. 2849-2861
    • Takajo, H.1    Takahashi, T.2    Ueda, R.3    Taninaka, M.4
  • 83
    • 84893999618 scopus 로고    scopus 로고
    • note
    • The corresponding mask is certainly much easier to implement.
  • 84
    • 84975571733 scopus 로고
    • Phase retrieval stagnation problems and solutions
    • J. R. Fienup and C. C. Wackerman, "Phase retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986).
    • (1986) J. Opt. Soc. Am. A , vol.3 , pp. 1897-1907
    • Fienup, J.R.1    Wackerman, C.C.2
  • 85
    • 84893992567 scopus 로고    scopus 로고
    • note
    • The algorithms discussed here for solving problem (25) can be viewed in the broader context of finding a zero of the sum of two maximal monotone operators. Good starting points are Refs. 86-88.
  • 86
    • 0043100459 scopus 로고    scopus 로고
    • Fejér-monotonicity in convex optimization
    • C. A. Floudas and P. M. Pardalos, eds. (Kluwer, New York)
    • P. L. Combettes, "Fejér-monotonicity in convex optimization," in Encyclopedia of Optimization, C. A. Floudas and P. M. Pardalos, eds. (Kluwer, New York, 2001), Vol. 2, pp. 106-114.
    • (2001) Encyclopedia of Optimization , vol.2 , pp. 106-114
    • Combettes, P.L.1
  • 87
    • 0004177997 scopus 로고
    • Ph.D. thesis (Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass.), available as Rep. LIDS-TH-1877 (Laboratory for Information and Decision Sciences, MIT)
    • J. Eckstein, "Splitting methods for monotone operators with applications to parallel optimization," Ph.D. thesis (Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1989), available as Rep. LIDS-TH-1877 (Laboratory for Information and Decision Sciences, MIT).
    • (1989) Splitting Methods for Monotone Operators with Applications to Parallel Optimization
    • Eckstein, J.1
  • 88
    • 0027113845 scopus 로고
    • On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators
    • J. Eckstein and D. P. Bertsekas, "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators," Math. Program. (Ser. A) 55, 293-318 (1992).
    • (1992) Math. Program. (Ser. A) , vol.55 , pp. 293-318
    • Eckstein, J.1    Bertsekas, D.P.2
  • 89
    • 84894009949 scopus 로고    scopus 로고
    • note
    • 90
  • 90
    • 0000256894 scopus 로고
    • Nonexpansive projections and resolvents of accretive operators in Banach spaces
    • R. B. Bruck and S. Reich, "Nonexpansive projections and resolvents of accretive operators in Banach spaces," Houston J. Math. 3, 459-470 (1977).
    • (1977) Houston J. Math. , vol.3 , pp. 459-470
    • Bruck, R.B.1    Reich, S.2
  • 91
    • 0030246542 scopus 로고    scopus 로고
    • On projection algorithms for solving convex feasibility problems
    • H. H. Bauschke and J. M. Borwein, "On projection algorithms for solving convex feasibility problems," SIAM Rev. 38, 367-426 (1996).
    • (1996) SIAM Rev. , vol.38 , pp. 367-426
    • Bauschke, H.H.1    Borwein, J.M.2
  • 94
    • 84948499836 scopus 로고
    • An algorithm for restricted least squares regression
    • R. L. Dykstra, "An algorithm for restricted least squares regression," J. Am. Stat. Assoc. 78, 837-842 (1983).
    • (1983) J. Am. Stat. Assoc. , vol.78 , pp. 837-842
    • Dykstra, R.L.1
  • 95
    • 0002436271 scopus 로고
    • A method for finding projections onto the intersection of convex sets in Hubert spaces
    • Springer, Berlin
    • J. P. Boyle and R. L. Dykstra, "A method for finding projections onto the intersection of convex sets in Hubert spaces," in Advances in Order Restricted Statistical Inference (Springer, Berlin, 1986), pp. 28-47.
    • (1986) Advances in Order Restricted Statistical Inference , pp. 28-47
    • Boyle, J.P.1    Dykstra, R.L.2
  • 96
    • 84894010481 scopus 로고    scopus 로고
    • note
    • 85 Dykstra's algorithm can be interpreted as a tight version of the Peaceman-Rachford algorithm. See page 77 in Ref. 87 for further information. Let us also note that in the standard linear case, the Peaceman-Rachford and Douglas-Rachford algorithms can be viewed from a unifying standpoint (see Section 7.4 in Ref. 97).
  • 98
    • 0037976260 scopus 로고
    • Dykstra's alternating projection algorithm for two sets
    • H. H. Bauschke and J. M. Borwein, "Dykstra's alternating projection algorithm for two sets," J. Approx. Theory 79, 418-443 (1994).
    • (1994) J. Approx. Theory , vol.79 , pp. 418-443
    • Bauschke, H.H.1    Borwein, J.M.2
  • 99
    • 0042391294 scopus 로고    scopus 로고
    • Dykstra's algorithm with Bregman projections: A convergence proof
    • H. H. Bauschke and A. S. Lewis, "Dykstra's algorithm with Bregman projections: a convergence proof," Optimization 48, 409-427 (2000).
    • (2000) Optimization , vol.48 , pp. 409-427
    • Bauschke, H.H.1    Lewis, A.S.2
  • 100
    • 0001886656 scopus 로고
    • A cyclic projection algorithm via duality
    • N. Gaffke and R. Mathar, "A cyclic projection algorithm via duality," Metrika 36, 29-54 (1989).
    • (1989) Metrika , vol.36 , pp. 29-54
    • Gaffke, N.1    Mathar, R.2
  • 101
    • 0023860607 scopus 로고
    • A successive projection method
    • S.-P. Han, "A successive projection method," Math. Program. (Ser. A) 40, 1-14 (1988).
    • (1988) Math. Program. (Ser. A) , vol.40 , pp. 1-14
    • Han, S.-P.1
  • 102
    • 0009208155 scopus 로고    scopus 로고
    • Two generalizations of Dykstra's cyclic projections algorithm
    • H. Hundal and F. Deutsch, "Two generalizations of Dykstra's cyclic projections algorithm," Math. Program. (Ser. A) 77, 335-355 (1997).
    • (1997) Math. Program. (Ser. A) , vol.77 , pp. 335-355
    • Hundal, H.1    Deutsch, F.2
  • 103
    • 0027579732 scopus 로고
    • Signal recovery by best feasible approximation
    • P. L. Combettes, "Signal recovery by best feasible approximation," IEEE Trans. Image Process. 2, 269-271 (1993).
    • (1993) IEEE Trans. Image Process. , vol.2 , pp. 269-271
    • Combettes, P.L.1
  • 104
    • 84893987214 scopus 로고    scopus 로고
    • note
    • The Douglas-Rachford algorithm was originally developed as a linear implicit iterative method to solve partial differential equations in Ref. 105 (see also Chap. 7 in Ref. 97). It was extended to an operator splitting method for finding a zero of the sum of two maximal monotone operators by Lions and Mercier in Ref. 106. When it is applied to the normal cone maps of the constraint sets, one obtains a method for solving problem (25). See Refs. 86-88 and 106 for further information.
  • 105
    • 84967782959 scopus 로고
    • On the numerical solution of heat conduction problems in two or three space variables
    • J. Douglas and H. H. Rachford, "On the numerical solution of heat conduction problems in two or three space variables," Trans. Am. Math. Soc. 82, 421-439 (1956).
    • (1956) Trans. Am. Math. Soc. , vol.82 , pp. 421-439
    • Douglas, J.1    Rachford, H.H.2
  • 106
    • 0000345334 scopus 로고
    • Splitting algorithms for the sum of two nonlinear operators
    • SIAM (Soc. Ind. Appl. Math.)
    • P.-L. Lions and B. Mercier, "Splitting algorithms for the sum of two nonlinear operators," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 16, 964-979 (1979).
    • (1979) J. Numer. Anal. , vol.16 , pp. 964-979
    • Lions, P.-L.1    Mercier, B.2
  • 107
    • 84893994753 scopus 로고    scopus 로고
    • note
    • kn ⇀ u.
  • 108
    • 84894005553 scopus 로고    scopus 로고
    • note
    • If we had used the literal update rule for the HIO algorithm, the present observation would change only in one respect: the set A would be replaced with S(n) (see Remark 4.1 and Ref. 79) and hence vary with n.
  • 109
    • 0002130882 scopus 로고
    • Projections on convex sets in Hubert space and spectral theory
    • E. H. Zarantonello, ed. (Academic, New York)
    • E. H. Zarantonello, "Projections on convex sets in Hubert space and spectral theory," in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed. (Academic, New York, 1971), pp. 237-424.
    • (1971) Contributions to Nonlinear Functional Analysis , pp. 237-424
    • Zarantonello, E.H.1
  • 111
    • 84893991498 scopus 로고    scopus 로고
    • note
    • n) of points in M converges weakly to a point x, then x may not be in M.
  • 112
    • 84894009913 scopus 로고    scopus 로고
    • note
    • Bx), as is the case when dim ℋ < +∞ (or when B is a closed affine subspace). Note, however, that the projector onto a closed convex set may fail to be weakly continuous. An example is on page 245 in Ref. 109.


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