H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).

[CrossRef]

D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).

[CrossRef]

H. H. Bauschke, J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1, 185–212 (1993).

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).

[CrossRef]

P.-L. Lions, B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16, 964–979 (1979).

[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).

[CrossRef]

H. H. Bauschke, J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1, 185–212 (1993).

H. H. Bauschke, J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1, 185–212 (1993).

J. P. Boyle, R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in Hilbert spaces,” Lect. Notes Stat. 37, 28–47 (1986).

D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).

[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).

[CrossRef]

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).

[CrossRef]

J. C. Dainty, 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.

J. P. Boyle, R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in Hilbert spaces,” Lect. Notes Stat. 37, 28–47 (1986).

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bound on object support from autocorrelation support and application to phase retrival,” J. Opt. Soc. Am. A 7, 3–13 (1990).

[CrossRef]

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

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

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J. C. Dainty, 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.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

A. Levi, 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).

[CrossRef]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 277–320.

P.-L. Lions, B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16, 964–979 (1979).

[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).

[CrossRef]

D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).

[CrossRef]

D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).

[CrossRef]

P.-L. Lions, B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16, 964–979 (1979).

[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

A. Levi, 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).

[CrossRef]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 277–320.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).

[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).

[CrossRef]

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).

[CrossRef]

A. Levi, 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).

[CrossRef]

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bound on object support from autocorrelation support and application to phase retrival,” J. Opt. Soc. Am. A 7, 3–13 (1990).

[CrossRef]

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

[CrossRef]

J. P. Boyle, R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in Hilbert spaces,” Lect. Notes Stat. 37, 28–47 (1986).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).

[CrossRef]

H. H. Bauschke, J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1, 185–212 (1993).

P.-L. Lions, B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16, 964–979 (1979).

[CrossRef]

D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).

[CrossRef]

H. Stark, ed., Image Recovery: Theory and Application (Academic, Orlando, Fla.1987).

Typically, 0.5≤β≤1.

It was pointed out in Remark 4.1 of Ref. 9that a more literal reformulation of Eq. (13) leads to (31)(∀t∈X)xn+1(t)=(PM(xn))(t)if t∈D or (PM(xn))(t)=0xn(t)-β(PM(xn))(t)otherwise.Under the assumption that the zero crossings of PM(xn)outside of Dare negligible, Eq. (31) reduces to Eq. (14). In digital computing, this assumption is justified by the fact that the probability of obtaining zero numbers is virtually zero.

If m=0,then x=0is the unique solution of the corresponding phase-retrieval problem.

For the BIO algorithm, this was already pointed out in Remark 5.6 of Ref. 9.

We stress that monitoring the sequences (PM(xn))n∈Nand (PS+PM(xn))n∈Nis well motivated from the convex consistent setting. Replace the nonconvex set Mand its corresponding projector PMwith the convex set Band the corre-sponding projector PB.If S+∩B≠∅,then, using the results in Ref. 9, one can prove that ES+(xn)→0with equality precisely when PB(xn)∈S+∩B.However, if S+∩B=∅,which is likely a better approximation of the geometry of the phase-retrieval problem, then minimizing ES+(xn)corresponds to finding a displacement vector for S+and Bin the sense of Ref. 19. If the problem is feasible but ES+(xn)is positive, then the algorithm has stagnated, and the value of ES+(xn)is an indicator of the quality of the stagnation point.

J. C. Dainty, 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.

In general, the object-domain space need not be restricted to real-valued signals. For a review of a phase-retrieval application in which the iterates are complex valued, see Ref. 12.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 277–320.