M. H. Hayes, "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform," IEEE Trans. Acoust. Speech Signal Process. ASSP-30,140–154 (1982).

M. I. Sezan and H. Stark, "Image restoration by the method of projections onto convex sets—Part II," IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part I," Optik 61, 247–262 (1982).

K. L. Garden and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part II," Optik 62, 131–142 (1982).

W. R. Fright and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part III," Optik 62, 219–230 (1982).

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

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets—Part I," IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," J Math. Anal. Appl. 83, 554–565 (1981).

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).

W. A. Pearlman and R. M. Gray, "Source coding of the discrete Fourier transform," IEEE Trans. Inf. Theory IT-24, 683–692 (1978).

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 894–702 (1978).

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

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).

W. R. Fright and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part III," Optik 62, 219–230 (1982).

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part I," Optik 61, 247–262 (1982).

K. L. Garden and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part II," Optik 62, 131–142 (1982).

H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Bates, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.

W. R. Fright and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part III," Optik 62, 219–230 (1982).

K. L. Garden and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part II," Optik 62, 131–142 (1982).

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

W. A. Pearlman and R. M. Gray, "Source coding of the discrete Fourier transform," IEEE Trans. Inf. Theory IT-24, 683–692 (1978).

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).

M. H. Hayes, "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform," IEEE Trans. Acoust. Speech Signal Process. ASSP-30,140–154 (1982).

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," J Math. Anal. Appl. 83, 554–565 (1981).

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).

A. V. Oppenheim, "The importance of phase in signals," Proc. IEEE 69, 529–541 (1981).

W. A. Pearlman and R. M. Gray, "Source coding of the discrete Fourier transform," IEEE Trans. Inf. Theory IT-24, 683–692 (1978).

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).

T. Quatieri, "Phase estimation with application to speech analysis," Sc.D. Dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1979).

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).

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

M. I. Sezan and H. Stark, "Image restoration by the method of projections onto convex sets—Part II," IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

M. I. Sezan and H. Stark, "Image restoration by the method of projections onto convex sets—Part II," IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," J Math. Anal. Appl. 83, 554–565 (1981).

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets—Part I," IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets—Part I," IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 894–702 (1978).

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).

M. H. Hayes, "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform," IEEE Trans. Acoust. Speech Signal Process. ASSP-30,140–154 (1982).

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 894–702 (1978).

W. A. Pearlman and R. M. Gray, "Source coding of the discrete Fourier transform," IEEE Trans. Inf. Theory IT-24, 683–692 (1978).

M. I. Sezan and H. Stark, "Image restoration by the method of projections onto convex sets—Part II," IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets—Part I," IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," J Math. Anal. Appl. 83, 554–565 (1981).

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part I," Optik 61, 247–262 (1982).

K. L. Garden and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part II," Optik 62, 131–142 (1982).

W. R. Fright and R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension, Part III," Optik 62, 219–230 (1982).

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

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).

An image is regarded here as a signal in two variables.

H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Bates, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.

A. V. Oppenheim, "The importance of phase in signals," Proc. IEEE 69, 529–541 (1981).

A Fourier transform pair is denoted by *f*(*x*) ↔ *F*(ω), and the Fourier transform is often abbreviated as FT. The FT operator is denoted by the operation ℱ.

A subset ɞ of ℋ is called convex of ƒ ε ɞ, g ε ɞ implies that αƒ + (1 - α)*g* ε ɞ for all α in the range 0 ≤ α ≤ 1.

Here is a demonstration of this: let ɞ denote the set and assume that ƒ ε ɞ. Then-ƒ ε ɞ, and if ɞ is convex so is (1/2)ƒ + 1/2(-ƒ). But the latter is the zero signal, which is not in ɞ. Hence ɞ is not convex.

Weak or inner-product convergence to ƒ implies that (ƒ_{k}, g) → (ƒ, *g*) for any *g* ℋ ℋ A more-familiar notion is strong convergence, which in a normed space is ‖ƒ-ƒ_{k}‖ → 0. In some cases ƒ_{n}, in Eq. (3) converges strongly to ƒ. This point is discussed in greater detail by Youla.^{11}

To a scale factor.

The error defined in Eq. (26) is consistent with that given by Youla,^{10} Hayes,^{5} and Hayes *et al.*^{4} Some authors, e.g., Fienup,^{19} in order to avoid using a definition involving an unknown ƒ, use other definitions.

Tisnonexpansiveiffor every *x,y εℋ*, ‖*T*_{y} - T_{x}‖ ≤ ‖*x - y*‖. A proof that *T*_{1} or, more generally, *T* as given in Eq. (4) is nonexpansive is furnished by Youla.^{11}

Re[ƒ] denotes the real part of ƒ.

T. Quatieri, "Phase estimation with application to speech analysis," Sc.D. Dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1979).

Corollary 1 of Ref. 11: Let ɞ be a closed convex subset of ℋThen, for any *x* εℋ, *y* ε ɞ, Re[(*x* - *P*_{ɞ}*x,y* - *P*_{ɞ}*x*)] ≤ 0.