Abstract

Phase problems occur in many scientific disciplines, particularly those involving remote sensing using a wave field. Although there has been much interest in phase retrieval in optics and in imaging in general over the past decade, phase retrieval has a much longer history in x-ray crystallography, and a variety of powerful and practical techniques have been developed. The nature of crystallography means that crystallographic phase problems are distinct from those in other imaging contexts, but there are a number of commonalities. Here the principles of phase retrieval in crystallography are outlined and are compared and contrasted with phase retrieval in general imaging. Uniqueness results are discussed, but the emphasis is on phase-retrieval algorithms and areas in which results in one discipline have, and may, contribute to the other.

© 1990 Optical Society of America

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    [CrossRef]
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1989

K. Namba, R. Pattanayek, G. Stubbs, “Visualization of protein-nucleic acid interactions in a virus: refined structure of intact tobacco mosaic virus at 2.9 Å resolution by x-ray fiber diffraction,” J. Mol. Biol. 208, 307–325 (1989).
[CrossRef] [PubMed]

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

1988

D. J. Granrath, “A positivity-based constraint on the phase variance of bispectral coefficients,” Opt. Commun. 67, 107–111 (1988).
[CrossRef]

1987

D. A. Marvin, R. K. Bryan, C. Nave, “Pf1 Inovirus. Electron density distribution calculated by a maximum entropy algorithm from native fiber diffraction data to 3 Å resolution and single isomorphous replacement data to 5 Å resolution,” J. Mol. Biol. 193, 315–343 (1987).
[CrossRef] [PubMed]

K. Namba, G. Stubbs, “Isomorphous replacement in fiber diffraction using limited numbers of heavy-atom derivatives,” Acta Crystallogr. A43, 64–69 (1987).

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. A43, 533–539 (1987).

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

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

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus at its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

R. H. T. Bates, “Aspects of the Erlangen-bispectrum,” Optik 76, 23–26 (1987).

M. M. Woolfson, “Direct methods—from birth to maturity,” Acta Crystallogr. A43, 593–612 (1987).

1986

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

E. Arnold, M. G. Rossman, “Effect of errors, redundancy, and solvent content in the molecular replacement procedure for the structure determination of biological macromolecules,” Proc. Natl. Acad. Sci. USA 83, 5489–5493 (1986).
[CrossRef] [PubMed]

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

S. W. Wilkins, D. Stuart, “Statistical geometry. IV. Maximum-entropy-based extension of multiple isomorphously phased x-ray data to 4 Å resolution for α-lactalbumin,” Acta Crystallogr. A42, 197–202 (1986).

1985

A. K. Livesey, J. Skilling, “Maximum entropy theory,” Acta Crystallogr. A41, 113–122 (1985).

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula,” Acta Crystallogr. A41, 286–290 (1985).

W. A. Hendrickson, J. L. Smith, S. Sheriff, “Direct phase determination based on anomalous scattering,” Meth. Enzymol. 115, 41–55 (1985).
[CrossRef]

J. L. C. Sanz, T. S. Huang, “On the stability and sensitivity of multidimensional signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 997–1004 (1985).
[CrossRef]

1984

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vision Graph. Image Process. 25, 205–217 (1984).
[CrossRef]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

G. Bricogne, “Maximum entropy and the foundations of direct methods,” Acta Crystallogr. A40, 410–445 (1984).

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

1983

1982

R. Nityananda, R. Narayan, “Maximum entropy image reconstruction—a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419–450 (1982).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

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

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).
[CrossRef]

G. Stubbs, L. Makowski, “Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fiber diffraction data,” Acta Crystallogr. A38, 417–425 (1982).

L. Makowski, “The use of continuous diffraction data as a phase constraint. II. Application to fiber diffraction data,” J. Appl. Crystallogr. 15, 546–557 (1982).
[CrossRef]

J. M. Beckers, “Differential speckle interferometry,” Opt. Acta 29, 361–362 (1982).
[CrossRef]

1980

L. Makowski, D. L. D. Caspar, D. A. Marvin, “Filamentous bacteriophage Pf1 structure determined at 7 Å resolution by refinement of models for the α-helical subunit,” J. Mol. Biol. 140, 149–181 (1980).
[CrossRef] [PubMed]

R. H. T. Bates, F. M. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

1979

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

G. L. Rogers, “The stellar interferometry of a star cluster with a prominent variable,” Opt. Commun. 30, 1–4 (1979).
[CrossRef]

1978

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

P. J. C. Smith, S. Arnott, “LALS: a linked-atom least-squares reciprocal space refinement system incorporating stereochemical restraints to supplement sparse diffraction data,” Acta Crystallogr. A34, 3–11 (1978).

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

S. E. Hull, M. J. Irwin, “On the application of phase relationships to complex structures. XIV. The additional use of statistical information in tangent-formula refinement,” Acta Crystallogr. A34, 863–870 (1978).

1977

A. Fortier, H. Hauptman, “Sextets in P 1: the joint probability distribution of thirty-one structure factors,” Acta Crystallogr. A33, 694–696 (1977).

1976

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Orionis,” Astrophys. J. 207, 174–180 (1976).
[CrossRef]

1975

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

G. J. Stubbs, R. Diamond, “The phase problem for cylindrically averaged diffraction patterns. Solution by isomorphous replacement and application to tobacco mosaic virus,” Acta Crystallogr. A31, 709–718 (1975).

1974

G. Bricogne, “Geometric sources of redundancy in intensity data and their use for phase determination,” Acta Crystallogr. A30, 395–405 (1974).

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

1973

R. H. T. Bates, P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–320 (1973).

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

1972

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

1970

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

W. A. Hendrickson, E. E. Lattman, “Representation of phase probability distributions for simplified combination of independent phase information,” Acta Crystallogr. B26, 136–143 (1970).

G. Tsoucaris, “A new method for phase determination. The ‘maximum determinant rule,” Acta Crystallogr. A26, 492–499 (1970).

G. Germain, P. Main, M. M. Woolfson, “On the application of phase relationships to complex structures. II. Getting a good start,” Acta Crystallogr. B26, 274–285 (1970).

1966

B. W. Mathews, “The extension of the isomorphous replacement method to include anomalous scattering measurements,” Acta Crystallogr. 20, 82–86 (1966).
[CrossRef]

P. Main, M. G. Rossmann, “Relationships among structure factors due to identical molecules in different crystallographic environments,” Acta Crystallogr. 21, 67–72 (1966).
[CrossRef]

1964

I. L. Karle, J. Karle, “An application of the symbolic addition method to the structure of L-arginine dihydrate,” Acta Crystallogr. 17, 835–841 (1964).
[CrossRef]

1963

M. G. Rossmann, D. M. Blow, “The determination of phases by the conditions of non-crystallographic symmetry,” Acta Crystallogr. 16, 39–44 (1963).
[CrossRef]

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

1962

M. G. Rossmann, D. M. Blow, “The detection of sub-units within the crystallographic asymmetric unit,” Acta Crystallogr. 15, 24–31 (1962).
[CrossRef]

1959

D. M. Blow, F. H. C. Crick, “The treatment of errors in the isomorphous replacement method,” Acta Crystallogr. 12, 794–802 (1959).
[CrossRef]

1958

D. M. Blow, “The structure of haemoglobin VII. Determination of phase angles in the non-centrosymmetric [100] zone,” Proc. R. Soc. London Ser. A 247, 302–336 (1958).
[CrossRef]

1956

J. Karle, H. Hauptman, “A theory of phase determination for the four types of non-centrosymmetric space groups 1P 222, 2P 22, 3P12, 3P22,” Acta Crystallogr. 9, 635–651 (1956).
[CrossRef]

1955

W. Cochran, “Relations between the phases of structure factors,” Acta Crystallogr. 8, 473–478 (1955).
[CrossRef]

1954

D. W. Green, V. M. Ingram, M. F. Perutz, “The structure of haemoglobin IV. Sign determination by the isomorphous replacement method,” Proc. R. Soc. London Ser. A 255, 287–307 (1954).

1952

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

1950

J. Karle, H. Hauptman, “The phases and magnitudes of the structure factors,” Acta Crystallogr. 3, 181–187 (1950).
[CrossRef]

1948

D. Harker, J. S. Kasper, “Phases of Fourier coefficients directly from crystal diffraction data,” Acta Crystallogr. 1, 70–75 (1948).
[CrossRef]

Aime, C.

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Argos, P.

P. Argos, M. G. Rossman, “Molecular replacement method,” in Theory and Practice of Direct Methods in Crystallography, M. F. C. Ladd, R. A. Palmer, eds. (Plenum, New York, 1980), pp. 361–417.
[CrossRef]

Arnold, E.

E. Arnold, M. G. Rossman, “Effect of errors, redundancy, and solvent content in the molecular replacement procedure for the structure determination of biological macromolecules,” Proc. Natl. Acad. Sci. USA 83, 5489–5493 (1986).
[CrossRef] [PubMed]

Arnott, S.

R. P. Millane, A. K. Mitra, S. Arnott, “Chondroitin 4-sulfate: comparison of the structures of the potassium and sodium salts,” J. Mol. Biol. 169, 903–920 (1983).
[CrossRef] [PubMed]

P. J. C. Smith, S. Arnott, “LALS: a linked-atom least-squares reciprocal space refinement system incorporating stereochemical restraints to supplement sparse diffraction data,” Acta Crystallogr. A34, 3–11 (1978).

S. Arnott, “Twenty years hard labor as a fiber diffractionist,” in Fiber Diffraction Methods, A. D. French, K. H. Gardner, eds., ACS Symp. Ser.131, 1–30 (1980).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Barakat, R.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Bates, R. H. T.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[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, “Aspects of the Erlangen-bispectrum,” Optik 76, 23–26 (1987).

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vision Graph. Image Process. 25, 205–217 (1984).
[CrossRef]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

R. H. T. Bates, F. M. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

R. H. T. Bates, P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–320 (1973).

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep.90, 203–297 (1982).
[CrossRef]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, Oxford, 1986).

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1986), Vol. 67, pp. 1–64.
[CrossRef]

Beckers, J. M.

J. M. Beckers, “Differential speckle interferometry,” Opt. Acta 29, 361–362 (1982).
[CrossRef]

Blow, D. M.

M. G. Rossmann, D. M. Blow, “The determination of phases by the conditions of non-crystallographic symmetry,” Acta Crystallogr. 16, 39–44 (1963).
[CrossRef]

M. G. Rossmann, D. M. Blow, “The detection of sub-units within the crystallographic asymmetric unit,” Acta Crystallogr. 15, 24–31 (1962).
[CrossRef]

D. M. Blow, F. H. C. Crick, “The treatment of errors in the isomorphous replacement method,” Acta Crystallogr. 12, 794–802 (1959).
[CrossRef]

D. M. Blow, “The structure of haemoglobin VII. Determination of phase angles in the non-centrosymmetric [100] zone,” Proc. R. Soc. London Ser. A 247, 302–336 (1958).
[CrossRef]

Blundell, T. L.

T. L. Blundell, L. N. Johnson, Protein Crystallography (Academic, New York, 1978).

Bricogne, G.

G. Bricogne, “Maximum entropy and the foundations of direct methods,” Acta Crystallogr. A40, 410–445 (1984).

G. Bricogne, “Geometric sources of redundancy in intensity data and their use for phase determination,” Acta Crystallogr. A30, 395–405 (1974).

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Bryan, R. K.

D. A. Marvin, R. K. Bryan, C. Nave, “Pf1 Inovirus. Electron density distribution calculated by a maximum entropy algorithm from native fiber diffraction data to 3 Å resolution and single isomorphous replacement data to 5 Å resolution,” J. Mol. Biol. 193, 315–343 (1987).
[CrossRef] [PubMed]

Cady, F. M.

R. H. T. Bates, F. M. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

Caspar, D. L. D.

L. Makowski, D. L. D. Caspar, D. A. Marvin, “Filamentous bacteriophage Pf1 structure determined at 7 Å resolution by refinement of models for the α-helical subunit,” J. Mol. Biol. 140, 149–181 (1980).
[CrossRef] [PubMed]

Cochran, W.

W. Cochran, “Relations between the phases of structure factors,” Acta Crystallogr. 8, 473–478 (1955).
[CrossRef]

Cocke, J. W.

J. W. Cocke, “The Cauchy/Schwarz inequality as a constraint in power spectrum/autocorrelation analysis and image reconstruction,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 46–49 (1985).
[CrossRef]

Crick, F. H. C.

D. M. Blow, F. H. C. Crick, “The treatment of errors in the isomorphous replacement method,” Acta Crystallogr. 12, 794–802 (1959).
[CrossRef]

Crowther, R. A.

R. A. Crowther, “Fast rotation function,” in The Molecular Replacement Method, M. G. Rossmann, ed. (Gordon & Breach, New York, 1972), pp. 173–178.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Debaerdemaeker, T.

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula,” Acta Crystallogr. A41, 286–290 (1985).

Diamond, R.

G. J. Stubbs, R. Diamond, “The phase problem for cylindrically averaged diffraction patterns. Solution by isomorphous replacement and application to tobacco mosaic virus,” Acta Crystallogr. A31, 709–718 (1975).

Fienup, J. R.

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus at its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

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

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

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low resolution image,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

Fortier, A.

A. Fortier, H. Hauptman, “Sextets in P 1: the joint probability distribution of thirty-one structure factors,” Acta Crystallogr. A33, 694–696 (1977).

Frieden, B. R.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

Gallwitz, U.

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

Gerchberg, R. W.

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

Germain, G.

G. Germain, P. Main, M. M. Woolfson, “On the application of phase relationships to complex structures. II. Getting a good start,” Acta Crystallogr. B26, 274–285 (1970).

Giacovazzo, C.

C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).

Gough, P. T.

R. H. T. Bates, P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–320 (1973).

Granrath, D. J.

D. J. Granrath, “A positivity-based constraint on the phase variance of bispectral coefficients,” Opt. Commun. 67, 107–111 (1988).
[CrossRef]

Green, D. W.

D. W. Green, V. M. Ingram, M. F. Perutz, “The structure of haemoglobin IV. Sign determination by the isomorphous replacement method,” Proc. R. Soc. London Ser. A 255, 287–307 (1954).

Haralick, R. M.

X. Zhuang, E. Ostevold, R. M. Haralick, “The principle of maximum entropy in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 157–193.

Harker, D.

D. Harker, J. S. Kasper, “Phases of Fourier coefficients directly from crystal diffraction data,” Acta Crystallogr. 1, 70–75 (1948).
[CrossRef]

Harvey, J. W.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Orionis,” Astrophys. J. 207, 174–180 (1976).
[CrossRef]

Hauptman, H.

A. Fortier, H. Hauptman, “Sextets in P 1: the joint probability distribution of thirty-one structure factors,” Acta Crystallogr. A33, 694–696 (1977).

J. Karle, H. Hauptman, “A theory of phase determination for the four types of non-centrosymmetric space groups 1P 222, 2P 22, 3P12, 3P22,” Acta Crystallogr. 9, 635–651 (1956).
[CrossRef]

J. Karle, H. Hauptman, “The phases and magnitudes of the structure factors,” Acta Crystallogr. 3, 181–187 (1950).
[CrossRef]

Hayes, M. H.

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).
[CrossRef]

Hendrickson, W. A.

W. A. Hendrickson, J. L. Smith, S. Sheriff, “Direct phase determination based on anomalous scattering,” Meth. Enzymol. 115, 41–55 (1985).
[CrossRef]

W. A. Hendrickson, E. E. Lattman, “Representation of phase probability distributions for simplified combination of independent phase information,” Acta Crystallogr. B26, 136–143 (1970).

W. A. Hendrickson, J. H. Konnert, “Incorporation of stereochemical information into crystallographic refinement,” in Computing in Crystallography, R. Diamond, S. Ramaseshan, K. Venkatesan, eds. (Indian Academy of Science, Bangalore, 1980), pp. 13.01–13.26.

Holmes, K. C.

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

Huang, T. S.

J. L. C. Sanz, T. S. Huang, “On the stability and sensitivity of multidimensional signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 997–1004 (1985).
[CrossRef]

Hull, S. E.

S. E. Hull, M. J. Irwin, “On the application of phase relationships to complex structures. XIV. The additional use of statistical information in tangent-formula refinement,” Acta Crystallogr. A34, 863–870 (1978).

Hunt, B. R.

Ingram, V. M.

D. W. Green, V. M. Ingram, M. F. Perutz, “The structure of haemoglobin IV. Sign determination by the isomorphous replacement method,” Proc. R. Soc. London Ser. A 255, 287–307 (1954).

Irwin, M. J.

S. E. Hull, M. J. Irwin, “On the application of phase relationships to complex structures. XIV. The additional use of statistical information in tangent-formula refinement,” Acta Crystallogr. A34, 863–870 (1978).

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Jensen, L. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Johnson, L. N.

T. L. Blundell, L. N. Johnson, Protein Crystallography (Academic, New York, 1978).

Kadiri, S.

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Karle, I. L.

I. L. Karle, J. Karle, “An application of the symbolic addition method to the structure of L-arginine dihydrate,” Acta Crystallogr. 17, 835–841 (1964).
[CrossRef]

Karle, J.

I. L. Karle, J. Karle, “An application of the symbolic addition method to the structure of L-arginine dihydrate,” Acta Crystallogr. 17, 835–841 (1964).
[CrossRef]

J. Karle, H. Hauptman, “A theory of phase determination for the four types of non-centrosymmetric space groups 1P 222, 2P 22, 3P12, 3P22,” Acta Crystallogr. 9, 635–651 (1956).
[CrossRef]

J. Karle, H. Hauptman, “The phases and magnitudes of the structure factors,” Acta Crystallogr. 3, 181–187 (1950).
[CrossRef]

Kasper, J. S.

D. Harker, J. S. Kasper, “Phases of Fourier coefficients directly from crystal diffraction data,” Acta Crystallogr. 1, 70–75 (1948).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Konnert, J. H.

W. A. Hendrickson, J. H. Konnert, “Incorporation of stereochemical information into crystallographic refinement,” in Computing in Crystallography, R. Diamond, S. Ramaseshan, K. Venkatesan, eds. (Indian Academy of Science, Bangalore, 1980), pp. 13.01–13.26.

Kowalczyk, A. M.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low resolution image,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

Labeyrie, A.

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

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

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

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Lattman, E. E.

W. A. Hendrickson, E. E. Lattman, “Representation of phase probability distributions for simplified combination of independent phase information,” Acta Crystallogr. B26, 136–143 (1970).

Levi, A.

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

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Livesey, A. K.

A. K. Livesey, J. Skilling, “Maximum entropy theory,” Acta Crystallogr. A41, 113–122 (1985).

Lohmann, A. W.

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Lynds, C. R.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Orionis,” Astrophys. J. 207, 174–180 (1976).
[CrossRef]

Main, P.

G. Germain, P. Main, M. M. Woolfson, “On the application of phase relationships to complex structures. II. Getting a good start,” Acta Crystallogr. B26, 274–285 (1970).

P. Main, M. G. Rossmann, “Relationships among structure factors due to identical molecules in different crystallographic environments,” Acta Crystallogr. 21, 67–72 (1966).
[CrossRef]

Makowski, L.

L. Makowski, “The use of continuous diffraction data as a phase constraint. II. Application to fiber diffraction data,” J. Appl. Crystallogr. 15, 546–557 (1982).
[CrossRef]

G. Stubbs, L. Makowski, “Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fiber diffraction data,” Acta Crystallogr. A38, 417–425 (1982).

L. Makowski, D. L. D. Caspar, D. A. Marvin, “Filamentous bacteriophage Pf1 structure determined at 7 Å resolution by refinement of models for the α-helical subunit,” J. Mol. Biol. 140, 149–181 (1980).
[CrossRef] [PubMed]

Mandelkow, E.

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

Martin, F.

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Marvin, D. A.

D. A. Marvin, R. K. Bryan, C. Nave, “Pf1 Inovirus. Electron density distribution calculated by a maximum entropy algorithm from native fiber diffraction data to 3 Å resolution and single isomorphous replacement data to 5 Å resolution,” J. Mol. Biol. 193, 315–343 (1987).
[CrossRef] [PubMed]

L. Makowski, D. L. D. Caspar, D. A. Marvin, “Filamentous bacteriophage Pf1 structure determined at 7 Å resolution by refinement of models for the α-helical subunit,” J. Mol. Biol. 140, 149–181 (1980).
[CrossRef] [PubMed]

Mathews, B. W.

B. W. Mathews, “The extension of the isomorphous replacement method to include anomalous scattering measurements,” Acta Crystallogr. 20, 82–86 (1966).
[CrossRef]

McCallum, B. C.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, Oxford, 1986).

Millane, R. P.

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

R. P. Millane, A. K. Mitra, S. Arnott, “Chondroitin 4-sulfate: comparison of the structures of the potassium and sodium salts,” J. Mol. Biol. 169, 903–920 (1983).
[CrossRef] [PubMed]

R. P. Millane, “Redundancy in multi-dimensional phase retrieval and blind deconvolution,” submitted to IEEE Trans. Acoust. Speech Signal Process.

R. P. Millane, “The effect of structural redundancy on uniqueness of the macromolecular crystallographic phase problem,” submitted to Acta Crystallogr. A.

R. P. Millane, “X-ray fiber diffraction studies of polysaccharide structures,” in Computer Modeling of Carbohydrates, A. D. French, J. W. Brady, eds., ACS Symp. Ser. (to be published).

R. P. Millane, “Positivity and uniqueness in phase retrieval and blind deconvolution,” submitted to Opt. Commun.

R. P. Millane, “Speckle holography with a variable reference,” submitted to Appl. Opt.

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Crystallographic Computing 4: Techniques and New Technologies, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

Mitra, A. K.

R. P. Millane, A. K. Mitra, S. Arnott, “Chondroitin 4-sulfate: comparison of the structures of the potassium and sodium salts,” J. Mol. Biol. 169, 903–920 (1983).
[CrossRef] [PubMed]

Mnyama, D.

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1986), Vol. 67, pp. 1–64.
[CrossRef]

Namba, K.

K. Namba, R. Pattanayek, G. Stubbs, “Visualization of protein-nucleic acid interactions in a virus: refined structure of intact tobacco mosaic virus at 2.9 Å resolution by x-ray fiber diffraction,” J. Mol. Biol. 208, 307–325 (1989).
[CrossRef] [PubMed]

K. Namba, G. Stubbs, “Isomorphous replacement in fiber diffraction using limited numbers of heavy-atom derivatives,” Acta Crystallogr. A43, 64–69 (1987).

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. A43, 533–539 (1987).

Narayan, R.

R. Nityananda, R. Narayan, “Maximum entropy image reconstruction—a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419–450 (1982).
[CrossRef]

Nave, C.

D. A. Marvin, R. K. Bryan, C. Nave, “Pf1 Inovirus. Electron density distribution calculated by a maximum entropy algorithm from native fiber diffraction data to 3 Å resolution and single isomorphous replacement data to 5 Å resolution,” J. Mol. Biol. 193, 315–343 (1987).
[CrossRef] [PubMed]

Newsam, G.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Nityananda, R.

R. Nityananda, R. Narayan, “Maximum entropy image reconstruction—a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419–450 (1982).
[CrossRef]

Norton, W. A.

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

O’Neill, E. L.

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Ostevold, E.

X. Zhuang, E. Ostevold, R. M. Haralick, “The principle of maximum entropy in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 157–193.

Pattanayek, R.

K. Namba, R. Pattanayek, G. Stubbs, “Visualization of protein-nucleic acid interactions in a virus: refined structure of intact tobacco mosaic virus at 2.9 Å resolution by x-ray fiber diffraction,” J. Mol. Biol. 208, 307–325 (1989).
[CrossRef] [PubMed]

Perutz, M. F.

D. W. Green, V. M. Ingram, M. F. Perutz, “The structure of haemoglobin IV. Sign determination by the isomorphous replacement method,” Proc. R. Soc. London Ser. A 255, 287–307 (1954).

Petrov, R.

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Readhead, A. C. S.

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

Ricort, G.

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Rogers, G. L.

G. L. Rogers, “The stellar interferometry of a star cluster with a prominent variable,” Opt. Commun. 30, 1–4 (1979).
[CrossRef]

Rossman, M. G.

E. Arnold, M. G. Rossman, “Effect of errors, redundancy, and solvent content in the molecular replacement procedure for the structure determination of biological macromolecules,” Proc. Natl. Acad. Sci. USA 83, 5489–5493 (1986).
[CrossRef] [PubMed]

P. Argos, M. G. Rossman, “Molecular replacement method,” in Theory and Practice of Direct Methods in Crystallography, M. F. C. Ladd, R. A. Palmer, eds. (Plenum, New York, 1980), pp. 361–417.
[CrossRef]

Rossmann, M. G.

P. Main, M. G. Rossmann, “Relationships among structure factors due to identical molecules in different crystallographic environments,” Acta Crystallogr. 21, 67–72 (1966).
[CrossRef]

M. G. Rossmann, D. M. Blow, “The determination of phases by the conditions of non-crystallographic symmetry,” Acta Crystallogr. 16, 39–44 (1963).
[CrossRef]

M. G. Rossmann, D. M. Blow, “The detection of sub-units within the crystallographic asymmetric unit,” Acta Crystallogr. 15, 24–31 (1962).
[CrossRef]

Sanz, J. L. C.

J. L. C. Sanz, T. S. Huang, “On the stability and sensitivity of multidimensional signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 997–1004 (1985).
[CrossRef]

Saxton, W. O.

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

Sayre, D.

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

Sheriff, S.

W. A. Hendrickson, J. L. Smith, S. Sheriff, “Direct phase determination based on anomalous scattering,” Meth. Enzymol. 115, 41–55 (1985).
[CrossRef]

Skilling, J.

A. K. Livesey, J. Skilling, “Maximum entropy theory,” Acta Crystallogr. A41, 113–122 (1985).

Smith, J. L.

W. A. Hendrickson, J. L. Smith, S. Sheriff, “Direct phase determination based on anomalous scattering,” Meth. Enzymol. 115, 41–55 (1985).
[CrossRef]

Smith, P. J. C.

P. J. C. Smith, S. Arnott, “LALS: a linked-atom least-squares reciprocal space refinement system incorporating stereochemical restraints to supplement sparse diffraction data,” Acta Crystallogr. A34, 3–11 (1978).

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stark, H.

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

Stout, G. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Stuart, D.

S. W. Wilkins, D. Stuart, “Statistical geometry. IV. Maximum-entropy-based extension of multiple isomorphously phased x-ray data to 4 Å resolution for α-lactalbumin,” Acta Crystallogr. A42, 197–202 (1986).

Stubbs, G.

K. Namba, R. Pattanayek, G. Stubbs, “Visualization of protein-nucleic acid interactions in a virus: refined structure of intact tobacco mosaic virus at 2.9 Å resolution by x-ray fiber diffraction,” J. Mol. Biol. 208, 307–325 (1989).
[CrossRef] [PubMed]

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. A43, 533–539 (1987).

K. Namba, G. Stubbs, “Isomorphous replacement in fiber diffraction using limited numbers of heavy-atom derivatives,” Acta Crystallogr. A43, 64–69 (1987).

G. Stubbs, L. Makowski, “Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fiber diffraction data,” Acta Crystallogr. A38, 417–425 (1982).

Stubbs, G. J.

G. J. Stubbs, R. Diamond, “The phase problem for cylindrically averaged diffraction patterns. Solution by isomorphous replacement and application to tobacco mosaic virus,” Acta Crystallogr. A31, 709–718 (1975).

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

Tate, C.

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula,” Acta Crystallogr. A41, 286–290 (1985).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Tsoucaris, G.

G. Tsoucaris, “A new method for phase determination. The ‘maximum determinant rule,” Acta Crystallogr. A26, 492–499 (1970).

Wackerman, C. C.

Walther, A.

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Warner, P. J.

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Weigelt, G.

Wilkins, S. W.

S. W. Wilkins, D. Stuart, “Statistical geometry. IV. Maximum-entropy-based extension of multiple isomorphously phased x-ray data to 4 Å resolution for α-lactalbumin,” Acta Crystallogr. A42, 197–202 (1986).

Wilkinson, P. N.

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

Wirnitzer, B.

Woolfson, M. M.

M. M. Woolfson, “Direct methods—from birth to maturity,” Acta Crystallogr. A43, 593–612 (1987).

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula,” Acta Crystallogr. A41, 286–290 (1985).

G. Germain, P. Main, M. M. Woolfson, “On the application of phase relationships to complex structures. II. Getting a good start,” Acta Crystallogr. B26, 274–285 (1970).

Worden, S. P.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Orionis,” Astrophys. J. 207, 174–180 (1976).
[CrossRef]

Zhuang, X.

X. Zhuang, E. Ostevold, R. M. Haralick, “The principle of maximum entropy in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 157–193.

Acta Crystallogr.

M. M. Woolfson, “Direct methods—from birth to maturity,” Acta Crystallogr. A43, 593–612 (1987).

D. Harker, J. S. Kasper, “Phases of Fourier coefficients directly from crystal diffraction data,” Acta Crystallogr. 1, 70–75 (1948).
[CrossRef]

J. Karle, H. Hauptman, “The phases and magnitudes of the structure factors,” Acta Crystallogr. 3, 181–187 (1950).
[CrossRef]

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

W. Cochran, “Relations between the phases of structure factors,” Acta Crystallogr. 8, 473–478 (1955).
[CrossRef]

J. Karle, H. Hauptman, “A theory of phase determination for the four types of non-centrosymmetric space groups 1P 222, 2P 22, 3P12, 3P22,” Acta Crystallogr. 9, 635–651 (1956).
[CrossRef]

I. L. Karle, J. Karle, “An application of the symbolic addition method to the structure of L-arginine dihydrate,” Acta Crystallogr. 17, 835–841 (1964).
[CrossRef]

G. Germain, P. Main, M. M. Woolfson, “On the application of phase relationships to complex structures. II. Getting a good start,” Acta Crystallogr. B26, 274–285 (1970).

S. E. Hull, M. J. Irwin, “On the application of phase relationships to complex structures. XIV. The additional use of statistical information in tangent-formula refinement,” Acta Crystallogr. A34, 863–870 (1978).

A. Fortier, H. Hauptman, “Sextets in P 1: the joint probability distribution of thirty-one structure factors,” Acta Crystallogr. A33, 694–696 (1977).

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula,” Acta Crystallogr. A41, 286–290 (1985).

G. Bricogne, “Maximum entropy and the foundations of direct methods,” Acta Crystallogr. A40, 410–445 (1984).

G. Tsoucaris, “A new method for phase determination. The ‘maximum determinant rule,” Acta Crystallogr. A26, 492–499 (1970).

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

D. M. Blow, F. H. C. Crick, “The treatment of errors in the isomorphous replacement method,” Acta Crystallogr. 12, 794–802 (1959).
[CrossRef]

W. A. Hendrickson, E. E. Lattman, “Representation of phase probability distributions for simplified combination of independent phase information,” Acta Crystallogr. B26, 136–143 (1970).

B. W. Mathews, “The extension of the isomorphous replacement method to include anomalous scattering measurements,” Acta Crystallogr. 20, 82–86 (1966).
[CrossRef]

M. G. Rossmann, D. M. Blow, “The determination of phases by the conditions of non-crystallographic symmetry,” Acta Crystallogr. 16, 39–44 (1963).
[CrossRef]

P. Main, M. G. Rossmann, “Relationships among structure factors due to identical molecules in different crystallographic environments,” Acta Crystallogr. 21, 67–72 (1966).
[CrossRef]

M. G. Rossmann, D. M. Blow, “The detection of sub-units within the crystallographic asymmetric unit,” Acta Crystallogr. 15, 24–31 (1962).
[CrossRef]

G. Bricogne, “Geometric sources of redundancy in intensity data and their use for phase determination,” Acta Crystallogr. A30, 395–405 (1974).

A. K. Livesey, J. Skilling, “Maximum entropy theory,” Acta Crystallogr. A41, 113–122 (1985).

S. W. Wilkins, D. Stuart, “Statistical geometry. IV. Maximum-entropy-based extension of multiple isomorphously phased x-ray data to 4 Å resolution for α-lactalbumin,” Acta Crystallogr. A42, 197–202 (1986).

G. J. Stubbs, R. Diamond, “The phase problem for cylindrically averaged diffraction patterns. Solution by isomorphous replacement and application to tobacco mosaic virus,” Acta Crystallogr. A31, 709–718 (1975).

G. Stubbs, L. Makowski, “Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fiber diffraction data,” Acta Crystallogr. A38, 417–425 (1982).

K. Namba, G. Stubbs, “Isomorphous replacement in fiber diffraction using limited numbers of heavy-atom derivatives,” Acta Crystallogr. A43, 64–69 (1987).

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. A43, 533–539 (1987).

P. J. C. Smith, S. Arnott, “LALS: a linked-atom least-squares reciprocal space refinement system incorporating stereochemical restraints to supplement sparse diffraction data,” Acta Crystallogr. A34, 3–11 (1978).

Appl. Opt.

Astron. Astrophys.

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

R. H. T. Bates, P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–320 (1973).

C. Aime, S. Kadiri, F. Martin, R. Petrov, G. Ricort, “Measurement of submilliarcsecond speckle displacements using a cross spectrum analysis technique. Test on atmospheric dispersion,” Astron. Astrophys. 134, 354–359 (1984).

Astrophys. J.

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Orionis,” Astrophys. J. 207, 174–180 (1976).
[CrossRef]

Astrophys. J. Lett.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Biophys. J.

R. P. Millane, R. H. T. Bates, W. R. Fright, W. A. Norton, “Towards direct phase retrieval in macromolecular crystallography,” Biophys. J. 49, 60–62 (1986).
[CrossRef] [PubMed]

Comput. Vision Graph. Image Process.

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vision Graph. Image Process. 25, 205–217 (1984).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

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).
[CrossRef]

J. L. C. Sanz, T. S. Huang, “On the stability and sensitivity of multidimensional signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 997–1004 (1985).
[CrossRef]

J. Appl. Crystallogr.

L. Makowski, “The use of continuous diffraction data as a phase constraint. II. Application to fiber diffraction data,” J. Appl. Crystallogr. 15, 546–557 (1982).
[CrossRef]

J. Astrophys. Astron.

R. Nityananda, R. Narayan, “Maximum entropy image reconstruction—a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419–450 (1982).
[CrossRef]

J. Math. Phys.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

J. Mod. Opt.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

J. Mol. Biol.

R. P. Millane, A. K. Mitra, S. Arnott, “Chondroitin 4-sulfate: comparison of the structures of the potassium and sodium salts,” J. Mol. Biol. 169, 903–920 (1983).
[CrossRef] [PubMed]

D. A. Marvin, R. K. Bryan, C. Nave, “Pf1 Inovirus. Electron density distribution calculated by a maximum entropy algorithm from native fiber diffraction data to 3 Å resolution and single isomorphous replacement data to 5 Å resolution,” J. Mol. Biol. 193, 315–343 (1987).
[CrossRef] [PubMed]

K. Namba, R. Pattanayek, G. Stubbs, “Visualization of protein-nucleic acid interactions in a virus: refined structure of intact tobacco mosaic virus at 2.9 Å resolution by x-ray fiber diffraction,” J. Mol. Biol. 208, 307–325 (1989).
[CrossRef] [PubMed]

L. Makowski, D. L. D. Caspar, D. A. Marvin, “Filamentous bacteriophage Pf1 structure determined at 7 Å resolution by refinement of models for the α-helical subunit,” J. Mol. Biol. 140, 149–181 (1980).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meth. Enzymol.

W. A. Hendrickson, J. L. Smith, S. Sheriff, “Direct phase determination based on anomalous scattering,” Meth. Enzymol. 115, 41–55 (1985).
[CrossRef]

Mon. Not. R. Astron. Soc.

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Nature (London)

K. C. Holmes, G. J. Stubbs, E. Mandelkow, U. Gallwitz, “Structure of tobacco mosaic virus at 6.7 Å resolution,” Nature (London) 254, 192–196 (1975).
[CrossRef]

Opt. Acta

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

J. M. Beckers, “Differential speckle interferometry,” Opt. Acta 29, 361–362 (1982).
[CrossRef]

Opt. Commun.

D. J. Granrath, “A positivity-based constraint on the phase variance of bispectral coefficients,” Opt. Commun. 67, 107–111 (1988).
[CrossRef]

G. L. Rogers, “The stellar interferometry of a star cluster with a prominent variable,” Opt. Commun. 30, 1–4 (1979).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

R. H. T. Bates, F. M. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Opt. Lett.

Optik

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

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

R. H. T. Bates, “Aspects of the Erlangen-bispectrum,” Optik 76, 23–26 (1987).

Proc. Natl. Acad. Sci. USA

E. Arnold, M. G. Rossman, “Effect of errors, redundancy, and solvent content in the molecular replacement procedure for the structure determination of biological macromolecules,” Proc. Natl. Acad. Sci. USA 83, 5489–5493 (1986).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

D. W. Green, V. M. Ingram, M. F. Perutz, “The structure of haemoglobin IV. Sign determination by the isomorphous replacement method,” Proc. R. Soc. London Ser. A 255, 287–307 (1954).

D. M. Blow, “The structure of haemoglobin VII. Determination of phase angles in the non-centrosymmetric [100] zone,” Proc. R. Soc. London Ser. A 247, 302–336 (1958).
[CrossRef]

Other

W. A. Hendrickson, J. H. Konnert, “Incorporation of stereochemical information into crystallographic refinement,” in Computing in Crystallography, R. Diamond, S. Ramaseshan, K. Venkatesan, eds. (Indian Academy of Science, Bangalore, 1980), pp. 13.01–13.26.

C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).

P. Argos, M. G. Rossman, “Molecular replacement method,” in Theory and Practice of Direct Methods in Crystallography, M. F. C. Ladd, R. A. Palmer, eds. (Plenum, New York, 1980), pp. 361–417.
[CrossRef]

T. L. Blundell, L. N. Johnson, Protein Crystallography (Academic, New York, 1978).

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[CrossRef]

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Crystallographic Computing 4: Techniques and New Technologies, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

R. A. Crowther, “Fast rotation function,” in The Molecular Replacement Method, M. G. Rossmann, ed. (Gordon & Breach, New York, 1972), pp. 173–178.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, Oxford, 1986).

X. Zhuang, E. Ostevold, R. M. Haralick, “The principle of maximum entropy in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 157–193.

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep.90, 203–297 (1982).
[CrossRef]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

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

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1986), Vol. 67, pp. 1–64.
[CrossRef]

R. P. Millane, “Positivity and uniqueness in phase retrieval and blind deconvolution,” submitted to Opt. Commun.

R. P. Millane, “X-ray fiber diffraction studies of polysaccharide structures,” in Computer Modeling of Carbohydrates, A. D. French, J. W. Brady, eds., ACS Symp. Ser. (to be published).

R. P. Millane, “Redundancy in multi-dimensional phase retrieval and blind deconvolution,” submitted to IEEE Trans. Acoust. Speech Signal Process.

R. P. Millane, “The effect of structural redundancy on uniqueness of the macromolecular crystallographic phase problem,” submitted to Acta Crystallogr. A.

R. P. Millane, “Speckle holography with a variable reference,” submitted to Appl. Opt.

J. W. Cocke, “The Cauchy/Schwarz inequality as a constraint in power spectrum/autocorrelation analysis and image reconstruction,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 46–49 (1985).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low resolution image,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

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Figures (8)

Fig. 1
Fig. 1

The unit cell (thick lines) and the lattice (circles) in a crystal.

Fig. 2
Fig. 2

Illustration of the atomicity property in one dimension.

Fig. 3
Fig. 3

Relationship between shifted autocorrelations (——) and the Patterson map (– – –) in one dimension.

Fig. 4
Fig. 4

Distribution P(Φ) for B large (——) and small (– – –).

Fig. 5
Fig. 5

Relationships between the structure factors used to derive Eq. (45) for isomorphous replacement.

Fig. 6
Fig. 6

Illustration of the lack of closure of the phase triangle.

Fig. 7
Fig. 7

Relationships between the structure factors used to derive relation (52) for the Bijvoet difference.

Fig. 8
Fig. 8

One quadrant of reciprocal space showing the reciprocal lattice points (○) and the additional positions at which the amplitude of the continuous transform is effectively sampled (×) as a result of a noncrystallographic threefold rotation axis in a hypothetical two-dimensional lattice.

Equations (77)

Equations on this page are rendered with MathJax. Learn more.

F ( u ) = f ( x ) exp ( i 2 π u · x ) d x ,
f ( x ) = F ( u ) exp ( i 2 π x · u ) d u ,
F ( u ) = | F ( u ) | exp [ i ϕ ( u ) ] .
A ( x ) = f ( y ) f * ( x + y ) d y ,
F ( w ) = f ( x ) exp ( i 2 π w x ) d x ,
F ( w ) = exp ( A + B w ) n = 1 ( 1 w / w n ) ,
I ( w ) = exp [ 2 Re ( A ) + 2 Re ( B ) z ] n = 1 ( 1 w / w n ) ( 1 w / w n * ) ,
f n ( x ) = 1 [ | F ( u ) | exp ( i arg { [ f n ( x ) ] } ) ] ,
f n + 1 ( x ) = f n ( x ) if x Ω = 0 otherwise ,
f n + 1 ( x ) = f n ( x ) if x Ω = f n ( x ) γ f n ( x ) otherwise ,
| S m ( u ) | 2 = | F ( u ) | 2 | H m ( u ) | 2 ,
f ( x ) = δ ( x ) + g ( x ) ,
A ( x ) = δ ( x ) + g ( x ) + g * ( x ) + g ( x ) g ( x ) ,
S m * ( u ) S m ( u + α ) = F * ( u ) F ( u + α ) H m * ( u ) H m ( u + α ) .
μ m ( x ) = j a m j δ ( x x m j ) .
f LWH ( x ) = s m ( x ) μ m ( x ) .
f saa ( x ) = s m ( x x m ) .
f ˆ ( x 1 , x 2 ) = f ( x ) f ( x + x 1 ) f ( x + x 2 ) d x
F ˆ ( u 1 , u 2 ) = F ( u 1 ) F ( u 2 ) F ( u 1 u 2 ) ,
Ŝ m ( u 1 , u 2 ) = F ˆ ( u 1 , u 2 ) Ĥ ( u 1 , u 2 ) ,
F ˆ l m = F l F m F l m .
ϕ n = ϕ m + ϕ n m ψ n m , m .
exp ( i ϕ n ) = ( 2 / n ) m = 1 n / 2 exp [ i ( ϕ m + ϕ n m ψ n m , m ) ] .
f ( x ) e ( x ) ( x ) = e ( x ) n = δ ( x x n ) ,
f ( x , y , z ) = e ( x , y , z ) m , n , p = δ ( x m a ) δ ( y n b ) δ ( z p c ) ,
e ( x , y , z ) = n = 1 N e n ( x x n , y y n , z z n ) ,
F ( u , υ , w ) = E ( u , υ , w ) h , k , l = δ ( u h / a ) δ ( υ k / b ) δ ( w l / c ) ,
F hkl = F ( h / a , k / b , l / c ) = E ( h / a , k / b , l / c ) = E ( u ) L ( u ) ,
F hkl = 0 c 0 b 0 a f ( x , y , z ) exp [ i 2 π ( h x / a + k y / b + l z / c ) ] d x d y d z
F h = V f ( x ) exp ( i 2 π h · x ) d x ,
F hkl = n = 1 N f n exp [ i 2 π ( h x n / a + k y n / b + l z n / c ) ] ,
f ( x , y , z ) = ( 1 / V ) h , k , l = F hkl exp [ i 2 π ( h x / a + k y / b + l z / c ) ]
f ( x ) = ( 1 / V ) h F h exp ( i 2 π h · x ) ,
P ( x ) = ( 1 / V ) h | F h | 2 exp ( i 2 π h · x ) .
P ( x , y , z ) = h , k , l = A ( x h a , y k b , z l c ) ,
| F o F h 1 F h 2 F h n F h 1 F o F h n F h n h 1 F o | 0 .
F h = θ h k F k F h k ,
F h 2 ( 1 + F 2 h ) / 2 ,
F h F k F h k / F 0 ,
ϕ h ϕ k + ϕ h k ,
E h = F h / ( j = 1 N f j 2 ) 1 / 2 .
P ( Φ 3 ) = exp ( B cos ϕ 3 ) / [ 2 π I 0 ( B ) ] ,
Φ 3 = ϕ h ϕ k ϕ h k 0 ,
B = 2 N 1 / 2 | E h E k E h k | .
tan ϕ h = k | E k E h k | sin ( ϕ k + ϕ h k ) k | E k E h k | cos ( ϕ k + ϕ h k )
F h D = F h + F h H ,
ϕ h = ϕ h H + cos 1 [ ( | F h D | 2 | F h | 2 | F h H | 2 ) / ( 2 | F h | | F h H | ) ] .
P ( x ) = ( 1 / υ ) h ( | F h D | | F h | ) 2 exp ( i 2 π h · x ) .
P ( ϕ ) = exp [ ( ϕ ) 2 / 2 E 2 ] ,
P ( ϕ ) = exp { j [ j ( ϕ ) 2 / 2 E j 2 ] } .
tan ( ϕ best ) = 0 2 π P ( ϕ ) sin ϕ d ϕ / 0 2 π P ( ϕ ) cos ϕ d ϕ
m = 0 2 π P ( ϕ ) cos ( ϕ best ϕ ) d ϕ / 0 2 π P ( ϕ ) d ϕ .
f = f o + Δ f + i Δ f ,
| F h | | F h | 2 | F h A | sin ( ϕ h ϕ h A ) ,
| F h ( λ 1 ) | 2 | F h ( λ 2 ) | 2 = [ a ( λ 1 ) a ( λ 2 ) ] | F 0 h A | 2 + [ b ( λ 1 ) b ( λ 2 ) ] | F h 0 | | F 0 h A | cos ( ϕ h 0 ϕ 0 h A ) ,
F h ( λ ) = [ F h ( λ ) + F h ( λ ) ] / 2 ,
f ( x ) = f ( Cx + d ) , x U ,
f ( x ) = ( 1 / υ ) h F h exp [ i 2 π h · ( Cx + d ) ] ,
F p = h H hp F h ,
H hp = exp ( i 2 π h · d ) U exp [ i 2 π ( p hC ) · x ] d x .
R ( θ 1 , θ 2 , θ 3 ) = U P ( x ) P ( y ) d x ,
R ( θ 1 , θ 2 , θ 3 ) = p | F p | 2 n | F n | 2 G pn ,
G pn = U exp [ i 2 π ( n + pC ) · x ] d x
F l ( R , ψ ) = F ( R , ψ , l / c ) = j n f j J n ( 2 π R r j ) exp { i [ n ( ψ + π / 2 ) n ϕ j + 2 π l z j / c ] } ,
I l ( R ) = ( 1 / 2 π ) 0 2 π | F l ( R , ψ ) | 2 d ψ ,
I l ( R ) = n | G n l ( R ) | 2 ,
G n l ( R ) = j f j J n ( 2 π R r j ) exp [ i ( n ϕ j + 2 π l z j / c ) ] .
f ( r , ϕ , z ) = ( 1 / c ) l n g n l ( r ) exp [ i ( n ϕ 2 π l z / c ) ] ,
g n l ( r ) = 0 G n l ( R ) J n ( 2 π R r ) 2 π R d R ,
G n D = G n + G n H = ( A n + i B n ) + ( A n H + i B n H ) ,
I = n ( A n 2 + B n 2 )
I D = n [ ( A n + A n H ) 2 + ( B n + B n H ) 2 ] .
H i = | F hkl | 2 ,
ϕ v = ϕ u + ψ u , v u ,
a = [ | F ( u 1 + u 2 ) | 2 + | F ( u 1 ) | 2 + | F ( u 2 ) 2 1 ] / × [ 2 | F ( u 1 + u 2 ) | | F ( u 2 ) | | F ( u 2 ) | ] ,
f ( x , t ) = a ( t ) δ ( x ) + g ( x ) ,
A ( x , t 1 ) A ( x , t 2 ) = [ a 2 ( t 1 ) a 2 ( t 2 ) ] δ ( x ) + [ a ( t 1 ) a ( t 2 ) ] [ g ( x ) + g * ( x ) ] .

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