Abstract
The problem of phase retrieval arises in experimental uses of diffraction to determine intrinsic structure because the modulus of a Fourier transform is all that can usually be measured after diffraction occurs. For finite distributions, the phase retrieval problem can be solved by methods of factorization in suitable rings of polynomials; for continuous distributions with compact support, the methods of complex analysis are needed to solve the phase retrieval problem. These methods are discussed and examples are given for illustration.
Similar content being viewed by others
References
Ahlfors, L.: Complex analysis. New York: McGraw-Hill 1966
Arsac, J.: Transformation de Fourier et théorie des distributions. Paris: Dunod 1961
Avrami, M.: A method of direct determination of crystal structure from X-ray data. Z. Kristallogr.100, 381–393 (1939)
Born, M., Wolf, L.: Principles of optics. Oxford: Pergamon Press 1970
Buerger, M.J.: Interpoint distances in cyclotomic sets. Can. Mineral.16, 301–314 (1978)
Buerger, M.J.: Exploration of cyclotomic point sets in tautoeikomic complementary pairs. Z. Kristallogr.145, 371–411 (1977)
Buerger, M.J.: Proofs and generalizations of Patterson's theorems on homometric complementary pairs. Z. Kristallogr.143, 79–98 (1976)
Buerger, M.J.: The algebra and geometry of convolutions. Atti. Acad. Noz. Jincei. Mem. Cl. Sci. Math. Nat. Sez.II (8, 683–695 (1962)
Buerger, M.J.: An algebraic representation of the images of sets of points. Atti. Acad. Sci. Torino Cl. Sci. Fis. Math. Nat.96, 175–192 (1961/1962)
Buerger, M.J.: Vector sets. Acta Crystallogr.3, 87–97 (1950)
Bullough, R.K.: On homometric sets. II. Sets obtained by singular transformations. Acta Crystallogr.17, 295–308 (1964)
Bullough, R.K.: On homometric sets. I. Some general theorems. Acta Crystallogr.14, 257–268 (1961)
Burge, R.E., Fiddy, M.A., Greenway, A.H., Ross, G.: The phase problem. Proc. R. Soc. London A350, 191–212 (1976)
Caelli, T.M.: On generating spatial configurations with identical interpoint distance distributions. Comb. Math. VII, Proc., New Castle, Australia. In: Lecture Notes in Mathematics, Vol. 829. Berlin, Heidelberg, New York: Springer 1979, pp. 69–75
Carpenter, G.: Principles of crystal structure determination. New York: Benjamin 1969
Cartwright, M.L.: The zeros of certain integral functions. Q. J. Math.1, 38–59 (1930)
Champeney, D.C.: Fourier transforms and their physical applications. London: Academic Press 1973
Chung Chieh: Analysis of cyclotomic sets. Z. Kristallogr.150, 261–277 (1979)
Cowley, J.: Diffraction physics. New York: North-Holland 1975
Fowle, E.N.: The design of FM pulse compression signals. IEEE Trans. IT-10, 61–67 (1964)
Harger, R.O.: Realization of ambiguity functions. TEEE Trans. SET-9, 127–130 (1963)
Hartman, P., Winter, A.: On the maxima of the Patterson functions. Phys. Rev.81, 271–273 (1951)
Hille, E.: Analytic function theory. Vols. I, II. New York: Chelsea 1973 and 1976
Hosemann, R., Bagchi, S.N.: On homometric sets. Acta Crystallogr.7, 237–241 (1954)
Hosemann, R., Bagchi, S.N.: Direct analysis of diffraction by matter. New York: North-Holland 1962
Kline, M., Kay, I.: Electromagnetic theory and geometrical optics. New York: Wiley 1965
Ladd, M.F.C., Palmer, R.A.: Structure determination by X-ray crystallography. New York: Plenum Press 1977
Lipson, H., Cochran, W.: The determination of crystal structures. Ithaca: Cornell University Press 1966
McLachlan, D.: X-ray crystal structure. New York: McGraw-Hill 1957
Morgan, S.P.: On the mathematical question of phase reconstruction, Misc. 10, Restoration of atmospherically degraded images, Vol. 3, Woods Hole Summer Study, Nat. Acad. Sci. and NRC, pp. 53–60 (1966)
O'Neill, E.L., Walther, A.: The question of phase in image formation. Optica Acta10, 33–40 (1962)
Paley, R.E.A.C., Wiener, N.: Fourier transforms in the complex domain. AMS Colloquim Publ. Vol. 19. New York, 1934
Patterson, A.L.: Ambiguities in the X-ray analysis of crystal structure. Phys. Rev.65, 195–201 (1944)
Patterson, A.L.: Homometric structures. Nature143, 939–940 (1939)
Patterson, A.L.: A direct method for the determination of the components of interatomic distances in crystals. Z. Kristallogr.90, 517–542 (1935)
Pauling, L., Shappell, M.D.: The crystal structure of bixbyite and theC-modification of the sesquioxides. Z. Kristallogr.75, 128–142 (1930)
Rau, V.G. et al.: On the calculation of possible Patterson cyclotomic sets. Proc. Acad. Sci. USSR225, 1110–1113 (1980)
Readhead, A.C.S., Wilkinson, R.N.: The mapping of compact radio sources from VLBI data, Astrophys. J.223, 25–36 (1978)
Rosenblatt, J., Seymour, P.: The structure of homometric sets, SIAM J. Alg. Disc. Meth.3, 343–350 (1982)
Rossi, P., McMurdy, C.W., McCreery, R.L.: Diffractive spectroelectrochemistry; use of diffracted light for monitoring electrogenerated chromophores. J. Am. Chem. Soc.103, 2524–2529 (1981)
Rothbauer, R.: On the separation of the unknown parameters of the problem of crystal-structure analysis. Acta. Crystallogr. A36, 27–32 (1980)
Schwartz, L.: Théorie des distributions. Paris: Hermann 1966
Silver, S.: Microwave antenna theory and design. MIT Rad. Lab. Series, Vol. 12, pp. 279–282. New York: McGraw-Hill 1949
Stout, G., Jensen, L.: X-ray structure determination. London: MacMillan 1968
Titchmarsh, E.C.: The zeros of certain integral functions. Proc. London Math. Soc. (2)25, 283–307 (1926)
Toraldo Di Francia, G.: Resolving power and information. J. Opt. Soc. Am.45, 497–501 (1955)
Walther, A.: The question of phase retrieval in optics. Optica Acta10, 41–49 (1963)
Wiener, N.: Generalized harmonic analysis. Acta Math.55, 117–258 (1930)
International tables for X-ray crystallography. Vol.I. Birmingham, England: The Kunech Press 1965
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Partially supported by NSF Grant MCS 8218800
Rights and permissions
About this article
Cite this article
Rosenblatt, J. Phase retrieval. Commun.Math. Phys. 95, 317–343 (1984). https://doi.org/10.1007/BF01212402
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212402