Numerical investigation of the uniqueness of phase retrieval

Both a new iterative grid-search technique and the iterative Fourier-transform algorithm are used to illuminate the relationships among the ambiguous images nearest a given object, error metric minima, and stagnation points of phase-retrieval algorithms. Analytic expressions for the subspace of ambiguous solutions to the phase-retrieval problem are derived for 2 × 2 and 3 × 2 objects. Monte Carlo digital experiments using a reduced-gradient search of these subspaces are used to estimate the probability that the worst-case nearest ambiguous image to a given object has a Fourier modulus error of less than a prescribed amount. Probability distributions for nearest ambiguities are estimated for different object-domain constraints.

[1]  E. Wolf.  Is a Complete Determination of the Energy Spectrum of Light Possible from Measurements of the Degree of Coherence , 1962 .

[2]  E. Leith,et al.  Reconstructed Wavefronts and Communication Theory , 1962 .

[3]  A. Walther.  The Question of Phase Retrieval in Optics , 1963 .

[4]  L. Carlitz,et al.  The distribution of irreducible polynomials in several indeterminates , 1963 .

[5]  EDWARD M. HOFSTETTER,et al.  Construction of time-limited functions with specified autocorrelation functions , 1964, IEEE Trans. Inf. Theory.

[6]  L. Carlitz.  The distribution of irreducible polynomials in several indeterminates. II , 1965 .

[7]  R.H.T. Bates,et al.  Inferring phase information from modulus information in two-dimensional aperture synthesis , 1974 .

[8]  W. O. Saxton,et al.  Computer techniques for image processing in electron microscopy , 1979 .

[9]  J R Fienup,et al.  Reconstruction of an object from the modulus of its Fourier transform. , 1978, Optics letters.

[10]  J. Fienup.  Space Object Imaging Through The Turbulent Atmosphere , 1979 .

[11]  L. Norton-Wayne.  Computer Techniques for Image Processing in Electron Microscopy , 1979, Advances in Imaging and Electron Physics.

[12]  L. G. Sodin,et al.  On the ambiguity of the image reconstruction problem , 1979 .

[13]  J. R. Fienup,et al.  Noise Properties Of Images Reconstructed From Fourier Modulus , 1980, Other Conferences.

[14]  Wayne Lawton.  A Numerical Algorithm For 2-D Wavefront Reconstruction From Intensity Measurements In A Single Plane , 1980, Other Conferences.

[15]  A. Huiser,et al.  Ambiguity of the phase-reconstruction problem. , 1980, Optics letters.

[16]  Philip E. Gill,et al.  Practical optimization , 1981 .

[17]  M. Hayes,et al.  Reducible polynomials in more than one variable , 1982, Proceedings of the IEEE.

[18]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[19]  I. Manolitsakis.  Two‐dimensional scattered fields: A description in terms of the zeros of entire functions , 1982 .

[20]  Thomas S. Huang,et al.  Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude , 1983 .

[21]  Fernando Cukierman,et al.  Stability of unique Fourier-transform phase reconstruction , 1983 .

[22]  J. Fienup,et al.  Uniqueness of phase retrieval for functions with sufficiently disconnected support , 1983 .

[23]  M A Fiddy,et al.  Enforcing irreducibility for phase retrieval in two dimensions. , 1983, Optics letters.

[24]  G. Newsam,et al.  Necessary conditions for a unique solution to two‐dimensional phase recovery , 1984 .

[25]  James R. Fienup.  Experimental Evidence of the Uniqueness of Phase Retrieval from Intensity Data , 1984 .

[26]  Alan H. Greenaway,et al.  Phaseless Object Reconstruction , 1984 .

[27]  I. Stefanescu.  On the phase retrieval problem in two dimensions , 1985 .

[28]  J. C. Dainty,et al.  A note on Eisenstein's irreducibility criterion for two-dimensional sampled objects , 1985 .

[29]  James R. Fienup,et al.  Phase-retrieval stagnation problems and solutions , 1986 .

[30]  B J Brames.  Unique phase retrieval with explicit support information. , 1986, Optics letters.

[31]  Thomas R. Crimmins,et al.  Phase Retrieval for Discrete Functions with Support Constraints: Summary , 1986, Topical Meeting On Signal Recovery and Synthesis II.

[32]  J. R. Fienup,et al.  The Effects Of Tapered Illumination And Fourier Intensity Errors On Phase Retrieval , 1988, Optics & Photonics.

[33]  M. Nieto-Vesperinas,et al.  Performance of a simulated-annealing algorithm for phase retrieval , 1988 .