Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion

Previously it was shown that one can solve the phase-retrieval problem from two intensities observed at the Fourier-transform plane of an object in one dimension by using the Fourier-series expansion. In this paper, an improved method using the logarithmic Hilbert transform and the Fourier series expansion is proposed. It is proved from the distribution of zeros in the complex plane that the Fourier-transform phase of Hermitian object functions cannot be retrieved by using the previous method but can be retrieved by using the method in this paper. The results reconstructed by the present method are also shown in computer simulations.

[1]  Nobuharu Nakajima,et al.  Phase retrieval from two intensity measurements using the Fourier series expansion , 1987 .

[2]  R. H. T. Bates,et al.  Composite two-dimensional phase-restoration procedure , 1983 .

[3]  J. G. Walker.  The Phase Retrieval Problem , 1981 .

[4]  Aharon Levi,et al.  Image restoration by the method of generalized projections with application to restoration from magnitude , 1984 .

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

[6]  Henri H. Arsenault,et al.  Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem , 1985 .

[7]  J G Walker,et al.  Computer simulation of a method for object reconstruction from stellar speckle interferometry data. , 1982, Applied optics.

[8]  James R. Fienup,et al.  Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint , 1987 .

[9]  M. Fiddy,et al.  The phase problem , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  B. J. Hoenders.  On the solution of the phase retrieval problem , 1975 .

[11]  M. A. Fiddy,et al.  Analytic Fourier Optics , 1979 .

[12]  R. Gerchberg.  A practical algorithm for the determination of phase from image and diffraction plane pictures , 1972 .

[13]  Nicholas George,et al.  Stationary phase approximations in Fresnel-zone magnitude-only reconstructions , 1987 .

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

[15]  H. Ferwerda,et al.  The Phase Reconstruction Problem for Wave Amplitudes and Coherence Functions , 1978 .

[16]  N Nakajima,et al.  Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information , 1986 .

[17]  M. Fiddy,et al.  Phase retrieval using two intensity measurements in the complex plane. , 1981, Optics letters.

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

[19]  M. Fiddy,et al.  The Inverse Scattering Problem in Structural Determinations , 1980 .

[20]  T. Asakura,et al.  A New Approach to Two-dimensional Phase Retrieval , 1985 .

[21]  M A Fiddy,et al.  Solution of the two-dimensional phase-retrieval problem. , 1985, Optics letters.