Abstract
In this paper, we discuss how to partially determine the Fourier transform
given the data
References
[1] E. J. Akutowicz, On the determination of the phase of a Fourier integral. I, Trans. Amer. Math. Soc. 83 (1956), 179–192. 10.1090/S0002-9947-1956-0080802-2Search in Google Scholar
[2] E. J. Akutowicz, On the determination of the phase of a Fourier integral. II, Trans. Amer. Math. Soc. 84 (1957), 237–238. 10.1090/S0002-9939-1957-0084639-6Search in Google Scholar
[3] R. Barakat and G. Newsam, Necessary conditions for a unique solution to two-dimensional phase recovery, J. Math. Phys. 25 (1984), no. 11, 3190–3193. 10.1063/1.526089Search in Google Scholar
[4] R. P. Boas, Jr., Entire Functions, Academic Press, New York, 1954. Search in Google Scholar
[5] M. L. Cartwright, Integral Functions, Cambridge Tracts Math. Math. Phys. 44, Cambridge University, Cambridge, 1956. Search in Google Scholar
[6] T. R. Crimmins and J. R. Fienup, Uniqueness of phase retrieval for functions with sufficiently disconnected support, J. Opt. Soc. Amer. 73 (1983), no. 2, 218–221. 10.1364/JOSA.73.000218Search in Google Scholar
[7] V. Elser, Solution of the crystallographic phase problem by iterated projections, Acta Crystallogr. A 59 (2003), 201–209. 10.1107/S0108767303002812Search in Google Scholar
[8] M. A. Fiddy and A. H. Greenaway, Phase retrieval using zero information, Optics Commun. 29 (1979), 270–272. 10.1016/0030-4018(79)90097-XSearch in Google Scholar
[9] J. R. Fienup, Reconstruction of an object from the modulus of its fourier transform, Optics Letters 3 (1978), 27–29. 10.1364/OL.3.000027Search in Google Scholar PubMed
[10] E. M. Hofstetter, Construction of time-limited functions with specified autocorrelation functions, IEEE Trans. Inform. Theory 10 (1964), 119–126. 10.1109/TIT.1964.1053648Search in Google Scholar
[11] N. E. Hurt, Phase Retrieval and Zero Crossings. Mathematical Methods in Image Reconstruction, Math. Appl. 52, Kluwer Academic, Dordrecht, 1989. 10.1007/978-94-010-9608-9Search in Google Scholar
[12] P. Jaming, Phase retrieval techniques for radar ambiguity problems, J. Fourier Anal. Appl. 5 (1999), no. 4, 309–329. 10.1007/BF01259373Search in Google Scholar
[13] P. Jaming, Uniqueness results in an extension of Pauli’s phase retrieval problem, Appl. Comput. Harmon. Anal. 37 (2014), no. 3, 413–441. 10.1016/j.acha.2014.01.003Search in Google Scholar
[14] P. Jaming, K. Kellay and R. Perez, III, Phase retrieval for wide band signals, J. Fourier Anal. Appl. 26 (2020), no. 4, Paper No. 54. 10.1109/SampTA45681.2019.9030853Search in Google Scholar
[15] P. Jaming and S. Pérez-Esteva, The phase retrieval problem for solutions of the Helmholtz equation, Inverse Problems 33 (2017), no. 10, Article ID 105007. 10.1088/1361-6420/aa8640Search in Google Scholar
[16] M. V. Klibanov, P. E. Sacks and A. V. Tikhonravov, The phase retrieval problem, Inverse Problems 11 (1995), no. 1, 1–28. 10.1088/0266-5611/11/1/001Search in Google Scholar
[17] P. Koosis, The Logarithmic Integral. I, Cambridge University, Cambridge, 1997. Search in Google Scholar
[18] B. J. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monogr. 5, American Mathematical Society, Providence, 1972. Search in Google Scholar
[19] B. Y. Levin, Lectures on Entire Functions, Transl. Math. Monogr. 150, American Mathematical Society, Providence, 1996. 10.1090/mmono/150Search in Google Scholar
[20] J. N. McDonald, Phase retrieval and magnitude retrieval of entire functions, J. Fourier Anal. Appl. 10 (2004), no. 3, 259–267. 10.1007/s00041-004-0973-9Search in Google Scholar
[21] J. Miao, P. Charalambous, J. Kirz and D. Sayre, Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens, Nature 400 (1999), 342–344. 10.1038/22498Search in Google Scholar
[22] J. Rosenblatt, Phase retrieval, Comm. Math. Phys. 95 (1984), no. 3, 317–343. 10.1007/BF01212402Search in Google Scholar
[23] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, New York, 2007. Search in Google Scholar
[24] D. Sayre, Some implication of a theorem due to Shannon, Acta Crystallogr. 5 (1952), 843–843. 10.1107/S0365110X52002276Search in Google Scholar
[25] Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao and M. Negev, Phase retrieval with application to optical imaging, IEEE Signal Process. Mag. 32 (2015), 87–109. 10.1109/MSP.2014.2352673Search in Google Scholar
[26] J. Spence, U. Weierstall and M. Howells, Coherence and sampling requirements for diffractive imaging, Ultramicroscopy 101 (2004), 149–152. 10.1016/j.ultramic.2004.05.005Search in Google Scholar PubMed
[27] A. Walther, The question of phase retrieval in optics, Optica Acta 10 (1963), 41–49. 10.1080/713817747Search in Google Scholar
[28] J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang and L. A. Nagahara, Atomic resolution imaging of a carbon nanotube from diffraction intensities, Science 300 (2003), 1419–1421. 10.1126/science.1083887Search in Google Scholar PubMed
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps
- Perturbations on K-fusion frames
- Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition
- Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems
- Construction of complex potentials for multiply connected domain
- Solution of a transport equation with discontinuous coefficients
- Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets
- Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals
- Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method
- M-projective curvature tensor on an (LCS)2n+1-manifold
- An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel
- z-arcs in the thirty degrees sector
Articles in the same Issue
- Frontmatter
- Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps
- Perturbations on K-fusion frames
- Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition
- Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems
- Construction of complex potentials for multiply connected domain
- Solution of a transport equation with discontinuous coefficients
- Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets
- Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals
- Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method
- M-projective curvature tensor on an (LCS)2n+1-manifold
- An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel
- z-arcs in the thirty degrees sector