Abstract
The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumptions or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference intensity measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference intensity measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference intensity measurements with a known or an unknown reference which is unrelated to the unknown signal.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akutowicz, E.J.: On the determination of the phase of a Fourier integral I. Trans. Am. Math. Soc. 83(1), 179–192 (1956)
Akutowicz, E.J.: On the determination of the phase of a Fourier integral II. Proc. Am. Math. Soc. 8(2), 234–238 (1957)
Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization. SIAM J. Imaging Sci. 7(1), 35–66 (2014)
Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15(4), 488–501 (2009)
Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)
Bandeira, A.S., Chen, Y., Mixon, D.G.: Phase retrieval from power spectra of masked signals. Inf. Interf. 3(2), 83–102 (2014)
Beinert, R., Plonka, G.: Ambiguities in one-dimensional discrete phase retrieval from Fourier magnitudes. J. Fourier Anal. Appl. 21(6), 1169–1198 (2015)
Beinert, R., Plonka, G.: Enforcing uniqueness in one-dimensional phase retrieval by additional signal information in time domain (2016). arXiv:1604.04493v1
Boas Jr., R.P.: Entire Functions. Academic, New York (1954)
Bodmann, B.G., Hammen, N.: Stable phase retrieval with low-redundancy frames. Adv. Comput. Math. 41(2), 317–331 (2015)
Bruck, Y.M., Sodin, L.G.: On the ambiguity of the image reconstruction problem. Opt. Commun. 30(3), 304–308 (1979)
Burge, R.E., Fiddy, M.A., Greenaway, A.H., Ross, G.: The phase problem. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 350, 191–212 (1976)
Candès, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1), 199–225 (2013)
Dainty, J.C., Fienup, J.R.: Phase retrieval and image reconstruction for astronomy. In: Stark, H. (ed.) Image Recovery: Theory and Application, chap. 7, pp. 231–275. Academic, Orlando (1987)
Hauptman, H.A.: The phase problem of x-ray crystallography. Rep. Progr. Phys. 54(11), 1427–1454 (1991)
Hildebrand, F.B.: Introduction to Numerical Analysis, 2nd edn. Dover Publications, New York (1987)
Hofstetter, E.M.: Construction of time-limited functions with specified autocorrelation functions. IEEE T. Inf. Theory 10(2), 119–126 (1964)
Kim, W., Hayes, M.H.: Iterative phase retrieval using two Fourier transform intensities. In: Proceedings: ICASSP 90: 1990 international conference on acoustics, speech and signal processing: April 3–6, 1990, vol. 3, pp. 1563–1566. IEEE Signal Processing Society (1990)
Kim, W., Hayes, M.H.: Phase retrieval using two Fourier-transform intensities. J. Opt. Soc. Am. A 7(3), 441–449 (1990)
Kim, W., Hayes, M.H.: The phase retrieval problem in x-ray crystallography. In: Proceedings: ICASSP 91: 1991 international conference on acoustics, speech and signal processing: May 14–17, 1991, vol. 3, pp. 1765–1768. IEEE Signal Processing Society (1991)
Kim, W., Hayes, M.H.: Phase retrieval using a window function. IEEE Trans. Signal Process. 41(3), 1409–1412 (1993)
Klibanov, M.V., Sacks, P.E., Tikhonravov, A.V.: The phase retrieval problem. Inverse Probl. 11(1), 1–28 (1995)
Maretzke, S.: A uniqueness result for propagation-based phase contrast imaging from a single measurement (2014). arXiv:1409.4794v1
Markushevich, A.I.: Theory of functions of a complex variable, 2nd edn. Chelsea Publishing Co., New York (1977)
Millane, R.P.: Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7(3), 394–411 (1990)
Nawab, S., Quatieri, T.F., Lim, J.S.: Algorithms for signal reconstruction from short-time Fourier transform magnitude. In: Proceedings: ICASSP 83: IEEE international conference on acoustics, speech, and signal, vol. 8, pp. 800–803. IEEE (1983)
Nawab, S., Quatieri, T.F., Lim, J.S.: Signal reconstruction from short-time Fourier transform magnitude. IEEE T. Acoust. Speech ASSP-31(4), 986–998 (1983)
Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cliffs (1989)
Pohl, V., Yang, F., Boche, H.: Phaseless signal recovery in infinite dimensional spaces using structured modulations. J. Fourier Anal. Appl. 20(6), 1212–1233 (2014)
Prony, R.: Essai expérimental et analytique sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, á différentes températures. Journal de l’École polytechnique 2, 24–76 (1795)
Raz, O., Dudovich, N., Nadler, B.: Vectorial phase retrieval of 1-d signals. IEEE Trans. Signal Process. 61(7), 1632–1643 (2013)
Raz, O., Schwartz, O., Austin, D., Wyatt, A.S., Schiavi, A., Smirnova, O., Nadler, B., Walmsley, I.A., Oron, D., Dudovich, N.: Vectorial phase retrieval for linear characterization of attosecond pulses. Phys. Rev. Lett. 107(13), 133,902(5) (2011)
Seifert, B., Stolz, H., Donatelli, M., Langemann, D., Tasche, M.: Multilevel Gauss-Newton methods for phase retrieval problems. J. Phys. A 39(16), 4191–4206 (2006)
Seifert, B., Stolz, H., Tasche, M.: Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness. J. Opt. Soc. Am. B Opt. Phys 21(5), 1089–1097 (2004)
Walther, A.: The question of phase retrieval in optics. Opt. Acta. 10(1), 41–49 (1963)
Wood, J.W., Fiddy, M.A., Burge, R.E.: Phase retrieval using two intensity measurements in the complex plane. Opt. Lett. 6(11), 514–516 (1981)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic, New York (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beinert, R. One-Dimensional Phase Retrieval with Additional Interference Intensity Measurements. Results Math 72, 1–24 (2017). https://doi.org/10.1007/s00025-016-0633-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0633-9