Skip to main content

Analytic Phase Retrieval Based on Intensity Measurements

Abstract

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc \(\mathbb{D}\) by using a sample value f (a) and certain n-intensity measurements \(\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|\), where a1, ⋯, \({a_n} \in \mathbb{D}\), and \({E_{{a_1} \cdots {a_n}}}\) is the n-th term of the Gram-Schmidt orthogonalization of the Szegö kernels \({k_{{a_1}}}, \cdots ,{k_{{a_n}}}\), or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna’s inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2- and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements \(\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|\).

This is a preview of subscription content, access via your institution.

References

  1. Chen, Q H, Qian T, Tan L H. A Theory on Non-Constant Frequency Decompositions and Applications. Advancements in Complex Analysis. Cham: Springer, 2020: 1–37

    MATH  Google Scholar 

  2. Garnett J. Bounded Analytic Functions. 236. Springer Science & Business Media, 2007

  3. Li Y F, Zhou C. Phase retrieval of finite Blaschke projection. Mathematical Methods in the Applied Sciences, 2020, 43(15): 9090–9101

    MathSciNet  Article  Google Scholar 

  4. Qian T. Cyclic AFD Algorithm for best rational. Mathematical Methods in the Applied Sciences, 2014, 37(6): 846–859

    MathSciNet  Article  Google Scholar 

  5. Qian T. Sparse representations of random signals. Mathematical Methods in the Applied Sciences, 2021, accepted

  6. Qian T, Wang Y B. Adaptive Fourier series — A variation of greedy algorithm. Advances in Computational Mathematics, 2011, 34(3): 279–293

    MathSciNet  Article  Google Scholar 

  7. Qian T, Wang J Z, Mai W X. An enhancement algorithm for cyclic adaptive Fourier decomposition. Applied and Computational Harmonic Analysis, 2019, 47(2): 516–525

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tao Qian.

Additional information

Dedicated to the memory of Professor Jiarong YU

Tao Qian was funded by The Science and Technology Development Fund, Macau SAR (File no. 0123/2018/A3). You-Fa Li was supported by the Natural Science Foundation of China (61961003, 61561006, 11501132), Natural Science Foundation of Guangxi (2016GXNS-FAA380049) and the talent project of the Education Department of the Guangxi Government for one thousand Young-Middle-Aged backbone teachers. Wei Qu was supported by the Natural Science Foundation of China (12071035).

Electronic supplementary material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qu, W., Qian, T., Deng, G. et al. Analytic Phase Retrieval Based on Intensity Measurements. Acta Math Sci 41, 2123–2135 (2021). https://doi.org/10.1007/s10473-021-0619-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0619-x

Key words

  • phase retrieval
  • Hardy space of the unit disc
  • Szegö kernel
  • Takenaka-Malmquist system
  • Gram-Schmidt orthogonalization
  • adaptive Fourier decomposition

2010 MR Subject Classification

  • 30H10