Abstract.

This paper studies the phase-only reconstruction problem of recovering a complex-valued signal \(\textbf{x}\) in \(\mathbb{C}^d\) from the phase of \(\textbf{Ax}\) where \(\textbf{A}\) is a given measurement matrix in \(\mathbb{C}^{m\times d}\). The reconstruction, if possible, should be up to a positive scaling factor. By using the rank of discriminant matrices, uniqueness conditions are derived to characterize whether the underlying signal can be uniquely reconstructed. We are also interested in the problem of minimal measurement number. We show that at least \(2d\) but no more than \(4d-2\) measurements are needed for the reconstruction of all \(\textbf{x}\in \mathbb{C}^d\), whereas the minimal measurement number is exactly \(2d-1\) if we pursue the recovery of almost all signals. Moreover, when adapted to the phase-only reconstruction of \(\textbf{x}\in \mathbb{R}^d\), our uniqueness conditions are more practical and general than existing ones. Our theoretical results can be straightforwardly extended to affine phase-only reconstruction where the phase of \(\textbf{Ax}+\textbf{b}\) is observed for some \(\textbf{b}\in \mathbb{C}^m\).

Keywords

  1. phase-only reconstruction
  2. magnitude retrieval
  3. phase retrieval
  4. measurement matrix
  5. minimal measurement number

MSC codes

  1. 15A03
  2. 15A09
  3. 15A29

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Acknowledgments.

Part of the work was done when the first author was an undergraduate at Sun Yat-Sen University. He would like to thank Professor Anhua Wan and Professor Xiaoyong Fu for some enlightening discussions.

Supplementary Materials

PLEASE NOTE: These supplementary files have not been peer-reviewed.
Index of Supplementary Materials
Title of paper: Signal Reconstruction from Phase-only Measurements: Uniqueness Condition, Minimal Measurement Number and Beyond
Authors: Junren Chen and Michael K. Ng
File: Supplement.pdf
Type: PDF
Contents: Additional details for the proof of Proposition 3.1.

References

1.
R. Balan, P. Casazza, and D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal., 20 (2006), pp. 345–356.
2.
A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal., 37 (2014), pp. 106–125.
3.
I. Bartolini, P. Ciaccia, and M. Patella, Warp: Accurate retrieval of shapes using phase of Fourier descriptors and time warping distance, IEEE Trans. Pattern Anal. Mach. Intell., 27 (2005), pp. 142–147.
4.
J. Behar, M. Porat, and Y. Y. Zeevi, Image reconstruction from localized phase, IEEE Trans. Signal Process., 40 (1992), pp. 736–743.
5.
P. T. Boufounos, Sparse signal reconstruction from phase-only measurements, in Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA), Citeseer, 2013, pp. 256–259.
6.
E. J. Candes, X. Li, and M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61 (2015), pp. 1985–2007.
7.
J. Chen and M. K. Ng, Uniform Exact Reconstruction of Sparse Signals and Low-rank Matrices from Phase-only Measurements, preprint, https://arxiv.org/abs/2209.12824, 2022.
8.
J. Chen, C.-L. Wang, M. K. Ng, and D. Wang, High Dimensional Statistical Estimation under One-bit Quantization, preprint, https://arxiv.org/abs/2202.13157, 2022.
9.
A. Conca, D. Edidin, M. Hering, and C. Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal., 38 (2015), pp. 346–356.
10.
S. Dirksen and S. Mendelson, Non-Gaussian hyperplane tessellations and robust one-bit compressed sensing, J. Eur. Math. Soc., 23 (2021), pp. 2913–2947.
11.
C. Espy and J. Lim, Effects of additive noise on signal reconstruction from Fourier transform phase, IEEE Trans. Acoustics, Speech, Signal Process., 31 (1983), pp. 894–898.
12.
T. Feuillen, M. E. Davies, L. Vandendorpe, and L. Jacques, (\(\ell_1\),\(\ell_2\))-rip and projected back-projection reconstruction for phase-only measurements, IEEE Signal Proc. Lett., 27 (2020), pp. 396–400.
13.
J. R. Fienup, Reconstruction of an object from the modulus of its fourier transform, Opt. Lett., 3 (1978), pp. 27–29.
14.
J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Optics, 21 (1982), pp. 2758–2769.
15.
B. Gao, Q. Sun, Y. Wang, and Z. Xu, Phase retrieval from the magnitudes of affine linear measurements, Adv. Appl. Math., 93 (2018), pp. 121–141.
16.
M. Hayes, The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Trans. Acoustics, Speech, Signal Process., 30 (1982), pp. 140–154.
17.
M. Hayes, J. Lim, and A. Oppenheim, Signal reconstruction from phase or magnitude, IEEE Trans. Acoustics, Speech, Signal Process., 28 (1980), pp. 672–680.
18.
G. Hua and M. T. Orchard, Image inpainting based on geometrical modeling of complex wavelet coefficients, in Proceedings of the 2007 IEEE International Conference on Image Processing, Vol. 1, IEEE, 2007, pp. I-553–I-556.
19.
M. Huang, Y. Rong, Y. Wang, and Z. Xu, Almost everywhere generalized phase retrieval, Appl. Comput. Harmon. Anal., 50 (2021), pp. 16–33.
20.
L. Jacques and T. Feuillen, The importance of phase in complex compressive sensing, IEEE Trans. Inform. Theory, 67 (2021), pp. 4150–4161.
21.
V. Kishore, S. Mukherjee, and C. S. Seelamantula, Phasesense-signal reconstruction from phase-only measurements via quadratic programming, in Proceedings of the 2020 International Conference on Signal Processing and Communications (SPCOM), IEEE, 2020, pp. 1–5.
22.
K. Knudson, R. Saab, and R. Ward, One-bit compressive sensing with norm estimation, IEEE Trans. Inform. Theory, 62 (2016), pp. 2748–2758.
23.
C. Kughlin and D. Hines, The phase correlation image alignment method, in Proceedings of the IEEE International Conference on Cybernetics and Society, IEEE, 1975, pp. 163–165.
24.
A. Levi and H. Stark, Signal restoration from phase by projections onto convex sets, J. Opt. Soc. Amer., 73 (1983), pp. 810–822.
25.
Y. Li and A. Kurkjian, Arrival time determination using iterative signal reconstruction from the phase of the cross spectrum, IEEE Trans. Acoustics, Speech, Signal Process., 31 (1983), pp. 502–504.
26.
E. Loveimi and S. M. Ahadi, Objective evaluation of magnitude and phase only spectrum-based reconstruction of the speech signal, in Proceedings of the 4th International Symposium on Communications, Control and Signal Processing (ISCCSP), IEEE, 2010, pp. 1–4.
27.
C. Ma, Novel criteria of uniqueness for signal reconstruction from phase, IEEE Trans. Signal Process., 39 (1991), pp. 989–992.
28.
A. V. Oppenheim, M. H. Hayes, and J. S. Lim, Iterative procedures for signal reconstruction from phase, in 1980 Intl. Optical Computing Conf. I, Vol. 231, SPIE, 1980, pp. 121–129.
29.
A. V. Oppenheim and J. S. Lim, The importance of phase in signals, Proc. IEEE, 69 (1981), pp. 529–541.
30.
M. Porat and G. Shachor, Signal representation in the combined phase-spatial space: Reconstruction and criteria for uniqueness, IEEE Trans. Signal Proces., 47 (1999), pp. 1701–1707.
31.
T. Quatieri and A. Oppenheim, Iterative techniques for minimum phase signal reconstruction from phase or magnitude, IEEE Trans. Acoustics, Speech, Signal Process., 29 (1981), pp. 1187–1193.
32.
A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), pp. 883–890.
33.
T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, Blind deconvolution through digital signal processing, Proc. IEEE, 63 (1975), pp. 678–692.
34.
J. Sun, Q. Qu, and J. Wright, A geometric analysis of phase retrieval, Found. Comput. Math., 18 (2018), pp. 1131–1198.
35.
S. Urieli, M. Porat, and N. Cohen, Optimal reconstruction of images from localized phase, IEEE Trans. Image Process., 7 (1998), pp. 838–853.
36.
S. Wang, L. Zhang, Y. Li, J. Wang, and E. Oki, Multiuser mimo communication under quantized phase-only measurements, IEEE Trans. Commun., 64 (2016), pp. 1083–1099.
37.
J. Wu, J. Liu, Y. Kong, X. Han, L. Senhadji, and H. Shu, Phase-only Signal Reconstruction by MagnitudeCut, preprint, https://arxiv.org/abs/1603.00210, 2016.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1341 - 1365
ISSN (online): 1095-712X

History

Submitted: 7 July 2021
Accepted: 30 January 2023
Published online: 5 July 2023

Keywords

  1. phase-only reconstruction
  2. magnitude retrieval
  3. phase retrieval
  4. measurement matrix
  5. minimal measurement number

MSC codes

  1. 15A03
  2. 15A09
  3. 15A29

Authors

Affiliations

Junren Chen
Department of Mathematics, The University of Hong Kong, Hong Kong.
Department of Mathematics, The University of Hong Kong, Hong Kong.

Funding Information

Research Grants Council, University Grants Committee (RGC, UGC): GRF 12300519, 17201020, 17300021, C1013-21GF, C7004-21GF
Funding: The first author was supported by an HKPFS scholarship from the Hong Kong Research Grants Council (RGC), the second author was supported in part by the Hong Kong RGC GRF 12300519, 17201020, 17300021, C1013-21GF, C7004-21GF, and jointly by NSFC-RGC N-HKU76921.

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