×

A direct solver for the phase retrieval problem in ptychographic imaging. (English) Zbl 07318078

Summary: Measurements achieved with ptychographic imaging are a special case of diffraction measurements. They are generated by illuminating small parts of a sample with, e.g., a focused X-ray beam. By shifting the sample, a set of far-field diffraction patterns of the whole sample is then obtained. From a mathematical point of view those measurements are the squared modulus of the windowed Fourier transform of the sample. Thus, we have a phase retrieval problem for local Fourier measurements. A direct solver for this problem was introduced by M. A. Iwen et al. [SIAM J. Imaging Sci. 9, No. 4, 1655–1688 (2016; Zbl 1352.49035)] and improved by M. Iwen et al. [“Phase retrieval from local measurements in two dimensions”, in: Y. M. Lu (ed.) et al., Wavelets and sparsity XVII. Proc. Vol. 10394. San Diego, CA: SPIE. Article ID 103940X (2017; doi:10.1117/12.2274355)]. Motivated by the applied perspective of ptychographic imaging, we present a generalization of this method and compare the different versions in numerical experiments. The new method proposed herein turns out to be more stable, particularly in the case of missing data.

MSC:

15Bxx Special matrices
78Axx General topics in optics and electromagnetic theory
94Axx Communication, information
90Cxx Mathematical programming

Citations:

Zbl 1352.49035
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bauschke, H. H.; Combettes, P. L.; Luke, D. R., Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization, J. Opt. Soc. Am. A, 19, 7, 1334-1345 (2002)
[2] Candès, E. J.; Li, X.; Soltanolkotabi, M., Phase retrieval from coded diffraction patterns, Applied and Computational Harmonic Analysis, 39, 2, 277-299 (2015) · Zbl 1329.78056
[3] Candès, E. J.; Li, X.; Soltanolkotabi, M., Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61, 4, 1985-2007 (2015) · Zbl 1359.94069
[4] Candès, E. J.; Strohmer, T.; Voroninski, V., PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Commun. Pure Appl. Math., 66, 8, 1241-1274 (2013) · Zbl 1335.94013
[5] Chapman, H. N., Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution, Ultramicroscopy, 66, 3, 153-172 (1996)
[6] Eldar, Y. C.; Hammen, N.; Mixon, D. G., Recent advances in phase retrieval, IEEE Signal Process. Mag., 33, 5, 158-162 (2016)
[7] Fienup, J. R., Reconstruction of a complex-valued object from the modulus of its fourier transform using support constraint, J. Opt. Soc. Am. A, 4, 118-123 (1986)
[8] Gerchberg, R. W.; Saxton, W. O., A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 35, 237-246 (1972)
[9] Gross, D.; Krahmer, F.; Kueng, R., Improved recovery guarantees for phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 42, 1, 37-64 (2017) · Zbl 1393.94250
[10] Hegerl, R.; Hoppe, W., Dynamic theory of crystalline structure analysis by electron diffraction in homogeneous primary wave field, Ber. Bunsen-Ges. Phys. Chem., 74, 1148-1154 (1970)
[11] Iwen, M.; Preskitt, B.; Saab, R.; Viswanathan, A., Phase retrieval from local measurements in two dimensions, (Wavelets and Sparsity XVII, Vol. 10394 (2017), International Society for Optics and Photonics), 103940X
[12] Iwen, M. A.; Preskitt, B.; Saab, R.; Viswanathan, A., Phase retrieval from local measurements: improved robustness via eigenvector-based angular synchronization, Appl. Comput. Harmonic Anal., 48, 1, 415-444 (2020) · Zbl 07140142
[13] Iwen, M. A.; Viswanathan, A.; Wang, Y., Fast phase retrieval from local correlation measurements, SIAM J. Imaging Sci., 9, 4, 1655-1688 (2016) · Zbl 1352.49035
[14] Luke, D. R., Phase retrieval. What’s new?, SIAM SIAG/OPT Views News, 25, 1, 1-6 (2017)
[15] Maiden, A.; Johnson, D.; Li, P., Further improvements to the ptychographical iterative engine, Optica, 4, 7, 736-745 (2017)
[16] Maiden, A. M.; Rodenburg, J. M., An improved ptychographical phase retrieval algorithm for diffractive imaging, Ultramicroscopy, 109, 10, 1256-1262 (2009)
[17] Marchesini, S.; Tu, Y.-C.; Wu, H.-T., Alternating projection, ptychographic imaging and phase synchronization, Appl. Comput. Harmonic Anal., 41, 3, 815-851 (2016) · Zbl 1388.94015
[18] Perlmutter, M.; Merhi, S.; Viswanathan, A.; Iwen, M., Inverting spectrogram measurements via aliased wigner distribution deconvolution and angular synchronization (2019)
[19] Pfeiffer, F., X-ray ptychography, Nat. Photonics, 12, 9-17 (2018)
[20] Preskitt, B. P., Phase Retrieval from Locally Supported Measurements (2018), UC San Diego, (Ph.D thesis)
[21] Rodenburg, J. M., Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys. Elsevier, 74, 87-184 (2008)
[22] Rodenburg, J. M.; Bates, R. H.T., The theory of super-resolution electron microscopy via wigner-distribution deconvolution, Phil. Trans. R. Soc. Lond., 339, 1655, 521-553 (1992)
[23] Seiboth, F.; Schropp, A.; Scholz, M.; Wittwer, F.; Rödel, C.; Wünsche, M.; Ullsperger, T.; Nolte, S.; Rahomäki, J.; Parfeniukas, K.; Giakoumidis, S.; Vogt, U.; Wagner, U.; Rau, C.; Boesenberg, U.; Garrevoet, J.; Falkenberg, G.; Galtier, E. C.; Lee, H. J.; Nagler, B.; Schroer, C. G., Perfect X-ray focusing via fitting corrective glasses to aberrated optics, Nat. Commun., 8 (2017)
[24] Viswanathan, A.; Iwen, M., Fast angular synchronization for phase retrieval via incomplete information, Proc. SPIE, 9597, 9597-9597-8 (2015)
[25] Xu, R.; Soltanolkotabi, M.; Haldar, J. P.; Unglaub, W.; Zusman, J.; Levi, A. F.J.; Leahy, R. M., Accelerated Wirtinger flow: A fast algorithm for ptychography (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.