University of Göttingen - Institute for Numerical and Applied Mathematics
Research group for Mathematical Signal and Image Processing

Fresnel wavelets for coherent diffractive imaging

(09/2011-06/2019, second and third funding period)

DFG project C11 within the Collaborative Research Center 755 (Nanoscale photonic imaging)
Coherent diffractive imaging shows considerable promise to achieve very high spatial resolution. In practice, the most widely applied methods for in-line hologram reconstruction and phase retrieval are iterated projection algorithms. In this project, we aim to construct a new Fresnel wavelet frame that is suitable to represent optimally sparse Fresnel transformed images. We want to apply the new Fresnel wavelet frame for denoising of the measured hologram intensity and to derive new reconstruction schemes for phase retrieval that take the sparsity of the propagated wave field in the Fresnel wavelet frame into account.

Principal investigator: Gerlind Plonka-Hoch
Staff: Robert Beinert, Jakob Geppert, Stefan Loock

Corresponding publications
One-dimensional discrete-time phase retrieval
Robert Beinert, Gerlind Plonka
In: Salditt T., Egner A., Luke D. (eds) Nanoscale Photonic Imaging. Topics in Applied Physics, vol 134. Springer, Cham, pp. 603-627, 2020, preprint as download.
Sparse Power Factorization: Balancing peakiness and sample complexity
Dominik Stöger, Jakob Geppert, Felix Krahmer
Advances in Computational Mathematics 45(3), 1711--1728, 2019, preprint available at arXiv.
Sparse Power Factorization With Refined Peakiness Conditions
Dominik Stöger, Jakob Geppert, Felix Krahmer
2018 IEEE Statistical Signal Processing Workshop (SSP), 2018, See at IEEE Xplore.
Enforcing uniqueness in one-dimensional phase retrieval by additional signal information in time domain
Robert Beinert, Gerlind Plonka
Applied and Computational Harmonic Analysis 45, 505-525, 2018, preprint as download (arXiv).
Refined performance guarantees for Sparse Power Factorization
Jakob Geppert, Felix Krahmer, Dominik Stöger
2017 International Conference on Sampling Theory and Applications (SampTA), 2017, See at IEEE Xplore.
One-dimensional phase retrieval with additional interference measurements
Robert Beinert
Results Math. 72(1-2) (2017), 1–24, 2017, preprint as download (arXiv).
Non-negativity constraints in the one-dimensional discrete-time phase retrieval problem
Robert Beinert
Information and Inference: A Journal of the IMA 6, 213-224, 2017, preprint as download (arXiv).
Ambiguities in one-dimensional phase retrieval from magnitudes of a linear canonical transform
Robert Beinert
Z. Angew. Math. Mech. (ZAMM), 1–5, 2017, preprint as download (arXiv).
Iterative Phase Retrieval with Sparsity Constraints
Stefan Loock, Gerlind Plonka
PAMM 16(1), 835-836, 2016, DOI: 10.1002/pamm.201610406.
Phase Retrieval with Sparsity Constraints
Stefan Loock
Dissertation, 2016, published online on 29 June 2016.
Using sparsity information for iterative phase retrieval in x-ray propagation imaging
Anne Pein, Stefan Loock, Gerlind Plonka, Tim Salditt
Opt. Express 24(8), 8332-8343., 2016, open access.
Ambiguities in one-dimensional phase retrieval of structured functions
Robert Beinert, Gerlind Plonka
Proc. Appl. Math. Mech. Volume 15, Issue 1, pp. 653-654, October 2015, DOI: 10.1002/pamm.201510316, preprint as download.
Ambiguities in one-dimensional discrete phase retrieval from Fourier magnitudes
Robert Beinert, Gerlind Plonka
The Journal of Fourier Analysis and Applications 21(6), 1169-1198, 2015, preprint as download.
Phase retrieval for Fresnel measurements using a shearlet sparsity constraint
Stefan Loock, Gerlind Plonka
Inverse Problems 30 (2014) 055005, 2014, preprint as download.
How many Fourier samples are needed for real function reconstruction?
Gerlind Plonka, Marius Wischerhoff
Journal of Applied Mathematics and Computing, Volume 42, Issue 1-2, pp 117-137, 2013, revised preprint as download.

Corresponding software
pyShearLab - A Python 2D Shearlet Toolbox
A Python toolbox for the two-dimensional discrete shearlet transform based on ShearLab3D.
Stefan Loock
PRwSC - A Toolbox for Phase Retrieval with Sparsity Constraints
A toolbox for Matlab and Python (pyPRwSC) for the reconstruction of images from phase retrieval data using sparsity constraints.
Stefan Loock

Research Group for Mathematical Signal and Image Processing

Institute for Numerical and Applied Mathematics
Lotzestr. 16-18
37083 Göttingen