The Effect of the Range of a Modulating Phase Mask on the Retrieval of a Complex Object from Intensity Measurements

The authors have been supported by the postdoctoral project (1.1.1.2/16/I/001, 1.1.1.2/VIAA/1/16/199), the CAMART2 project (grant agreement ID 739508), the Latvian Investment and Development Agency (LIDA) project (KC-PI-2017/105), and the grant for the Latvian State Emeritus Scientists.In many fields of science, it is often impossible to preserve the information about the phase of the electromagnetic field, and only the information about the magnitude is available. This is known as the phase problem. Various algorithms have been proposed to recover the information about phase from intensity measurements. Nowadays, iterative algorithms of phase retrieval have become popular. Many of these algorithms are based on modulating the object under study with several masks and retrieving the missing information about the phase of an object by applying mathematical optimization methods. Several of these algorithms are able to retrieve not only the phase but also the magnitude of the object under study. In this study, we investigate the effect of the range of modulation of a mask on the accuracy of the retrieved magnitude and phase map. We conclude that there is a sharp boundary of the range of modulation separating the successfully retrieved magnitude and phase maps from those retrieved unsuccessfully. A decrease in the range of modulation affects the accuracy of the retrieved magnitude and phase map differently. © 2021 V. Karitans et al., published by Sciendo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.Postdoctoral project (1.1.1.2/16/I/001, 1.1.1.2/VIAA/1/16/199); the Latvian Investment and Development Agency (LIDA) project (KC-PI-2017/105), and the grant for the Latvian State Emeritus Scientists; Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508, project CAMART2

Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version

We consider smooth optimization problems with a Hermitian positive semi-definite fixed-rank constraint, where a quotient geometry with three Riemannian metrics $g^i(\cdot, \cdot)$ $(i=1,2,3)$ is used to represent this constraint. By taking the nonlinear conjugate gradient method (CG) as an example, we show that CG on the quotient geometry with metric $g^1$ is equivalent to CG on the factor-based optimization framework, which is often called the Burer--Monteiro approach. We also show that CG on the quotient geometry with metric $g^3$ is equivalent to CG on the commonly-used embedded geometry. We call two CG methods equivalent if they produce an identical sequence of iterates $\{X_k\}$. In addition, we show that if the limit point of the sequence $\{X_k\}$ generated by an algorithm has lower rank, that is $X_k\in \mathbb C^{n\times n}, k = 1, 2, \ldots$ has rank $p$ and the limit point $X_*$ has rank $r < p$, then the condition number of the Riemannian Hessian with metric $g^1$ can be unbounded, but those of the other two metrics stay bounded. Numerical experiments show that the Burer--Monteiro CG method has slower local convergence rate if the limit point has a reduced rank, compared to CG on the quotient geometry under the other two metrics. This slower convergence rate can thus be attributed to the large condition number of the Hessian near a minimizer

The effect of noise, a constant background, and bit depth on the phase retrieval of pure phase objects

Acknowledgements – The authors are supported by the postdoctoral project (1.1.1.2/16/I/001; 1.1.1.2/ VIAA/1/16/199, State Education Development Agency, Republic of Latvia).In the current study, we investigate the effect of uniform white noise, Poisson noise and a constant background on the phase retrieval of pure phase objects. We also study the influence of the aforementioned factors on phase retrieval at different bit depths of intensity measurements. An algorithm called PhaseLift is used for phase retrieval as it requires a small number of modulating masks and can retrieve the phase of an object from sparse intensity measurements of low bit depth. A test object is modulated by eight random masks generated from a single mask and the phase of the object is retrieved from coded diffraction patterns. Different levels of uniform white noise, Poisson noise and constant background are superimposed on the diffraction patterns and the root-mean-square error (RMSE) of the retrieved object is calculated at each level. The results suggest that Poisson noise and a constant background at the same level cause similar RMSE compared to uniform white noise. Lowering the bit depth from 18-bits to 14-bits resulted in the decrease of the RMSE caused by Poisson noise and a constant background. We conclude that the effects of noise and constant background can be reduced by lowering the bit depth. © 2021 WrocÅ‚aw University of Science and Technology. All rights reserved.State Education Development Agency Republic of Latvia 1.1.1.2/16/I/001; 1.1.1.2/ VIAA/1/16/199; The Institute of Solid State Physics, University of Latvia (Latvia), as the Centre of Excellence has received funding from the European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-Teaming Phase2 under grant agreement No. 739508, project CAMART2

Non-Convex Phase Retrieval from STFT Measurements

The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise