Oleh Melnyk

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5–9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

We consider blind ptychography, an imaging technique which aims to reconstruct an object of interest from a set of its diffraction patterns each obtained by a local illumination. As the distribution of the light within the illuminated region, called the window, is not known, it has to be estimated as well. For the recovery, a minimization of amplitude-base squared loss via gradient and stochastic gradient descent methods is considered. In particular, this includes extended Ptychographic Iterative Engine as a special case of stochastic gradient descent. We show that with a proper choice of step sizes, all methods converge to a critical point at a sublinear rate and discuss possibilities for larger step sizes.

In this paper, we propose mathematical models for reconstructing the optical flow in time-harmonic elastography. In this image acquisition technique, the object undergoes a special time-harmonic oscillation with known frequency so that only the spatially varying amplitude of the velocity field has to be determined. This allows for a simpler multi-frame optical flow analysis using Fourier analytic tools in time. We propose three variational optical flow models and show how their minimization can be tackled via Fourier transform in time. Numerical examples with synthetic as well as real-world data demonstrate the benefits of our approach.

Ptychography is a computational imaging technique that aims to reconstruct the object of interest from a set of diffraction patterns. Each of these is obtained by a localized illumination of the object, which is shifted after each illumination to cover its whole domain. Because in the resulting measurements the phase information is lost, ptychography gives rise to solving a phase retrieval problem. In this work, we consider ptychographic measurements contaminated by a background signal. Such a background is caused by imperfections in the experimental setup and appears as a signal that is added to the diffraction patterns. The background is assumed to be independent of the shift of the object, i.e., it is the same for all diffraction patterns. Two algorithms are provided, for arbitrary objects and for so-called phase objects that do not absorb the light but only scatter it. For the second type, a uniqueness of reconstruction is established for almost every object. Our approach is based on the Wigner distribution deconvolution, which lifts the object to a higher-dimensional matrix space where the recovery can be reformulated as a linear problem. The background only affects a few equations of the linear system that are therefore discarded. The lost information is then restored using redundancy in the higher-dimensional space.

Ptychography is a computational imaging technique that aims to reconstruct the object of interest from a set of diffraction patterns. Each of these is obtained by a localized illumination of the object, which is shifted after each illumination to cover its whole domain. As in the resulting measurements the phase information is lost, ptychography gives rise to solving a phase retrieval problem. In this work, we consider ptychographic measurements corrupted with background noise, a type of additive noise that is independent of the shift, i.e., it is the same for all diffraction patterns. Two algorithms are provided, for arbitrary objects and for so-called phase objects that do not absorb the light but only scatter it. For the second type, a uniqueness of reconstruction is established for almost every object. Our approach is based on the Wigner Distribution Deconvolution, which lifts the object to a higher-dimensional matrix space where the recovery can be reformulated as a linear problem. Background noise only affects a few equations of the linear system that are therefore discarded. The lost information is then restored using redundancy in the higher-dimensional space. Keywords: phase retrieval, ptychography, background noise, Wigner Distribution Deconvolution, uniqueness of reconstruction.

High-dimensional real-world systems can often be well characterized by a small number of simultaneous low-complexity interactions. The analysis of variance (ANOVA) decomposition and the anchored decomposition are typical techniques to find sparse additive decompositions of functions. In this paper, we are interested in a setting, where these decompositions are not directly sparse, but become so after an appropriate basis transform. Noting that the sparsity of those additive function decompositions is equivalent to the fact that most of its mixed partial derivatives vanish, we can exploit a connection to the underlying function graphs to determine an orthogonal transform that realizes the appropriate basis change. This is done in three steps: we apply singular value decomposition to minimize the number of vertices of the function graph, and joint block diagonalization techniques of families of matrices followed by sparse minimization based on relaxations of the zero 'norm' for minimizing the number of edges. For the latter one, we propose and analyze minimization techniques over the manifold of special orthogonal matrices. Various numerical examples illustrate the reliability of our approach for functions having, after a basis transform, a sparse additive decomposition into summands with at most two variables.

Ptychography is a lensless imaging technique, which considers reconstruction from a set of far-field diffraction patterns obtained by illuminating small overlapping regions of the specimen. In many cases, the distribution of light inside the illuminated region is unknown and has to be estimated along with the object of interest. This problem is referred to as blind ptychography. While in ptychography the illumination is commonly assumed to have a point spectrum, in this paper we consider an alternative scenario with a nontrivial light spectrum known as blind polychromatic ptychography. First, we show that nonblind polychromatic ptychography can be seen as a recovery from quadratic measurements. Then, a reconstruction from such measurements can be performed by a variant of the Amplitude Flow algorithm, which has guaranteed sublinear convergence to a critical point. Second, we address recovery from blind polychromatic ptychographic measurements by devising an alternating minimization version of Amplitude Flow and showing that it converges to a critical point at a sublinear rate.

In this paper, we focus on Stochastic Amplitude Flow (SAF) for phase retrieval, a stochastic gradient descent for the amplitude-based squared loss. While the convergence to a critical point of (nonstochastic) Amplitude Flow is well-understood, SAF is a much less studied algorithm. We close this gap by deriving the convergence guarantees for SAF based on the contributions for Amplitude Flow and analysis for stochastic gradient descent. These results are then applied to two more algorithms, which can be seen as instances of SAF. The first is an extension of the Kaczmarz method for phase retrieval. The second is Ptychographic Iterative Engine, which is a popular algorithm for ptychography, a special case of phase retrieval with the short-time Fourier transform. Keywords: phase retrieval, Amplitude Flow, stochastic gradient descent, ptychography, Ptychographic Iterative Engine, Kaczmarz method.

We propose algorithms based on an optimisation method for inverse multislice ptychography in, e.g. electron microscopy. The multislice method is widely used to model the interaction between relativistic electrons and thick specimens. Since only the intensity of diffraction patterns can be recorded, the challenge in applying inverse multislice ptychography is to uniquely reconstruct the electrostatic potential in each slice up to some ambiguities. In this conceptual study, we show that a unique separation of atomic layers for simulated data is possible when considering a low acceleration voltage. We also introduce an adaptation for estimating the illuminating probe. For the sake of practical application, we finally present slice reconstructions using experimental 4D scanning transmission electron microscopy (STEM) data.