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We investigate the eigenstructure of matrix formulations used for modeling scattering processes in materials by transmission electron microscopy (TEM). Considering dynamical scattering is fundamental in describing the interaction between an electron wave and the material under investigation. In TEM, both the Bloch wave formulation and the multislice method are commonly employed to model the scattering process, but comparing these models directly is challenging. Unlike the Bloch wave formulation, which represents the transmission function in terms of the scattering matrix, the traditional multislice method does not have a pure transmission function due to the entanglement between electron waves and the propagation function within the crystal. To address this, we propose a reformulation of the multislice method into a matrix framework, which we refer to as transmission matrix. This enables a direct comparison to the well-known scattering matrix, derived from the Bloch wave formulation, and analysis of their eigenstructures. We show theoretically that both matrices are equal, under the condition that the angles of the eigenvalues differ no more than modulo $2\pi n$ for integer $n$, and the eigenvectors of the transmission and scattering matrix are related in terms of a two-dimensional Fourier matrix. The characterization of both matrices in terms of physical parameters, such as total projected potentials, is also discussed, and we perform numerical simulations to validate our theoretical findings. Finally, we show that the determinant of the transmission matrix can be used to estimate the mean inner potential., Comment: 18 pages, 10 figures

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. Keywords: optical flow, elastography, Fourier transform, iteratively reweighted least squares, Horn--Schunck method, Comment: 29 pages, 8 figures

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 spare, 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., Comment: 46 pages, 10 figures, 8 tables

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.

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 unknown, it also has to be estimated as well. For the recovery, we consider gradient and stochastic gradient descent methods for the minimization of amplitude-base squared loss. In particular, this includes extended Ptychographic Iterative Engine as a special case of stochastic gradient descent. We show that all methods converge to a critical point at a sublinear rate with a proper choice of step sizes. We also discuss possibilities for larger step sizes., Comment: 39 pages

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., Comment: 31 pages

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, a 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 non-trivial light spectrum known as blind polychromatic ptychography. Firstly, we show that non-blind polychromatic ptychography can be seen as a recovery from quadratic measurements. Then, a reconstruction from such measurements can be performed by a variant of Amplitude Flow algorithm, which has guaranteed sublinear convergence to a critical point. Secondly, 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. Keywords: ptychography, phase retrieval, blind, alternating minimization, gradient descent., Comment: 33 pages, 11 figures

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.

We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit ball. We complement our analysis by numerical experiments comparing the solutions of noise-blind and noise-aware formulations. These experiments confirm that noise-blind optimization methods exhibit comparable robustness to noise-aware formulations. Keywords: low-rank matrix recovery, phase retrieval, quotient property, noise-blind, robustness, nuclear norm minimization, Comment: 39 pages, 4 figures

The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as a optimization problem on graphs involving the graph Laplacian matrix. We consider a general, weighted version of this problem, where the impact of the noise differs between different pairs of entries and some of the differences are erased completely; this version arises for example in ptychography. We study two common approaches for solving this problem, namely eigenvector relaxation and semidefinite convex relaxation. Although some recovery guarantees are available for both methods, their performance is either unsatisfying or restricted to the unweighted graphs. We close this gap, deriving recovery guarantees for the weighted problem that are completely analogous to the unweighted version., Comment: 20 pages, 5 figures

Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance for Gaussian windows. In this paper we develop a subspace completion algorithm, which enables stable reconstruction for a much wider choice of windows, including Gaussian windows. The combination with a regularization technique leads to improved conditioning and better noise robustness., Comment: 21 pages, 4 figures