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.

In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schrödinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of the Schrödinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using nonlinear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

A new X-ray computed tomography technique for the purpose of imaging fluidized beds is presented. It consists of an experimental set-up with three stationary X-ray source and flat panel detector pairs, a geometric calibration and data processing workflow, and an image reconstruction algorithm. The technique enables sparse-angular tomographic reconstruction in large 3D regions of fluidized beds at framerates up to 200 Hz, and therefore images bubbles along their whole trajectories through the volume. It allows for a unique analysis of bubble dynamics in fluidized beds, including bubble velocities, bubble transformations, i.e., time evolution of the bubble distributions in space, and bubble–bubble interactions. In this article, we first analyze the main limitation of the technique, the sparse angular resolution, through numerical simulations. We then test the experimental set-up through imaging a series of phantoms. Lastly, we demonstrate results from a Geldart B bubbling fluidized bed.

We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data.

Optical measurements often exhibit mixed Poisson–Gaussian noise statistics, which hampers the image quality, particularly under low signal-to-noise ratio (SNR) conditions. Computational imaging falls short in such situations when solely Poissonian noise statistics are assumed. In response to this challenge, we define a loss function that explicitly incorporates this mixed noise nature. By using a maximum-likelihood estimation, we devise a practical method to account for a camera readout noise in gradient-based ptychography optimization. Our results, based on both experimental and numerical data, demonstrate that this approach outperforms the conventional one, enabling enhanced image reconstruction quality under challenging noise conditions through a straightforward methodological adjustment.

Our understanding of the past behavior of the geomagnetic field arises from magnetic signals stored in geological materials, e.g., (volcanic) rocks. Bulk rock samples, however, often contain magnetic grains that differ in chemistry, size, and shape; some of them record the Earth's magnetic field well, others are unreliable. The presence of a small amount of adverse behaved magnetic grains in a sample may already obscure important information on the past state of the geomagnetic field. Recently it was shown that it is possible to determine magnetizations of individual grains in a sample by combining X-ray computed tomography and magnetic surface scanning measurements. Here we establish this new Micromagnetic Tomography (MMT) technique and make it suitable for use with different magnetic scanning techniques, and for both synthetic and natural samples. We acquired reliable magnetic directions by selecting subsets of grains in a synthetic sample, and we obtained rock-magnetic information of individual grains in a volcanic sample. This illustrates that MMT opens up entirely new venues of paleomagnetic and rock-magnetic research. MMT's unique ability to determine the magnetization of individual grains in a nondestructive way allows for a systematic analysis of how geological materials record and retain information on the past state of the Earth's magnetic field. Moreover, by interpreting only the contributions of known magnetically well-behaved grains in a sample, MMT has the potential to unlock paleomagnetic information from even the most complex, crucial, or valuable recorders that current methods are unable to recover.