Ptychography is an imaging technique, the goal of which is to reconstruct the object of interest from a set of diffraction patterns obtained by illuminating its small regions. When the distribution of light within the region is known, the recovery of the object from ptychographic measurements becomes a special case of the phase retrieval problem. In the other case, also known as blind ptychography, the recovery of both the object and the distribution is necessary.

One of the well-known reconstruction methods for blind ptychography is extended Ptychographic Iterative Engine. Despite its popularity, there was no known analysis of its performance. Based on the fact that it is a stochastic gradient descent method, we derive its convergence guarantees if the step sizes are chosen sufficiently small. The second considered method is a generalization of the Amplitude Flow algorithm for phase retrieval, a gradient descent scheme for the minimization of the amplitude-based squared loss. By applying an alternating minimization procedure, the blind ptychography is reduced to phase retrieval subproblems, which can be solved by performing a few steps of Amplitude Flow. The resulting procedure converges to a critical point at a sublinear rate.

In the recent years, the topic of high dynamic range (HDR) tomography has started to gather attention due to recent advances in the hardware technology. The issue is that registering high-intensity projections that exceed the dynamic range of the detector cause sensor saturation, which, in turn, leads to a loss of information. Inspired by the multi-exposure fusion strategy in computational photography, a common approach is to acquire multiple Radon Transform projections at different exposure levels that are algorithmically fused to facilitate HDR reconstructions.

As opposed to this, a single-shot alternative to the multi-exposure fusion approach has been proposed in our recent line of work which is based on the Modulo Radon Transform, a novel generalization of the conventional Radon transform. In this case, Radon Transform projections are folded via a modulo non-linearity, which allows HDR values to be mapped into the dynamic range of the sensor and, thus, avoids saturation or clipping. The folded measurements are then mapped back to their ambient range using reconstruction algorithms.

In this talk we introduce a novel Fourier domain recovery method, namely the OMP-FBP method, which is based on the Orthogonal Matching Pursuit (OMP) algorithm and Filtered Back Projection (FBP) formula. The proposed OMP-FBP method offers several advantages; it is agnostic to the modulo threshold or the number of folds, can handle much lower sampling rates than previous approaches and is empirically stable to noise and outliers. The effectivity of the OMP-FBP recovery method is illustrated by numerical experiments.

This talk is based on joint work with Ayush Bhandari (Imperial College London).

We discuss recent advances in phaseless sampling of the short-time Fourier transform (STFT). The main focus of the talk lies in the question if phaseless samples of the STFT contain enough information to recover signals belonging to infinite-dimensional function spaces. It turns out, that this problem differs from ordinary sampling in a rather fundamental and subtle way: if the sampling set is a lattice then uniqueness is unachievable, independent of the choice of the window function and the density of the lattice. Based on this discretization barrier, we present possible ways to still achieve unique recoverability from samples: lattice-uniqueness is possible if the signal class gets restricted to certain proper subspaces of $L^2(\mathbb R)$, such as the class of compactly-supported functions or shift-invariant spaces. Finally, we highlight that sampling on so-called square-root lattices achieves uniqueness in $L^2(\mathbb R)$.

Optical diffraction tomography (ODT) consists in the recovery of the three-dimensional scattering potential of a microscopic object rotating around its center from a series of illuminations with coherent light. Standard reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane. In common physical measurement setups, the collected data only consists in intensities; so only phaseless measurements are available. To overcome the loss of the required phases, we propose a new reconstruction approach for ODT based on three key ingredients. First, the light propagation is modeled using Born's approximation enabling us to use the Fourier diffraction theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier transform via total variation regularization utilizing a primal-dual iteration, which also yields a novel numerical inversion formula for ODT with known phase. The third ingredient is a hybrid input-output scheme. We achieve convincing numerical results showing that the computed 2D and 3D reconstructions are even comparable to the ones obtained with known phase. This indicate that ODT with phaseless data is possible.