We present general measurement schemes with which unique conversion of diffraction patterns into the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction can be guaranteed without the knowledge of relative orientations and locations.

This approach has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns.

We also discuss conditions for unique determination of a strong phase object from its phase projection data, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes.

Over the recent years, ptychography became a standard technique for high resolution scanning transmission electron microscopy. To achieve better and better resolutions, the mathematical model had to be refined several times. In the simplest approach the measurements can be understood as a discrete, phaseless short-time Fourier transform $$ I(s) = \left| \mathcal F [\phi \cdot \tau_s w]\right|^2. $$ Here, $\tau_s w$ is an (often unknown) window function, shifted by $s$, $\phi$ the object we would like to recover and $I(s)$ the measured intensity at position $s$. Typically, a few thousands of such measurements are recorded, where $s$ lies on a regular grid. However, for specimens thicker than a few nanometers, this approximation already breaks down. A more sophisticated interaction model $M(\phi, \tau_s w)$ is needed.

Furthermore, the incoherence of the microscope is a crucial limit and has to be considered as well. We end up with a model like $$ I(s) = \sum_j \left| \mathcal F [M( \phi, \tau_s w_j)]\right|^2. $$

In this talk we give an overview over these approaches and discuss their challenges. We also show reconstructions of experimental data.

Mapping a multi-dimensional function in a predefined bounded amplitude range can be achieved via a transformation known as the modulo nonlinearity. The recovery of the function from the modulo samples was addressed for the one-dimensional case as part of the Unlimited Sensing Framework (USF) based on uniform samples [1-3], but also based on neuroscience inspired time encoded samples [4]. Alternative analyses implemented de-noising of modulo data in one and multiple dimensions [5]. Extensions of the recovery to multi-dimensional inputs typically amount to a line-by-line analysis of the data on one-dimensional slices. Apart from enabling the reconstruction of a wider class of inputs, this approach does not show an inherent need to apply modulo for high dimensional inputs.

In this talk, we present a modulo sampling operator specifically tailored to multiple dimensional inputs, called multi-dimensional modulo-hysteresis [6]. It is shown that the model can use dimensions two and above to generate redundancy that can be exploited for robust input recovery. This redundancy is particularly made possible by the hysteresis parameter of the operator. A few properties of the new operator are proven, followed by a guaranteed input recovery approach. We demonstrate theoretically and via numerical examples that when the input is corrupted by Gaussian noise the reconstruction error drops asymptotically to 0 for high enough sampling rates, which was not possible for the one-dimensional scenario. We additionally extend the recovery guarantees to classes of non-bandlimited inputs from shift-invariant spaces and include additional simulations with different noise distributions. This work enables extensions to multi-dimensional inputs for neuroscience inspired sampling schemes [4], inherently known for their noisy characteristics.

[1] Bhandari, F. Krahmer, R. Raskar. On unlimited sampling and reconstruction. IEEE Trans. Sig. Proc. 69 (2020) 3827–3839. doi:10.1109/tsp.2020.3041955

[2] Bhandari, F. Krahmer, T. Poskitt. Unlimited sampling from theory to practice: Fourier-Prony recovery and prototype ADC. IEEE Trans. Sig. Proc. (2021) 1131-1141. doi:10.1109/TSP.2021.3113497.

[3] D. Florescu, F. Krahmer, A. Bhandari. The surprising benefits of hysteresis in unlimited sampling: Theory, algorithms and experiments. IEEE Trans. Sig. Proc. 70 (2022) 616–630. doi:10.1109/tsp.2022.3142507

[4] D. Florescu, A. Bhandari. Time encoding via unlimited sampling: theory, algorithms and hardware validation. IEEE Trans. Sig. Proc. 70 (2022) 4912-4924.

[5] H. Tyagi. Error analysis for denoising smooth modulo signals on a graph. Applied and Computational Harmonic Analysis 57 (2022) 151–184.

[6] D. Florescu, A. Bhandari. Multi-Dimensional Unlimited Sampling and Robust Reconstruction. arXiv preprint 2002 arXiv:2209.06426.

We consider the problem of recovering a function $f\in L^2(\mathbb{R})$ (up to a multiplicative phase factor) from phase-less samples of its short-time Fourier transform $V_g f$, where $$ V_g f(x,y) =\int_\mathbb{R} f(t) \overline{g(t-x)} e^{-2\pi i y t}\,dt, $$ with $g\in L^2(\mathbb{R})$ a window function. Recently established dicretization barriers state that in general $f$ is not uniquely determined given $|V_g f(\Lambda)|:=\{|V_g f(\lambda)|, \lambda\in\Lambda\}$ if $\Lambda\subseteq \mathbb{R}^2$ is a lattice (irrespectively of the choice of the window $g$ and the density of the lattice $\Lambda$). We show that these discretization barriers can be overcome by employing multiple window functions. More precisely, we prove that $$ \{|V_{g_1} f(\Lambda)|, |V_{g_2} f(\Lambda)|, |V_{g_3} f(\Lambda)|, |V_{g_4} f(\Lambda)|\} $$ uniquely determines $f\sim e^{i\theta}f$ when $g_1,\ldots,g_4$ are suitably chosen windows provided that $\Lambda$ has sufficient density.

Joint work with Philipp Grohs and Lukas Liehr.