Phase Retrieval with Application to Optical Imaging

This review article provides a contemporary overview of phase retrieval in optical imaging, linking the relevant optical physics to the information processing methods and algorithms. Its purpose is to describe the current state of the art in this area, identify challenges, and suggest vision and areas where signal processing methods can have a large impact on optical imaging and on the world of imaging at large, with applications in a variety of fields ranging from biology and chemistry to physics and engineering

Sparsity-regularized coded ptychography for robust and efficient lensless microscopy on a chip

In ptychographic imaging, the trade-off between the number of acquisitions and the resultant imaging quality presents a complex optimization problem. Increasing the number of acquisitions typically yields reconstructions with higher spatial resolution and finer details. Conversely, a reduction in measurement frequency often compromises the quality of the reconstructed images, manifesting as increased noise and coarser details. To address this challenge, we employ sparsity priors to reformulate the ptychographic reconstruction task as a total variation regularized optimization problem. We introduce a new computational framework, termed the ptychographic proximal total-variation (PPTV) solver, designed to integrate into existing ptychography settings without necessitating hardware modifications. Through comprehensive numerical simulations, we validate that PPTV-driven coded ptychography is capable of producing highly accurate reconstructions with a minimal set of eight intensity measurements. Convergence analysis further substantiates the robustness, stability, and computational feasibility of the proposed PPTV algorithm. Experimental results obtained from optical setups unequivocally demonstrate that the PPTV algorithm facilitates high-throughput, high-resolution imaging while significantly reducing the measurement burden. These findings indicate that the PPTV algorithm has the potential to substantially mitigate the resource-intensive requirements traditionally associated with high-quality ptychographic imaging, thereby offering a pathway toward the development of more compact and efficient ptychographic microscopy systems.Comment: 15 pages, 7 figure

Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers

Ptychography has risen as a reference X-ray imaging technique: it achieves resolutions of one billionth of a meter, macroscopic field of view, or the capability to retrieve chemical or magnetic contrast, among other features. A ptychographyic reconstruction is normally formulated as a blind phase retrieval problem, where both the image (sample) and the probe (illumination) have to be recovered from phaseless measured data. In this article we address a nonlinear least squares model for the blind ptychography problem with constraints on the image and the probe by maximum likelihood estimation of the Poisson noise model. We formulate a variant model that incorporates the information of phaseless measurements of the probe to eliminate possible artifacts. Next, we propose a generalized alternating direction method of multipliers designed for the proposed nonconvex models with convergence guarantee under mild conditions, where their subproblems can be solved by fast element-wise operations. Numerically, the proposed algorithm outperforms state-of-the-art algorithms in both speed and image quality.Comment: 23 page

Roadmap on optical security

Postprint (author's final draft

PDEs with Compressed Solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an $L^1$ form, such as the divisible sandpile problem and signum-Gordon. Addition of an $L^1$ term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving $L^1$ based problems.Comment: 21 pages, 15 figure