arXiv subjects

Lorenz Kuger

Publications and source records attributed to Lorenz Kuger.

4 recordsLinked to original sources

arXiv2026

Position-Blind Ptychography: Viability of image reconstruction via data-driven variational inference

In this work, we present and investigate the novel blind inverse problem of position-blind ptychography, i.e., ptychographic phase retrieval without any knowledge of scan positions, which then must be recovered jointly with the image. The motivation for this problem comes from single-particle diffractive X-ray imaging, where particles in random orientations are illuminated and a set of diffraction patterns is collected. If one uses a highly focused X-ray beam, the measurements would also become sensitive to the beam positions relative to each particle and therefore ptychographic, but these positions are also unknown. We investigate the viability of image reconstruction in a simulated, simplified 2-D variant of this difficult problem, using variational inference with modern data-driven image priors in the form of score-based diffusion models. We find that, with the right illumination structure and a strong prior, one can achieve reliable and successful image reconstructions even under measurement noise, in all except the most difficult evaluated imaging scenario.

eess.IV↗

arXiv2024

Analysis of Primal-Dual Langevin Algorithms

We analyze a recently proposed class of algorithms for the problem of sampling from probability distributions $μ^\ast$ in $\mathbb{R}^d$ with a Lebesgue density of the form $μ^\ast(x) \propto \exp(-f(Kx)-g(x))$, where $K$ is a linear operator and $f,g$ convex and non-smooth. The method is a generalization of the primal-dual hybrid gradient optimization algorithm to a sampling scheme. We give the iteration's continuous time limit, a stochastic differential equation in the joint primal-dual variable, and its mean field limit Fokker-Planck equation. Under mild conditions, the scheme converges to a unique stationary state in continuous and discrete time. Contrary to purely primal overdamped Langevin diffusion, the stationary state in continuous time does not have $μ^\ast$ as its primal marginal. Thus, further analysis is carried out to bound the bias induced by the partial dualization, and potentially correct for it in the diffusion. Time discretizations of the diffusion lead to implementable algorithms, but, as is typical in Langevin Monte Carlo methods, introduce further bias. We prove bounds for these discretization errors, which allow to give convergence results relating the produced samples to the target. We demonstrate our findings numerically first on small-scale examples in which we can exactly verify the theoretical results, and subsequently on typical examples of larger scale from Bayesian imaging inverse problems.

math.OC↗

arXiv2024

Proximal Langevin Sampling With Inexact Proximal Mapping

In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution. For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool. If the potential defining the distribution is non-smooth, these discretizations are usually of an implicit form leading to Langevin sampling algorithms that require the evaluation of proximal operators. For some of the potentials relevant in imaging problems this is only possible approximately using an iterative scheme. We investigate the behaviour of a proximal Langevin algorithm under the presence of errors in the evaluation of proximal mappings. We generalize existing non-asymptotic and asymptotic convergence results of the exact algorithm to our inexact setting and quantify the bias between the target and the algorithm's stationary distribution due to the errors. We show that the additional bias stays bounded for bounded errors and converges to zero for decaying errors in a strongly convex setting. We apply the inexact algorithm to sample numerically from the posterior of typical imaging inverse problems in which we can only approximate the proximal operator by an iterative scheme and validate our theoretical convergence results.

stat.CO↗

arXiv2022

On multiple scattering in Compton scattering tomography and its impact on fan-beam CT

The recent development of energy-resolving scintillation crystals opens the way to new types of applications and imaging systems. In the context of computerized tomography (CT), it enables to use the energy as a dimension of information supplementing the source and detector positions. It is then crucial to relate the energy measurements to the properties of Compton scattering, the dominant interaction between photons and matter. An appropriate model of the spectral data leads to the concept of Compton scattering tomography (CST). Multiple-order scattering constitutes the major difficulty of CST. It is, in general, impossible to know how many times a photon was scattered before being measured. In the literature, this nature of the spectral data has often been eluded by considering only the first-order scattering in models of the spectral data. This consideration, however, does not represent the reality as second- and higher-order scattering are a substantial part of the spectral measurement. In this work, we propose to tackle this difficulty by an analysis of the spectral data in terms of modeling and mapping properties. Due to the complexity of the multiple order scattering, we model and study the second-order scattering and extend the results to the higher orders by conjecture. The study ends up with a general reconstruction strategy based on the variations of the spectral data which is illustrated by simulations on a joint CST-CT fan beam scanner. We further show how the method can be extended to high energetic polychromatic radiation sources.

math.NA↗