arXiv subjects

Christina Brandt

Publications and source records attributed to Christina Brandt.

8 recordsLinked to original sources

arXiv2026

Reconstructions of Single Pixel X-Ray Transforms with Applications in Nuclear-Disarmament Verification

In nuclear arms control and disarmament processes, it is crucial to determine whether an object is a nuclear weapon or not without revealing sensitive information about it. At the MIT: Laboratory for Nuclear Security and Policy, such a nuclear verification method was developed, showcasing a transmission-based approach [1]. This method's essential part rests on a mathematical operation, the Single-Pixel X-Ray Transform: a cone of X-rays transmits an object and the remaining intensity is measured with a single-pixel detector. This transformation and the recovery of objects from dimensionless single-pixel measurements more generally has only been analyzed to a limited extent. In this work, we investigate some of the Single Pixel X-Ray Transform's mathematical properties. More specifically, we show that the Single Pixel X-ray transform is non-linear, continuous, Fréchet-differentiable and convex. We also introduce a method of reconstructing an object based only on a finite number of dimensionless, noisy Single Pixel X-Ray Transform measurement values. This method is based on Douglas-Rachford splitting and uses total variation denoising. We present an implementation for this method, focusing on rotational symmetric objects, as they allow the use of a one-dimensional direct total variation denoising algorithm [2].

math.NA↗

arXiv2025

Energy Time Ptychography for one-dimensional phase retrieval

Phase retrieval is at the heart of adaptive optics and modern high-resolution imaging. Without phase information, optical systems are limited to intensity-only measurements, hindering full reconstruction of object structures and wavefront dynamics essential for advanced applications. Here, we address a one-dimensional phase problem linking energy and time, which arises in X-ray scattering from ultrasharp nuclear resonances. We leverage the Mössbauer effect, where nuclei scatter radiation without energy loss to the lattice, and are sensitive to their magneto-chemical environments. Rather than using traditional spectroscopy with radioactive gamma-ray sources, we measure nuclear forward scattering of synchrotron X-ray pulses in the time domain, providing superior sensitivity and faster data acquisition. Extracting spectral information from a single measurement is challenging due to the missing phase information, typically requiring extensive modeling. Instead, we use multiple energetically overlapping measurements to retrieve both the transmission spectrum and the phase of the scattering response, similar to ptychographic phase retrieval in imaging. Our robust approach can overcome the bandwidth limitations of gamma-ray sources, opening new research directions with modern X-ray sources and Mössbauer isotopes.

physics.optics↗

arXiv2023

Dynamic image reconstruction with motion priors in application to 3D magnetic particle imaging

Various imaging modalities allow for time-dependent image reconstructions from measurements where its acquisition also has a time-dependent nature. Magnetic particle imaging (MPI) falls into this class of imaging modalities and it thus also provides a dynamic inverse problem. Without proper consideration of the dynamic behavior, motion artifacts in the reconstruction become an issue. More sophisticated methods need to be developed and applied to the reconstruction of the time-dependent sequences of images. In this context, we investigate the incorporation of motion priors in terms of certain flow-parameter-dependent PDEs in the reconstruction process of time-dependent 3D images in magnetic particle imaging. The present work comprises the method development for a general 3D+time setting for time-dependent linear forward operators, analytical investigation of necessary properties in the MPI forward operator, modeling aspects in dynamic MPI, and extensive numerical experiments on 3D+time imaging including simulated data as well as measurements from a rotation phantom and in-vivo data from a mouse.

math.NA↗

arXiv2023

Field-free Line Magnetic Particle Imaging: Radon-based Artifact Reduction with Motion Models

Magnetic particle imaging is a promising medical imaging technique. Applying changing magnetic fields to tracer material injected into the object under investigation results in a change in magnetization. Measurement of related induced voltage signals enables reconstruction of the particle distribution. For the field-free line scanner the scanning geometry is similar to the one in computerized tomography. We make use of these similarities to derive a forward model for dynamic particle concentrations. We validate our theoretical findings for synthetic data. By utilizing information about the object's dynamics in terms of a diffeomorphic motion model, we are able to jointly reconstruct the particle concentration and the corresponding dynamic Radon data without or with reduced motion artifacts. Thereby, we apply total variation regularization for the concentration and an optional sparsity constraint on the Radon data.

eess.IV↗

arXiv2022

Radon-based Image Reconstruction for MPI using a continuously rotating FFL

Magnetic particle imaging is a relatively new tracer-based medical imaging technique exploiting the non-linear magnetization response of magnetic nanoparticles to changing magnetic fields. If the data are generated by using a field-free line, the sampling geometry resembles the one in computerized tomography. Indeed, for an ideal field-free line rotating only in between measurements it was shown that the signal equation can be written as a convolution with the Radon transform of the particle concentration. In this work, we regard a continuously rotating field-free line and extend the forward operator accordingly. We obtain a similar result for the relation to the Radon data but with two additive terms resulting from the additional time-dependencies in the forward model. We jointly reconstruct particle concentration and corresponding Radon data by means of total variation regularization yielding promising results for synthetic data.

eess.IV↗

arXiv2021

Modeling Magnetic Particle Imaging for Dynamic Tracer Distributions

Magnetic Particle Imaging (MPI) is a promising tracer-based, functional medical imaging technique which measures the non-linear magnetization response of magnetic nanoparticles to a dynamic magnetic field. For image reconstruction, system matrices from time-consuming calibration scans are used predominantly. Finding modeled forward operators for magnetic particle imaging, which are able to compete with measured matrices in practice, is an ongoing topic of research. The existing models for magnetic particle imaging are by design not suitable for arbitrary dynamic tracer concentrations. Neither modeled nor measured system matrices account for changes in the concentration during a single scanning cycle. In this paper we present a new MPI forward model for dynamic concentrations. A static model will be introduced briefly, followed by the changes due to the dynamic behavior of the tracer concentration. Furthermore, the relevance of this new extended model is examined by investigating the influence of the extension and example reconstructions with the new and the standard model.

eess.IV↗

arXiv2017

Mathematical Analysis of the 1D Model and Reconstruction Schemes for Magnetic Particle Imaging

Magnetic particle imaging (MPI) is a promising new in-vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.

math.NA↗

arXiv2012

An Analysis of One-Dimensional Schelling Segregation

We analyze the Schelling model of segregation in which a society of n individuals live in a ring. Each individual is one of two races and is only satisfied with his location so long as at least half his 2w nearest neighbors are of the same race as him. In the dynamics, randomly-chosen unhappy individuals successively swap locations. We consider the average size of monochromatic neighborhoods in the final stable state. Our analysis is the first rigorous analysis of the Schelling dynamics. We note that, in contrast to prior approximate analyses, the final state is nearly integrated: the average size of monochromatic neighborhoods is independent of n and polynomial in w.

cs.GT↗