arXiv subjects

Benedikt Wirth

Publications and source records attributed to Benedikt Wirth.

At least 19 recordsLinked to original sources

arXiv2025

Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces. We expect that this approach will be useful for obtaining completeness results for other manifolds of mappings.

math.DG↗

arXiv2025

Geodesic Calculus on Implicitly Defined Latent Manifolds

Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data.

cs.LG↗

arXiv2025

On MAP estimates and source conditions for drift identification in SDEs

We consider the inverse problem of identifying the drift in an SDE from $n$ observations of its solution at $M+1$ distinct time points. We derive a corresponding MAP estimate, we prove differentiability properties as well as a so-called tangential cone condition for the forward operator, and we review the existing theory for related problems, which under a slightly stronger tangential cone condition would additionally yield convergence rates for the MAP estimate as $n\to\infty$. Numerical simulations in 1D indicate that such convergence rates indeed hold true.

math.NA↗

arXiv2025

Computer-Assisted Proofs for Geometric Optimization: From Crystallization to Carbon Nanotubes

We present a framework based on computer-assisted proofs that turns geometry optimization simulations for atomistic structures into mathematical proofs. Starting from a numerically computed approximation of a local minimizer or saddle point, we use validated numerical computations to prove the existence of a critical point of the potential energy close to this approximation. We demonstrate this framework in two settings. In the first, we study capped carbon nanotubes modeled as minimizers of carbon interatomic potentials (harmonic, Tersoff, and a Huber potential) and obtain proven bounds on tube diameter, bond lengths, and bond angles. In particular, we show that caps induce diameter oscillations along the tube. As a second application, we consider a finite Lennard-Jones crystal in a face-centered cubic (fcc) lattice and provide computer-assisted proofs of a local minimizer representing the perfect crystal, a local minimizer with a single vacancy defect, and a saddle point that connects two single-vacancy configurations on the energy landscape.

physics.comp-ph↗

arXiv2025

Discrete Geodesic Calculus in the Space of Sobolev Curves

The Riemannian manifold of curves with a Sobolev metric is an important and frequently studied model in the theory of shape spaces. Various numerical approaches have been proposed to compute geodesics, but so far elude a rigorous convergence theory. By a slick modification of a temporal Galerkin discretization we manage to preserve coercivity and compactness properties of the continuous model and thereby are able to prove convergence for the geodesic boundary value problem. Likewise, for the numerical analysis of the geodesic initial value problem we are able to exploit the geodesic completeness of the underlying continuous model for the error control of a time-stepping approximation. In fact, we develop a convergent discretization of a comprehensive Riemannian calculus that in addition includes parallel transport, covariant differentiation, the Riemann curvature tensor, and sectional curvature, all important tools to explore the geometry of the space of curves. Selected numerical examples confirm the theoretical findings and show the qualitative behaviour. To this end, a low-dimensional submanifold of Sobolev curves with explicit formulas for ground truth covariant derivatives and curvatures are considered.

math.NA↗

arXiv2024

Convergence of spectral discretization for the flow of diffeomorphisms

The Large Deformation Diffeomorphic Metric Mapping (LDDMM) or flow of diffeomorphism is a classical framework in the field of shape spaces and is widely applied in mathematical imaging and computational anatomy. Essentially, it equips a group of diffeomorphisms with a right-invariant Riemannian metric, which allows to compute (Riemannian) distances or interpolations between different deformations. The associated Euler--Lagrange equation of shortest interpolation paths is one of the standard examples of a partial differential equation that can be approached with Lie group theory (by interpreting it as a geodesic ordinary differential equation on the Lie group of diffeomorphisms). The particular group $\mathcal D^m$ of Sobolev diffeomorphisms is by now sufficiently understood to allow the analysis of geodesics and their numerical approximation. We prove convergence of a widely used Fourier-type space discretization of the geodesic equation. It is based on a regularity estimate, for which we also provide a new proof: Geodesics in $\mathcal D^m$ preserve any higher order Sobolev regularity of their initial velocity.

math.NA↗

arXiv2024

Convergence of Poisson point processes and of optimal transport regularization with application in variational analysis of PET reconstruction

Poisson distributed measurements in inverse problems often stem from Poisson point processes that are observed through discretized or finite-resolution detectors, one of the most prominent examples being positron emission tomography (PET). These inverse problems are typically reconstructed via Bayesian methods. A natural question then is whether and how the reconstruction converges as the signal-to-noise ratio tends to infinity and how this convergence interacts with other parameters such as the detector size. In this article we carry out a corresponding variational analysis for the exemplary Bayesian reconstruction functional from [arXiv:2311.17784,arXiv:1902.07521], which considers dynamic PET imaging (i.e.\ the object to be reconstructed changes over time) and uses an optimal transport regularization.

math.ST↗

arXiv2024

Formulas for the $h$-mass on $1$-currents with coefficients in $\mathbb{R}^m$

We consider the minimization of the $h$-mass over normal $1$-currents in $\mathbb{R}^n$ with coefficients in $\mathbb{R}^m$ and prescribed boundary. This optimization is known as multi-material transport problem and used in the context of logistics of multiple commodities, but also as a relaxation of nonconvex optimal transport tasks such as so-called branched transport problems. The $h$-mass with norm $h$ can be defined in different ways, resulting in three functionals $\mathcal{M}_h,|\cdot|_H$, and $\mathbb{M}_h$, whose equality is the main result of this article: $\mathcal{M}_h$ is a functional on $1$-currents in the spirit of Federer and Fleming, norm $|\cdot|_H$ denotes the total variation of a Radon measure with respect to $H$ induced by $h$, and $\mathbb{M}_h$ is a mass on flat $1$-chains in the sense of Whitney. On top we introduce a new and improved notion of calibrations for the multi-material transport problem: we identify calibrations with (weak) Jacobians of optimizers of the associated convex dual problem, which yields their existence and natural regularity.

math.OC↗

arXiv2024

Motion simulation of radio-labeled cells in whole-body positron emission tomography

Cell tracking is a subject of active research gathering great interest in medicine and biology. Positron emission tomography (PET) is well suited for tracking radio-labeled cells in vivo due to its exceptional sensitivity and whole-body capability. For validation, ground-truth data are desirable that realistically mimic the flow of cells in a clinical situation. This study develops a workflow (CeFloPS) for simulating moving radio-labeled cells in a human phantom. From the XCAT phantom, the blood vessels are reduced to nodal networks along which cells can move and distribute to organs and tissues. The movement is directed by the blood flow, which is calculated in each node using the Hagen-Pooiseuille equation and Kirchhoff's laws assuming laminar flow. Organs are voxelized and movement of cells from artery entry to vein exit is generated via a biased 3D random walk. The probabilities of cells moving or remaining in tissues are derived from rate constants of tracer kinetic-based compartment modeling. PET listmode data is generated using the Monte-Carlo simulation framework GATE based on the definition of a large-body PET scanner with cell paths as moving radioactive sources and the XCAT phantom providing attenuation data. From the flow simulation of 100,000 cells, 100 sample cells were further processed by GATE and listmode data was reconstructed into images for comparison. As demonstrated by comparisons of simulated and reconstructed cell distributions, CeFloPS is capable of simulating cell behavior in whole-body PET. It achieves this simulation in a way that is anatomically and physiologically reasonable, thereby providing valuable data for the development and validation of cell tracking algorithms.

physics.med-ph↗

arXiv2024

Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.

math.NA↗

arXiv2024

Active Axial Motion Compensation in Multiphoton-Excited Fluorescence Microscopy

In living organisms, the natural motion caused by the heartbeat, breathing, or muscle movements leads to the deformation of tissue caused by translation and stretching of the tissue structure. This effect results in the displacement or deformation of the plane of observation for intravital microscopy and causes motion-induced aberrations of the resulting image data. This, in turn, places severe limitations on the time during which specific events can be observed in intravital imaging experiments. These limitations can be overcome if the tissue motion can be compensated such that the plane of observation remains steady. We have developed a mathematical shape space model that can predict the periodic motion of a cylindrical tissue phantom resembling blood vessels. This model is then used to rapidly calculate the future position of the plane of observation of a confocal multiphoton fluorescence microscope. The focal plane is continuously adjusted to the calculated position with a piezo-actuated objective lens holder. We demonstrate active motion compensation for non-harmonic axial displacements of the vessel phantom with a field of view up to 400 $μ$m $\times$ 400 $μ$m, vertical amplitudes of more than 100 $μ$m, and at a rate of 0.5 Hz.

physics.optics↗

arXiv2024

Convergent autoencoder approximation of low bending and low distortion manifold embeddings

Autoencoders, which consist of an encoder and a decoder, are widely used in machine learning for dimension reduction of high-dimensional data. The encoder embeds the input data manifold into a lower-dimensional latent space, while the decoder represents the inverse map, providing a parametrization of the data manifold by the manifold in latent space. A good regularity and structure of the embedded manifold may substantially simplify further data processing tasks such as cluster analysis or data interpolation. We propose and analyze a novel regularization for learning the encoder component of an autoencoder: a loss functional that prefers isometric, extrinsically flat embeddings and allows to train the encoder on its own. To perform the training it is assumed that for pairs of nearby points on the input manifold their local Riemannian distance and their local Riemannian average can be evaluated. The loss functional is computed via Monte Carlo integration with different sampling strategies for pairs of points on the input manifold. Our main theorem identifies a geometric loss functional of the embedding map as the $Γ$-limit of the sampling-dependent loss functionals. Numerical tests, using image data that encodes different explicitly given data manifolds, show that smooth manifold embeddings into latent space are obtained. Due to the promotion of extrinsic flatness, these embeddings are regular enough such that interpolation between not too distant points on the manifold is well approximated by linear interpolation in latent space as one possible postprocessing.

math.NA↗

arXiv2023

Exact reconstruction and reconstruction from noisy data with anisotropic total variation

It is well-known that point sources with sufficient mutual distance can be reconstructed exactly from finitely many Fourier measurements by solving a convex optimization problem with Tikhonov-regularization (this property is sometimes termed superresolution). In case of noisy measurements one can bound the reconstruction error in unbalanced Wasserstein distances or weak Sobolev-type norms. A natural question is to what other settings the phenomenon of superresolution extends. We here keep the same measurement operator, but replace the regularizer to anisotropic total variation, which is particularly suitable for regularizing piecewise constant images with horizontal and vertical edges. Under sufficient mutual distance between the horizontal and vertical edges we prove exact reconstruction results and $L^1$ error bounds in terms of the measurement noise.

math.OC↗

arXiv2023

A Bayesian model for dynamic mass reconstruction from PET listmode data

Positron emission tomography (PET) is a classical imaging technique to reconstruct the mass distribution of a radioactive material. If the mass distribution is static, this essentially leads to inversion of the X-ray transform. However, if the mass distribution changes temporally, the measurement signals received over time (the so-called listmode data) belong to different spatial configurations. We suggest and analyse a Bayesian approach to solve this dynamic inverse problem that is based on optimal transport regularization of the temporally changing mass distribution. Our focus lies on a rigorous derivation of the Bayesian model and the analysis of its properties, treating both the continuous as well as the discrete (finitely many detectors and time binning) setting.

math.OC↗

arXiv2023

Parametrizing Product Shape Manifolds by Composite Networks

Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural network approximation. We show that this is indeed possible for shape spaces with a special product structure, namely those smoothly approximable by a direct sum of low-dimensional manifolds. Our proposed architecture leverages this structure by separately learning approximations for the low-dimensional factors and a subsequent combination. After developing the approach as a general framework, we apply it to a shape space of triangular surfaces. Here, typical examples of data manifolds are given through datasets of articulated models and can be factorized, for example, by a Sparse Principal Geodesic Analysis (SPGA). We demonstrate the effectiveness of our proposed approach with experiments on synthetic data as well as manifolds extracted from data via SPGA.

cs.LG↗

arXiv2022

Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions

We consider the shape and topology optimization problem to design a structure that minimizes a weighted sum of material consumption and (linearly) elastic compliance under a fixed given boundary load. As is well-known, this problem is in general not well-posed since its solution typically requires the use of infinitesimally fine microstructure. Therefore we examine the effect of singularly perturbing the problem by adding the structure perimeter to the cost. For a uniaxial and a shear load in two space dimensions, corresponding energy scaling laws were already derived in the literature. This work now derives the scaling law for the case of a uniaxial load in three space dimensions, which can be considered the simplest three-dimensional setting. In essence, it is expected (and confirmed in this article) that for a uniaxial load the compliance behaves almost like the dissipation in a scalar flux problem so that lower bounds from pattern analysis in superconductors can directly be applied. The upper bounds though require nontrivial modifications of the constructions known from superconductors. Those become necessary since in elasticity one has the additional constraint of torque balance.

math.OC↗

arXiv2022

A non-convex variational model for joint polyenergetic CT reconstruction, sensor denoising and material decomposition

Computed Tomography (CT) is widely used in engineering and medicine for imaging the interior of objects, patients, or animals. If the employed X-ray source is monoenergetic, image reconstruction essentially means the inversion of a ray transform. Typical X-ray sources are however polyenergetic (i.e. emit multiple wavelengths, each with different attenuation behaviour), and ignoring this fact may lead to artefacts such as beam hardening. An additional difficulty in some settings represents the occurrence of two different types of noise, the photon counting effect on the detector and the electronic noise generated e.g. by CCD cameras. We propose a novel variational image reconstruction model that takes both noise types and the polyenergetic source into account and moreover decomposes the reconstruction into different materials based on their different attenuation behaviour. In addition to a detailed mathematical analysis of the model we put forward a corresponding iterative algorithm including its convergence analysis. Numerical reconstructions of phantom data illustrate the feasibility of the approach.

math.OC↗

arXiv2022

Duality in branched transport and urban planning

In recent work arXiv:2109.07820 we have shown the equivalence of the widely used nonconvex (generalized) branched transport problem with a shape optimization problem of a street or railroad network, known as (generalized) urban planning problem. The argument was solely based on an explicit construction and characterization of competitors. In the current article we instead analyse the dual perspective associated with both problems. In more detail, the shape optimization problem involves the Wasserstein distance between two measures with respect to a metric depending on the street network. We show a Kantorovich$\unicode{x2013}$Rubinstein formula for Wasserstein distances on such street networks under mild assumptions. Further, we provide a Beckmann formulation for such Wasserstein distances under assumptions which generalize our previous result in arXiv:2109.07820. As an application we then give an alternative, duality-based proof of the equivalence of both problems under a growth condition on the transportation cost, which reveals that urban planning and branched transport can both be viewed as two bilinearly coupled convex optimization problems.

math.OC↗