arXiv subjects

Stefan Sommer

Publications and source records attributed to Stefan Sommer.

At least 55 records · Page 3Linked to original sources

arXiv2022

Diffusion Mean Estimation on the Diagonal of Product Manifolds

Computing sample means on Riemannian manifolds is typically computationally costly as exemplified by computation of the Fréchet mean which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fréchet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.

stat.ME↗

arXiv2022

Atlas Generative Models and Geodesic Interpolation

Generative neural networks have a well recognized ability to estimate underlying manifold structure of high dimensional data. However, if a single latent space is used, it is not possible to faithfully represent a manifold with topology different from Euclidean space. In this work we define the general class of Atlas Generative Models (AGMs), models with hybrid discrete-continuous latent space that estimate an atlas on the underlying data manifold together with a partition of unity on the data space. We identify existing examples of models from various popular generative paradigms that fit into this class. Due to the atlas interpretation, ideas from non-linear latent space analysis and statistics, e.g. geodesic interpolation, which has previously only been investigated for models with simply connected latent spaces, may be extended to the entire class of AGMs in a natural way. We exemplify this by generalizing an algorithm for graph based geodesic interpolation to the setting of AGMs, and verify its performance experimentally.

cs.LG↗

arXiv2021

Diffusion bridges for stochastic Hamiltonian systems and shape evolutions

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalisation as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solve inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models thereby substantially expanding the applicability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.

math.NA↗

arXiv2021

Bridge Simulation and Metric Estimation on Lie Groups

We present a simulation scheme for simulating Brownian bridges on complete and connected Lie groups. We show how this simulation scheme leads to absolute continuity of the Brownian bridge measure with respect to the guided process measure. This result generalizes the Euclidean result of Delyon and Hu to Lie groups. We present numerical results of the guided process in the Lie group $\SO(3)$. In particular, we apply importance sampling to estimate the metric on $\SO(3)$ using an iterative maximum likelihood method.

math.ST↗

arXiv2021

Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.

cs.LG↗

arXiv2021

Prototyping Autonomous Robotic Networks on Different Layers of RAMI 4.0 with Digital Twins

In this decade, the amount of (industrial) Internet of Things devices will increase tremendously. Today, there exist no common standards for interconnection, observation, or the monitoring of these devices. In context of the German "Industrie 4.0" strategy the Reference Architectural Model Industry 4.0 (RAMI 4.0) was introduced to connect different aspects of this rapid development. The idea is to let different stakeholders of these products speak and understand the same terminology. In this paper, we present an approach using Digital Twins to prototype different layers along the axis of the RAMI 4.0, by the example of an autonomous ocean observation system developed in the project ARCHES.

cs.SE↗

arXiv2021

Developing an Underwater Network of Ocean Observation Systems with Digital Twin Prototypes -- A Field Report from the Baltic Sea

During the research cruise AL547 with RV ALKOR (October 20-31, 2020), a collaborative underwater network of ocean observation systems was deployed in Boknis Eck (SW Baltic Sea, German exclusive economic zone (EEZ)) in the context of the project ARCHES (Autonomous Robotic Networks to Help Modern Societies). This network was realized via a Digital Twin Prototype approach. During that period different scenarios were executed to demonstrate the feasibility of Digital Twins in an extreme environment such as underwater. One of the scenarios showed the collaboration of stage IV Digital Twins with their physical counterparts on the seafloor. This way, we address the research question, whether Digital Twins represent a feasible approach to operate mobile ad hoc networks for ocean and coastal observation.

cs.SE↗

arXiv2021

Diffusion Means and Heat Kernel on Manifolds

We introduce diffusion means as location statistics on manifold data spaces. A diffusion mean is defined as the starting point of an isotropic diffusion with a given diffusivity. They can therefore be defined on all spaces on which a Brownian motion can be defined and numerical calculation of sample diffusion means is possible on a variety of spaces using the heat kernel expansion. We present several classes of spaces, for which the heat kernel is known and sample diffusion means can therefore be calculated. As an example, we investigate a classic data set from directional statistics, for which the sample Fréchet mean exhibits finite sample smeariness.

stat.ME↗

arXiv2021

Currents and K-functions for Fiber Point Processes

Analysis of images of sets of fibers such as myelin sheaths or skeletal muscles must account for both the spatial distribution of fibers and differences in fiber shape. This necessitates a combination of point process and shape analysis methodology. In this paper, we develop a K-function for shape-valued point processes by embedding shapes as currents, thus equipping the point process domain with metric structure inherited from a reproducing kernel Hilbert space. We extend Ripley's K-function which measures deviations from spatial homogeneity of point processes to fiber data. The paper provides a theoretical account of the statistical foundation of the K-function and its extension to fiber data, and we test the developed K-function on simulated as well as real data sets. This includes a fiber data set consisting of myelin sheaths, visualizing the spatial and fiber shape behavior of myelin configurations at different debts.

stat.ME↗

arXiv2020

PADDIT: Probabilistic Augmentation of Data using Diffeomorphic Image Transformation

For proper generalization performance of convolutional neural networks (CNNs) in medical image segmentation, the learnt features should be invariant under particular non-linear shape variations of the input. To induce invariance in CNNs to such transformations, we propose Probabilistic Augmentation of Data using Diffeomorphic Image Transformation (PADDIT) -- a systematic framework for generating realistic transformations that can be used to augment data for training CNNs. We show that CNNs trained with PADDIT outperforms CNNs trained without augmentation and with generic augmentation in segmenting white matter hyperintensities from T1 and FLAIR brain MRI scans.

cs.CV↗

arXiv2019

Simulation of Conditioned Diffusions on the Flat Torus

Diffusion processes are fundamental in modelling stochastic dynamics in natural sciences. Recently, simulating such processes on complicated geometries has found applications for example in biology, where toroidal data arises naturally when studying the backbone of protein sequences, creating a demand for efficient sampling methods. In this paper, we propose a method for simulating diffusions on the flat torus, conditioned on hitting a terminal point after a fixed time, by considering a diffusion process in R 2 which we project onto the torus. We contribute a convergence result for this diffusion process, translating into convergence of the projected process to the terminal point on the torus. We also show that under a suitable change of measure, the Euclidean diffusion is locally a Brownian motion.

math.PR↗

arXiv2018

Generalizations of Ripley's K-function with Application to Space Curves

The intensity function and Ripley's K-function have been used extensively in the literature to describe the first and second moment structure of spatial point sets. This has many applications including describing the statistical structure of synaptic vesicles. Some attempts have been made to extend Ripley's K-function to curve pieces. Such an extension can be used to describe the statistical structure of muscle fibers and brain fiber tracks. In this paper, we take a computational perspective and construct new and very general variants of Ripley's K-function for curves pieces, surface patches etc. We discuss the method from [Chiu, Stoyan, Kendall, & Mecke 2013] and compare it with our generalizations theoretically, and we give examples demonstrating the difference in their ability to separate sets of curve pieces.

stat.ME↗

arXiv2018

Stochastic Image Deformation in Frequency Domain and Parameter Estimation using Moment Evolutions

Modelling deformation of anatomical objects observed in medical images can help describe disease progression patterns and variations in anatomy across populations. We apply a stochastic generalisation of the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework to model differences in the evolution of anatomical objects detected in populations of image data. The computational challenges that are prevalent even in the deterministic LDDMM setting are handled by extending the FLASH LDDMM representation to the stochastic setting keeping a finite discretisation of the infinite dimensional space of image deformations. In this computationally efficient setting, we perform estimation to infer parameters for noise correlations and local variability in datasets of images. Fundamental for the optimisation procedure is using the finite dimensional Fourier representation to derive approximations of the evolution of moments for the stochastic warps. Particularly, the first moment allows us to infer deformation mean trajectories. The second moment encodes variation around the mean, and thus provides information on the noise correlation. We show on simulated datasets of 2D MR brain images that the estimation algorithm can successfully recover parameters of the stochastic model.

math.ST↗

arXiv2018

An Average of the Human Ear Canal: Recovering Acoustical Properties via Shape Analysis

Humans are highly dependent on the ability to process audio in order to interact through conversation and navigate from sound. For this, the shape of the ear acts as a mechanical audio filter. The anatomy of the outer human ear canal to approximately 15-20 mm beyond the Tragus is well described because of its importance for customized hearing aid production. This is however not the case for the part of the ear canal that is embedded in the skull, until the typanic membrane. Due to the sensitivity of the outer ear, this part, referred to as the bony part, has only been described in a few population studies and only ex-vivo. We present a study of the entire ear canal including the bony part and the tympanic membrane. We form an average ear canal from a number of MRI scans using standard image registration methods. We show that the obtained representation is realistic in the sense that it has acoustical properties almost identical to a real ear.

cs.CV↗

arXiv2018

Math in the Black Forest: Workshop on New Directions in Shape Analysis

These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical foundations in infinite-dimensional Riemannian geometry, geometric statistics, and geometric stochastics, and with applications in medical imaging, evolutionary development, and fluid dynamics. The workshop is the 6th instance of a series of workshops on the same topic.

math.DG↗

arXiv2018

A Geometric Framework for Stochastic Shape Analysis

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker-Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker-Planck equation to sample moments of the data, while the second approach employs an Expectation-Maximisation based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

cs.CV↗

arXiv2018

String Methods for Stochastic Image and Shape Matching

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.

cs.CV↗

arXiv2018

An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces.

math.ST↗