Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images

We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Our software---CHUNKYEuler---is available as open source on Bitbucket. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams

Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems

Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to distribute, it is hard to generalize to multifiltrations and is computationally prohibitive for big data-sets. In this paper we study the concept of Euler Characteristics Curves, for one parameter filtrations and Euler Characteristic Profiles, for multi-parameter filtrations. While being a weaker invariant in one dimension, we show that Euler Characteristic based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations and practical applicability for big data problems. In addition we show that the Euler Curves and Profiles enjoys certain type of stability which makes them robust tool in data analysis. Lastly, to show their practical applicability, multiple use-cases are considered.Comment: 32 pages, 19 figures. Added remark on multicritical filtrations in section 4, typos correcte

Euler characteristic surfaces

We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images

Mathematics in Medical Diagnostics - 2022 Proceedings of the 4th International Conference on Trauma Surgery Technology

The 4th event of the Giessen International Conference Series on Trauma Surgery Technology took place on April, the 23rd 2022 in Warsaw, Poland. It aims to bring together practical application research, with a focus on medical imaging, and the TDA experts from Warsaw. This publication contains details of our presentations and discussions

Two-Dimensional Core-Collapse Supernova Models with Multi-Dimensional Transport

We present new two-dimensional (2D) axisymmetric neutrino radiation/hydrodynamic models of core-collapse supernova (CCSN) cores. We use the CASTRO code, which incorporates truly multi-dimensional, multi-group, flux-limited diffusion (MGFLD) neutrino transport, including all relevant $\mathcal{O}(v/c)$ terms. Our main motivation for carrying out this study is to compare with recent 2D models produced by other groups who have obtained explosions for some progenitor stars and with recent 2D VULCAN results that did not incorporate $\mathcal{O}(v/c)$ terms. We follow the evolution of 12, 15, 20, and 25 solar-mass progenitors to approximately 600 milliseconds after bounce and do not obtain an explosion in any of these models. Though the reason for the qualitative disagreement among the groups engaged in CCSN modeling remains unclear, we speculate that the simplifying ``ray-by-ray' approach employed by all other groups may be compromising their results. We show that ``ray-by-ray' calculations greatly exaggerate the angular and temporal variations of the neutrino fluxes, which we argue are better captured by our multi-dimensional MGFLD approach. On the other hand, our 2D models also make approximations, making it difficult to draw definitive conclusions concerning the root of the differences between groups. We discuss some of the diagnostics often employed in the analyses of CCSN simulations and highlight the intimate relationship between the various explosion conditions that have been proposed. Finally, we explore the ingredients that may be missing in current calculations that may be important in reproducing the properties of the average CCSNe, should the delayed neutrino-heating mechanism be the correct mechanism of explosion.Comment: ApJ accepted version. Minor changes from origina

Euler Characteristic Tools For Topological Data Analysis

In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieve state-of-the-art performance in supervised tasks at a very low computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.Comment: 39 page

Integration of Particle-Gas Systems with Stiff Mutual Drag Interaction

Numerical simulation of numerous mm/cm-sized particles embedded in a gaseous disk has become an important tool in the study of planet formation and in understanding the dust distribution in observed protoplanetary disks. However, the mutual drag force between the gas and the particles can become so stiff, particularly because of small particles and/or strong local solid concentration, that an explicit integration of this system is computationally formidable. In this work, we consider the integration of the mutual drag force in a system of Eulerian gas and Lagrangian solid particles. Despite the entanglement between the gas and the particles under the particle-mesh construct, we are able to devise a numerical algorithm that effectively decomposes the globally coupled system of equations for the mutual drag force and makes it possible to integrate this system on a cell-by-cell basis, which considerably reduces the computational task required. We use an analytical solution for the temporal evolution of each cell to relieve the time-step constraint posed by the mutual drag force as well as to achieve the highest degree of accuracy. To validate our algorithm, we use an extensive suite of benchmarks with known solutions in one, two, and three dimensions, including the linear growth and the nonlinear saturation of the streaming instability. We demonstrate numerical convergence and satisfactory consistency in all cases. Our algorithm can for example be applied to model the evolution of the streaming instability with mm/cm-sized pebbles at high mass loading, which has important consequences for the formation scenarios of planetesimals.Comment: Accepted for publication in the Astrophysical Journal Supplement Series. 21 pages, 15 figures. Fixed cross references for equation

Topology, Metrics and Data: Computational Methods and Applications.

PhD Theses.The eld of topological data analysis (TDA) combines computational geometry and algebraic topology notions for analyzing data. This thesis presents methods and e cient algorithms that extend the TDA toolset. After introducing the needed background information about Euler characteristic curves and persistent homology, the former objects are extended to bi-dimensional ltrations. The result are Euler characteristic surfaces, which capture insights about data over a pair of parameters. Moreover, algorithms to compute these objects are described for both image and point data. Persistent homology in `1 metric is also studied. It is proven that in this setting Alpha and Cech ltration are not equivalent in general. On the other hand, two new ltrations | Alpha ag and Minibox | are de ned and proven equivalent to Cech ltrations in homological dimensions zero and one. Algorithms for nding Minibox edges are described, and Minibox ltrations are empirically shown to speed up the computation of Cech persistence diagrams with computational experiments. Then a new family of summary functions of persistence diagrams is de ned, which is related to persistence landscapes. These are called cumulative landscapes and are used to vectorize the information contained in persistence diagrams. In particular, discretizations of these functions and their Fourier coe cients are used to obtain feature vectors that can be applied in supervised classi cation problems. The e ectiveness of these feature vectors for the classi cation of data is compared against vectors obtained using persistence landscapes on two open-source datasets. Finally, a novel method is described for the analysis of high-dimensional genomics data. Optimized metrics are de ned on genomic vectors making use of a loss function. These are used in combination with a distance-based classi cation method, showing good performance compared to standard machine learning algorithms. Moreover, the structure of the given optimized metrics helps identify coordinates of the genomic vectors, which are most important for the classi cation task under study

Lattice Boltzmann Methods for Particulate Flows with Medical and Technical Applications

Particulate flows appear in numerous medical and technical applications. The main aim of this thesis is to contribute models and numerical schemes towards an accurate as well as efficient simulation of a huge number of arbitrarily shaped particles. We therefore develop holistic mesoscopic models and simulation approaches using the Lattice Boltzmann Method, that on massively parallel machines efficiently solve a variety of problems of particulate flows