Paweł Dłotko

Paweł Dłotko, Davide Gurnari

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.

Rafał Topolnicki, Paweł Dłotko, Maciej Matyka

Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

Paweł Dłotko, Ruben Specogna

A topology preserving skeleton is a synthetic representation of an object that retains its topology and many of its significant morphological properties. The process of obtaining the skeleton, referred to as skeletonization or thinning, is a very active research area. It plays a central role in reducing the amount of information to be processed during image analysis and visualization, computer-aided diagnosis or by pattern recognition algorithms. This paper introduces a novel topology preserving thinning algorithm which removes \textit{simple cells}---a generalization of simple points---of a given cell complex. The test for simple cells is based on \textit{acyclicity tables} automatically produced in advance with homology computations. Using acyclicity tables render the implementation of thinning algorithms straightforward. Moreover, the fact that tables are automatically filled for all possible configurations allows to rigorously prove the generality of the algorithm and to obtain fool-proof implementations. The novel approach enables, for the first time, according to our knowledge, to thin a general unstructured simplicial complex. Acyclicity tables for cubical and simplicial complexes and an open source implementation of the thinning algorithm are provided as additional material to allow their immediate use in the vast number of practical applications arising in medical imaging and beyond.

James A. D. Binnie, Paweł Dłotko, John Harvey et al.

It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the data, hence the complexity of the dataset at hand. A great variety of dimension estimators have been developed to find the intrinsic dimension of the data but there is little guidance on how to reliably use these estimators. This survey reviews a wide range of dimension estimation methods, categorising them by the geometric information they exploit: tangential estimators which detect a local affine structure; parametric estimators which rely on dimension-dependent probability distributions; and estimators which use topological or metric invariants. The paper evaluates the performance of these methods, as well as investigating varying responses to curvature and noise. Key issues addressed include robustness to hyperparameter selection, sample size requirements, accuracy in high dimensions, precision, and performance on non-linear geometries. In identifying the best hyperparameters for benchmark datasets, overfitting is frequent, indicating that many estimators may not generalise well beyond the datasets on which they have been tested.

Ciara Frances Loughrey, Nick Orr, Anna Jurek-Loughrey et al.

Mapper algorithm can be used to build graph-based representations of high-dimensional data capturing structurally interesting features such as loops, flares or clusters. The graph can be further annotated with additional colouring of vertices allowing location of regions of special interest. For instance, in many applications, such as precision medicine, Mapper graph has been used to identify unknown compactly localized subareas within the dataset demonstrating unique or unusual behaviours. This task, performed so far by a researcher, can be automatized using hotspot analysis. In this work we propose a new algorithm for detecting hotspots in Mapper graphs. It allows automatizing of the hotspot detection process. We demonstrate the performance of the algorithm on a number of artificial and real world datasets. We further demonstrate how our algorithm can be used for the automatic selection of the Mapper lens functions.

Bartosz Zieliński, Michał Lipiński, Mateusz Juda et al.

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.