Mateusz Juda

Jagiellonian University | UJ · Institute of Computer Science

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...

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs are compact 2D representations formed by multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper, we introduce persisten...

Topological data analysis, such as persistent homology has shown beneficial properties for machine learning in many tasks. Topological representations, such as the persistence diagram (PD), however, have a complex structure (multiset of intervals) which makes it difficult to combine with typical machine learning workflows. We present novel compact...

Methods from computational topology are becoming more and more popular in computer vision and have shown to improve the state-of-the-art in several tasks. In this paper, we investigate the applicability of topological descriptors in the context of 3D surface analysis for the classification of different surface textures. We present a comprehensive s...

Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simpl...

We investigate topological descriptors for 3D surface analysis, i.e. the classification of surfaces according to their geometric fine structure. On a dataset of high-resolution 3D surface reconstructions we compute persistence diagrams for a 2D cubical filtration. In the next step we investigate different topological descriptors and measure their a...

We investigate topological descriptors for 3D surface analysis, i.e. the classification of surfaces according to their geometric fine structure. On a dataset of high-resolution 3D surface reconstructions we compute persistence diagrams for a 2D cubical filtration. In the next step we investigate different topological descriptors and measure their a...

We present a detailed description of a fundamental group algorithm based on Forman's combinatorial version of Morse theory. We use this algorithm in a classification problem of prime knots up to 14 crossings.

We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the -skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how...

We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined wit...

Recent work on algebraic-topological methods for verifying coverage in planar sensor networks relied exclusively on centralized computation: a limiting constraint for large networks. This paper presents a distributed algorithm for homology computation over a sensor network, for purposes of verifying coverage. The techniques involve reduction and co...

We consider the class of weak (p-2)-faceless p-pseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with ℤ 2 coefficients may be computed in time O(n) and the ℤ 2 homology generators in time O(nm) where n denotes the cardinality of the p-pseudomanifold on input and m is the number of homology genera...

Two implementations of a homology algorithm based on the Forman's discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms.