TY - CONF
AB - Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.
AU - Krcál, Marek
AU - Pilarczyk, Pawel
ID - 1237
TI - Computation of cubical Steenrod squares
VL - 9667
ER -
TY - JOUR
AB - We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.
AU - Harker, Shaun
AU - Kokubu, Hiroshi
AU - Mischaikow, Konstantin
AU - Pilarczyk, Pawel
ID - 1252
IS - 4
JF - Proceedings of the American Mathematical Society
TI - Inducing a map on homology from a correspondence
VL - 144
ER -
TY - JOUR
AB - We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.
AU - Golmakani, Ali
AU - Luzzatto, Stefano
AU - Pilarczyk, Pawel
ID - 1254
IS - 2
JF - Experimental Mathematics
TI - Uniform expansivity outside a critical neighborhood in the quadratic family
VL - 25
ER -
TY - JOUR
AB - We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.
AU - Held, Martin
AU - Huber, Stefan
AU - Palfrader, Peter
ID - 1272
IS - 5
JF - Computer-Aided Design and Applications
TI - Generalized offsetting of planar structures using skeletons
VL - 13
ER -
TY - JOUR
AB - Aiming at the automatic diagnosis of tumors using narrow band imaging (NBI) magnifying endoscopic (ME) images of the stomach, we combine methods from image processing, topology, geometry, and machine learning to classify patterns into three classes: oval, tubular and irregular. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.
AU - Dunaeva, Olga
AU - Edelsbrunner, Herbert
AU - Lukyanov, Anton
AU - Machin, Michael
AU - Malkova, Daria
AU - Kuvaev, Roman
AU - Kashin, Sergey
ID - 1289
IS - 1
JF - Pattern Recognition Letters
TI - The classification of endoscopy images with persistent homology
VL - 83
ER -
TY - JOUR
AB - We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.
AU - Durst, Sebastian
AU - Kegel, Marc
AU - Klukas, Mirko D
ID - 1292
IS - 2
JF - Acta Mathematica Hungarica
TI - Computing the Thurston–Bennequin invariant in open books
VL - 150
ER -
TY - JOUR
AB - Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls.
AU - Edelsbrunner, Herbert
AU - Iglesias Ham, Mabel
ID - 1295
JF - Electronic Notes in Discrete Mathematics
TI - Multiple covers with balls II: Weighted averages
VL - 54
ER -
TY - JOUR
AB - In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.
AU - Akopyan, Arseniy
AU - Balitskiy, Alexey
ID - 1330
IS - 2
JF - Israel Journal of Mathematics
TI - Billiards in convex bodies with acute angles
VL - 216
ER -
TY - JOUR
AB - We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.
AU - Akopyan, Arseniy
AU - Balitskiy, Alexey
AU - Karasev, Roman
AU - Sharipova, Anastasia
ID - 1360
IS - 10
JF - Proceedings of the American Mathematical Society
TI - Elementary approach to closed billiard trajectories in asymmetric normed spaces
VL - 144
ER -
TY - JOUR
AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1408
IS - 1
JF - Discrete & Computational Geometry
TI - On computability and triviality of well groups
VL - 56
ER -