Relaxed Disk Packing

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted to CCCG 201

Sphere Packing with Limited Overlap

The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice.Comment: 12 pages, 3 figures, submitted to SOCG 201

IST Austria Thesis

We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications

Multiple covers with balls I: Inclusion–exclusion

Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software

Relaxed disk packing

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations

Algorithm to Compute a Minimal Length Basis of Representative Cocycles of Cohomology Generators

An algorithm to compute a minimal length basis of representative cocycles of cohomology generators for 2D images is proposed. We based the computations on combinatorial pyramids foreseeing its future extension to 3D objects. In our research we are looking for a more refined topological description of deformable 2D and 3D shapes, than they are the often used Betti numbers. We define contractions on the object edges toward the inner of the object until the boundaries touch each other, building an irregular pyramid with this purpose. We show the possible use of the algorithm seeking the minimal cocycles that connect the convex deficiencies on a human silhouette. We used minimality in the number of cocycle edges in the basis, which is a robust description to rotations and noise

Invariant Representative Cocycles of Cohomology Generators using Irregular Graph Pyramids

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.Comment: Special issue on Graph-Based Representations in Computer Visio

Irregular graph pyramids and representative cocycles of cohomology generators

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns ‘quantities’ to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. Extension to nD and application in the context of pattern recognition are discussed

Introducing the H2020 AQUACROSS project: Knowledge, assessment, and management for AQUAtic biodiversity and ecosystem services aCROSS EU policies

The AQUACROSS project was an unprecedented effort to unify policy concepts, knowledge, and management of freshwater, coastal, and marine ecosystems to support the cost-effective achievement of the targets set by the EU Biodiversity Strategy to 2020. AQUACROSS aimed to support EU efforts to enhance the resilience and stop the loss of biodiversity of aquatic ecosystems as well as to ensure the ongoing and future provision of aquatic ecosystem services. The project focused on advancing the knowledge base and application of Ecosystem-Based Management. Through elaboration of eight diverse case studies in freshwater and marine and estuarine aquatic ecosystem across Europe covering a range of environmental management problems including, eutrophication, sustainable fisheries as well as invasive alien species AQUACROSS demonstrated the application of a common framework to establish cost-effective measures and integrated Ecosystem-Based Management practices. AQUACROSS analysed the EU policy framework (i.e. goals, concepts, time frames) for aquatic ecosystems and built on knowledge stemming from different sources (i.e. WISE, BISE, Member State reporting within different policy processes, modelling) to develop innovative management tools, concepts, and business models (i.e. indicators, maps, ecosystem assessments, participatory approaches, mechanisms for promoting the delivery of ecosystem services) for aquatic ecosystems at various scales of space and time and relevant to different ecosystem types

Track A Basic Science

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/138319/1/jia218438.pd