Digital Topology Java Applet

We present here a java applet, accessible through the World Wide Web, which allows to apply to a binary digital image a series of topological algorithms for image processing

The complexity of the normal surface solution space

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2 tables; v2: added minor clarification

Boundaries and Topological Algorithms

This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edges of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modeled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning

The Topological Processor for the future ATLAS Level-1 Trigger: from design to commissioning

The ATLAS detector at LHC will require a Trigger system to efficiently select events down to a manageable event storage rate of about 400 Hz. By 2015 the LHC instantaneous luminosity will be increased up to 3 x 10^34 cm-2s-1, this represents an unprecedented challenge faced by the ATLAS Trigger system. To cope with the higher event rate and efficiently select relevant events from a physics point of view, a new element will be included in the Level-1 Trigger scheme after 2015: the Topological Processor (L1Topo). The L1Topo system, currently developed at CERN, will consist initially of an ATCA crate and two L1Topo modules. A high density opto-electroconverter (AVAGO miniPOD) drives up to 1.6 Tb/s of data from the calorimeter and muon detectors into two high-end FPGA (Virtex7-690), to be processed in about 200 ns. The design has been optimized to guarantee excellent signal in- tegrity of the high-speed links and low latency data transmission on the Real Time Data Path (RTDP). The L1Topo receives data in a standalone protocol from the calorimeters and muon detectors to be processed into several VHDL topological algorithms. Those algorithms perform geometrical cuts, correlations and calculate complex observables such as the invariant mass. The output of such topological cuts is sent to the Central Trigger Processor. This talk focuses on the relevant high-density design characteristic of L1Topo, which allows several hundreds optical links to processed (up to 13 Gb/s each) using ordinary PCB material. Relevant test results performed on the L1Topo prototypes to characterize the high-speed links latency (eye diagram, bit error rate, margin analysis) and the logic resource utilization of the algorithms are discussed.Comment: 5 pages, 6 figure

The Pandora multi-algorithm approach to automated pattern recognition of cosmic-ray muon and neutrino events in the MicroBooNE detector.

The development and operation of liquid-argon time-projection chambers for neutrino physics has created a need for new approaches to pattern recognition in order to fully exploit the imaging capabilities offered by this technology. Whereas the human brain can excel at identifying features in the recorded events, it is a significant challenge to develop an automated, algorithmic solution. The Pandora Software Development Kit provides functionality to aid the design and implementation of pattern-recognition algorithms. It promotes the use of a multi-algorithm approach to pattern recognition, in which individual algorithms each address a specific task in a particular topology. Many tens of algorithms then carefully build up a picture of the event and, together, provide a robust automated pattern-recognition solution. This paper describes details of the chain of over one hundred Pandora algorithms and tools used to reconstruct cosmic-ray muon and neutrino events in the MicroBooNE detector. Metrics that assess the current pattern-recognition performance are presented for simulated MicroBooNE events, using a selection of final-state event topologies

Optimizing the double description method for normal surface enumeration

Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces, and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.Comment: 27 pages, 12 figures; v2: Removed the 3^n bound from Section 3.3, fixed the projective equation in Lemma 4.4, clarified "most triangulations" in the introduction to section 5; v3: replace -ise with -ize for Mathematics of Computation (note that this changes the title of the paper

Algorithms for recognizing knots and 3-manifolds

This is a survey paper on algorithms for solving problems in 3-dimensional topology. In particular, it discusses Haken's approach to the recognition of the unknot, and recent variations.Comment: 17 Pages, 7 figures, to appear in Chaos, Fractals and Soliton

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A