200 research outputs found
Broadcasting Automata and Patterns on Z^2
The Broadcasting Automata model draws inspiration from a variety of sources
such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and
nature, employing many of the same pattern forming methods that can be seen in
the superposition of waves and resonance. Algorithms for broad- casting
automata model are in the same vain as those encountered in distributed
algorithms using a simple notion of waves, messages passed from automata to au-
tomata throughout the topology, to construct computations. The waves generated
by activating processes in a digital environment can be used for designing a
vari- ety of wave algorithms. In this chapter we aim to study the geometrical
shapes of informational waves on integer grid generated in broadcasting
automata model as well as their potential use for metric approximation in a
discrete space. An explo- ration of the ability to vary the broadcasting radius
of each node leads to results of categorisations of digital discs, their form,
composition, encodings and gener- ation. Results pertaining to the nodal
patterns generated by arbitrary transmission radii on the plane are explored
with a connection to broadcasting sequences and ap- proximation of discrete
metrics of which results are given for the approximation of astroids, a
previously unachievable concave metric, through a novel application of the
aggregation of waves via a number of explored functions
Introducing Quantum Ricci Curvature
Motivated by the search for geometric observables in nonperturbative quantum
gravity, we define a notion of coarse-grained Ricci curvature. It is based on a
particular way of extracting the local Ricci curvature of a smooth Riemannian
manifold by comparing the distance between pairs of spheres with that of their
centres. The quantum Ricci curvature is designed for use on non-smooth and
discrete metric spaces, and to satisfy the key criteria of scalability and
computability. We test the prescription on a variety of regular and random
piecewise flat spaces, mostly in two dimensions. This enables us to quantify
its behaviour for short lattices distances and compare its large-scale
behaviour with that of constantly curved model spaces. On the triangulated
spaces considered, the quantum Ricci curvature has good averaging properties
and reproduces classical characteristics on scales large compared to the
discretization scale.Comment: 43 pages, 27 figure
Pattern formations with discrete waves and broadcasting sequences
This thesis defines the Broadcasting Automata model as an intuitive and complete method of distributed pattern formation, partitioning and distributed geometric computation. The system is examined within the context of Swarm Robotics whereby large numbers of minimally complex robots may be deployed in a variety of circumstances and settings with goals as diverse as from toxic spill containment to geological survey. Accomplishing these tasks with such simplistic machines is complex and has been deconstructed in to sub-problems considered to be signif- icant because, when composed, they are able to solve much more complex tasks. Sub-problems have been identified, and studied as pattern formation, leader elec- tion, aggregation, chain formation, hole avoidance, foraging, path formation, etc. The Broadcasting Automata draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood sequences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. To this end the thesis gives an in depth analysis of the primitive tools of the Broadcasting Automata model, nodal patterns, where waves from a variety of transmitters can in linear time construct partitions and patterns with results per- taining to the numbers of different patterns and partitions, along with the number of those that differ, are given. Using these primitives of the model a variety of algorithms are given including leader election, through the location of the centre of a discrete disc, and a solution to the Firing Squad Synchronisation problem. These problems are solved linearly.An exploration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and generation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and approximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions. Broadcasting Automata aims to place itself as a robust and complete linear time and large scale system for the construction of patterns, partitions and geometric computation. Algorithms and methodologies are given for the solution of problems within Swarm Robotics and an extension to neighbourhood sequences. It is also hoped that it opens up a new area of research that can expand many older and more mature works
Local Features, Structure-from-motion and View Synthesis in Spherical Video
This thesis addresses the problem of synthesising new views from spherical video or image sequences. We propose an interest point detector and feature descriptor that allows us to robustly match local features between pairs of spherical images and use this as part of a structure-from-motion pipeline that allows us to estimate camera pose from a spherical video sequence. With pose estimates to hand, we propose methods for view stabilisation and novel viewpoint synthesis.
In Chapter 3 we describe our contribution in the area of feature detection and description in spherical images. First, we present a novel representation for spherical
images which uses a discrete geodesic grid composed of hexagonal pixels. Second, we extend the BRISK binary descriptor to the sphere, proposing methods for multiscale
corner detection, sub-pixel position and sub-octave scale refinement and descriptor construction in the tangent space to the sphere.
In Chapter 4 we describe our contributions in the area of spherical structure-from-motion. We revisit problems from multiview geometry in the context of spherical images. We propose methods suited to spherical camera geometry for the
spherical-n-point problem and calibrated spherical reconstruction. We introduce a new probabilistic interpretation of spherical structure-from-motion which uses the von Mises-Fisher distribution in spherical feature point positions. This model provides an alternate objective function that we use in bundle adjustment.
In Chapter 5 we describe our contributions in the area of view synthesis from spherical images. We exploit the camera pose estimates made by our pipeline and use these in two view synthesis applications. The first is view stabilisation where we remove the effect of viewing direction changes, often present in first person video. Second, we propose a method for synthesising novel viewpoints
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
Inference and experimental design for percolation and random graph models.
The problem of optimal arrangement of nodes of a random weighted graph is
studied in this thesis. The nodes of graphs under study are fixed, but their edges
are random and established according to the so called edge-probability function.
This function is assumed to depend on the weights attributed to the pairs of graph
nodes (or distances between them) and a statistical parameter. It is the purpose
of experimentation to make inference on the statistical parameter and thus to
extract as much information about it as possible. We also distinguish between two
different experimentation scenarios: progressive and instructive designs.
We adopt a utility-based Bayesian framework to tackle the optimal design
problem for random graphs of this kind. Simulation based optimisation methods,
mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain
the solution. We study optimal design problem for the inference based on partial
observations of random graphs by employing data augmentation technique.
We prove that the infinitely growing or diminishing node configurations asymptotically
represent the worst node arrangements. We also obtain the exact solution
to the optimal design problem for proximity graphs (geometric graphs) and numerical
solution for graphs with threshold edge-probability functions.
We consider inference and optimal design problems for finite clusters from bond
percolation on the integer lattice Zd and derive a range of both numerical and
analytical results for these graphs. We introduce inner-outer plots by deleting
some of the lattice nodes and show that the ‘mostly populated’ designs are not
necessarily optimal in the case of incomplete observations under both progressive
and instructive design scenarios.
Finally, we formulate a problem of approximating finite point sets with lattice
nodes and describe a solution to this problem
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