Beta-Skeletons have Unbounded Dilation

A fractal construction shows that, for any beta>0, the beta-skeleton of a point set can have arbitrarily large dilation. In particular this applies to the Gabriel graph.Comment: 8 pages, 9 figure

Minimum-weight triangulation is NP-hard

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures. This revision contains a few improvements in the expositio

Configraphics:

This dissertation reports a PhD research on mathematical-computational models, methods, and techniques for analysis, synthesis, and evaluation of spatial configurations in architecture and urban design. Spatial configuration is a technical term that refers to the particular way in which a set of spaces are connected to one another as a network. Spatial configuration affects safety, security, and efficiency of functioning of complex buildings by facilitating certain patterns of movement and/or impeding other patterns. In cities and suburban built environments, spatial configuration affects accessibilities and influences travel behavioural patterns, e.g. choosing walking and cycling for short trips instead of travelling by cars. As such, spatial configuration effectively influences the social, economic, and environmental functioning of cities and complex buildings, by conducting human movement patterns. In this research, graph theory is used to mathematically model spatial configurations in order to provide intuitive ways of studying and designing spatial arrangements for architects and urban designers. The methods and tools presented in this dissertation are applicable in: arranging spatial layouts based on configuration graphs, e.g. by using bubble diagrams to ensure certain spatial requirements and qualities in complex buildings; and analysing the potential effects of decisions on the likely spatial performance of buildings and on mobility patterns in built environments for systematic comparison of designs or plans, e.g. as to their aptitude for pedestrians and cyclists. The dissertation reports two parallel tracks of work on architectural and urban configurations. The core concept of the architectural configuration track is the ‘bubble diagram’ and the core concept of the urban configuration track is the ‘easiest paths’ for walking and cycling. Walking and cycling have been chosen as the foci of this theme as they involve active physical, cognitive, and social encounter of people with built environments, all of which are influenced by spatial configuration. The methodologies presented in this dissertation have been implemented in design toolkits and made publicly available as freeware applications

Constructing Simplicial Complexes over Topological Spaces

The first step in topological data analysis is often the construction of a simplicial complex. This complex approximates the lost topology of a sampled point set. Current techniques often assume that the input is embedded in a metric -- often Euclidean -- space, and make significant use of the underlying geometry for efficient computation. Consequently, these techniques do not extend to non-Euclidean or non-metric spaces. In this thesis, we present an oracle-based framework for constructing simplicial complexes over arbitrary topological spaces. The framework consists of an oracle and an algorithm that builds the simplicial complex by making calls to the oracle. We compare different algorithmic approaches for the construction, as well as alternate ways of representing the simplicial complex in the computation. Finally, we demonstrate the utility of our framework as a tool for approaching problems of diverse nature by presenting three applications: to multiword search in Google, to the computational analysis of a language and to the study of protein structure

Personalised body counter calibration using anthropometric parameters

This book describes the development of a new method for personalisation of efficiency factors in partial body counting. Its achieved goal is the quantification of uncertainties in those factors due to variation in anatomy of the measured persons, and their reduction by correlation with anthropometric parameters. The method was applied to a detector system at the In Vivo Measurement Laboratory at Karlsruhe Institute of Technology using Monte Carlo simulation and computational phantoms

Studies of several tetrahedralization problems

The main purpose of decomposing an object into simpler components is to simplify a problem involving the complex object into a number of subproblems having simpler components. In particular, a tetrahedralization is a partition of the input domain in R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations have applications in the finite element method, mesh generation, computer graphics, and robotics. This thesis investigates four problems in tetrahedralizations and triangulations. The first problem is on the computational complexity of tetrahedralization detections. We present an O(nm log n) algorithm to determine whether a set of line segments .C is the edge set of a tetrahedralization, where m is the number of segments and n is the number of endpoints in .C. We show that it is NP-complete to decide whether .C contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations. After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized in O(n²) time. The third problem is on the tetrahedralization of two nested convex polyhedra. We give a method to tetrahedralize the region between two nested convex polyhedra into a linear number of tetrahedra without introducing Steiner points. This result answers an open problem raised by Bern [16]. The fourth problem is on the lower bound for β-skeletons belonging to minimum weight triangulations. We prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value, the β-skeleton of a point set may not always be a subgraph of the minimum weight triangulation of this point set. This result settles Keil's conjecture [62]

Craniofacial growth and development in modern humans and Neanderthals

This thesis assesses craniofacial growth, development and the dynamics of developmental interactions among cranial regions in modern humans and Neanderthals. To these ends, virtual segmentation, landmarking and Geometric Morphometrics (GM) are applied to an ontogenetic series of the whole crania of 68 H. sapiens and 12 H. neanderthalensis. First, the ontogenetic shape and form changes in the cranial vault, base and face are explored, and the locations and magnitudes of these changes are discussed. Secondly, allometric scaling is tested for differences among different age classes in the three regions of the cranium. In addition, the degree of covariation among these and how it changes over time is investigated.The study then focuses on interactions among facial regions. First, similar analyses as those used in the study of the cranium are applied to compare growth, development and covariation among parts of the face in different age classes. Additionally, a sample of 227 modern humans from 0 to 6 years of age is analysed using path analysis, to investigate the cascade of interactions and relative contributions of soft tissue and skeletal elements to the overall growth and development of the face. Last, the facial morphology of H. sapiens is compared to that of H. neanderthalensis and their ontogenetic trajectories are tested for divergence. Novel method registration-free colour maps are used to visualise regional changes during growth and development and to compare the morphologies of the two species. Covariation among facial elements is also compared to assess potential differences in developmental interactions. In modern humans, the results show that allometry and covariation change significantly among age classes and between cranial regions during ontogeny and that covariation is stronger in younger subadults than in older subadults and adults. Among modern humans, significantly divergent trajectories are observed between age classes during ontogeny in all three cranial regions. In the modern human face, allometric scaling also differs among age stages in each region. Interestingly, covariation among facial regions becomes progressively non-significant with time, with the exception of those including the nose and maxilla. Path analysis in modern humans shows a large contribution of the proxy used for nasal septum to the overall facial development. Soft tissues contribute only locally to the development of some skeletal elements of the face. Major aspects of the differences between adult modern humans and Neanderthals are already present in the youngest individuals. However, additional differences arise through differences in the degree of change in facial size and significantly divergent allometric trajectories. Analyses of covariation among Neanderthal facial regions suffer from small sample size but, where significant, suggest that the interactions among cranial components are similar to those in modern humans, with some differences

A probabilistic reasoning and learning system based on Bayesian belief networks

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