1,197 research outputs found
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Visualising the structure of architectural open spaces based on shape analysis
This paper proposes the application of some well known two-dimensional
geometrical shape descriptors for the visualisation of the structure of
architectural open spaces. The paper demonstrates the use of visibility
measures such as distance to obstacles and amount of visible space to calculate
shape descriptors such as convexity and skeleton of the open space. The aim of
the paper is to indicate a simple, objective and quantifiable approach to
understand the structure of open spaces otherwise impossible due to the complex
construction of built structures.Comment: 10 pages, 9 figure
Robust and flexible multi-scale medial axis computation
The principle of the multi-scale medial axis (MMA) is important in that any object is detected at a blurring scale proportional to the size of the object. Thus it provides a sound balance between noise removal and preserving detail. The robustness of the MMA has been reflected in many existing applications in object segmentation, recognition, description and registration. This thesis aims to improve the computational aspects of the MMA. The MMA is obtained by computing ridges in a “medialness” scale-space derived from an image. In computing the medialness scale-space, we propose an edge-free medialness algorithm, the Concordance-based Medial Axis Transform (CMAT). It not only depends on the symmetry of the positions of boundaries, but also is related to the symmetry of the intensity contrasts at boundaries. Therefore it excludes spurious MMA branches arising from isolated boundaries. In addition, the localisation accuracy for the position and width of an object, as well as the robustness under noisy conditions, is preserved in the CMAT. In computing ridges in the medialness space, we propose the sliding window algorithm for extracting locally optimal scale ridges. It is simple and efficient in that it can readily separate the scale dimension from the search space but avoids the difficult task of constructing surfaces of connected maxima. It can extract a complete set of MMA for interfering objects in scale-space, e.g. embedded or adjacent objects. These algorithms are evaluated using a quantitative study of their performance for 1-D signals and qualitative testing on 2-D images
Zoom invariant vision of figural shape: The mathematics of cores
Believing that figural zoom invariance and the cross-figural boundary linking implied by medial loci are important aspects of object shape, we present the mathematics of and algorithms for the extraction of medial loci directly from image intensities. The medial loci called cores are defined as generalized maxima in scale space of a form of medial information that is invariant to translation, rotation, and in particular, zoom. These loci are very insensitive to image disturbances, in strong contrast to previously available medial loci, as demonstrated in a companion paper. Core-related geometric properties and image object representations are laid out which, together with the aforementioned insensitivities, allow the core to be used effectively for a variety of image analysis objectives.
Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams
In a cold-dark-matter universe, cosmological structure formation proceeds in
rough analogy to origami folding. Dark matter occupies a three-dimensional
'sheet' of free- fall observers, non-intersecting in six-dimensional
velocity-position phase space. At early times, the sheet was flat like an
origami sheet, i.e. velocities were essentially zero, but as time passes, the
sheet folds up to form cosmic structure. The present paper further illustrates
this analogy, and clarifies a Lagrangian definition of caustics and streams:
caustics are two-dimensional surfaces in this initial sheet along which it
folds, tessellating Lagrangian space into a set of three-dimensional regions,
i.e. streams. The main scientific result of the paper is that streams may be
colored by only two colors, with no two neighbouring streams (i.e. streams on
either side of a caustic surface) colored the same. The two colors correspond
to positive and negative parities of local Lagrangian volumes. This is a severe
restriction on the connectivity and therefore arrangement of streams in
Lagrangian space, since arbitrarily many colors can be necessary to color a
general arrangement of three-dimensional regions. This stream two-colorability
has consequences from graph theory, which we explain. Then, using N-body
simulations, we test how these caustics correspond in Lagrangian space to the
boundaries of haloes, filaments and walls. We also test how well outer caustics
correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS.
9 pages, 5 figure
Multiscale medial shape-based analysis of image objects
pre-printMedial representation of a three-dimensional (3-D) object or an ensemble of 3-D objects involves capturing the object interior as a locus of medial atoms, each atom being two vectors of equal length joined at the tail at the medial point. Medial representation has a variety of beneficial properties, among the most important of which are 1) its inherent geometry, provides an object-intrinsic coordinate system and thus provides correspondence between instances of the object in and near the object(s); 2) it captures the object interior and is, thus, very suitable for deformation; and 3) it provides the basis for an intuitive object-based multiscale sequence leading to efficiency of segmentation algorithms and trainability of statistical characterizations with limited training sets. As a result of these properties, medial representation is particularly suitable for the following image analysis tasks; how each operates will be described and will be illustrated by results: 1) segmentation of objects and object complexes via deformable models; 2) segmentation of tubular trees, e.g., of blood vessels, by following height ridges of measures of fit of medial atoms to target images; 3) object-based image registration via medial loci of such blood vessel trees; 4) statistical characterization of shape differences between control and pathological classes of structures. These analysis tasks are made possible by a new form of medial representation called m-reps, which is described
Automatic Generation of the Axial Lines of Urban Environments to Capture What We Perceive
Based on the concepts of isovists and medial axes, we developed a set of
algorithms that can automatically generate axial lines for representing
individual linearly stretched parts of open space of an urban environment. Open
space is the space between buildings, where people can freely move around. The
generation of the axial lines has been a key aspect of space syntax research,
conventionally relying on hand-drawn axial lines of an urban environment, often
called axial map, for urban morphological analysis. Although various attempts
have been made towards an automatic solution, few of them can produce the axial
map that consists of the least number of longest visibility lines, and none of
them really works for different urban environments. Our algorithms provide a
better solution than existing ones. Throughout this paper, we have also argued
and demonstrated that the axial lines constitute a true skeleton, superior to
medial axes, in capturing what we perceive about the urban environment.
Keywords: Visibility, space syntax, topological analysis, medial axes, axial
lines, isovistsComment: 13 pages, 9 figures submitted to International Journal of
Geographical Information Science. With version 2, the concept of bucket has
been explained and illustrated in more detail. With version 3, better
formating and finetun
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