9,290 research outputs found
Extracting curve-skeletons from digital shapes using occluding contours
Curve-skeletons are compact and semantically relevant shape descriptors, able to summarize both topology and pose of a wide range of digital objects. Most of the state-of-the-art algorithms for their computation rely on the type of geometric primitives used and sampling frequency. In this paper we introduce a formally sound and intuitive definition of curve-skeleton, then we propose a novel method for skeleton extraction that rely on the visual appearance of the shapes. To achieve this result we inspect the properties of occluding contours, showing how information about the symmetry axes of a 3D shape can be inferred by a small set of its planar projections. The proposed method is fast, insensitive to noise, capable of working with different shape representations, resolution insensitive and easy to implement
Skeletons of stable maps II: Superabundant geometries
We implement new techniques involving Artin fans to study the realizability
of tropical stable maps in superabundant combinatorial types. Our approach is
to understand the skeleton of a fundamental object in logarithmic
Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is
used to examine the structure of the locus of realizable tropical curves and
derive 3 principal consequences. First, we prove a realizability theorem for
limits of families of tropical stable maps. Second, we extend the sufficiency
of Speyer's well-spacedness condition to the case of curves with good
reduction. Finally, we demonstrate the existence of liftable genus 1
superabundant tropical curves that violate the well-spacedness condition.Comment: 17 pages, 1 figure. v2 fixes a minor gap in the proof of Theorem D
and adds details to the construction of the skeleton of a toroidal Artin
stack. Minor clarifications throughout. To appear in Research in the
Mathematical Science
Patch-type Segmentation of Voxel Shapes using Simplified Surface Skeletons
We present a new method for decomposing a 3D voxel shape into disjoint segments using the shape’s simplified surface-skeleton. The surface skeleton of a shape consists of 2D manifolds inside its volume. Each skeleton point has a maximally inscribed ball that touches the boundary in at least two contact points. A key observation is that the boundaries of the simplified fore- and background skeletons map one-to-one to increasingly fuzzy, soft convex, respectively concave, edges of the shape. Using this property, we build a method for segmentation of 3D shapes which has several desirable properties. Our method segments both noisy shapes and shapes with soft edges which vanish over low-curvature regions. Multiscale segmentations can be obtained by varying the simplification level of the skeleton. We present a voxel-based implementation of our approach and illustrate it on several realistic examples.
Essential skeletons of pairs and the geometric P=W conjecture
We construct weight functions on the Berkovich analytification of a variety
over a trivially-valued field of characteristic zero, and this leads to the
definition of the Kontsevich-Soibelman skeletons and the essential skeletons of
pairs. We prove that the weight functions determine a metric on the
pluricanonical bundles which coincides with Temkin's canonical metric in the
smooth case. The weight functions are defined in terms of log discrepancies,
which makes the Kontsevich-Soibelman and essential skeletons computable: this
allows us to relate the essential skeleton to its discretely-valued
counterpart, and explicitly describe the closure of the Kontsevich-Soibelman
skeletons. As a result, we employ these techniques to compute the dual boundary
complexes of certain character varieties: this provides the first evidence for
the geometric P=W conjecture in the compact case, and the first application of
Berkovich geometry in non-abelian Hodge theory.Comment: Sections 1.6-1.7 rewritten and minor changes in Sections 6-
The fully connected N-dimensional skeleton: probing the evolution of the cosmic web
A method to compute the full hierarchy of the critical subsets of a density
field is presented. It is based on a watershed technique and uses a probability
propagation scheme to improve the quality of the segmentation by circumventing
the discreteness of the sampling. It can be applied within spaces of arbitrary
dimensions and geometry. This recursive segmentation of space yields, for a
-dimensional space, a succession of -dimensional subspaces that
fully characterize the topology of the density field. The final 1D manifold of
the hierarchy is the fully connected network of the primary critical lines of
the field : the skeleton. It corresponds to the subset of lines linking maxima
to saddle points, and provides a definition of the filaments that compose the
cosmic web as a precise physical object, which makes it possible to compute any
of its properties such as its length, curvature, connectivity etc... When the
skeleton extraction is applied to initial conditions of cosmological N-body
simulations and their present day non linear counterparts, it is shown that the
time evolution of the cosmic web, as traced by the skeleton, is well accounted
for by the Zel'dovich approximation. Comparing this skeleton to the initial
skeleton undergoing the Zel'dovich mapping shows that two effects are competing
during the formation of the cosmic web: a general dilation of the larger
filaments that is captured by a simple deformation of the skeleton of the
initial conditions on the one hand, and the shrinking, fusion and disappearance
of the more numerous smaller filaments on the other hand. Other applications of
the N dimensional skeleton and its peak patch hierarchy are discussed.Comment: Accepted for publication in MNRA
Self-Attractive Random Walks: The Case of Critical Drifts
Self-attractive random walks undergo a phase transition in terms of the
applied drift: If the drift is strong enough, then the walk is ballistic,
whereas in the case of small drifts self-attraction wins and the walk is
sub-ballistic. We show that, in any dimension at least 2, this transition is of
first order. In fact, we prove that the walk is already ballistic at critical
drifts, and establish the corresponding LLN and CLT.Comment: Final version sent to the publisher. To appear in Communications in
Mathematical Physic
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