370 research outputs found
Discrete bisector function and Euclidean skeleton in 2D and 3D
International audienceWe propose a new definition and an exact algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient
Distance, granulometry, skeleton
In this chapter, we present a series of concepts and operators based on the notion of distance. As often with mathematical morphology, there exists more than one way to present ideas, that are simultaneously equivalent and complementary. Here, our problem is to present methods to characterize sets of points based on metric, geometry and topology considerations. An important concept is that of the skeleton, which is of fundamental importance in pattern recognition, and has many practical application
Skeletal representations of orthogonal shapes
In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version
Fast and robust curve skeletonization for real-world elongated objects
We consider the problem of extracting curve skeletons of three-dimensional,
elongated objects given a noisy surface, which has applications in agricultural
contexts such as extracting the branching structure of plants. We describe an
efficient and robust method based on breadth-first search that can determine
curve skeletons in these contexts. Our approach is capable of automatically
detecting junction points as well as spurious segments and loops. All of that
is accomplished with only one user-adjustable parameter. The run time of our
method ranges from hundreds of milliseconds to less than four seconds on large,
challenging datasets, which makes it appropriate for situations where real-time
decision making is needed. Experiments on synthetic models as well as on data
from real world objects, some of which were collected in challenging field
conditions, show that our approach compares favorably to classical thinning
algorithms as well as to recent contributions to the field.Comment: 47 pages; IEEE WACV 2018, main paper and supplementary materia
Elliptical Distance Transforms And Object Splitting
The classical morphological method to separate fused objects in binary images is to use the watershed transform on the complement of the distance transform of the binary image. This method assumes roughly disk-like objects and cannot separate objects when they are fused together beyond a certain point. In this paper we revisit the issue by assuming that fused objects are unions of ellipses rather than mere disks. The problem is recast in terms of finding the constituent primary grains given a boolean model of ellipses. To this end, we modify the well-known pseudo-Euclidean distance transform algorithm to generate arbitrary elliptical distance transforms to reduce the dimension of the problem and we present a goodness-of-fit measure that allows us to select ellipses. The results of the methods are given on both synthetic sample boolean models and real data
On a Linear Program for Minimum-Weight Triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time
constant-factor approximation algorithm, and a variety of effective polynomial-
time heuristics that, for many instances, can find the exact MWT. Linear
programs (LPs) for MWT are well-studied, but previously no connection was known
between any LP and any approximation algorithm or heuristic for MWT. Here we
show the first such connections: for an LP formulation due to Dantzig et al.
(1985): (i) the integrality gap is bounded by a constant; (ii) given any
instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201
Generalized offsetting of planar structures using skeletons
We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams
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