Thin discrete triangular meshes

Barneva, Reneta P.; Brimkov, Valentin E.; Nehlig, Philippe

doi:10.1016/s0304-3975(98)00346-6

Cited by 26 publications

(28 citation statements)

References 19 publications

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“…, c l } of voxels that satisfy the following two axioms: (A1) c i is α-adjacent to c j iff i = j ± 1(modulo l), and (B1) γ is one-dimensional with respect to α-adjacency. Then a digital manifold is recursively defined as follows: 6 The definition applies to digital analogs of hole-free hypersurfaces, i.e., ones without tunnels ((n − 1)-gaps). In fact, in the framework of the approach of [14], a hypersurface with tunnels can be an (n − 2)-dimensional set of n-cells (see Fig.…”

Section: Digital Surfacesmentioning

confidence: 99%

“…6). It is known that these assumptions do not restrict the generality (see, e.g., [4,6,19]). The last one guarantees that each pixel of the integer grid in Ox 1 x 2 is a projection of exactly one voxel from P (0, a 1 , a 2 , a 3 , |a 3 |).…”

Section: Theorem 5 ([4 Proposition 9]) Letmentioning

confidence: 99%

“…If a 3 < a 1 + a 2 , then L is thicker than naive and thinner than standard. (6,7,16) restricted to a (10, 14)-cube. All level lines have identical shape since gcd(6, 7) = 1.…”

Section: Fact 7 Let a Naive Planementioning

confidence: 99%

“…Studies about gaps and tunnels in digital lines, planes, and polyhedral surfaces are available, e.g., in [1][2][3][5][6][7][8]11,12]. A recent work [15] provides the formula g = v − 2(p + c − h) + b, where g is the number of gaps, v the number of vertices, p the number of pixels, h the number of holes, c the number of connected components, and b the number of 2 × 2 grid squares in a two-dimensional digital picture.…”

Section: Introductionmentioning

confidence: 99%

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Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Brimkov

2009

Discrete Applied Mathematics

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a b s t r a c tIn this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension. Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces (in particular planes) and curves.Published by Elsevier B.V.

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Section: Digital Surfacesmentioning

confidence: 99%

Section: Theorem 5 ([4 Proposition 9]) Letmentioning

confidence: 99%

“…If a 3 < a 1 + a 2 , then L is thicker than naive and thinner than standard. (6,7,16) restricted to a (10, 14)-cube. All level lines have identical shape since gcd(6, 7) = 1.…”

Section: Fact 7 Let a Naive Planementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Brimkov

2009

Discrete Applied Mathematics

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“…Certain advantages of the analytical definitions are discussed in the literature (see, e.g., [2,6]). Their main worth seems to lie in the fact that one can study an object in terms of a few parameters that define it.…”

Section: Introductionmentioning

confidence: 99%

Connectivity of discrete planes

Brimkov

2004

Theoretical Computer Science

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Studying connectivity of discrete objects is a major issue in discrete geometry and topology. In the present work we deal with connectivity of discrete planes in the framework of Reveillès analytical definition [11]. Accordingly, a discrete plane is a set P (a, b, c, µ, ω) of integer points (x, y, z) satisfying the Diophantine inequalities 0 ≤ ax + by + cz + µ < ω. The parameter µ ∈ Z estimates the plane intercept while ω ∈ N is the plane thickness. Given three integers (plane coefficients) a, b, and c with 0 ≤ a ≤ b ≤ c, one can seek the maximal ω for which the discrete plane P (a, b, c, µ, ω) is disconnected. We call this remarkable topological invariable the connectivity number of P (a, b, c, µ, ω) and denote it Ω(a, b, c). Despite several attempts over the last ten years to determine the connectivity number, this is still an open question. In the present paper we propose a solution to the problem. For this, we first investigate some combinatorial properties of discrete planes. These structural results facilitate the deeper understanding of the discrete plane structure. On this basis, we obtain a series of results, in particular, we provide an explicit solution to the problem under certain conditions. We also obtain exact upper and lower bounds on Ω(a, b, c) and design an O(a log b) algorithm for its computation.

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Polygonalisation and Polyhedralisation by Optimisation

Linh

Imiya

Torii

Geometric Properties for Incomplete Data

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Thin discrete triangular meshes

Cited by 26 publications

References 19 publications

Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Connectivity of discrete planes

Polygonalisation and Polyhedralisation by Optimisation

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