Minimal Set of Constraints for 2d Constrained Delaunay Reconstruction

Devillers, Olivier; Estkowski, Regina; Gandoin, Pierre-Marie; Hurtado, Ferrán; Ramos, Pedro; Sacristán, Vera

doi:10.1142/s0218195903001244

Cited by 9 publications

(18 citation statements)

References 5 publications

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“…Let G be a triangulation, v 1 v 2 be an edge in G (but not an edge of the convex hull of G), and (v 1 , v 2 , v 3 ) and (v 1 , v 2 , v 4 ) be the triangles adjacent to v 1 v 2 in G. We say that v 1 v 2 is a locally Delaunay edge if the circle through {v 1 , v 2 , v 3 } does not contain v 4 or equivalently if the circle through {v 1 , v 2 , v 4 } does not contain v 3 . Every edge of the convex hull of G is also considered to be locally Delaunay [3].…”

Section: Definition 2 (Euclidean Visibility Graph Of I) the Euclideamentioning

confidence: 99%

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Essential Constraints of Edge-Constrained Proximity Graphs

Bose

Carufel

Shaikhet

et al. 2017

JGAA

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Abstract. Given a plane forest F = (V, E) of |V | = n points, we find the minimum set S ⊆ E of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F . We present an O(n log n)-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, β-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set S ⊆ E of edges of a given plane graph I = (V, E) such that I ⊆ CG β (V, S) for 1 ≤ β ≤ 2, where CG β (V, S) is the constraint β-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.

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Section: Definition 2 (Euclidean Visibility Graph Of I) the Euclideamentioning

confidence: 99%

“…In particular, Devillers et al [3] investigate how to compute the minimum set S ⊆ E of a given plane triangulation T = (V, E), such that T is a constrained Delaunay triangulation (DT ) of the graph (V, S). They show that S and V is the only information that needs to be stored.…”

Section: Introductionmentioning

confidence: 99%

Essential Constraints of Edge-Constrained Proximity Graphs

Bose

Carufel

Shaikhet

et al. 2017

JGAA

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“…Every edge of the convex hull of G is also considered to be locally Delaunay [3]. This definition is equivalent to the classical definition used for example by Chew in [1]: CDT (I) is the unique triangulation of V such that each edge e is either in E or there exists a circle C with the following properties:…”

Section: Definition 3 (Constrained Visibility Graph Of I)mentioning

confidence: 99%

Essential Constraints of Edge-Constrained Proximity Graphs

Bose

Carufel

Shaikhet

et al. 2016

Lecture Notes in Computer Science

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Given a plane forest F = (V, E) of |V | = n points, we find the minimum set S ⊆ E of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F . We present an O(n log n)-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, β-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set S ⊆ E of edges of a given plane graph I = (V, E) such that I ⊆ CG β (V, S) for 1 ≤ β ≤ 2, where CG β (V, S) is the constraint β-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.

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“…Devillers and Gandoin [36] also mention the compression of GIS as an application of their geometric coder, and develop an edge coder well suited for this context, which results in compression rates as low as 0.2 bits per vertex for the complete connectivity. Besides, Devillers et al show that the minimal set to be transmitted to guarantee the exact reconstruction of the initial model by constrained Delaunay triangulation is constituted by the edges that are nonlocally Delaunay [42].…”

Section: Compression and Reconstructionmentioning

confidence: 99%

Reconstruction Algorithms as a Suitable Basis for Mesh Connectivity Compression

Chaine

Gandoin

Roudet

2009

IEEE Trans. Automat. Sci. Eng.

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During a highly productive period running from 1995 to about 2002, the research in lossless compression of surface meshes mainly consisted in a hard battle for the best bitrates. However, for a few years, compression rates seem stabilized around 1.5 bit per vertex for the connectivity coding of usual triangular meshes, and more and more work is dedicated to remeshing, lossy compression, or gigantic mesh compression, where memory access and CPU optimizations are the new priority. However, the size of 3D models keeps growing, and many application fields keep requiring lossless compression. In this paper, we present a new contribution for single-rate lossless connectivity compression, which first brings improvement over current state of the art bitrates, and second, does not constraint the coding of the vertex positions, offering therefore a good complementarity with the best performing geometric compression methods. The initial observation having motivated this work is that very often, most of the connectivity part of a mesh can be automatically deduced from its geometric part using reconstruction algorithms. This has already been used within the limited framework of projectable objects (essentially, terrain models and GIS), but finds here its first generalization to arbitrary triangular meshes, without any limitation regarding the topological genus, the number of connected components, the manifoldness or the regularity. This can be obtained by constraining and guiding a Delaunay-based reconstruction algorithm so that it outputs the initial mesh to be coded. The resulting rates seem extremely competitive when the meshes are fully included in Delaunay, and are still good compared to the state-of-the-art in the case of scanned models.Note to Practitioners-A 3D triangle mesh is composed of a geometric part (the vertex coordinates) and a connectivity part (the description of the triangles). In this paper, we show how to reencode such surface meshes in order to obtain near zero connectivity cost for some class of surface meshes (and very good rates in the general case), while guaranteeing in the same time state-of-the-art geometry encoding cost. This method can be useful in all application areas where the mesh size is a bottleneck (typically network or storage applications). The best results are obtained for meshes made from 3D scans (in contrast to CAD meshes). The main current limitations of the method are the computing times (about 1 s per 1000 points part of the mesh, for compression which can be done offline) and the memory footprint.

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Minimal Set of Constraints for 2d Constrained Delaunay Reconstruction

Cited by 9 publications

References 5 publications

Essential Constraints of Edge-Constrained Proximity Graphs

Essential Constraints of Edge-Constrained Proximity Graphs

Essential Constraints of Edge-Constrained Proximity Graphs

Reconstruction Algorithms as a Suitable Basis for Mesh Connectivity Compression

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