Formal Study of Plane Delaunay Triangulation

Dufourd, Jean-François; Bertot, Yves

doi:10.1007/978-3-642-14052-5_16

Cited by 17 publications

(11 citation statements)

References 29 publications

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“…The naive algorithm is unsatisfactory as it does not provide a good way to find the triangle inside of which a new point may occur. This can be improved by using Delaunay triangulations, as already studied formally in [6] and a well-known algorithm of "visibility" walk in the triangulation [4], which can be proved to have guarantees to terminate only when the triangulation satisfies the Delaunay criterion. This is the planned future work.…”

Section: Resultsmentioning

confidence: 99%

“…A first attempt with convex hulls was provided by Pichardie and Bertot [17] where the only data structure used was that of lists but the question of non general positions (where points may be aligned) was also studied. Notable work is provided by Dufourd and his colleagues [3,5,6,1]. In particular, Dufourd advocated the use of hypermaps to represent many of the data-structures of computational geometry.…”

Section: Related Workmentioning

confidence: 99%

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Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

Bertot

2018

Lecture Notes in Computer Science

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This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex notions are those of boundary and separating edges. When performing proofs about this algorithm, questions of symmetry appear and this exposition attempts to give an account of how these symmetries can be handled. All this work relies on formal developments made with Coq and the mathematical components library. An Abstract Description of TriangulationGiven a set of points, a triangulating algorithm returns a collection of triangles that must cover the space between these points (the convex hull), have no overlap, and such that all the points of the input set are vertices of at least one triangle. When the input points represent obstacles, the triangulation can help construct safe routes between these obstacles, thanks to Delaunay triangulations and Voronoï diagrams.

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Section: Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

Bertot

2018

Lecture Notes in Computer Science

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“…The most extensive among these is due to the formalization of four colour theorem [8,9] which considers only planar graphs. The work by Dufourd and Bertot on formalizing plane Delaunay triangulation [5] utilises a similar notion of graphs based on hypermaps. In a recent work [4] aimed towards formalizing the graph minor theorem for treewidth two Doczkal et al developed a general library for graphs using the Coq Proof Assistant.…”

Section: Related Workmentioning

confidence: 99%

A constructive formalization of the weak perfect graph theorem

Singh

Natarajan

2020

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs

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The Perfect Graph Theorems are important results in graph theory describing the relationship between clique number ω(G) and chromatic number χ(G) of a graph G. A graph G is called perfect if χ(H) = ω(H) for every induced subgraph H of G. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for working with finite simple graphs. We model finite simple graphs in the Coq Proof Assistant by representing its vertices as a finite set over a countably infinite domain. We argue that this approach provides a formal framework in which it is convenient to work with different types of graph constructions (or expansions) involved in the proof of the Lovász Replication Lemma (LRL), which is also the key result used in the proof of Weak Perfect Graph Theorem. Finally, we use this setting to develop a constructive formalization of the Weak Perfect Graph Theorem.

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“…The triangulation with linear interpolation algorithm uses the values of multiple neighboring pixels. The Delaunay triangulation [7][8] of the point set is first computed with the z-values of the vertices determining the tilt of the triangles.…”

Section: Noise Fixingmentioning

confidence: 99%

A novel method to suppress noise in marine radar images based on pulse-pulse correlation

Ding

Chen

et al. 2011

SPIE Proceedings

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As the X-band marine radar often suffers from interference of electromagnetic waves of the same frequency transmitted by radars in its vicinity, the acquired images frequently contain co-channel interference noise. The noise degrades the quality of the marine radar images and is unfavorable to the processing and interpretation of the marine radar images. To suppress the noise in marine radar images, a novel method based on pulse-pulse correlation is proposed. This method includes three steps: threshold segmentation, noise extraction and noise fixing. In the threshold segmentation step, the threshold T is calculated based on the K distribution sea clutter model. In the noise extraction step, a 3×3 window is applied. By using the window, the pixels of noise can be extracted, and at the same time the pixels of non-noise can be discarded. In the noise fixing step, the strategy of piecewise interpolation is applied. At the region near to the image center, the triangulation with linear interpolation algorithm is applied; at the region far from the image center, the nearest neighbor algorithm is applied. The real X band marine radar image was used to test the performance of the proposed method. The obtained results show that the proposed method is able to reduce the co-channel interference noise from the marine radar images significantly and keep the information of objects in the images such as ships and islands. Besides, the proposed method can be fast in speed of operation.

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Formal Study of Plane Delaunay Triangulation

Cited by 17 publications

References 29 publications

Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

A constructive formalization of the weak perfect graph theorem

A novel method to suppress noise in marine radar images based on pulse-pulse correlation

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