Topic
Delaunay triangulation
About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.
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TL;DR: An algorithm based on spatial tessellation and approximation of each triangle patch in the Delaunay triangulation by a bivariate polynomial is advanced to construct a high resolution (HR) high quality image from a set of low resolution (LR) frames.
Abstract: An algorithm based on spatial tessellation and approximation of each triangle patch in the Delaunay (1934) triangulation (with smoothness constraints) by a bivariate polynomial is advanced to construct a high resolution (HR) high quality image from a set of low resolution (LR) frames. The high resolution algorithm is accompanied by a site-insertion algorithm for update of the initial HR image with the availability of more LR frames till the desired image quality is attained. This algorithm, followed by post filtering, is suitable for real-time image sequence processing because of the fast expected (average) time construction of Delaunay triangulation and the local update feature.
187 citations
TL;DR: A very novel hybrid approach, which compares triangles rather than blocks, or single points, and objects are modeled as a set of connected triangles built onto these points to detect copy-move forgery.
Abstract: Copy–move forgery is one of the most common types of tampering for digital images Detection methods generally use block-matching approaches, which first divide the image into overlapping blocks and then extract and compare features to find similar ones, or point-based approaches, in which relevant keypoints are extracted and matched to each other to find similar areas In this paper, we present a very novel hybrid approach, which compares triangles rather than blocks, or single points Interest points are extracted from the image, and objects are modeled as a set of connected triangles built onto these points Triangles are matched according to their shapes (inner angles), their content (color information), and the local feature vectors extracted onto the vertices of the triangles Our methods are designed to be robust to geometric transformations Results are compared with a state-of-the-art block matching method and a point-based method Furthermore, our data set is available for use by academic researchers
186 citations
TL;DR: In this paper, the authors describe the minimal triangulations of polygonal domains and the restricted triangulation problem of finding a T of minimal weight among those containing s (S is a subset of E where no two edges of S cross each other).
Abstract: Publisher Summary This chapter describes the minimal triangulations of polygonal domains. If V is a set of n distinct points (vertices) M1, M2,. . . , Mn in the plane, no three points are collinear. This assumption is not essential but simplifies the explanations. Let E be the family of ½n(n − 1) line segments (edges) joining the vertices of V. A triangulation T of V is a maximal subset of E in which no two edges cross each other. Minimal weight triangulation (MWT) is a triangulation on V for which s(T) is minimal. The restricted triangulation problem consists of finding a T of minimal weight among those containing s (S is a subset of E where no two edges of S cross each other. Then there exists some triangulation T such that S ⊂ T.). Triangulation of a convex polygon and triangulation of a simple polygon domain are discussed in the chapter.
186 citations
01 Sep 2003
TL;DR: In this paper, a set of points that are evenly spaced across the surface is automatically found, and then a geodesic Delaunay triangulation of the points is created, using a Voronoi diagram construction based on Fast Marching.
Abstract: In this paper, we present a method for remeshing triangulated manifolds by using geodesic path calculations and distance maps. Our work builds on the Fast Marching algorithm, which has been extended to arbitrary meshes by Sethian and Kimmel. First, a set of points that are evenly spaced across the surface is automatically found. A geodesic Delaunay triangulation of the set of points is then created, using a Voronoi diagram construction based on Fast Marching. At last, we use the distance information to find a simple parameterization of the manifold. Marching algorithm makes this method computationally inexpensive, and gives very good results. Examples are shown for synthetic and real surfaces.
183 citations
IBM1
TL;DR: It is shown that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulated, and the weighted sum of squares of the edge lengths is the smallest for Delaunays triangulations.
Abstract: In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.
181 citations