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.

Fractal image compression based on Delaunay triangulation and vector quantization

Abstract: Presents a new scheme for fractal image compression based on adaptive Delaunay triangulation. Such a partition is computed on an initial set of points obtained with a split and merge algorithm in a grey level dependent way. The triangulation is thus fully flexible and returns a limited number of blocks allowing good compression ratios. Moreover, a second original approach is the integration of a classification step based on a modified version of the Lloyd algorithm (vector quantization) in order to reduce the encoding complexity. The vector quantization algorithm is implemented on pixel histograms directly generated from the triangulation. The aim is to reduce the number of comparisons between the two sets of blocks involved in fractal image compression by keeping only the best representative triangles in the domain blocks set. Quality coding results are achieved at rates between 0.25-0.5 b/pixel depending on the nature of the original image and on the number of triangles retained.

A new method for accurate estimation of velocity field statistics

Abstract: We introduce two new methods to obtain reliable velocity field statistics from N-body simulations, or indeed from any general density and velocity fluctuation field sampled by discrete points, These methods, the Voronoi tessellation method and Delaunay tessellation method, are based on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity held is sampled. In the Voronoi method the velocity is supposed to be uniform within the Voronoi polyhedra, whereas the Delaunay method constructs a velocity field by linear interpolation between the four velocities at the locations defining each Delaunay tetrahedron. The most important advantage of these methods is that they provide an optimal estimator for determining the statistics of volume-averaged quantities, as opposed to the available numerical methods that mainly concern mass-averaged quantities. As the major share of the related analytical work on velocity field statistics has focused on volume-averaged quantities, the availability of appropriate numerical estimators is of crucial importance for checking the validity of the analytical perturbation calculations. In addition, it allows us to study the statistics of the velocity field in the highly non-linear clustering regime. Specifically we describe in this paper how to estimate, in both the Voronoi and the Delaunay methods, the value of the volume-averaged expansion scalar theta = H(-1)del .upsilon (the divergence of the peculiar velocity, expressed in units of the Hubble constant H), as well as the value of the shear and the vorticity of the velocity field, at an arbitrary position. The evaluation of these quantities on a regular grid leads to an optimal estimator for determining the probability distribution function (PDF) of the volume-averaged expansion scalar, shear and vorticity. Although in most cases both the Voronoi and the Delaunay methods lead to equally good velocity field estimates, the Delaunay method may be slightly preferable. In particular it performs considerably better at small radii. Note that it is more CPU-time intensive while its requirement for memory space is almost a factor 8 lower than the Voronoi method. As a test we here apply our estimator to that of an N-body simulation of such structure formation scenarios. The PDF:; determined from the simulations agree very well with the analytical predictions. An important benefit of the present method is that, unlike previous methods, it is capable of probing accurately the velocity field statistics in regions of very low density, which in N-body simulations are typically sparsely sampled, In a forthcoming paper we will apply the newly developed tool to a plethora of structure formation scenarios, of both Gaussian and non-Gaussian initial conditions, in order to see to what extent the velocity field PDFs are sensitive discriminators, highlighting fundamental physical differences between the scenarios.

Triangulations and applications

Abstract: This book will serve as a valuable source of information about triangulations for the graduate student and researcher. With emphasis on computational issues, it presents the basic theory necessary to construct and manipulate triangulations. In particular, the book gives a tour through the theory behind the Delaunay triangulation, including algorithms and software issues. It also discusses various data structures used for the representation of triangulations.

Image-consistent surface triangulation

Abstract: Given a set of 3D points that we know lie on the surface of an object, we can define many possible surfaces that pass through all of these points. Even when we consider only surface triangulations, there are still an exponential number of valid triangulations that all fit the data. Each triangulation will produce a different faceted surface connecting the points. Our goal is to overcome this ambiguity and find the particular surface that is closest to the true object surface. We do not know the true surface but instead we assume that we have a set of images of the object. We propose selecting a triangulation based on its consistency with this set of images of the object. We present an algorithm that starts with an initial rough triangulation and refines the triangulation until it obtains a surface that best accounts for the images of the object. Our method is thus able to overcome the surface ambiguity problem and at the same time capture sharp corners and handle concave regions and occlusions. We show results for a few real objects.