Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr

3D Compression: from A to Zip a first complete example

Imagens invadiram a maioria das publicacações e comunicacões contemporâneas. Esta expansão acelerou-se com o desenvolvimento de métodos eficientes de compressão da imagem. Hoje o processo da criação de imagens é baseado nos objetos multidimensionais gerados por CAD, simulações físicas, representações de dados ou soluções de problemas de otimização. Esta variedade das fontes motiva o desenho de esquemas de compressão adaptados a classes específicas de modelos. O lançamento recente do Google Sketch’up com o seu armazém de modelos 3D acelerou a passagem das imagens bidimensionais às tridimensionais. Entretanto, este o tipo de sistemas requer um acesso rápido aos modelos 3D, possivelmente gigantes, que é possível somente usando de esquemas eficientes da compressão. Esse trabalho faz parte de um tutorial ministrado no Sibgrapi 2007.Images invaded most of contemporary publications and communications. This expansion has accelerated with the development of efficient schemes dedicated to image compression. Nowadays, the image creation process relies on multidimensional objects generated from computer aided design, physical simulations, data representation or optimisation problem solutions. This variety of sources motivates the design of compression schemes adapted to specific class of models. The recent launch of Google Sketch’up and its 3D models warehouse has accelerated the shift from two-dimensional images to three-dimensional ones. However, these kind of systems require fast access to eventually huge models, which is possible only through the use of efficient compression schemes. This work is part of a tutorial given at the XXth Brazilian Symposium on Computer Graphics and Image Processing (Sibgrapi 2007)

Schnyder woods for higher genus triangulated surfaces

The final version of this extended abstract has been published in "Discrete and Computational Geometry (2009)"International audienceSchnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertex-spanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity

A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1-dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n)

Vector field reconstruction from sparse samples with applications

Abstract. We present a novel algorithm for 2D vector field reconstruction from sparse set of points–vectors pairs. Our approach subdivides the domain adaptively in order to make local piecewise polynomial approximations for the field. It uses partition of unity to blend those local approximations together, generating a global approximation for the field. The flexibility of this scheme allows handling data from very different sources. In particular, this work presents important applications of the proposed method to velocity and acceleration fields ’ analysis, in particular for fluid dynamics visualization

Volume-Enclosing Surface Extraction

In this paper we present a new method, which allows for the construction of triangular isosurfaces from three-dimensional data sets, such as 3D image data and/or numerical simulation data that are based on regularly shaped, cubic lattices. This novel volume-enclosing surface extraction technique, which has been named VESTA, can produce up to six different results due to the nature of the discretized 3D space under consideration. VESTA is neither template-based nor it is necessarily required to operate on 2x2x2 voxel cell neighborhoods only. The surface tiles are determined with a very fast and robust construction technique while potential ambiguities are detected and resolved. Here, we provide an in-depth comparison between VESTA and various versions of the well-known and very popular Marching Cubes algorithm for the very first time. In an application section, we demonstrate the extraction of VESTA isosurfaces for various data sets ranging from computer tomographic scan data to simulation data of relativistic hydrodynamic fireball expansions.Comment: 24 pages, 33 figures, 4 tables, final versio