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)

Efficient edgebreaker for surfaces of arbitrary topology

Abstract. The typical surfaces models handled by contemporary Computer Graphics applications have millions of triangles and numerous connected component, handles and boundaries. Edgebreaker and Spirale Reversi are examples of efficient schemes to compress and decompress their connectivity. A surprisingly simple linear–time implementation has been proposed for triangulated surfaces homeomorphic to a sphere and was subsequently extended to surfaces with handles. Here, we further extend its scope to surfaces with multiple components, handles, and multiple boundaries. The result is a simple and efficient compression/decompression solution for the broad class of orientable manifold surfaces

Efficient Edgebreaker for surfaces of arbitrary topology

The typical surfaces models handled by contemporary Computer Graphics applications have millions of triangles and numerous connected component, handles and bounding loops. Edgebreaker and Spirale Reversi are examples of efficient schemes to compress and decompress their connectivity. A surprisingly simple lineartime implementation has been proposed for triangulated surfaces homeomorphic to a sphere and was subsequently extended to surfaces with handles. Here, we further extend its scope to surfaces with multiple components, handles, and multiple bounding loops. The result is a simple and efficient compression/decompression solution for the broad class of orientable manifold surfaces