Succinct representation of triangulations with a boundary

We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with $m$ faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g-1)lg m bits (which is negligible as long as g << m/lg m)

Un modelo distribuido para calcular descriptores locales de malla 3D basados en k-rings

In order to facilitate 3D object processing, it is common to use high-level representations such as local descriptors that are usually computed using defined neighborhoods. K-rings, a technique to define them, is widely used by several methods. In this work, we propose a model for the distributed computation of local descriptors over 3D triangular meshes, using the concept of k-rings. In our experiments, we measure the performance of our model on huge meshes, evaluating the speedup, the scalability, and the descriptor computation time. We show the optimal configuration of our model for the cluster we implemented and the linear growth of computation time regarding the mesh size and the number of rings. We used the Harris response, which describes the saliency of the object, for our tests.Para facilitar el procesamiento de objetos 3D, es común utilizar representaciones de alto nivel, como los descriptores locales que generalmente se calculan utilizando vecindarios definidos. K-rings es una técnica para definirlos y es ampliamente utilizada por varios métodos. En este trabajo, proponemos un modelo para el cálculo distribuido de descriptores locales sobre mallas triangulares 3D, utilizando el concepto de anillos k. En nuestros experimentos, medimos el rendimiento de nuestro modelo en mallas enormes, evaluando la aceleración, la escalabilidad y el tiempo de cálculo del descriptor. Mostramos la configuración óptima de nuestro modelo para el clúster que implementamos y el crecimiento lineal del tiempo de cálculo con respecto al tamaño de la malla y el número de anillos. Usamos la respuesta de Harris, que describe la prominencia del objeto, para nuestras pruebas

ESQ: Editable SQuad Representation for Triangle Meshes

International audienceWe consider the problem of designing space efficient solutions for representing the connectivity information of manifold triangle meshes. Most mesh data structures are quite redundant, storing a large amount of information in order to efficiently support mesh traversal operators. Several compact data structures have been proposed to reduce storage cost while supporting constant-time mesh traversal. Some recent solutions are based on a global re-ordering approach, which allows to implicitly encode a map between vertices and faces. Unfortunately, these compact representations do not support efficient updates, because local connectivity changes (such as edge-contractions, edge-flips or vertex insertions) require re-ordering the entire mesh. Our main contribution is to propose a new way of designing compact data structures which can be dynamically maintained. In our solution, we push further the limits of the re-ordering approaches: the main novelty is to allow to re-order vertex data (such as vertex coordinates), and to exploit this vertex permutation to easily maintain the connectivity under local changes. We describe a new class of data structures, called Editable SQuad (ESQ), offering the same navigational and storage performance as previous works, while supporting local editing in amortized constant time. As far as we know, our solution provides the most compact dynamic data structure for triangle meshes. We propose a linear-time and linear-space construction algorithm, and provide worst-case bounds for storage and time cost.Cet article traite de la conception de structure de données usant peu de mémoire pour représenter des surfaces manifold triangulées. La plupart des structures utilisées sont largement redondantes pour permettre un parcours efficace des adjacences entre triangles. Par ailleurs il existe des structures compactes, basées sur une renumérotation qui code de manière implicite une correspondance entre faces et sommets. Malheureusement, ces structures ne permettent pas de modifier la triangulation car des opérations telles que insertion suppression ou bascule d'arête nécessite de renuméroter toute la triangulation. Nous proposons une nouvelle méthode de conception de structures de données compactes permettant une mise à jour dynamique en adaptant l'idée de renumérotation. Nous introduisons Editab SQuad (ESQ), une nouvelle famille de structures de données qui a les mêmes performances de stockage et de temps d'accés que les précédents travaux tout en permettant des modifications locales en temps constant amorti

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

Optimal succinct representations of planar maps

This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations

Succinct Indexes

This thesis defines and designs succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. As opposed to succinct (integrated data/index) encodings, the main advantage of succinct indexes is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this thesis, we present succinct indexes for various data types, namely strings, binary relations, multi-labeled trees and multi-labeled graphs, as well as succinct text indexes. For strings, binary relations and multi-labeled trees, when the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data. Using our techniques, we improve several previous results. We design succinct representations for strings and binary relations that are more compact than previous results, while supporting access/rank/select operations efficiently. Our high-order entropy compressed text index provides more efficient support for searches than previous results that occupy essentially the same amount of space. Our succinct representation for labeled trees supports more operations than previous results do. We also design the first succinct representations of labeled graphs. To design succinct indexes, we also have some preliminary results on succinct data structure design. We present a theorem that characterizes a permutation as a suffix array, based on which we design succinct text indexes. We design a succinct representation of ordinal trees that supports all the navigational operations supported by various succinct tree representations. In addition, this representation also supports two other encodings schemes of ordinal trees as abstract data types. Finally, we design succinct representations of planar triangulations and planar graphs which support the rank/select of edges in counter clockwise order in addition to other operations supported in previous work, and a succinct representation of k-page graph which supports more efficient navigation than previous results for large values of k

Succinct representation of triangulations with a boundary. WADS

Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g − 1) lg m bits (which is negligible as long as g ≪ m / lg m).