Summary. 3D meshes are widely used in graphic and simulation applications for approximating 3D objects. When representing complex shapes in a raw data format, meshes consume a large amount of space. Applications calling for compact storage and fast transmission of 3D meshes have motivated the multitude of algorithms developed to efficiently compress these datasets. In this paper we survey recent developments in compression of 3D surface meshes. We survey the main ideas and intuition behind techniques for single-rate and progressive mesh coding. Where possible, we discuss the theoretical results obtained for asymptotic behavior or optimality of the approach. We also list some open questions and directions for future research. 1

Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R−L of the support of the one-dimensional ISE, or precisely: n −1/4 rn law − → (8/9) 1/4 r. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks (e (n) , ˆ W (n) ) to the Brownian snake description (e, ˆ W) of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity. 1.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

We discuss space-efficient encoding schemes for planar graphs and maps. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. They are near-optimal in number of bits per edge. 1 Introduction In this paper we discuss space-efficient binary encoding schemes for several classes of unlabeled connected planar graphs and maps. In encoding a graph we must encode the incidences among vertexes and edges. By maps we understand topological equivalence classes of planar embeddings of planar graphs. In encoding a map we are required to encode the topology of the embedding i.e., incidences among faces, edges, and vertexes, as well as the graph. Each map is an embedding of a unique graph, but a given graph may have multiple embeddings. Hence maps must require more bits to encode than graphs in some average sense. There are a number of recent results on space-efficient encoding. A standard adjacency list encoding of an unlabeled graph G requires...

We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups.

We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular the result answers a question of Bender and Canfield in [BC94] and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4-regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arques ([Arq87]) of the generating function of planar maps with respect to vertices and faces.

We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2-connected and has approximately 2n edges. We present a rst attack on the problem of describing what the random planar graph looks like.

We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3-connected components. For 3-connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.

In this paper we present a data structure improving region segmentation of 2D images. This data structure provides an efficient access to the set of pixel of one region. It also provides topological informations like the frontier of a region, the neighbours of a region or the set of regions included in one region. Thanks to this data structure different segmentation algorithms can be combined to perform the segmentation of an image. Interactive refinement or merge of regions can also be performed efficiently. Keywords Segmentation, inter-pixel boundary, topological map. I. introduction The problem of extracting objects from a complex image has been widely studied for the last fifty years. It quickly appeared that this problem cannot be solved without an priori knowledge of the objects to be recognized. Segmentation algorithms can thus be categorized in two classes: domain-dependent algorithms which attempt to recognize specific objects in a scene -- for instance tumors in chest radi...

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