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49
Recent advances in compression of 3D meshes
 In Advances in Multiresolution for Geometric Modelling
, 2003
"... 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 multit ..."
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Cited by 83 (3 self)
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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 singlerate 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
Guaranteed 3.67V bit encoding of planar triangle graphs
 11TH CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY (CCCG'’99
, 1999
"... We present a new representation that is guaranteed to encode any planar triangle graph of V vertices in less than 3.67V bits. Our code improves on all prior solutions to this well studied problem and lies within 13% of the theoretical lower limit of the worst case guaranteed bound. It is based on a ..."
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Cited by 61 (14 self)
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We present a new representation that is guaranteed to encode any planar triangle graph of V vertices in less than 3.67V bits. Our code improves on all prior solutions to this well studied problem and lies within 13% of the theoretical lower limit of the worst case guaranteed bound. It is based on a new encoding of the CLERS string produced by Rossignacs Edgebreaker compression [Rossignac99]. The elegance and simplicity of this technique makes it suitable for a variety of 2D and 3D triangle mesh compression applications. Simple and fast compression/decompression algorithms with linear time and space complexity are available.
Compact Representations of Separable Graphs
 In Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree qu ..."
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Cited by 44 (11 self)
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We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.
Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker
 Journal of Computational Geometry, Theory and Applications
, 1999
"... The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string a sequence of t symbols from the set {C, L,E,R,S}, each represented by a 1, 2 or ..."
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Cited by 44 (14 self)
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The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string a sequence of t symbols from the set {C, L,E,R,S}, each represented by a 1, 2 or 3 bit code. We show here that, in practice, the string can be further compressed to between 0.91t and 1.26t bits using an entropy code. These results improve over the 2.3t bits code proposed by Keeler and Westbrook (1995) and over the various 3D triangle mesh compression techniques published recently (Gumhold and Strasser, 1998; Itai and Rodeh, 1982; Naor, 1990; Touma and Gotsman, 1988; Turan, 1984), which exhibit either larger constants or cannot guarantee a linear worst case storage complexity. The decompression proposed by Rossignac (1999) is complicated and exhibits a nonlinear time complexity. The main contribution reported here is a simpler and efficient decompression algorithm, calle...
Connectivity Compression for Irregular Quadrilateral Meshes
, 2000
"... Applications that require Internet access to remote 3D datasets are often limited by the storage costs of 3D models. Several compression methods are available to address these limits for objects represented by triangle meshes. Many CAD and VRML models, however, are represented as quadrilateral mesh ..."
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Cited by 42 (11 self)
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Applications that require Internet access to remote 3D datasets are often limited by the storage costs of 3D models. Several compression methods are available to address these limits for objects represented by triangle meshes. Many CAD and VRML models, however, are represented as quadrilateral meshes or mixed triangle/quadrilateral meshes, and these models may also require compression. We present an algorithm for compressing such quadrilateral meshes, and we demonstrate that in general they may be encoded with fewer bits than triangle meshes with the same number of vertices. By preserving and exploiting the original quad structure, our approach achieves encodings 3080% smaller than an approach based on randomly splitting quads into triangles. We present both a code with a proven worstcase cost of 2.67 bits per vertex for meshes without valencetwo vertices and entropycoding results for typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the regularity of the mesh....
Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 42 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Orderly Spanning Trees with Applications to Graph Encoding and Graph Drawing
 In 12 th Symposium on Discrete Algorithms (SODA
, 2001
"... The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar ..."
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Cited by 33 (6 self)
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The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a lineartime algorithm that obtains an orderly pair (H
A Fast General Methodology For InformationTheoretically Optimal Encodings Of Graphs
, 1999
"... . We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtai ..."
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Cited by 28 (3 self)
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. We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most fi(n)+o(fi(n)) bits for any continuous superadditive function fi(n) so that there are at most 2 fi(n)+o(fi(n)) distinct nnode graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from f1; : : : ; ` 1 g for ` 1 n; (5) the edges of G are labeled with labels from f1; : : : ; ` 2 ...
An InformationTheoretic Upper Bound on Planar Graphs Using WellOrderly Maps
, 2011
"... This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but a ..."
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Cited by 26 (3 self)
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This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an informationtheoretic upper bound on planar graphs, we introduce a definition of a quasicanonical embedding for planar graphs: wellorderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute wellorderly maps and implying an efficient coding of planar graphs.
New Bounds on The Encoding of Planar Triangulations
, 2000
"... Compact encodings of the connectivity of planar triangulations is a very important subject not only in graph theory but also in computer graphics. In 1962 Tutte determined the number of different planar triangulations. From his results follows that the encoding of the connectivity of planar triangul ..."
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Cited by 25 (3 self)
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Compact encodings of the connectivity of planar triangulations is a very important subject not only in graph theory but also in computer graphics. In 1962 Tutte determined the number of different planar triangulations. From his results follows that the encoding of the connectivity of planar triangulations with three border edges and vertices consumes in the asymptotic limit for � at least � � � � bits. Currently the best compression method with guaranteed upper bounds is based on the encoding of � �Edgebreaker strings and consumes no more than �� � bits per vertex. In this report we improve these results to �� � bits per vertex. We also present a new coding scheme for the split indices in a different encoding method the CutBorder Machine. We describe an encoding with an upper bound of �� � bits per vertex. Finally, we introduce a CutBorder data structure which allows for linear coding and decoding algorithms.