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42
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 70 (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 59 (13 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.
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 41 (13 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...
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 39 (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 36 (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
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 queri ..."
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Cited by 36 (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.
An InformationTheoretic Upper Bound of Planar Graphs Using Triangulation
, 2003
"... We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n ..."
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Cited by 24 (5 self)
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We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled nnode planar graphs have at least 1.70n edges and at most 2.54n edges.
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 24 (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 ...
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 21 (2 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.
Planar graphs, via wellorderly maps and trees
 In 30 th International Workshop, Graph  Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science
, 2004
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequ ..."
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Cited by 20 (4 self)
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1