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Edgebreaker: A Simple Compression for Surfaces with Handles
Jarek Rossignac
♥
, Hélio Lopes
♠
, Alla Safanova
♦
, Geovan Tavares
♠
, Andrzej Szymczak
♥
♥
Georgia Institute of Technology, College of Computing and GVU Center, Atlanta, GA. USA
♠
Pontifical Catholic University (PUCRio), Department of Mathematics,
Rio de Janeiro, RJ, Brazil
♦
Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA
{jarek@cc.gatech.edu, lopes@mat.pucrio.br, asafanov@cs.cmu.edu, geovan@mat.pucrio.br, andrzej@cc.gatech.edu}
ABSTRACT
The Edgebreaker is an efficient scheme for compressing
triangulated surfaces. A surprisingly simple implementation
of Edgebreaker has been proposed for surfaces homeomorphic
to a sphere. It uses the CornerTable data structure, which
represents the connectivity of a triangulated surface by two
tables of integers, and encodes them with less than 2 bits per
triangle. We extend this simple formulation to deal with
triangulated surfaces with handles and present the detailed
pseudocode for the encoding and decoding algorithms (which
take one page each). We justify the validity of the proposed
approach
using
the mathematical
formulation
of the
Handlebody
theory
for surfaces,
which
explains
the
topological changes that occur when two boundary edges of a
portion of a surface are identified.
Keywords
Triangle meshes; Connectivity
Graph; 3D Compression;
Handlebody Theory.
1.
INTRODUCTION
The Edgebreaker compression and decompression algorithms
[13] may be used to encode the connectivity of any simply
connected manifold triangle mesh with a guaranteed worst case
code of 1.80 bits per triangle [4]. In practice, the Edgebreaker
encoding may often be further compressed
to less than one bit
per triangle through the use of Entropy codes [14]. But the true
value of Edgebreaker lies in the efficiency and surprising
simplicity of the algorithms [15], which fit in a couple of
pages and use only a few arrays of integers as sole data
structure. Because of its simplicity, Edgebreaker is viewed as
the emerging standard for 3D compression [17] and may
provide an alternative for the current MPEG4 standard that is
based on Rossignac's previous work with Gabriel Taubin [19].
This simple
implementation of Edgebreaker, as a state
machine, is publicly available through the web [3] and has
been recently enhanced to support meshes with an arbitrary
number of handles (also called throughholes). We provide in
this paper a detailed description of this extension, imbedding
it in a theoretic setting of the Handlebody theory, and
including formal proofs.
An
important
topological
property
of
boundary
representations (abbreviated BReps) [1] is the EulerPoincaré
formula, dated from the turn of the century [11], which states
that an orientable connected triangulated surface
S
without
boundary is uniquely identified by its Euler characteristic
χ
(
S
)
=V
−
E+F,
where
V,E
and
F
indicate respectively the
number of vertices, edges and faces of S.
The
Euler
characteristic classifies
S
according to the Euler formula that is
χ
(
S)=
2
−
2
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 Fall '08
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