# Handles - Edgebreaker A Simple Compression for Surfaces...

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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 (PUC-Rio), Department of Mathematics, Rio de Janeiro, RJ, Brazil Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA 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 Corner-Table 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 MPEG-4 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 through-holes). 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 B-Reps) [1] is the Euler-Poincaré 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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