In this paper, we propose a valence-driven, single-resolution encoding technique for lossless compression of triangle mesh connectivity. Building upon a valence-based approach pioneered by Touma and Gotsman 22, we design a new valence-driven conquest for arbitrary meshes that always guarantees smaller compression rates than the original method. Furthermore, we provide a novel theoretical entropy study of our technique, hinting the optimality of the valence-driven approach. Finally, we demonstrate the practical efficiency of this approach (in agreement with the theoretical prediction) on a series of test meshes, resulting in the lowest compression ratios published so far, for both irregular and regular meshes, small or large.

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

... this paper we introduce a connectivity encoding method which extends these ideas to 2manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and dual mesh in a symmetric fashion. It generates two sequences of symbols, vertex valences and face degrees, and encodes them separately using two context-based arithmetic coders. This allows us to exploit vertex and/or face regularity if present. When the mesh exhibits perfect face regularity (e.g., a pure triangle or quad mesh) and/or perfect vertex regularity (valence six or four respectively) the corresponding bit rate vanishes to zero asymptotically. For triangle meshes, our technique is equivalent to earlier valence driven approaches. We report compression results for a corpus of standard meshes. In all cases we are able to show coding gains over earlier coders, sometimes as large as 50%. Remarkably, we even slightly gain over coders specialized to triangle or quad meshes. A theoretical analysis reveals that our approach is near-optimal as we achieve the Tutte entropy bound for arbitrary planar graphs of 2 bits per edge in the worst case.

We present a simple linear time algorithm for decoding Edgebreaker encoded triangle meshes in a single traversal. The Edgebreaker encoding technique, introduced in [5], encodes the connectivity of triangle meshes homeomorphic to a sphere with a guaranteed # bits per triangle or less. The encoding algorithm visits every triangle of the mesh in a depthfirst order. The original decoding algorithm [5] recreates the triangles in the same order they have been visited by the encoding algorithm and exhibits a worst case time complexity of ### # #. More recent work [6] uses the same traversal order and improves the worst case to ####. However, for meshes with handles multiple traversals are needed during both encoding and decoding. We introduce here a simpler decoding technique that performs a single traversal and recreates the triangles in reverse order.

In this paper we present a coder for polygon mesh connectivity that delivers the best connectivity compression rates meshes reported so far. Our coder is an extension of the vertex-based coder for triangle mesh connectivity by Touma and Gotsman [26]. We code polygonal connectivity as a sequence of face and vertex degrees and exploit the correlation between them for mutual predictive compression. Because low-degree vertices are likely to be surrounded by high-degree faces and vice versa, we predict vertex degrees based on neighboring face degrees and face degrees based on neighboring vertex degrees.

We survey recent developments in compact representations of 3D mesh data. This includes: Methods to reduce the complexity of meshes by simplification, thereby reducing the number of vertices and faces in the mesh; Methods to resample the geometry in order to optimize the vertex distribution; Methods to compactly represent the connectivity data (the graph structure defined by the edges) of the mesh; Methods to compactly represent the geometry data (the vertex coordinates) of a mesh.

Among progressive 3D mesh compression algorithms, the kd-tree-based algorithm proposed by Gandoin and Devillers [1] is one of the state-of-the-art algorithms. Based on the observation that this geometry coder has a large amount of overhead at high kd-tree levels, we propose an octree-based geometry coder that demands a less amount of coding bits at high octree levels by applying selective cell subdivision at high tree levels, leading to a better rate-distortion performance for the low bit rate coding. Experimental results show that, compared with the kd-tree-based coder, the proposed 3D geometry coder performs better for an expanded tree of a level less than or equal to 8 but slightly worse for the full tree expansion with 12-bit quantization.

This article presents an algebraic analysis of a mesh-compression technique called high-pass quantization [Sorkine et al. 2003]. In high-pass quantization, a rectangular matrix based on the mesh topological Laplacian is applied to the vectors of the Cartesian coordinates of a polygonal mesh. The resulting vectors, called δ-coordinates, are then quantized. The applied matrix is a function of the topology of the mesh and the indices of a small set of mesh vertices (anchors) but not of the location of the vertices. An approximation of the geometry can be reconstructed from the quantized δ-coordinates and the spatial locations of the anchors. In this article, we show how to algebraically bound the reconstruction error that this method generates. We show that the small singular value of the transformation matrix can be used to bound both the quantization error and the rounding error which is due to the use of floating-point arithmetic. Furthermore, we prove a bound on this singular value. The bound is a function of the topology of the mesh and of the selected anchors. We also propose a new anchor-selection algorithm, inspired by this bound. We show experimentally that the method is effective and that the computed upper bound on the error is not too pessimistic.

We propose a divide and conquer algorithm for the single resolution encoding of triangle mesh connectivity. Starting from a boundary edge we grow a zig-zag strip which divides the mesh into two submeshes which are encoded separately in a recursive process. We introduce a novel data structure for triangle mesh encoding, a binary tree with positive integer weights assigned to its nodes. The length of the initial strip is stored in the root of the binary tree, while the encoding of the left and right submesh are stored in the left and right subtree, respectively. We £nd a simple criterion determining which objects of this data structure correspond to triangle meshes. As the algorithm implicitly traverses the triangles of the mesh, it can be classi£ed into the family of Edgebreaker like encoding schemes. Hence, the compression ratios, both in the form of theoretical upper bounds and practical results are similar to the Edgebreaker’s, while the simplicity and ¤exibility of the algorithm makes it particularly suitable for applications where the connectivity encoding is only a small part of the problem at hand.