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GVU Technical Report Series

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Author: Rossignac, Jarek × Author: Szymczak, Andrzej ×

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Out-of-Core Compression and Decompression of Large n-dimensional Scalar Fields

2003 , Ibarria, Lorenzo (Lawrence) , Lindstrom, Peter , Rossignac, Jarek , Szymczak, Andrzej

We present a simple method for compressing very large and regularly sampled scalar fields. Our method is particularly attractive when the entire data set does not fit in memory and when the sampling rate is high relative to the feature size of the scalar field in all dimensions. Although we report results for R³ and R⁴ data sets, the proposed approach may be applied to higher dimensions. The method is based on the new Lorenzo predictor, introduced here, which estimates the value of the scalar field at each sample from the values at processed neighbors. The predicted values are exact when the n-dimensional scalar field is an implicit polynomial of degree n — 1. Surprisingly, when the residuals (differences between the actual and predicted values) are encoded using arithmetic coding, the proposed method often outperforms wavelet compression in an L ∞ sense. The proposed approach may be used both for lossy and lossless compression and is well suited for out-of-core compression and decompression, because a trivial implementation, which sweeps through the data set reading it once, requires maintaining only a small buffer in core memory, whose size barely exceeds a single (n — 1)-dimensional slice of the data.

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Connectivity Compression for Irregular Quadrilateral Meshes

1999 , King, Davis , Szymczak, Andrzej , Rossignac, Jarek

Many 3D models used in engineering, scientific, and visualization applications are represented by an irregular mesh of bounding quadrilaterals. We propose a scheme for compressing the connectivity of irregular quadrilateral meshes in 0.26-1.7 bits/quad, a 25-45% savings over randomly splitting quads into triangles and applying triangle mesh compression. Our approach is an extension of the Edgebreaker compression approach and of the Wrap&Zip decompression technique.

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Connectivity Compression for Irregular Quadrilateral Meshes

1999 , King, Davis , Rossignac, Jarek , Szymczak, Andrzej

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 encoding the connectivity of such quadrilateral meshes, and we demonstrate that by preserving and exploiting the original quad structure, our approach achieves encodings 30 - 80% smaller than an approach based on randomly splitting quads into triangles. We present both a code with a proven worst-case cost of 3 bits per vertex (or 2.75 bits per vertex for meshes without valence-two vertices) and entropy-coding results for typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the regularity of the mesh. Our method may be implemented by a rule for a particular splitting of quads into triangles and by using the compression and decompression algorithms introduced in [Rossignac99] and [Rossignac&Szymczak99]. We also present extensions to the algorithm to compress meshes with holes and handles and meshes containing triangles and other polygons as well as quads.

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Edgebreaker: A Simple Compression for Surfaces with Handles

2002 , Rossignac, Jarek , Lopes, Helio , Safanova, Alla , Tavares, Geovan , Szymczak, Andrzej

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.

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Wrap&zip: Linear decoding of planar triangle graphs

1999 , Rossignac, Jarek , Szymczak, Andrzej

The Edgebreaker compression technique, introduced by Rossignac, encodes any unlabeled triangulated planar graph of t triangles using a string of 2t bits. The string contains a sequence of t letters from the set {C, L, E, R, S} and 50% of these letters are C. Exploiting constraints on the sequence, we show that the string may in practice be further compressed to 1.6t bits using model independent codes and even more using model specific entropy codes. These results improve over the 2.3t bits needed by Keeler and Westbrook and over the various 3D triangle mesh compression techniques published recently, which all exhibit larger constants or non-linear worst case storage costs. As in Edgebreaker, we compress the mesh using a spiraling triangle-spanning tree and generate the same sequence of letters. Edgebreaker's decompression uses a look-ahead procedure to identify the third vertex of split triangles (S letter) by counting letter occurrences in the remaining part of the sequences. We introduce here a new decompression technique, which eliminates this look-ahead and thus exhibits a linear asymptotic time complexity. Wrap&zip converts the string into the corresponding triangle-spanning tree and assigns orientations to each one of its free edges. During that "wrapping" process, whenever two consecutive edges point to the same vertex, it glues them together, possibly continuing the "zip" along the next pair of edges that just became adjacent. By labeling the vertices according to the order in which they first appear in the triangle-spanning tree, this compression approach may be used to encode the connectivity (incidence of labeled graphs) of three-dimensional triangle meshes that are homeomorphic to a sphere. Being able to decompress connectivity prior to vertex locations is essential for the most advanced geometry compression schemes, which use connectivity to predict the location of a vertex from the location of its previously decoded neighbors.

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Grow & Fold: Compression of Tetrahedral Meshes

1999 , Szymczak, Andrzej , Rossignac, Jarek

Standard representations of irregular finite element meshes combine vertex data (sample coordinates and node values) and connectivity (tetrahedron-vertex incidence). Connectivity specifies how the samples should be interpolated. It may be encoded for each tetrahedron as four vertex-references, which together occupy 128 bits. For simple 3D meshes with a single scalar value per node, non-compressed vertex data occupies 14 times less storage. Our Grow & Fold format reduces the connectivity storage down to 7 bits per tetrahedron: 3 of these are used to encode the presence of children in a tetrahedron spanning tree; the other 4 constrain sequences of "folding" operations, so that they produce the connectivity graph of the original mesh. Additional bits must be used for each handle in the mesh and for each topological lock in the tree. By storing vertex data in an order defined by the tree, we avoid the need to store tetrahedron-vertex references, and facilitate variable coding techniques for the vertex data. We provide the details of simple, loss-less compression and decompression algorithms and describe implementation results.

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Edgebreaker on a Corner Table: A Simple Technique for Representing and Compressing Triangulated Surfaces

2001 , Rossignac, Jarek , Safanova, Alla , Szymczak, Andrzej

A triangulated surface S with V vertices is sometimes stored as a list of T independent triangles, each described by the 3 floating-point coordinates of its vertices. This representation requires about 576V bits and provides no explicit information regarding the adjacency between neighboring triangles or vertices. A variety of boundary-graph data structures may be derived from such a representation in order to make explicit the various adjacency and incidence relations between triangles, edges, and vertices. These relations are stored to accelerate algorithms that visit the surface in a systematic manner and access the neighbors of each vertex or triangle. Instead of these complex data structures, we advocate a simple Corner Table, which explicitly represents the triangle/vertex incidence and the triangle/triangle adjacency of any manifold or pseudo-mainfold triangle mesh, as two tables of integers. The Corner Table requires about 12VlogxV bits and must be accompanied by a vertex table, which requires 96V bits, of Floats are used. The Corner Table may be derived from the list of independent triangles. For meshes homeomorphic to a sphere, it may be compressed to less than 4V bits by storing the "clers" sequence of triangle-labels from the set {C,L,E,$,S}. Further compression to 3.6V bits may be guaranteed by using context-based codes for the clers symbols. Entropy codes reduce the storage for large meshes to less than 2V bits. Meshes with more complex topologies may require O(log2V) additional bits per handle of hole. We present here a publicly available, simple, state-machine implementation of the Edgebreaker compression, which traverses the corner table, computes the CLERS symbols, and constructs an ordered list of vertex references. Vertices are encoded, in the order in which they appear on the list, as corrective displacements between their predicted and actual locations. Quantizing vertex coordinates to 12 bits and predicting each vertex as a linear combinations of its previously encoded neighbors leads to short displacements, for which entropy codes drop the total vertex location storage for heavily sampled typical meshes below 16V bits.

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Implant Sprays: Compression of Progressive Tetrahedral Mesh Connectivity

1999 , Pajarola, Renato B. , Rossignac, Jarek , Szymczak, Andrzej

Irregular tetrahedral meshes, which are popular in many engineering and scientific applications, often contain a large number of vertices. A mesh of V vertices and T tetrahedra requires 48-V bits or less to store the vertex coordinates, 4-T-log₂(V) bits to store the tetrahedra-vertex incidence relations, also called connectivity information, and k-V bits to store the k-bit value samples associated with the vertices. Given that T is 5 to 7 times larger than V and that V often exceeds 32², the storage space required for the connectivity is larger than 300-V bits and thus dominates the overall storage cost. Our "implants spray" compression approach introduced in this paper reduces this cost to about 30-V bits or less - a 10:1 compression ratio. Furthermore, implant spray supports the progressive refinement of a crude model through a series of vertex-splits operations.

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