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ItemCompressed Progressive Meshes(Georgia Institute of Technology, 1999) Pajarola, Renato B. ; Rossignac, JarekMost systems that support the visual interaction with 3D models use shape representations based on triangle meshes. The size of these representations imposes limits on applications, where complex 3D models must be accessed remotely. Techniques for simplifying and compressing 3D models reduce the transmission time. Multiresolution formats provide quick access to a crude model and then refine it progressively. Unfortunately, compared to the best nonprogressive compression methods, previously proposed progressive refinement techniques impose a signitifant overhead when the full resolution model must be downloaded. The CPM (Compressed Progressive Meshes) appreach proposed here eliminates this overhead. It uses a new "patching" technique, which refines the topology of the mesh in batches, which each increase the number of vertices by up to 50%. Less than 4 bits per triangle encode where and how the topological refinements should be applied. We estimate the position of new vertices from the positions of their topological neighbors in the less refined mesh using a new estimator that leads to representations of vertex coordinates that are 50% more compact than previously reported progressive geometry compression techniques.

ItemOptimal Bit Allocation in 3D Compression(Georgia Institute of Technology, 1999) King, Davis ; Rossignac, JarekTo use 3D models on the Internet or in other bandwidthlimited applications, it is often necessary to compress their triangle mesh representations. We consider the problem of balancing two forms of lossy mesh compression: reduction of the number of vertices by simplification, and reduction of the number of bits of resolution used per vertex coordinate via quantization. Let A be a triangle mesh approximation for an original model O. Suppose that A has V vertices, each represented using B bits per coordinate. Given a file size F for A, what are the optimal values of B and V? Given a desired error level E, what are estimates of B and V, and how many total bits are needed? We develop answers to these questions by using a shape complexity measure K that allows us to express the optimal value of B for a general model in terms of V and K alone. We give formulas linking B, V, F, E and K, and we provide a simple algorithm for estimating the optimal B and V for an existing triangle mesh with a given file size F.

ItemGuaranteed 3.67V Bit Encoding of Planar Triangle Graphs(Georgia Institute of Technology, 1999) King, Davis ; Rossignac, JarekWe 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 Rossignac's 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.

ItemConnectivity Compression for Irregular Quadrilateral Meshes(Georgia Institute of Technology, 1999) King, Davis ; Rossignac, Jarek ; Szymczak, AndrzejApplications 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 worstcase cost of 3 bits per vertex (or 2.75 bits per vertex for meshes without valencetwo vertices) and entropycoding 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.

ItemWrap&zip: Linear decoding of planar triangle graphs(Georgia Institute of Technology, 1999) Rossignac, Jarek ; Szymczak, AndrzejThe 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 nonlinear worst case storage costs. As in Edgebreaker, we compress the mesh using a spiraling trianglespanning tree and generate the same sequence of letters. Edgebreaker's decompression uses a lookahead 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 lookahead and thus exhibits a linear asymptotic time complexity. Wrap&zip converts the string into the corresponding trianglespanning 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 trianglespanning tree, this compression approach may be used to encode the connectivity (incidence of labeled graphs) of threedimensional 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.

ItemImplant Sprays: Compression of Progressive Tetrahedral Mesh Connectivity(Georgia Institute of Technology, 1999) Pajarola, Renato B. ; Rossignac, Jarek ; Szymczak, AndrzejIrregular 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 48V bits or less to store the vertex coordinates, 4Tlog₂(V) bits to store the tetrahedravertex incidence relations, also called connectivity information, and kV bits to store the kbit 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 300V bits and thus dominates the overall storage cost. Our "implants spray" compression approach introduced in this paper reduces this cost to about 30V bits or less  a 10:1 compression ratio. Furthermore, implant spray supports the progressive refinement of a crude model through a series of vertexsplits operations.

ItemConnectivity Compression for Irregular Quadrilateral Meshes(Georgia Institute of Technology, 1999) King, Davis ; Szymczak, Andrzej ; Rossignac, JarekMany 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 0.261.7 bits/quad, a 2545% 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.

ItemMatchmaker: Manifold Breps for Nonmanifold rsets(Georgia Institute of Technology, 1999) Rossignac, Jarek ; Cardoze, David Enrique FabregaMany solid modeling construction techniques produce nonmanifold rsets (solids). With each nonmanifold model N we can associate a family of manifold solid models that are infinitely close to N in the geometric sense. For polyhedral solids, each nonmanifold edge of N with 2k incident faces will be replicated k times in any manifold model M of that family. Furthermore, some nonmanifold vertices of N must also be replicated in M, possibly several times. M can be obtained by defining, in N, a single adjacent face TA(E,F) for each pair (E,F) that combines an edge E and an incident face F. The adjacency relation satisfies TA(E,TA(E,F))=F. The choice of the map A defines which vertices of N must be replicated in M and how many times. The resulting manifold representation of a nonmanifold solid may be encoded using simpler and more compact datastructures, especially for triangulated model, and leads to simpler and more efficient algorithms, when it is used instead of a nonmanifold representation for a variety of tasks, such as simplification, compression, interference detection or rendering. Most choices of the map A lead to invalid (selfintersecting) boundaries and to unnecessary vertex replications for M. We propose an efficient algorithm, called Matchmaker, which computes a map A, such that there exists an infinitely small perturbation of the vertices and edges of M that produces a valid (non selfintersecting) boundary of a manifold solid. Furthermore, our approach avoids most unnecessary vertex replications.

ItemBlist: A Boolean List Formulation of CSG Trees(Georgia Institute of Technology, 1999) Rossignac, JarekSet membership classification algorithms visit nodes of a CSG tree through a recursive divideandconquer process, which stores intermediate results in a stack, whose depth equals the height, H, of the tree. During this process, the candidate sets is usually subdivided into uniform cells, whose interior is disjoint from primitives' boundaries. Cells inside the CSG object are identified by combining the binary results of classifying them against the primitives. In parallel systems, which allocate a different process to each leaf of the tree, and in algorithms that classify large collections of regularly spaced candidate sets (points, pixels, voxels, rays, or crosssections) against the primitives using forward differences, a separate stack is associated with each candidate or cell. Our new representation for CSG trees, called Blist, distributes the merging operation to the primitives and reduces the storage requirement for each cell to log(H+1) bits. Blist can represent any Boolean expression as a list of primitives, each containing a reference to the primitive's description (type, parameter, transformation), a sign, a stamp, and a name. During set membership classification, a label is attached to each cell and passed to the successive primitives in the Blist. When the name written on the label matches the primitive's name, the cell is classified against the primitive. If the result matches the primitive's sign, the name stored in the primitive's stamp is put on the label  if not, a zero name is used. The elimination of the intermediate CSG nodes and of the recursive merging operations make the Blist architecture particularly well suited for parallel hardware configurations. We provide a simple algorithm for converting CSG expressions to Blists. It uses rotations on the positive form of the tree to reduce the number of bits needed for each label.

ItemSolid Modeling(Georgia Institute of Technology, 1999) Rossignac, Jarek ; Requicha, Aristides A. G.A solid model is a digital representation of the geometry of an existing or envisioned physical object. Solid models are used in many industries, from entertainment to health care. They play a major role in the discretepart manufacturing industries, where precise models of parts and assemblies are created using solid modeling software or more general computeraided design (CAD) systems. Solid modeling is an interdisciplinary field that involves a growing number of areas. Its objectives evolved from a deep understanding of the practices and requirements of the targeted application domains. Its formulation and rigor are based on mathematical foundations derived from general and algebraic topology, and from Euclidean, differential, and algebraic geometry. The computational aspects of solid modeling deal with efficient data structures and algorithms, and benefit from recent developments in the field of computational geometry. Efficient processing is essential, because the complexity of industrial models is growing faster than the performance of commercial workstations. Techniques for modeling and analyzing surfaces and for computing their intersections are important in solid modeling. This area of research, sometimes called computer aided geometric design, has strong ties with numerical analysis and differential geometry. Graphic userinterface (GUI) techniques also play a crucial role in solid modeling, since they determine the overall usability of the modeler and impace the user's productivity. There have always been strong symbiotic links and overlaps between the solid modeling community and the computer graphics community. Solid modeling interfaces are based on efficient threedimensional (3D) graphics techniques, whereas research in 3D graphics focuses on fast or photorealistic rendering of complex scenes, often composed of solid models, and on realistic or artistic animations of nonrigid objects. A similar symbiotic relation with computer vision is regaining popularity, as many research efforts in vision are modelbased and attempt to extract 3D models from images or video sequences of existing parts or scenes. These efforts are particularly important for solid modeling, because the cost of manually designing solid models of existing objects or scenes far excees the other costs (hardware, software, maintenance, and training) associated with solid modeling. Finally, the growing complexity of solid models and the growing need for collaboration, reusability of design, and interoperability of software require expertise in distributed databases, constraint management systems, optimization techniques, object linking standards, and internet protocols. This report provides a brief overview of the solid modeling field, its fundamental technologies, and some important applications.