Optimal Bit Allocation in 3D Compression

To use 3D models on the Internet or in other bandwidth-limited 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

Compressed Progressive Meshes

Most 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. Multi-resolution formats provide quick access to a crude model and then refine it progressively. Unfortunately, compared to the best non-progressive 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

Solid Modeling

To appear in the Encyclopedia of Electrical and Electronics Engineering, Ed. J. Webster, John Wiley & Sons, 1999.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 discrete-part manufacturing industries, where precise models of parts and assemblies are created using solid modeling software or more general computer-aided 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 user-interface (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 three-dimensional (3D) graphics techniques, whereas research in 3D graphics focuses on fast or photo-realistic rendering of complex scenes, often composed of solid models, and on realistic or artistic animations of non-rigid objects. A similar symbiotic relation with computer vision is regaining popularity, as many research efforts in vision are model-based 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

OrthoMap: Homeomorphism-guaranteeing normal-projection map between surfaces

Consider two (n—1)-dimensional manifolds, S and Sʹ in Rn. We say that they are projection-homeomorphic when the closest projection of each one onto the other is a homeomorphism. We give tight conditions under which S and Sʹ are projection-homeomorphic. These conditions involve the local feature size for S and for Sʹ and the Hausdorff distance between them. Our results hold for arbitrary n

Edge-Sharpener: Recovering Sharp Features in Triangulations of Non-adaptively Re-meshed Surfaces

3D scanners, iso-surface extraction procedures, and several recent geometric compression schemes sample surfaces of 3D shapes in a regular fashion, without any attempt to align the samples with the sharp edges and corners of the original shape. Consequently, the interpolating triangle meshes chamfer these sharp features and thus exhibit significant errors. The new Edge-Sharpener filter introduced here identifies the chamfer edges and subdivides them and their incident triangles by inserting new vertices and by forcing these vertices to lie on intersections of planes that locally approximate the smooth surfaces that meet at these sharp features. This post-processing significantly reduces the error produced by the initial sampling process. For example, we have observed that the L2 error introduced by the SwingWrapper remeshing-based compressor can be reduced down to a fifth by executing Edge-Sharpener after decompression, with no additional informatio

ITR/PE+SY digital clay for shape input and display

Issued as final reportNational Science Foundatio

Design/analysis/transmission/simulation of shapes and animations

Presented on August 28, 2009 at 2:00 pm in room 2443 of the Klaus Advanced Computing Building on the Georgia Tech campus.Jarek Rossignac (IC, http://www.gvu.gatech.edu/~jarek/) will present an overview of his recent research activities (with collaborators and students) and open challenges in shape and animation processing. These include: - SOT: Compact representation of tetrahedral meshes - J-splines: C^4 subdivision curves, surfaces, and animation - SAM: Steady interpolating affine motion - OCTOR: Exceptions in steady patterns - Pearling: Realtime segmentation of tubular structures in images and 3D medical scans - Surgem: Heart surgery planning and optimization based on blood flow simulation - APL: Aquatic Propulsion Lab, tools for designing and simulating swimming strategies - Ball map: Tangent-ball correspondence and compatibility between pairs of shapes - Ball-morph: Interpolation and applications to entertainment and medical surface reconstruction

Structured Topological Complexes: A Feature-Based API for Non-Manifold Topologies

Much of recent research in solid modeling was focused on the extension of boundary representation schemes to non-manifold models and, more recently, on the definition of high-level operators for creating such models. We propose here a very simple, yet general formalism, which subsumes and simplifies many previous approaches. Our Structural Topological Complexes (STC) extend our previous work on Selective Geometric Complexes (SGC) and on Constructive Non-Regularized Geometry (CNRG) and provide the foundations for a new generation of representations--both constructive and evaluated--and of APIs that are independent of the particular geometric domain and even of the particular approximation scheme for geometric primitives

Optimized Blist Form (OBF)

Tech Report GIT-GVU-07-10 (original number: GIT-GVU-06-18) revised on May 23, 2007.Any Boolean expressions may be converted into positive-form, which has only union and intersection operators. Let E be a positive-form expression with n literals. Assume that the truth-values of the literals are read one at a time. The numbers s(n) of steps (operations) and b(n) of working memory bits (footprint) needed to evaluate E depend on E and on the evaluation technique. A recursive evaluation performs s(n)=n–1 steps but requires b(n)=log(n)+1 bits. Evaluating the disjunctive form of E uses only b(n)=2 bits, but may lead to an exponential growth of s(n). We propose a new Optimized Blist Form (OBF) that requires only s(n)=n steps and b(n)=⌈log2j⌉ bits, where j=⌈log2(2n/3+2)⌉. We provide a simple and linear cost algorithm for converting positive-form expressions to their OBF. We discuss three applications: (1) Direct CSG rendering, where a candidate surfel stored at a pixel is classified against an arbitrarily complex Boolean expression using a footprint of only 6 stencil bits; (2) the new Logic Matrix (LM), which evaluates any positive form logical expression of n literals in a single cycle and uses a matrix of at most n×j wire/line connections; and (3) the new Logic Pipe (LP), which uses n gates that are connected by a pipe of ⌈log2j⌉ lines and when receiving a staggered stream of input vectors produces a value of a logical expression at each cycle

Edgebreaker: Connectivity Compression for Triangle Meshes

Edgebreaker is a simple scheme for compressing the triangle/vertex incidence graphs (sometimes called connectivity or topology) of three-dimensional triangle meshes. Edgebreaker improves upon the worst case storage required by previously reported schemes, most of which require O(nlogn) bits to store the incidence graph of a mesh of n triangles. Edgebreaker requires only 2n bits or less for simple meshes and can also support fully general meshes by using additional storage per handle and hole. Edgebreaker's compression and decompression processes perform the same traversal of the mesh from one triangle to an adjacent one. At each stage, compression produces an op-code describing the topological relation between the current triangle and the boundary of the remaining part of the mesh. Decompression uses these op-codes to reconstruct the entire incidence graph. Because Edgebreaker's compression and decompression are independent of the vertex locations, they may be combined with a variety of vertex-compressing techniques that exploit topological information about the mesh to better estimate vertex locations. Edgebreaker may be used to compress the connectivity of an entire mesh bounding a 3D polyhedron or the connectivity of a triangulated surface patch whose boundary needs not be encoded. Its superior compression capabilities, the simplicity of its implementation, and its versatility make Edgebreaker particularly suitable for the emerging 3D data exchange standards for interactive graphic applications. The paper also offers a comparative survey of the rapidly growing field of geometric compression