Jaroslaw R. Rossignac

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 storage required by previously reported schemes, most of which can guarantee only an O(t log(t)) storage cost for the incidence graph of a mesh of t triangles. Edgebreaker requires at most 2t bits for any mesh homeomorphic to a sphere and supports fully general meshes by using additional storage per handle and hole. For large meshes, entropy coding yields less than 1.5 bits per triangle. Edgebreakers compression and decompression processes perform identical traversals 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 Edgebreakers 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 need not be encoded. The paper also offers a comparative survey of the rapidly growing field of geometric compression.

Compressed progressive meshes

Most systems that support visual interaction with 3D models use shape representations based on triangle meshes. The size of these representations imposes limits on applications for which 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 significant overhead when the full resolution model must be downloaded. The CPM (compressed progressive meshes) approach proposed here eliminates this overhead. It uses a new technique, which refines the topology of the mesh in batches, which each increase the number of vertices by up to 50 percent. Less than an amortized total of 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 percent more compact than previously reported progressive geometry compression techniques.

Offsetting operations in solid modelling

Abstract The range of operations on solids supported by current geometric modelling systems is very limited. Typically, solids represented in a modeller can be transformed by rigid motions and combined by Boolean operations. This paper introduces another family of transformations, called solid offsetting, which map solids into solids. Offset solids are expanded or contracted versions of an original object. Offsetting operations are potentially useful for tolerance analysis, clearance testing, design-rule checking in VLSI, modelling of etching and coating processes, cutter path generation for numerically-controlled machine tools, collision free path planning for robot motions, and for constant-radius rounding and filleting (‘blending’) of solids. This paper discusses mathematical properties of solid offsetting, associated representations and algorithms, support of offsetting operations in solid modellers, and applications. Results of an experimental implementation are presented.

Dynapack: space-time compression of the 3D animations of triangle meshes with fixed connectivity

Dynapack exploits space-time coherence to compress the consecutive frames of the 3D animations of triangle meshes of constant connectivity. Instead of compressing each frame independently (space-only compression) or compressing the trajectory of each vertex independently (time-only compression), we predict the position of each vertex v of frame f from three of its neighbors in frame f and from the positions of v and of these neighbors in the previous frame (space-time compression). We introduce here two extrapolating spacetime predictors: the ELP extension of the Lorenzo predictor, developed originally for compressing regularly sampled 4D data sets, and the Replica predictor. ELP may be computed using only additions and subtractions of points and is a perfect predictor for portions of the animation undergoing pure translations. The Replica predictor is slightly more expensive to compute, but is a perfect predictor for arbitrary combinations of translations, rotations, and uniform scaling. For the typical 3D animations that we have compressed, the corrections between the actual and predicted value of the vertex coordinates may be compressed using entropy coding down to an average ranging between 1.37 and 2.91 bits, when the quantization used ranges between 7 and 13 bits. In comparison, space-only compression yields a range of 1.90 to 7.19 bits per coordinate and time-only compressions yields a range of 1.77 to 6.91 bits per coordinate. The implementation of the Dynapack compression and decompression is trivial and extremely fast. It perform a sweep through the animation, only accessing two consecutive frames at a time. Therefore, it is particularly well suited for realtime and out-of-core compression, and for streaming decompression.

Depth-Buffering Display Techniques for Constructive Solid Geometry

Solid modelers based on constructive solid geometry (CSG) typically generate shaded displays directly from CSG by using ray-casting techniques, which do not require informatin on the faces, edges, and vertices that bound a solid. This article describes an alternative-a simple new algorithm based on a depth-buffering or z-buffering approach. The z-buffer display algorithm operates directly on CSG, does not require explicit boundary data, and is easier to implement than ray casting. Ray-casting and z-beffering algorithms have comparable performances, but z-buffering is often faster for objects with complex surfaces, because it avoids expensive curve/surface intersection calculations. Because of their simplicity, depth-buffering algorithms for CSG are well-suited to hardware implementations, and may lead to machines simpler than those now being built for ray casting.

FlowFixer: using BFECC for fluid simulation

Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation by using the level set method. We show that it can be applied to reduce dissipation and diffusion encountered in various advection steps in uid simulation such as velocity, smoke density and image advections. BFECC can be implemented easily on top of the r st order upwinding or semi-Lagrangian integration of advection equations, while providing second order accuracy both in space and time. When applied to level set evolution, BFECC reduces volume loss signi cantly . We combine these techniques with variable density projection and show that they yield a realistic animations of two-phase ows. We demonstrate the bene ts of this approach on the image advection and on the simulation of smoke, of bubbles in water, and of a highly dynamic interaction between water, a solid, and air.

Issues on feature-based editing and interrogation of solid models

Abstract Operations that create additive or subtractive volume features, such as bosses or slots, simplify the computer aided design of mechanical parts. Surface features, whether extracted automatically or selected interactively, group functionally related boundary elements, and thus provide an expedient interface between CAD systems and analysis or manufacturing applications. Despite much progress in CAD, design remains an iterative process and involves error-prone modifications of previous solutions. Features should in principle offer a high level vocabulary for characterizing errors and for specifying how they should be corrected. This paper points out the semantic ambiguities of simplistic feature-based commands for editing models. It recommends procedural models for editing volume features, and corrective volumes for editing surface features. It shows how space decomposition techniques and CSG expressions based on active zones reduce the cost of executing an editing command. Error detection may be automated by supporting intentional features, which correspond to the desired characteristics of the model, and by endowing them with domain dependent validity criteria expressed in terms of associated geometric elements. The paper demonstrates that validity may be tested by simply interrogating a mixed-dimensional geometric structure which is used to represent not only the model, but also the interactions between the geometric elements associated with intentional features.

Active zones in CSG for accelerating boundary evaluation, redundancy elimination, interference detection, and shading algorithms

Solids defined by Boolean combinations of solid primitives may be represented in constructive solid geometry (CSG) as binary trees. Most CSG-based algorithms (e.g., for boundary evaluation, graphic shading, interference detection) do various forms of set-membership classification by traversing the tree associated with the solid. These algorithms usually generate intermediate results that do not contribute to the final result, and hence may be regarded as redundant and a source of inefficiency. To reduce such inefficiencies, we associate with each primitive A in a tree S an active zone Z that represents the region of space where changes to A affect the solid represented by S, and we use a representation of Z instead of S for set-membership classification. In the paper we develop a mathematical theory of active zones, prove that they correspond to the intersection of certain nodes of the original trees, and show how they lead to efficient new algorithms for boundary evaluation, for detecting and eliminating redundant nodes in CSG trees, for interference (null-set) detection, and for graphic shading.

Constraints in constructive solid geometry

The success of solid modelling in industrial design depends on facilities for specifying and editing parameterized models of solids through user-friendly interaction with a graphical front-end. Systems based on a dual representation, which combines Constructive Solid Geometry (CSG) and Boundary representation (BRep), seem most suitable for modelling mechanical parts. Typically they accept a CSG-compatible input (Boolean combinations of solid primitives) and offer facilities for parameterizing and editing part definitions. The user need not specify the topology of the boundary, but often has to solve three-dimensional trigonometric problems to compute the parameters of rigid motions that specify the positions of primitive solids. A front-end that automatically converts graphical input into rigid motions may be easily combined with boundary-oriented input, but its integration in dual systems usually complicates the editing process and limits the possibilities of parameterizing solid definitions. This report proposes a solution based on three main ideas: (1) enhance the semantics of CSG representations with rigid motions that operate on arbitrary collections of sub-solids regardless of their position in the CSG tree, (2) store rigid motions in terms of unevaluated constraints on graphically selected boundary features, (3) evaluate constraints independently, one at a time, in user-specified order. The third idea offers an alternative to known approaches, which convert all constraints into a large system of simultaneous equations to be solved by iterative numerical methods. The resulting front-end is inadequate for solving problems where multiple constraints must be met simultaneously, but provides a powerful tool for specifying and interactively editing parameterized models of mechanical parts and mechanisms. The order in which constraints are evaluated may also be used as a language for specifying the sequence of assembly and set-up operations. An implementation under way is based on the interpreter of a new object oriented programming language, enhanced with geometric classes. Constraint evaluation results in the activation of methods which compute rigid motions from surface information. The set of available methods may be extended by the users, and methods may be integrated in higher level functions whose algorithmic nature simplifies the treatment of degenerate cases. Graphic interaction is provided through a geometrical engine which lets the user manipulate shaded images produced efficiently from the CSG representation of solid models.

Advections with Significantly Reduced Dissipation and Diffusion

Back and forth error compensation and correction (BFECC) was recently developed for interface computation using a level set method. We show that BFECC can be applied to reduce dissipation and diffusion encountered in a variety of advection steps, such as velocity, smoke density, and image advections on uniform and adaptive grids and on a triangulated surface. BFECC can be implemented trivially as a small modification of the first-order upwind or semi-Lagrangian integration of advection equations. It provides second-order accuracy in both space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We demonstrate the benefits of this approach on image advection and on the simulation of smoke, bubbles in water, and the highly dynamic interaction between water, a solid, and air. We also apply BFECC to dye advection to visualize vector fields