The planar COTS map proposed here takes the unit square to a region R bounded by four log-spiral edges. It is Corner-operated (controlled by the four corners of R ) and Tran-similar (it maps translations to similarities). The tiles of the COTS map of a regular pattern are similar to each other. It may facilitate intuitive design and algorithmic optimization of procedural models of complex, possibly multi-resolution, lattices, because it affords constant-cost algorithms for Point-in-Lattice testing and for Total-Area-Calculations. We provide simple, closed-form expressions for evaluating the COTS map and its inverse from the positions of its corners. We conjecture that the COTS map may be useful in a variety of applications in Engineering, Architecture, and Art, and we provide a few illustrative examples of its possibilities. We compare it to related, previously proposed, planar maps and discuss several variations and extensions.

We present Grouper: an all-in-one compact file format, random-access data structure, and streamable representation for large triangle meshes. Similarly to the recently published SQuad representation, Grouper represents the geometry and connectivity of a mesh by grouping vertices and triangles into fixed-size records, most of which store two adjacent triangles and a shared vertex. Unlike SQuad, however, Grouper interleaves geometry with connectivity and uses a new connectivity representation to ensure that vertices and triangles can be stored in a coherent order that enables memory-efficient sequential stream processing. We present a linear-time construction algorithm that allows streaming out Grouper meshes using a small memory footprint while preserving the initial ordering of vertices. As a part of this construction, we show how the problem of assigning vertices and triangles to groups reduces to a well-known NP-hard optimization problem, and present a simple yet effective heuristic solution that performs well in practice. Our array-based Grouper representation also doubles as a triangle mesh data structure that allows direct access to vertices and triangles. Storing only about two integer references per triangle--i.e., less than the three vertex references stored with each triangle in a conventional indexed mesh format--Grouper answers both incidence and adjacency queries in amortized constant time. Our compact representation enables data-parallel processing on multicore computers, instant partitioning and fast transmission for distributed processing, as well as efficient out-of-core access. We demonstrate the versatility and performance benefits of Grouper using a suite of example meshes and processing kernels.

Abstract Several design and animation techniques use a one‐dimensional proxy C (a spine curve in 3D) to control the deformation or behavior of a digital model of a 3D shape S. We propose a modification of these “skinning” techniques that ensures local volume preservation, which is important for the physical plausibility of digital simulations. In the proposed “fleshing” techniques, as input, we consider a smooth spine C0, a model S0of a solid that lies “sufficiently close” to C0, and a deformed version C1of C0that is “not overly bent”. (We provide a precise characterization of these restrictions.) As output, we produce a bijective mapping M, that maps any point X of S onto a point M(X) of M(S). M satisfies two properties: (1) The closest projection of X on C0and of M(X) on C1have the same arc length parameter. (2) U and M(U) have the same volume, where U is any subset of S. We provide three different closed form expressions for radial, normal and binormal fleshing and discuss the details of their practical real‐time implementation.

Abstract The SQuad data structure represents the connectivity of a triangle mesh by its “S table” of about 2 rpt (integer references per triangle). Yet it allows for a simple implementation of expected constant‐time, random‐access operators for traversing the mesh, including in‐order traversal of the triangles incident upon a vertex. SQuad is more compact than the Corner Table (CT), which stores 6 rpt, and than the recently proposed SOT, which stores 3 rpt. However, in‐core access is generally faster in CT than in SQuad, and SQuad requires rebuilding the S table if the connectivity is altered. The storage reduction and memory coherence opportunities it offers may help to reduce the frequency of page faults and cache misses when accessing elements of a mesh that does not fit in memory. We provide the details of a simple algorithm that builds the S table and of an optimized implementation of the SQuad operators.

We propose LR ( Laced Ring )---a simple data structure for representing the connectivity of manifold triangle meshes. LR provides the option to store on average either 1.08 references per triangle or 26.2 bits per triangle. Its construction, from an input mesh that supports constant-time adjacency queries, has linear space and time complexity, and involves ordering most vertices along a nearly-Hamiltonian cycle. LR is best suited for applications that process meshes with fixed connectivity, as any changes to the connectivity require the data structure to be rebuilt. We provide an implementation of the set of standard random-access, constant-time operators for traversing a mesh, and show that LR often saves both space and traversal time over competing representations.

We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA). Steady affine motions, for which ARA=0, include translations, rotations, screws, and the golden spiral. To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C , we propose the Steady Affine Morph (SAM), defined as A t ∘ B with A = C ∘ B -1 . A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count, n , the series of uniformly sampled poses, A i/n ∘ B , of a SAM form a regular pattern which may be easily controlled by changing B , C , or n , and where consecutive poses are related by the same affinity A 1/n . Although a real matrix A t does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities A . Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes A t , when it exists, using closed-form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation.

An Expanded Boolean Expression (EBE) does not contain any XOR or EQUAL operators. The occurrence of each variable is a different literal. We provide a linear time algorithm that converts an EBE of n literals into a logically equivalent Ordered Boolean List (OBL) and show how to use the OBL to evaluate the EBE in n steps and O(log log n) space, if the values of the literals are each read once in the order prescribed by the OBL. (An evaluation workspace of 5 bits suffices for all EBEs of up to six billion literals.) The primary application is the SIMD architecture, where the same EBE is evaluated in parallel for different input vectors when rendering solid models on the GPU directly from their Constructive Solid Geometry (CSG) representation. We compare OBL to the Reduced Ordered Binary Decision Diagram (ROBDD) and suggest possible applications of OBL to logic verification and to circuit design.

We define b-compatibility for planar curves and propose three ball morphing techniques between pairs of b-compatible curves. Ball-morphs use the automatic ball-map correspondence, proposed by Chazal et al., from which we derive different vertex trajectories (linear, circular, and parabolic). All three morphs are symmetric, meeting both curves with the same angle, which is a right angle for the circular and parabolic. We provide simple constructions for these ball-morphs and compare them to each other and other simple morphs (linear-interpolation, closest-projection, curvature-interpolation, Laplace-blending, and heat-propagation) using six cost measures (travel-distance, distortion, stretch, local acceleration, average squared mean curvature, and maximum squared mean curvature). The results depend heavily on the input curves. Nevertheless, we found that the linear ball-morph has consistently the shortest travel-distance and the circular ball-morph has the least amount of distortion.

Abstract The generation of inbetween frames that interpolate a given set of key frames is a major component in the production of a 2D feature animation. Our objective is to considerably reduce the cost of the inbetweening phase by offering an intuitive and effective interactive environment that automates inbetweening when possible while allowing the artist to guide, complement, or override the results.Tightinbetweens, which interpolate similar key frames, are particularly time‐consuming and tedious to draw. Therefore, we focus on automating these high‐precision and expensive portions of the process. We have designed a set of user‐guided semi‐automatic techniques that fit well with current practice and minimize the number of required artist‐gestures. We present a novel technique for stroke interpolation from only two keys which combines a stroke motion constructed from logarithmic spiral vertex trajectories with a stroke deformation based on curvature averaging and twisting warps. We discuss our system in the context of a feature animation production environment and evaluate our approach with real production data.

The Corner Table (CT) promoted by Rossignac et al. provides a simple and efficient representation of triangle meshes, storing 6 integer references per triangle (3 vertex references in the V table and 3 references to opposite corners in the O table that accelerate access to adjacent triangles). The Compact Half Face (CHF) proposed by Lage et al. extends CT to tetrahedral meshes, storing 8 references per tetrahedron (4 in the V table and 4 in the O table). We call it the Vertex Opposite Table (VOT) and propose a sorted variation, SVOT, which does not require any additional storage and yet provides, for each vertex, a reference to an incident corner from which an incident tetrahedron may be recovered and the star of the vertex may be traversed at a constant cost per visited element. We use a set of powerful wedge-based operators for querying and traversing the mesh. Finally, inspired by tetrahedral mesh encoding techniques used by Weiler et al. and by Szymczak and Rossignac, we propose our Sorted O Table (SOT) variation, which eliminates the V table completely and hence reduces storage requirements by 50% to only 4 references and 9 bits per tetrahedron, while preserving the vertex-to-incident-corner references and supporting our wedge operators with a linear average cost.

We define b-compatibility for planar curves and propose three ball morphing techniques (b-morphs) between pairs of b-compatible curves. B-morphs use the automatic ball-map correspondence, proposed by Chazal et al. [12], from which they derive vertex trajectories (Linear, Circular, Parabolic). All are symmetric, meeting both curves with the same angle, which is a right angle for the Circular and Parabolic. We provide simple constructions for these b-morphs using the maximal disks in the finite region bounded by the two curves. We compare the b-morphs to each other and to other simple morphs (Linear Interpolation (LI), Closest Projection (CP), Curvature Interpolation (CI), Laplace Blending (LB), Heat Propagation (HP)) using seven measures of quality deficiency (travel distance, distortion, stretch, local acceleration, surface area, average curvature, maximal curvature). We conclude that the ratios of these measures depends heavily on the test case, especially for LI, CI, and LB, which compute correspondence from a uniform geodesic parameterization. Nevertheless, we found that the Linear b-morph has consistently the shortest travel distance and that the Circular b-morph has the least amount of distortion.

Hierarchies of patterns of features, of sub-assemblies, or of CSG sub-expressions are used in architectural and mechanical CAD to eliminate laborious repetitions from the design process. Yet, often the placement, shape, or even existence of a selection of the repeated occurrences in the pattern must be adjusted. The specification of a desired selection of occurrences in a hierarchy of patterns is often tedious (involving repetitive steps) or difficult (requiring interaction with an abstract representation of the hierarchy graph). The OCTOR system introduced here addresses these two drawbacks simultaneously, offering an effective and intuitive solution, which requires only two mouse-clicks to specify any one of a wide range of possible selections. It does not require expanding the graph or storing an explicit list of the selected occurrences and is simple to compute. It is hence well suited for a variety of CAD applications, including CSG, feature-based design, assembly mock-up, and animation. We discuss a novel representation of a selection, a technology that makes it possible to use only two mouse-clicks for each selection, and the persistence of these selections when the hierarchy of patterns is edited.

Abstract We explore the automatic recovery of solids from their binary volumetric discretizations. In particular, we propose an approach, called Pressing, for smoothing isosurfaces extracted from binary volumes while recovering their large planar regions (flats). Pressing yields a surface that is guaranteed to contain the samples of the volume classified as interior and exclude those classified as exterior. It uses global optimization to identify flats and constrained bilaplacian smoothing to eliminate sharp features and high frequencies from the rest of the isosurface. It recovers sharp edges between flat regions and between flat and smooth regions. Hence, the resulting isosurface is usually a very accurate approximation of the original solid. Furthermore, the segmentation of the isosurface into flat and curved faces and the sharp/smooth labelling of their edges may be valuable for shape recognition, simplification, compression and various reverse engineering and manufacturing applications.

Many scientific, imaging, and geospatial applications produce large high-precision scalar fields sampled on a regular grid. Lossless compression of such data is commonly done using predictive coding, in which weighted combinations of previously coded samples known to both encoder and decoder are used to predict subsequent nearby samples. In hierarchical, incremental, or selective transmission, the spatial pattern of the known neighbors is often irregular and varies from one sample to the next, which precludes prediction based on a single stencil and fixed set of weights. To handle such situations and make the best use of available neighboring samples, we propose a local spectral predictor that offers optimal prediction by tailoring the weights to each configuration of known nearby samples. These weights may be precomputed and stored in a small lookup table. We show that predictive coding using our spectral predictor improves compression for various sources of high-precision data. 1.

Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time.

Abstract-To eliminate the need to evaluate the intersection curves in explicit representations of surface cutouts or of trimmed faces in BReps of CSG solids, we advocate using Constructive Solid Trimming (CST). A CST face is the intersection of a surface with a Blist representation of a trimming CSG volume. We propose a new GPU-based CSG rendering algorithm that trims the boundary of each primitive using a Blist of its active zone. This approach is faster than the previously reported Blister approach, eliminates occasional speckles of wrongly colored pixels, and provides additional capabilities: painting on surfaces, rendering semitransparent CSG models, and highlighting selected features in the BReps of CSG models.

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.

We use the abbreviations tet and tri for tetrahedron and triangle. TetStreamer encodes a Delaunay tet mesh in a back-to-front visibility order and streams it from a server to a client (volumetric visualizer). During decompression, the server performs the view-dependent back-to-front sorting of the tets by identifying and deactivating one free tet at a time. A tet is free when all its back faces are on the sheet. The sheet is a tri mesh separating active and inactive tets. It is initialized with the back-facing boundary of the mesh. It is compressed using EdgeBreaker and transmitted first. It is maintained by both the server and the client and advanced towards the viewer passing one free tet at a time. The client receives a compressed bit stream indicating where to attach free tets to the sheet. It renders each free tet and updates the sheet by either flipping a concave edge, removing a concave valence-3 vertex, or inserting a new vertex to split a tri. TetStreamer compresses the connectivity of the whole let mesh to an average of about 1.7 bits per tet. The footprint (in-core memory required by the client) needs only to hold the evolving sheet, which is a small fraction of the storage that would be required by the entire tet-mesh. Hence, TetStreamer permits us to receive, decompress, and visualize or process very large meshes on clients with a small in-core memory. Furthermore, it permits us to use volumetric visualization techniques, which require that the mesh be processed in view-dependent back-to-front order, at no extra memory, performance or transmission cost.

Consider two (n - 1)-dimensional manifolds, S and S' 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 S' are projection-homeomorphic. These conditions involve the local feature size for S and for S' and the Hausdorff distance between them. Our results hold for arbitrary n.

In contrast to machined mechanical parts, the 3D shapes encountered in biomedical or styling applications contain many tubular parts, protrusions, engravings, embossings, folds, and smooth bends. It is difficult to design and edit such features using the parameterized operations or even free-form deformations available in CAD or animation systems. The Bender tool proposed here complements previous solutions by allowing a designer holding a 6 DoF 3D tracker in each hand to control the position and orientation of the ends of a stretchable virtual ribbon, which is used to grab the shape in its vicinity and to deform it in realtime, as the designer continues to move, bend, and twist the ribbon. To ensure realtime performance and intuitive control of the ribbon, we model its centerline as a circular biarc and perform adaptive refinement of the triangle-mesh approximation of the surface. To produce a natural and predictable warp, we use the initial and final shapes of the ribbon to define a one-parameter family of screw-motions. The deformation of a surface point is computed by finding its locally closest projection, or projections, on the biarc and by applying the corresponding screws, weighted by a function that decays with the distance to the projection. The combination of these solutions leads to an easy-to-use and effective tool for the direct manipulation of organic or stylized shapes.

Given a planar set S of arbitrary topology and a radius r, we show how to construct an r-tightening of S, which is a set whose boundary has a radius of curvature everywhere greater than or equal to r and which only differs from S in a morphologically-defined tolerance zone we call the mortar. The mortar consists of the thin or highly curved parts of S, such as corners, gaps, and small connected components, while the boundary of a tightening consists of minimum-length loops through the mortar. Tightenings are defined independently of shape representation, and it may be possible to find them using a variety of algorithms. We describe how to approximately compute tightenings for sets represented as binary images using constrained, level-set curvature flow.

In this paper, we present a packet-loss resilient system for the transmission of progressively compressed three-dimensional (3D) models. It is based on a joint source and channel coding approach that trades off geometry precision for increased error resiliency to optimize the decoded model quality on the client side. We derive a theoretical framework for the overall system by which the channel packet loss behavior and the channel bandwidth can be directly related to the decoded model quality at the receiver. First, the 3D model is progressively compressed into a base mesh and a number of refinement layers. Then, we assign optimal forward error correction code rates to protect these layers according to their importance to the decoded model quality. Experimental results show that with the proposed unequal error protection approach, the decoded model quality degrades more gracefully (compared to either no error protection or equal error protection methods) as the packet-loss rate increases.

In this article, we propose an error-resilient transmission method for progressively compressed 3D models. The proposed method is scalable with respect to both channel bandwidth and channel packet-loss rate. We jointly design source and channel coders using a statistical measure that (i) calculates the number of both source and channel coding bits, and (ii) distributes the channel coding bits among the transmitted refinement levels in order to maximize the expected decoded model quality. In order to keep the total number of bits before and after applying error protection the same, we transmit fewer triangles in the latter case to accommodate the channel coding bits. When the proposed method is used to transmit a typical model over a channel with a 10% packet-loss rate, the distortion (measured using the Hausdorff distance between the original and the decoded models) is reduced by 50% compared to the case when no error protection is applied.

By combining depth peeling with a linear formulation of a Boolean expression called Blist, the Blister algorithm renders an arbitrary CSG model of n primitives in at most k steps, where k is the number of depth-layers in the arrangement of the primitives. Each step starts by rendering each primitive to produce candidate surfels on the next depth-layer. Then, it renders the primitives again, one at a time, to classify the candidate surfels against the primitive and to evaluate the Boolean expression directly on the GPU. Since Blist does not expand the CSG expression into a disjunctive (sum-of-products) form, Blister has O(kn) time complexity. We explain the Blist formulation while providing algorithms for CSG-to-Blist conversion and Blist-based parallel surfel classification. We report real-time performance for nontrivial CSG models. On hardware with an 8-bit stencil buffer, we can render all possible CSG expressions with 3909 primitives.

Various acquisition, analysis, visualization, and compression approaches sample surfaces of 3D shapes in a uniform fashion without any attempt to align the samples with sharp edges or to adapt the sampling density to the surface curvature. Consequently, triangle meshes that interpolate these samples usually chamfer sharp features and exhibit a relatively large error in their vicinity. We present two new filters that improve the quality of these resampled models. EdgeSharpener restores the sharp edges by splitting the chamfer edges and forcing the new vertices to lie on intersections of planes extending the smooth surfaces incident upon these chamfers. Bender refines the resulting triangle mesh using an interpolating subdivision scheme that preserves the sharpness of the recovered sharp edges while bending their polyline approximations into smooth curves. A combined Sharpen&Bend postprocessing significantly reduces the error produced by feature-insensitive sampling processes. For example, we have observed that the mean-squared distortion introduced by the SwingWrapper remeshing-based compressor can often be reduced by 80 percent executing EdgeSharpener alone after decompression. For models with curved regions, this error may be further reduced by an additional 60 percent if we follow the EdgeSharpening phase by Bender.

Abstract The computation of the largest planar region approximating a 3D object is an important problem with wide applications in modeling and rendering. Given a voxelization of the 3D object, we propose an efficient algorithm to solve a discrete version of this problem. The input of the algorithm is the set of grid edges connecting the interior and the exterior of the object (called sticks). Using a voting‐based approach, we compute the plane that slices the largest number of sticks and is orientation‐compatible with these sticks. The robustness and efficiency of our approach rests on the use of two different parameterizations of the planes with suitable properties. The first of these is exact and is used to retrieve precomputed local solutions of the problem. The second one is discrete and is used in a hierarchical voting scheme to compute the global maximum. This problem has diverse applications that range from finding object signatures to generating simplified models. Here we demonstrate the merits of the algorithm for efficiently computing an optimized set of textured impostors for a given polygonal model. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling

Wavemesh is a powerful scheme for 3D triangular mesh processing. In sharp contrast with other approaches using wavelets for mesh compression which apply only to meshes having subdivision connectivity, wavemesh can simplify, approximate, and compress meshes even if they do not respect this constraint. Results clearly indicate that wavemesh outperforms the other approaches in terms of progressive lossless compression. We propose in this paper an improvement for our scheme : higher efficiency for meshes with large subdivision connectivity sets, as shown by experimental results. Also, in some cases, the enhanced wavemesh can even perform better than monoresolution approaches in terms of connectivity compression.

The NSF Digital Clay project is focused on the design, prototyping, integration, and validation of a computer-controlled physical device capable of taking any of a wide range of possible shapes in response to changes in a digital 3D model or to changes in the pressure exercised upon it by human hands. Although it clearly is a natural and unavoidable evolution of 3D graphical user interfaces, its unprecedented capabilities constitute a major leap in technologies and paradigms for 3D display, for 3D input, and for collaborative 3D design. In this paper, we provide an overview of the Digital Clay project and discuss the challenges, design choices, and initial solutions for a new finger sculpting interface designed for the Digital Clay and prototyped using conventional 3D I/O hardware.

Abstract We present a simple method for compressing very large and regularly sampled scalar fields. Our method is particularlyattractive when the entire data set does not fit in memory and when the sampling rate is high relative to thefeature size of the scalar field in all dimensions. Although we report results for and data sets, the proposedapproach may be applied to higher dimensions. The method is based on the new Lorenzo predictor, introducedhere, which estimates the value of the scalar field at each sample from the values at processed neighbors. The predictedvalues 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 beused 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 asmall buffer in core memory, whose size barely exceeds a single ( n −1)‐ dimensional slice of the data. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Compression, scalar fields,out‐of‐core.

Abstract We present a novel Cell‐To‐Cell Visibility (C2CV) algorithm, which given two polyhedra, AandBand a connectedand oriented manifold triangle mesh, S offers a simple, fast and conservative test for detecting when A and B areoccluded from each other by S. Previously disclosed C2CV algorithms either relied on costly occlusion fusion orwere restricted to convex or “apparently convex” occluders, which makes them inappropriate for scenes wherepotential occluders are arbitrary triangulated surfaces, such as the body of a car or a portion of a terrain. Thesimplicity of our C2CV algorithm, named ShieldTester, stems from a new Occlusion Theorem, introduced herewhich permits to establish occlusion by computing the intersection of S with a single ray from a vertex ofAtoa vertex ofB. ShieldTester may be used to establish that pairs of cells in a subdivision of space are hidden fromeach other by a relatively large surface occluder, so that when the viewer is in one cell, the objects in the othercell need not be displayed. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Computational Geometryand Object Modeling: Occlussion Culling, Visibility Test, Triangle Meshes

We focus on the lossy compression of manifold triangle meshes. Our SwingWrapper approach partitions the surface of an original mesh M into simply connected regions, called triangloids . From these, we generate a new mesh M ′ . Each triangle of M ′ is an approximation of a triangloid of M . By construction, the connectivity of M ′ is fairly regular and can be compressed to less than a bit per triangle using EdgeBreaker or one of the other recently developed schemes. The locations of the vertices of M ′ are compactly encoded with our new prediction technique, which uses a single correction parameter per vertex. SwingWrapper strives to reach a user-defined output file size rather than to guarantee a given error bound. For a variety of popular models, a rate of 0.4 bits/triangle yields an L 2 distortion of about 0.01% of the bounding box diagonal. The proposed solution may also be used to encode crude meshes for adaptive transmission or for controlling subdivision surfaces.

A free-form deformation that warps a surface or solid may be specified in terms of one or several point-displacement constraints that must be interpolated by the deformation. The Twister approach introduced here, adds the capability to impose an orientation change, adding three rotational constraints, at each displaced point. Furthermore, it solves for a space warp that simultaneously interpolates two sets of such displacement and orientation constraints. With a 6 DoF magnetic tracker in each hand, the user may grab two points on or near the surface of an object and simultaneously drag them to new locations while rotating the trackers to tilt, bend, or twist the shape near the displaced points. Using a new formalism based on a weighted average of screw displacements , Twister computes in realtime a smooth deformation, whose effect decays with distance from the grabbed points, simultaneously interpolating the 12 constraints. It is continuously applied to the shape, providing realtime graphic feedback. The two-hand interface and the resulting deformation are intuitive and hence offer an effective direct manipulation tool for creating or modifying 3D shapes.

In this paper, we present a packet-loss resilient 3-D graphics transmission system that is scalable with respect to both channel bandwidth and channel error characteristics. The algorithm trades off source coding efficiency for increased bit-stream error resilience to optimize the decoded mesh quality on the client side. It uses the Compressed Progressive Mesh (CPM) algorithm to generate a hierarchical bit-stream representing different levels of details (LODs). We assign optimal forward error correction (FEC) code rates to protect different parts of the bit-stream differently. These optimal FEC code rates are determined theoretically via a distortion function that accounts for: the channel packet loss rate, the nature of the encoded 3-D mesh and the error protection bit-budget. We present experimental results, which show that with our unequal error protection (UEP) optimal approach, the decoded mesh quality degrades more gracefully (compared to either no error protection (NEP) or equal error protection (EEP) methods) as the packet loss rate increases.

In this paper, we present an unequal error protection method for packet-loss resilient transmission of progressively compressed 3D meshes. The proposed method is based on a source and channel coding approach where we set up a theoretical framework for the overall system by which the channel packet loss behavior and the channel bandwidth can be directly related to the decoded mesh quality at the receiver. In particular, we develop a statistical distortion measure and optimize it to compute the best combination of (i) the number of triangles to transmit, (ii) the total number of channel coding bits, and (iii) the distribution of these error-protection bits among the transmitted layers in order to maximize the expected decoded mesh quality at the receiver. The proposed method differs from the earlier approaches in two major aspects: (i) determination of the number of channel coding bits (C) and (ii) the approach of reducing the source rate in order to accommodate for channel coding bits. When the proposed method is used to transmit a typical 3D mesh over a channel with a 10% packet loss rate, the distortion (measured using the Hausdorff distance between the original and the decoded overly sampled meshes) is reduced by 50% compared to the case when no error protection is applied.

We propose a protocol for efficient streaming of 3-D animations over lossy channels. In order to improve the expected quality on the client's side, we first transmit a crude model of the 3-D mesh, of its texture, and of its immediate evolution. We endow this data with significant error-protection against transmission error. Then we transmit a series of upgrades that refine the accuracy of the model and/or of the animation. These are encoded with lower levels of error protection that is proportional to their impact on the quality of the upgraded animation. We propose the following types of upgrade chunks: selection of a subset of vertices, adjustments of the accuracy of the positions of the selected vertices, connectivity refinements, motion adjustments of the selected subset of vertices, adjustments of the accuracy of the texture coordinates of the selected vertices, and upgrade of the quality of the texture. Finally, given the allocated bit-budget, we determine the optimal number of both source and channel coding bits assigned for each chunk to maximize the animation quality on the client's side. The optimization takes into account both the bandwidth and the error characteristics of the channel.

Digitized cell animations are typically composed of frames, which contain a small number of regions, which each contain pixels of the same color and exhibit a significant level of shape coherence through time. To exploit this coherence, we treat the stack of frames as a 3D volume and represent the evolution of each region by the bounding surface of the 3D volume V that it sweeps out. To reduce transmission costs, we triangulate and simplify the bounding surface and then encode it using the Edgebreaker compression scheme. To restore a close approximation of the original animation, the client player decompresses the surface and produces the successive frames by intersecting V with constant-time planes. The intersection is generated in real-time with standard graphics hardware through an improved capping (i.e. solid clipping) technique, which correctly handles overlapping facets. We have tested this approach on real and synthetic black and white animations and report compression ratios that improve upon those produced using the MPEG, MRLE, and GZIP compression standards for an equivalent quality result.

Edgebreaker is a simple technique for compressing three-dimensional triangle meshes. We introduce here a new formulation of Edgebreaker, which leads to a very simple implementation. We describe it in terms of a simple data structure, which we call the Corner Table. It represents the connectivity of any manifold mesh as two tables, V and O, such that for a corner c, which is the association of a triangle with a vertex, V[c] is an integer reference to the vertex of c and O[c] is an integer reference to the opposite corner. For meshes that are homeomorphic to a sphere, Edgebreaker encodes these two tables with less than 2 bits per triangle. It compresses vertex locations using Touma and Gottsman's parallelogram predictor. We also present a new decompression, inspired by the Wrap&Zip decompression technique developed in collaboration with Andrzej Szymczak. We call it Zip&Wrap, because it works in the inverse direction from Wrap&Zip and zips cracks in the reconstructed mesh sooner. The detailed source code for the compression and the decompression algorithms fits on a single page. A further improvement of the codebook of Edgebreaker, developed with D. King, guarantees no more than 1.73 bits per triangle for the connectivity. Entropy encoding reduces this cost in practice to less than a bit per triangle when the mesh is large. Through minor modifications, the Edgebreaker algorithm has been adapted to manifold meshes with holes and handles, to non-triangle meshes, and to non-manifold meshes. A Corner-Table implementation of these is described elsewhere.

Most visibility culling algorithms require convexity of occluders. Occluder synthesis algorithms attempt to construct large convex occluders inside bulky non-convex sets. Occluder fusion algorithms generate convex occluders that are contained in the umbra cast by a group of objects given an area light. In this paper we prove that convexity requirements can be shifted from the occluders to their umbra with no loss of efficiency, and use this property to show how some special non-planar, non-convex closed polylines that we call “hoops” can be used to compute occlusion efficiently for objects that have no large interior convex sets and were thus rejected by previous approaches.

An ideal triangle mesh compression technology would simultaneously support the following objectives: (1) progressive refinements of the received mesh during decompression, (2) nearly optimal compression ratios for both geometry and connectivity, and (3) in-line, real-time decompression algorithms for hardware or software implementations. Because these three objectives impose contradictory constraints, previously reported efforts have focused primarily on one (sometimes two) of these objectives. The SQUEEZE technique introduced in this paper addresses all three constraints simultaneously, and attempts to provide the best possible compromise. For a mesh of T triangles, SQUEEZE compresses the connectivity to 3.7T bits, which is competitive with the best progressive compression techniques reported so far. The geometric prediction error encoding technique introduced in this paper leads to a geometry compression that is improved by 20% over that of previous schemes. Our initial implementation on a 300-MHz CPU achieved a decompression rate of up to 46,000 triangles per second. SQUEEZE downloads a model through a number of successive refinement stages, providing the benefit of progressivity.

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.

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/spl middot/T/spl middot/log/sub 2/(V) bits to store the tetrahedra-vertex incidence relations, also called connectivity information, and kV 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 32/sup 3/, 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 the 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.

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. Edgebreaker's 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 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 need not be encoded. The paper also offers a comparative survey of the rapidly growing field of geometric compression.

Interactive 3D visualization and Internet-based access to information are already common. Their combination is-or soon will be-the primary vehicle for accessing remote databases in fundamental areas of manufacturing, architecture, petroleum, urban planning, tourism, defense, medicine, electronic commerce, and entertainment. Unfortunately, whether based on precise 3D geometry or involving combinations of shapes and images, the complexity of 3D graphic models of airplanes, cities, or virtual stores significantly exceeds the limits of what can be quickly downloaded over popular connections and what can be rendered on personal workstations during interactive exploration. The author reviews recent progress in the compression and simplification of 3D models and in the progressive transmission of these models for interactive graphic exploration, which may combine traditional 3D graphics with image-based rendering. The author proposes an architecture for a 3D server capable of supporting a large number of independent client-users accessing interactively various subsets of a possibly distributed database of complex 3D models.

The virtual-reality modeling language (VRML) is rapidly becoming the standard file format for transmitting three-dimensional (3-D) virtual worlds across the Internet. Static and dynamic descriptions of 3-D objects, multimedia content, and a variety of hyperlinks can be represented in VRML files. Both VRML browsers and authoring tools for the creations of VRML files are widely available for several different platforms. In this paper, we describe the topologically assisted geometric compression technology included in our proposal for the VRML compressed binary format. This technology produces significant reduction of file sizes and, subsequently, of the time required for transmission of such filed across the Internet. Compression ratios of 50:1 or more are achieved for large models. The proposal also includes a binary encoding to create compact, rapidly parsable binary VRML files. The proposal is currently being evaluated by the Compressed Binary Format Working Group of the VRML consortium as a possible extension of the VRML standard. In the topologically assisted compression scheme, a polyhedron is represented using two interlocking trees: a spanning tree of vertices and a spanning tree of triangles. The connectivity information represented in other compact schemes, such as triangular strips and generalized triangular meshes, can be directly derived from this representation. Connectivity information for large models is compressed with storage requirements approaching one bit per triangle. A variable-length, optionally lossy compression technique is used for vertex positions, normals, colors, and texture coordinates. The format supports all VRML property binding conventions.

The abundance and importance of complex 3-D data bases in major industry segments, the affordability of interactive 3-D rendering for office and consumer use, and the exploitation of the Internet to distribute and share 3-D data have intensified the need for an effective 3-D geometric compression technique that would significantly reduce the time required to transmit 3-D models over digital communication channels, and the amount of memory or disk space required to store the models. Because the prevalent representation of 3-D models for graphics purposes is polyhedral and because polyhedral models are in general triangulated for rendering, this article introduces a new compressed representation for complex triangulated models and simple, yet efficient, compression and decompression algorithms. In this scheme, vertex positions are quantized within the desired accuracy, a vertex spanning tree is used to predict the position of each vertex from 2,3, or 4 of its ancestors in the tree, and the correction vectors are entropy encoded. Properties, such as normals, colors, and texture coordinates, are compressed in a similar manner. The connectivity is encoded with no loss of information to an average of less than two bits per triangle. The vertex spanning tree and a small set of jump edges are used to split the model into a simple polygon. A triangle spanning tree and a sequence of marching bits are used to encode the triangulation of the polygon. Our approach improves on Michael Deering's pioneering results by exploiting the geometric coherence of several ancestors in the vertex spanning tree, preserving the connectivity with no loss of information, avoiding vertex repetitions, and using about three fewer bits for the connectivity. However, since decompression requires random access to all vertices, this method must be modified for hardware rendering with limited onboard memory. Finally, we demonstrate implementation results for a variety of VRML models with up to two orders of magnitude compression.

The manufacturing industry has invested vast amounts of resources in the deployment and use of solid modeling technology. Although expensive to generate and potentially very valuable in many product related activities, 3D models have rarely been exploited to support product management, documentation, collaborative review, and promotion, because they were only accessible to trained designers equipped with expensive graphics workstations. Intranet access, popular 3D exchange formats, and affordable 3D graphics chips permit to download and view 3D models using a personal computer. Although these basic capabilities are revolutionizing the entertainment and marketing industry and have reduced the cost of a design station, they are of little help to non-designers in the manufacturing industry. The author articulates a vision where 3D data is available and exploited at all phases of a product life cycle. The paper investigates the shortcomings of the current technology, identifies the fundamental research issues, and reviews recent advances in 3D data compression, in the automatic generation of levels-of-detail for interactive rendering, and in the innovative exploitation of 3D input devices for an intuitive and effective navigation.

In the past, access to 3D databases was restricted to few specialists having the appropriate CAD skills, software, and graphics hardware. The availability of inexpensive graphics support on personal computers, the Internet’s impact on private and commercial communication, and the emergence of multimedia standards provide the basis for linking CAD databases with other personal productivity and communication tools and for making them accessible to everyone at home, in schools, in hospitals, or in the industry. For example, employees that have no design expertise, customers, and suppliers would benefit from having an easy access to the 3D databases of a company for: collaborative design review, 3D‐based multi‐media problem reports, collaborative problem solving and tracking, online training and documentation, internet‐based part purchasing and subcontracting, demonstration to customers, or advertising. This presentation will address three of the key issues that have so far limited the non‐specialist’s access to 3D databases. First‐time or occasional non‐expert users need to become instant experts in 3D navigation through Virtual Environments or in the interactive manipulation of digital 3D models, so that they may immediately focus on their tasks, and not waste precious time learning and fighting an unnatural user interface. Immersive VR is not the panacea – other more effective techniques show promise. The data complexity found in commercial CAD databases, especially in the automotive, aerospace, and construction industries, significantly exceeds the capabilities of any interactive graphics system. This situation is not likely to change, since the growth of the complexity and availability of 3D models outpaces the performance improvement of personal computers. Research on the automatic simplification of 3D models and on the use of levels of detail to accelerate the rendering of distant portions of the scene is growing rapidly. The still limited bandwidth of internet communication channels prohibits a pervasive access to large amounts of 3D data. Recent 3D compression techniques reduce the storage requirements for polyhedral 3D models by two orders of magnitudes.

Abstract Floating point round‐off causes erroneous and inconsistent decisions in geometric modelling algorithms. These errors lead to the generation of topologically invalid boundary models for CSG objects and significantly reduce the reliability of CAD applications. Previously known methods that guarantee topological consistency by relying on arbitrary precision rational arithmetic or on symbol‐manipulation techniques are too expensive for practical purposes. This paper presents a new solution which takes as input a “fixed precision” regularized Boolean combination of linear half‐spaces and produces a polyhedral boundary model that has the exact topology of the corresponding solid. Each half‐space is represented by four homogeneous coefficients infixed precision format (La bits for the three direction cosines and Ld bits for the constant term, i.e. the distance from the origin). Exact answers to all topological and ordering questions are computed using a fixed length, 3 La+ Ld+ 2 bits, integer format. This new guaranteed tight limit on the number of bits necessary for performing intermediate calculations is achieved by expressing all of the topological decisions based on geometric computations in terms of the signs of 4 by 4 determinants of the input coefficients. The coordinates of intersection vertices are not required for making the correct topological decisions and hence vertices and lines are represented implicitly in terms of planes.

Abstract We propose a new algorithm for automatically computing approximations of a given polyhedral object at different levels of details. The application for this algorithm is the display of very complex scenes. where many objects are seen with a range of varying levels of detail. Our approach is similar to the region‐merging method used for image segmentation. We iteratively collapse edges, based on a measure of the geometric deviation from the initial shape. When edges are merged in the right order, this strategy produces a continuum of valid approximations of the original object, which can be used for faster rendering at vastly different scales.

The objective of solid modeling is to represent, manipulate and reason about the 3D shape of solid physical objects by computer. Such representations should be unambiguous. Solid modeling's major application areas include design, manufacturing, computer vision, graphics and virtual reality. The field draws on diverse sources, including numerical analysis, symbolic algebraic computation, approximation theory, applied mathematics, point set topology, algebraic geometry, computational geometry and databases. In this article, we begin with some mathematical foundations of the field. We next review the major representation schemata of solids. Then, major layers of abstraction in a typical solid modeling system are characterized. The lowest level of abstraction comprises a substratum of basic service algorithms. At an intermediate level of abstraction there are algorithms for larger, more conceptual operations. Finally, a yet higher level of abstraction presents to the user a functional view that is typically targeted towards solid design. We look at some applications and at user interaction concepts. The classical design paradigms of solid modeling concentrated on obtaining one specific final shape. Those paradigms are becoming supplanted by feature-based, constraint-based design paradigms that are oriented more toward the design process and define classes of shape instances. These new paradigms venture into territory that has yet to be explored systematically. Concurrent with this paradigm shift, there is also a shift in the system architecture towards modularized confederations of plug-compatible functional components.

Abstract We present a new, simple, yet efficient algorithm for triangulating multiply‐connected polygons. The algorithm requires sorting only local concave minima (sags). The order in which triangles are created mimics a flooding process of the interior of the polygon. At each stage, the algorithm analyses the positions and neighborhoods of two vertices only, and possibly checks for active sags, so as to determine which of five possible actions to take. Actions are based on a local decomposition of the polygon into monotonic regions, or gorges (raise the water level in the current gorge, spill into an adjacent gorge, jump to the other bank of a filled gorge, divide a gorge into two, and fill a gorge to its top). The implementation is extremely simple and numerically robust for a large class of polygons. It has been tested on millions of cases as a preprocessing step of a walkthrough and inspection program for complex mechanical and architectural scenes. Extensive experimental results indicate that the observed complexity in terms of the number of vertices, remains under in all cases.

Abstract The metamorphosis between two user‐specified objects offers an intuitive metaphor for designing animations of deforming shapes. We present a new technique for interactively editing such deformations and for animating them in realtime. Besides the starting and ending shapes, our approach offers easy to use additional control over the deformations. The new Bezier Interpolating Polyhedron (BIP) provides a graphics representation of such a deforming object formulated mathematically as a point describing a Bezier curve in the space of all polyhedra. We replace, in the Bezier formulation, the traditional control points by arbitrary polyhedra and the vector addition by the Minkowski sum. BIPs are composed of Animated GRaphic ELement (AGRELs), which are faces with constant orientation, but with parametrized vertices represented by Bezier curves. AGRELs were designed to efficiently support smooth realtime animation on commercially available rendering hardware. We provide a tested algorithm for automatically computing BIPs from the sequence of control polyhedra and demonstrate its applications to animation design.

To reduce the cost of correcting design errors, assemblies of mechanical parts are modeled using CAD systems and verified electronically before the designs are sent to manufacturing.Shaded images are insufficient for examining the internal structures of assemblies and for detecting interferences.Thus, designers must rely on expensive numerical techniques that compute geometric representations of cross-sections and of intersections of solids.The solid-clipping approach presented here bypasses these geometric calculations and offers realtime rendering of cross-sections and interferences for solids represented by their facetted boundaries.In its simplest form, the technique is supported by contemporary highend graphics workstations.Its variations, independently developed elsewhere, have already been demonstrated.Our implementation is based on the concept of a cutvolume interactively manipulated to remove obstructing portions of the assembly and reveal its internal structure.For clarity, faces of the cut-volume which intersect a single solid are hatched and shaded with the color of that solid.Interference areas between two or more solids are highlighted.Furthermore, to help users find the first occurrence of an interference along a search direction, we have developed an adaptive subdivision search based on a projective approach which guarantees a sufficient condition for object disjointness.The additional performance cost for solid-clipping and interference highlighting is comparable to the standard rendering cost.An efficient implementation of the disjointness test requires a minor extension of the graphics functions currently supported on commercial hardware.

Computer programs that simulate the deformations of geometric shapes have played a key role in the increasing popularity of software tools for artistic animation. Previously published techniques for specifying and animating deformations are either limited in their domain or ill suited for interactive editing and visualization. This is because the effects of alterations performed by the animator on the model's parameters may not always be anticipated, and because realtime animation may only be produced by visualizing pre-computed sequences of 3D frames, which are obtained by a slow process and require vast amounts of storage. To support an interactive environment for animation design, we have developed a new, simple, and efficient animation primitive: a Parameterized Interpolating Polyhedron, or PIP for short. PIPs are easily specified and edited by providing their initial and final shapes, which may be any polyhedra, and need not have corresponding boundary elements. PIPs may be efficiently animated on standard graphic hardware because a PIP is a smoothly varying family of polyhedra bounded by faces that evolve with time. The faces have constant orientations and vertices that each move on a straight line between a vertex of the initial shape and a vertex of the final one. The cost of recalculating the time dependant information of a PIP is small in comparison to the display cost. We provide simple and efficient algorithms, based on Minkowski sum operations, for computing PIPs. When both the initial and final shapes are convex, the resulting faces are the true boundary of the deforming object, otherwise subsets of the resulting faces may lie inside the object. In both cases, correct images are automatically generated using standard depth-buffer hardware. The tools we have developed are convenient for interactively designing animation sequences that show the metamorphosis of 3D shapes. They may also be used to simulate the geometric effect of a variety of manufacturing operations, and for interactively selecting the optimal compromise between two or more shapes. They are being integrated in the LAMBADA design and inspection environment for animated assemblies, where deformations and rigid-body motions may be easily combined and synchronized using a hierarchical representation.

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.