Abstract: This paper deals with modeling point sets with attributes. A point set in a geometric space of an arbitrary dimension is a geometric model of a real/abstract object or process under consideration. An attribute is a mathematical model of an object property of arbitrary nature (material, photometric, physical, statistical, etc.) defined at any point of the point set. We provide a brief survey of different modeling techniques related to point sets with attributes. It spans such different areas as solid modeling, heterogeneous objects modeling, scalar fields or "implicit surface" modeling and volume graphics. Then, on the basis of this survey we formulate requirements to a general model of hypervolumes (multidimensional point sets with multiple attributes). A general hypervolume model and its components such as objects, operations, and relations are introduced and discussed. A function representation (FRep) is used as the basic model for the point set geometry and attributes represented independently using real-valued scalar functions of several variables. Each function defining the geometry or an attribute is evaluated at the given point by a procedure traversing a constructive tree structure with primitives in the leaves and operations in the nodes of the tree. This reflects the constructive nature of the symmetric approach to modeling geometry and associated attributes in multidimensional space. To demonstrate a particular application of the proposed general model, we consider in detail the problem of texturing, introduce a model of constructive hypervolume texture, and then discuss its implementation, as well as the special modeling language we used for modeling hypervolume objects.

Abstract: This paper describes a technique to animate three-dimensional sampled volumes. The technique gives the animator the ability to treat volumes as if they were standard polygonal models and to use all of the standard animation/motion capture tools on volumetric data. A volumetric skeleton is computed from a volumetric model using a multi-resolution thinning procedure. The volumetric skeleton is centered in the object and accurately represents the shape of the object. The thinning process is reversible in that the volumetric model can be reconstructed from the volumetric skeleton. The volumetric skeleton is then connected and imported into a standard graphics animation package for animation. The animated skeleton is used for reconstruction, which essentially recreates a deformed volume around the deformed skeleton. Polygons are never computed and the entire process remains in the volumetric domain. This technique is demonstrated on one of the most complex 3D datasets, the Visible Male, resulting in actual "human animation,"

Abstract: We propose a sculpture metaphor based on a multiresolution volumetric representation. It allows the user to model both precise and coarse features while maintaining interactive updates and display rates. The modelled surface is an iso-surface of a scalar field, which is sampled on an adaptive hierarchical grid that dynamically subdivides or undivides itself. Field modifications are transparent to the user: The user feels as if he were directly interacting with the surface via a tool that either adds or removes "material." Meanwhile, the tool modifies the scalar field around the surface, its size and shape automatically guiding the underlying grid subdivision. In order to give an interactive feedback whatever the tool's size, tools are applied in an adaptive way, the grid being always updated from coarse to fine levels. This maintains interactive rates even for large tool sizes. It also enables the user to continuously apply a tool, with an immediate coarse-scale feedback of the multiple actions being provided. A dynamic level-of-detail (LOD) mechanism ensures that the iso-surface is displayed at interactive rates regardeless of the zoom value; surface elements, generated and stored at each level of resolution, are displayed depending on their size on the screen. The system may switch to a coarser surface display during user actions, thus always ensuring interactive visual feedback. Two applications illustrate the use of this system: First, complex shapes with both coarse and fine features can be sculpted from scratch. Second, we show that the system can be used to edit models that have been converted from a mesh representation.

Abstract: Multiresolution motion analysis has gained considerable research interest as a unified framework to facilitate a variety of motion editing tasks. Within this framework, motion data are represented as a collection of coefficients that form a coarse-to-fine hierarchy. The coefficients at the coarsest level describe the global pattern of a motion signal, while those at fine levels provide details at successively finer resolutions. Due to the inherent nonlinearity of the orientation space, the challenge is to generalize multiresolution representations for motion data that contain orientations as well as positions. Our goal is to develop a multiresolution analysis method that guarantees coordinate-invariance without singularity. To do so, we employ two novel ideas: hierarchical displacement mapping and motion filtering. Hierarchical displacement mapping provides an elegant formulation to describe positions and orientations in a coherent manner. Motion filtering enables us to separate motion details level-by-level to build a multiresolution representation in a coordinate-invariant way. Our representation facilitates multiresolution motion editing through level-wise coefficient manipulation that uniformly addresses issues raised by motion modification, blending, and stitching.

Abstract: Superquadrics are a family of parametric shapes which can model a diverse set of objects. They have received significant attention because of their compact representation and robust methods for recovery of 3D models. However, their assumption of intrinsical symmetry fails in modeling numerous real-world examples such as the human body, animals, and other naturally occurring objects. In this paper, we present a novel approach, which is called extended superquadric, to extend superquadric's representation power with exponent functions. An extended superquadric model can be deformed in any direction because it extends the exponents of superquadrics from constants to functions of the latitude and longitude angles in the spherical coordinate system. Thus, extended superquadrics can model more complex shapes than superquadrics. It also maintains many desired properties of superquadrics such as compactness, controllability, and intuitive meaning, which are all advantageous for shape modeling, recognition, and reconstruction. In this paper, besides the use of extended superquadrics for modeling, we also discuss the recovery of extended superquadrics from 3D information (reconstruction). Experiments on both realistic modeling and extended superquadric fitting are presented. Our results are very encouraging and indicate that the use of extended superquadric has potential benefits for the generation of synthetic images for computer graphics and that extended superquadric also is a promising paradigm for shape representation and recovery in computer vision.

Abstract: Camera calibration is the estimation of parameters (both intrinsic and extrinsic) associated with a camera being used for imaging. Given the world coordinates of a number of precisely placed points in a 3D space, camera calibration requires the measurement of the 2D projection of those scene points on the image plane. While the coordinates of the points in space can be known precisely, the image coordinates that are determined from the digital image are often inaccurate and hence noisy. In this paper, we look at the statistics of the behavior of the camera calibration parameters, which are important for stereo matching, when the image plane measurements are corrupted by noise. We derive analytically the behavior of the camera calibration matrix under noisy conditions and further show that the elements of the camera calibration matrix have a Gaussian distribution if the noise introduced into the measurement system is Gaussian. Under certain approximations we derive relationships between the camera calibration parameters and the noisy camera calibration matrix and compare it with Monte Carlo simulations.

Abstract: Although the Hausdorff distance is a popular device to measure the differences between sets, it is not natural for some specific classes of sets, especially for the medial axis transform which is defined as the set of all pairs of the centers and the radii of the maximal balls contained in another set. In spite of its many advantages and possible applications, the medial axis transform has one great weakness, namely its instability under the Hausdorff distance when the boundary of the original set is perturbed. Though many attempts have been made for the resolution of this phenomenon, most of them are heuristic in nature and lack precise error analysis. In this paper, we show that this instability can be remedied by introducing a new metric called the hyperbolic Hausdorff distance, which is most natural for measuring the differences between medial axis transforms. Using the hyperbolic Hausdorff distance, we obtain error bounds, which make the operation of medial axis transform almost an isometry. By various examples, we also show that the bounds obtained are sharp. In doing so, we show that bounding both the Hausdorff distance between domains and the Hausdorff distance between their boundaries is necessary and sufficient for bounding the hyperbolic Hausdorff distance between their medial axis transforms. These results drastically improve the previous results and open a new way to practically control the Hausdorff distance error of the domains under its medial axis transform error, and vice versa.

Abstract: A solid is a connected orientable compact subset of R3 which is a 3-manifold with boundary. Moreover, its boundary consists of finitely many components, each of which is a subset of the union of finitely many almost smooth surfaces. Motivated by numerical robustness issues, we consider a finite collection of boxes, with faces parallel to the coordinate planes, which covers the boundary of the solid itself. An interval solid is the union of this collection and the solid. In this paper we develop sufficient conditions on the collection of the boxes and a 3-manifold, so that the union of the collection and the manifold is homeomorphic to the manifold itself. Finally, we outline an approach for constructing an interval solid, using interval arithmetic, homeomorphic to the solid.

Abstract: We present an algorithm that computes the convex hull of multiple rational curves in the plane. The problem is reformulated as one of finding the zero-sets of polynomial equations in one or two variables; using these zero-sets we characterize curve segments that belong to the boundary of the convex hull. We also present a preprocessing step that can eliminate many redundant curve segments.

Abstract: Voxelization is the transformation of geometric surfaces into voxels. Up to date this process has been done essentially using incremental algorithms. Incremental algorithms have the reputation of being efficient but they lack an important property: robustness. The voxelized representation should envelop its continuous model. However, without robust methods this cannot be guaranteed. This article describes novel techniques of robust voxelization and visualization of implicit surfaces. First of all our recursive subdivision voxelization algorithm is reviewed. This algorithm was initially inspired by Duff's image space subdivision method. Then, we explain the algorithm to voxelize implicit surfaces defined in spherical or cylindrical coordinates. Next, we show a new technique to produce infinite replications of implicit objects and their voxelization method. Afterward, we comment on the parallelization of our voxelization procedure. Finally we present our voxel visualization algorithm based on point display. Our voxelization algorithms can be used with any data structure, thanks to the fact that a voxel is only stored once the last subdivision level is reached. We emphasize the use of the octree, though, because it is a convenient way to store the discrete model hierarchically. In a hierarchy the discrete model refinement is simple and possible from any previous voxelized scene thanks to the fact that the voxelization algorithms are robust.

Abstract: Given several algebraic surfaces and corresponding auxiliary planes, a scheme for constructing a piecewise algebraic surface to blend the given surfaces is presented along the intersection curves of the given surfaces and their corresponding auxiliary planes. The algorithm starts with the suitable partition of the 3D space into tetrahedrons or prisms in which the algebraic surface patches are defined. Then a smooth piecewise algebraic surface of low degree is constructed which meets the initial surfaces with a certain order of geometric continuity. Examples are provided to demonstrate the detailed construction process and to compare our method with previous approaches. The results show some advantages of the new blending method.

Abstract: This paper describes a system for building 3D models of indoor scenes from sets of noisy laser range images. It addresses several important aspects of this problem, namely, preprocessing, which includes image segmentation and planar model fitting; view registration, which is the method of determining the rigid transformation that describes the relative pose of the camera platform between views; and reconstruction, which is the subsequent integration or fusion of separate range images into a single 3D model. Our proposed strategy is to use a statistical sensor model. We thus account for noise properties of the data at each stage in the reconstruction process, which produces reliable results even in the presence of significant measurement noise. We give an empirical analysis of a plane-based registration method and present results using real range data that demonstrate the performance of the entire reconstruction system.

Abstract: This paper proposes a camera-based real-time system for building a three dimensional (3D) human head model. The proposed system is first trained in a semi-automatic way to locate the user's facial area and is then used to build a 3D model based on the front and profile views of the user's face. This is achieved by directing the user to position his or her face and profile in a highlighted area, which is used to train a neural network to distinguish the background from the face. With a blink from the user, the system is then capable of accurately locating a set of characteristic feature points on the front and profile views of the face, which are used for the adaptation of a generic 3D face model. This adaptation procedure is initialized with a rigid transformation of the model aiming to minimize the distances of the 3D model feature nodes from the calculated 3D coordinates of the 2D feature points. Then, a nonrigid transformation ensures that the feature nodes are displaced optimally close to their exact calculated positions, dragging their neighbors in a way that deforms the facial model in a natural looking manner. A male hair model is created using a 3D ellipsoid, which is truncated and merged with the adapted face model. A cylindrical texture map is finally built from the two image views covering the whole area of the head by exploiting the inherent face symmetry. The final result is a complete, textured model of a specific person's head.

Abstract: This paper presents a coherent computational framework to efficiently, and more so robustly, evaluate, interrogate, and compute a whole variety of characteristic curves on freeform parametric rational surfaces represented as (piecewise) polynomial or rational functions. These characteristic curves are expressed as zero sets of bivariate rational functions and include silhouette curves and isoclines from a prescribed viewing direction and/or point, reflection lines and reflection ovals, and highlight lines. This zero set formulation allows for a better treatment of singular cases while these characteristic curves are crucial for various applications, from visualization through interrogation to design and manufacturing.

Abstract: Explosions are among the most common special effects employed in the entertainment industry. Producing real explosions, however, is costly and potentially dangerous and special effects technicians have been turning to software to generate explosion sequences. Software tools typically allow artists to re-create the appearance of an explosion but ignore physical processes. More accurate techniques, used in scientific research, produce physically correct results but demand substantial computational resources and do not provide support for artistic expression. This paper discusses an explosion animation system, X-Sim, that incorporates physical principles relevant to the generation of an explosion in a compressible medium. Diverse explosion effects can be created automatically while still allowing the animator to adjust the processes involved. Computational issues concerning X-Sim are addressed and a multiresolution scheme is introduced that focuses computation on turbulent regions of the environment. Results generated with X-Sim, evaluated qualitatively and quantitatively, indicate that the system can generate explosion phenomena with reasonable accuracy. Animations produced with this system also point to its flexibility and utility.

Abstract: Defects in boundary representation models often lead to system errors in modeling software and associated applications. This paper analyzes the model rectification problem of manifold boundary models, and argues that a rectify-by-reconstruction approach is needed in order to reach the global optimal solution. The restricted face boundary reconstruction problem is shown to be NP-hard. Based on this, the solid boundary reconstruction problem is also shown to be NP-hard.

Abstract: Image-morphing techniques are often employed to generate fluid transitions between two reference images. Many algorithms have been proposed in this area. Existing warp functions used in morphing algorithms are one-to-one. Consequently, when some features are visible in only one of the reference images, “ghosting” problems will appear in some in-between images. In addition, since the control features have global effects, there is no way to control different objects in the scene separately. In this paper, two new morphing approaches are introduced. In the first approach, called region-based morphing, regions are specified and morphed separately. In the second approach, called layer-based morphing, regions are defined in different layers. Hidden information is recovered and is used to solve the visibility problems.

Abstract: Traditionally most camera-based position estimation systems use only a few points to calibrate cameras In this paper, we investigate a novel and alternate approach for 3D position estimation by using a larger number of points arranged in a 3D grid We present an implementation of the active-space indexing mechanism which uses three cameras Given the corresponding points in camera images, a precise estimation of the position can be obtained The active-space indexing method can be also used as a spatial filter to eliminate the large number of possible corresponding pairs from consideration This capability, unique only to the active-space indexing method, provides a tractable algorithm to the otherwise intractable situation

Abstract: We describe an efficient algorithm for coding the connectivity information of general polygon meshes. In contrast to most existing algorithms which are suitable only for triangular meshes, and pay a penalty for treatment of nontriangular faces, this algorithm codes the connectivity information in a direct manner. Our treatment of the special case of triangular meshes is shown to be equivalent to the Edgebreaker algorithm. Using our methods, any triangle mesh may be coded in no more than 2 bits/triangle (approximately 4 bits/vertex), a quadrilateral mesh in no more than 3.5 bits/quad (approximately 3.5 bits/vertex), and the most common case of a quad mesh with few triangles in no more than 4 bits/polygon.

Abstract: Tools for assisting with editing human motion have become one of the most active research areas in the field of computer animation. Not surprisingly, the area has demonstrated some stunning successes in both research and practice. This paper explores the range of constraint-based techniques used to alter motions while preserving specific spatial features. We examine a variety of methods, defining a taxonomy of these methods that is categorized by the mechanism employed to enforce temporal constraints. We pay particular attention to a less explored category of techniques that we term per-frame inverse kinematics plus filtering, and we show how these methods may provide an easier to implement while retaining the benefits of other approaches.

Abstract: In this paper, we develop a hairstyle modeling and animation technique specifically designed for human hairs, and we report several experimental results. Using simplified cantilever beam model and one-dimensional projective differential equations of angular momenta, we give a practical solution to the problem of enormous complexity. Even though our hair animation algorithm is an approximate solution, it includes all the relevant dynamic elements such as gravity, wind, inertia, air-resistance, hair-to-head, and hair-to-hair friction forces. Collision is an important element that makes a collection of hair strands look like hair. We develop an accurate but efficient hair-to-head and hair-to-hair collision detection and treatment algorithm. The algorithm produces quite realistic results; still it runs at an interactive speed. An interesting contribution of our algorithm is that it unifies hairstyle modeling and animation into a single equation, so that (1) hairstyling can be done under the effects of gravity and other internal or external forces, and (2) original hairstyle is more or less restored even after the initial hair is tangled by the application of external forces or head movements.

Abstract: We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ?uCu generated by a moving circle. Given two ringed surfaces ?uCu1 and ?vCv2, we formulate the condition Cu1 ? Cv2 ? ? (i.e., that the intersection of the two circles Cu1 and Cv2 is nonempty) as a bivariate equation ?(u, v)=0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution of ?(u, v)=0 to the intersection point Cu1 ? Cv2. Thus it is trivial to construct the intersection curve once we have computed the zero-set of ?(u, v)=0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation ?(u, v)=0 is reduced to a decomposable form, f(u)=g(v) or ?f(u)?g(v)?=|r(u)|, which can be solved more efficiently than the general case.