Wujun Che

Lines of curvature and umbilical points for implicit surfaces

This paper develops a method to analyze and compute the lines of curvature and their differential geometry defined on implicit surfaces. With our technique, we can explicitly obtain the analytic formulae of the associated geometric attributes of an implicit surface, e.g. torsion of a line of curvature and Gaussian curvature. Additionally, it can be used to directly derive the closed formulae of principal directions and corresponding principal curvature of an implicit surface. We also present a novel criterion for non-umbilical points and umbilical points on an implicit surface.

Technical Section: Meshless quadrangulation by global parameterization

Point cloud is a basic description of discrete shape information. Parameterization of unorganized points is important for shape analysis and shape reconstruction of natural objects. In this paper we present a new algorithm for global parameterization of an unorganized point cloud and its application to the meshing of the cloud. Our method is guided by principal directions so as to preserve the intrinsic geometric properties. After initial estimation of principal directions, we develop a kNN(k-nearest neighbor) graph-based method to get a smooth direction field. Then the point cloud is cut to be topologically equivalent to a disk. The global parameterization is computed and its gradients align well with the guided direction field. A mixed integer solver is used to guarantee a seamless parameterization across the cut lines. The resultant parameterization can be used to triangulate and quadrangulate the point cloud simultaneously in a fully automatic manner, where the shape of the data is of any genus.

Billboards for Tree Simplification and Real-Time Forest Rendering

Modeling and rendering of large forest scenes are a challenging task, whose geometric complexity surpasses even the capabilities of current graphics hardware. A tree model simplification tool is developed in this paper to automatically generate both hybrid polygon/image-based tree models for close distance viewing and cross billboards for far distance viewing. In the hybrid model, the foliage of the tree is replaced by a set of arbitrarily oriented billboards while the trunk and branches remains polygon models, so that the 3D perception is enhanced and weak parallax of usual image based approaches is avoided. A fast forest rendering system is also developed. We present techniques in the system for rendering forest consisting of many highly detailed plant models. This forest rendering system allows us to render a large forest scene in real-time. Walk-through and flyover a forest are also simulated, illustrating applications for computer games and interactive visualization of landscapes.

Realistic real-time rendering for large-scale forest scenes

Fast rendering of a large-scale forest landscape scene is important in many applications, as video games, Internet graphics applications, landscape or cityscape scene design and visualization, and virtual forestry. A challenge in virtual reality is realistic rendering of large scale scenes consisting of complex plant models. A series of level of detail tree models are usually constructed to compress the overall forest complexity in view-dependent forest navigation. In this paper a new leaf modeling method is presented to have leaf models match leaf textures, so that the visual effect and model complexity can be balanced well. In addition, vertex buffer objects and tree clipping operation allow rendering a large forest containing thousands of trees in real-time. The experiments show that these techniques can be easily used in applications such as video games and interactive navigation of landscapes. Walk-through and flyover a forest are both feasible using our techniques.

Computing lines of curvature for implicit surfaces

Lines of curvature are important intrinsic characteristics of a curved surface used in a wide variety of geometric analysis and processing. Although their differential attributes have been examined in detail, their global geometric distribution and topological pattern are very difficult to compute over the whole surface because of umbilical points and unstable numerical computation. No studies have yet been carried out on this problem, especially for an implicit surface. In this paper, we present a scheme for computing and visualizing the lines of curvature defined on an implicit surface. A key structure is introduced, conveying significant structure information about lines of curvature to facilitate their investigation, rather than computing their whole net. Our current framework is confined to a collection of manageable structures, consisting of algorithms to locate some seed umbilical points, to compute the lines of curvature through them, and finally to assemble this structure. The numerical implementations are provided in detail and a novel evaluation function measuring the violation of umbilical points in an implicit surface, i.e. indicating how much a point is to be umbilical, is also presented. This paper is the continuation of [Che, W.J., Paul, J.-C., Zhang, X.P., 2007. Lines of curvature and umbilical points for implicit surfaces. Computer Aided Geometric Design 24 (7), 395-409].

SMI 2011: Full Paper: Direct quad-dominant meshing of point cloud via global parameterization

In this paper, we present a new algorithm for quad-dominant meshing of unorganized point clouds based on periodic global parameterization. Our meshing method is guided by principal directions so as to preserve the intrinsic geometric properties. We use local Delaunay triangulation to smooth the initial principal directions and adapt the global parameterization to point clouds. By optimizing the fairness measure we can find the two scalar functions whose gradients best align with the guided principal directions. To handle the redundant vertices in the iso-lines due to overlapped triangles, an approach is specially designed to clean the iso-lines. Our approach is fully automatic and applicable to a surface of arbitrary genus. We also show an application of our method in curve skeleton extraction from incomplete point cloud data.

Smooth Transition Between Different Plant Leaves Models

Foliage modeling and rendering is an important topic in virtual environment. Texture mapping and foliage simplification are usual solutions to forest navigation keeping leaf realism and with a fast speed. These two techniques are still used separately, so that their common advantages are not unified. A new technique is presented in this paper on a smooth transition between a textured leaf model and the one without texture by defining two functions: a leaf vein quadratic interpolation function and an implicit leaf contour function. The new foliage representation is a mixture of models with textures and models without textures, which is a balance between materials and geometry. Experiments demonstrate that our transition algorithm can be used as a continuous multiresolution representation of foliage, keeping high realism and high data compression in rendering plant foliage in large scenes.

Ridge extraction of a smooth 2-manifold surface based on vector field

This paper presents a general scheme to compute ridges on a smooth 2-manifold surface from the standpoint of a vector field. A ridge field is introduced. Starting with an initial ridge, which may or may not be umbilical, a ridge line is then traced by calculating an associated integral curve of this field in conjunction with a new projection procedure to prevent it from diverging. This projection is the first that can optimize a ridge guess to lie on a ridge line uniquely and accurately. In order to follow this scheme, we not only develop practical ridge formulae but also address their corresponding computational procedures for an analytical surface patch, especially for an implicit surface. In contrast to other existing methods, our new approach is mathematically sound and characterized by considering the full geometric structures and topological patterns of ridges on a generic smooth surface. The resulting ridges are accurate in the numerical sense and meet the requirement of high accuracy with complete topology. Although the objective of this paper is to develop a mathematically sound framework for ridges on a smooth surface, we give a comprehensive review of relevant works on both meshes and smooth surfaces for readers.

Global parameterization and quadrilateral meshing of point cloud

Point data are basic media for shape information acquisition and representation. A new approach is presented for global parameterization of unorganized point data and application to the meshing of point models with noises. While most of recent researches focus on quadrangulation of mesh models, it is extended to point models in this work, so as to loosely reconstruct the shape model by quadrilateral meshing of curvature isolines. The new approach is guided by principal directions, so as to preserve intrinsic geometric properties. A robust method is applied to estimate curvatures in presence of noise and outliers based on fitting surface normals. The parameterization of [Ray et al. 2006] is then adapted to point data by local Delaunay triangulation. Isolines are extracted by discarding and merging the redundant segments in each local triangle. This method is totally automatic, and a high-quality quadrilateral dominated mesh can be generated, as shown in Figure 1.

Fast and symmetry-aware quadrangulation

A variety of methods have been developed to produce quad meshes of high quality; however, there are still limitations in those state-of-the-art algorithms. First, there is little consideration in the symmetry of a mesh shape, which is important for further use of the quad mesh, such as UV layout. Second, most of current methods are time consuming for large models. To address these problems, we propose a new quadrangulation method to preserve shape symmetry. A hierarchical approach is also presented to accelerate the whole process. Our method achieves the symmetry of the final quad mesh by controlling the positions of singular points. After an initial construction of a cross field, we detect the symmetric relations among singularities and refine their positions to make them better reflect symmetries of the model. By taking the singularities as constraints, a cross field is then recomputed as the guided direction field for a global parameterization. To reduce the time spent on quadrangulation of a large model, we first simplify it while preserving its most significant geometric features. A coarse quadrangulation is then performed on the simplified mesh, and the final one is obtained by improving the coarse one in terms of the original mesh.