Quad/Triangle Subdivision

In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Designers often want the added flexibility of having both quads and triangles in their models. It is also well known that triangle meshes generate poor limit surfaces when using a quad scheme, while quad‐only meshes behave poorly with triangular schemes. Our new scheme is a generalization of the well known Catmull‐Clark and Loop subdivision algorithms. We show that our surfaces are C ¹ everywhere and provide a proof that it is impossible to construct such a C ² scheme at the quad/triangle boundary. However, we provide rules that produce surfaces with bounded curvature at the regular quad/triangle boundary and provide optimal masks that minimize the curvature divergence elsewhere. We demonstrate the visual quality of our surfaces with several examples. ACM CSS: I.3.5 Computer Graphics—Curve, surface, solid, and object representations     

A new fast normal-based interpolating subdivision scheme by cubic Bézier curves

In this paper, we propose a new fast normal-based interpolating subdivision scheme for curve and surface design. Different from the 4-points interpolating subdivision scheme, it is based on cubic Bezier curves and the normal vectors are used to generate a circle. Both a convex edge and an inflexion edge can be subdivided into convex sub-edges and then generate smooth curves. Under proper angle conditions, this subdivision scheme converges and the limit curve will be \(\hbox {G}^{1}\) smoothness. When applying it to subdivide surface on triangle/quadrilateral meshes, we use the normal vectors and have no need to consider the meshes neighboring to the current surface elements. Such advantage leads to that the subdivision scheme has fast rendering speed without changing the topology of the meshes. Subdivision examples and results by our scheme are illustrated and meantime is compared with those generated by other well-known schemes. It shows that this scheme can generate a more smooth curve based on both a convex edge and an inflexion edge, and the limit surface has better smoothness than those of other interpolating schemes.     

A New Class of Guided C2 Subdivision Surfaces Combining Good Shape with Nested Refinement

Converting quadrilateral meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distribution of recent G‐spline constructions with the refinability of subdivision surfaces. This avoids the complex refinement of G‐spline constructions and the poor shape of standard subdivision. Guided subdivision can then be used both to generate the surface and hierarchically compute functions on the surface. Specifically, we present a C² subdivision algorithm of polynomial degree bi‐6 and a curvature bounded algorithm of degree bi‐5. We prove that the common eigenstructure of this class of subdivision algorithms is determined by their guide and demonstrate that their eigenspectrum (speed of contraction) can be adjusted without harming the shape. For practical implementation, a finite number of subdivision steps can be completed by a high‐quality cap. Near irregular points this allows leveraging standard polynomial tools both for rendering of the surface and for approximately integrating functions on the surface.     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

Evolving Guide Subdivision

To overcome the well-known shape deficiencies of bi-cubic subdivision surfaces, Evolving Guide subdivision (EG subdivision) generalizes C² bi-quartic (bi-4) splines that approximate a sequence of piecewise polynomial surface pieces near extraordinary points. Unlike guided subdivision, which achieves good shape by following a guide surface in a two-stage, geometry-dependent process, EG subdivision is defined by five new explicit subdivision rules. While formally only C¹ at extraordinary points, EG subdivision applied to an obstacle course of inputs generates surfaces without the oscillations and pinched highlight lines typical for Catmull-Clark subdivision. EG subdivision surfaces join C² with bi-3 surface pieces obtained by interpreting regular sub-nets as bi-cubic tensor-product splines and C² with adjacent EG surfaces. The EG subdivision control net surrounding an extraordinary node can have the same structure as Catmull-Clark subdivision: two rings of 4-sided facets around each extraordinary nodes so that extraordinary nodes are separated by at least one regular node.     

RGB Subdivision

We introduce the RGB subdivision: an adaptive subdivision scheme for triangle meshes, which is based on the iterative application of local refinement and coarsening operators, and generates the same limit surface of the Loop subdivision, independently on the order of application of local operators. Our scheme supports dynamic selective refinement, as in Continuous Level Of Detail models, and it generates conforming meshes at all intermediate steps. The RGB subdivision is encoded in a standard topological data structure, extended with few attributes, which can be used directly for further processing. We present an interactive tool that permits to start from a base mesh and use RGB subdivision to dynamically adjust its level of detail.     

Interpolating an Unlimited Number of Curves Meeting at Extraordinary Points on Subdivision Surfaces*

Interpolating curves by subdivision surfaces is one of the major constraints that is partially addressed in the literature. So far, no more than two intersecting curves can be interpolated by a subdivision surface such as Doo‐Sabin or Catmull‐Clark surfaces. One approach that has been used in both of theses surfaces is the polygonal complex approach where a curve can be defined by a control mesh rather than a control polygon. Such a definition allows a curve to carry with it cross derivative information which can be naturally embodied in the mesh of a subdivision surface. This paper extends the use of this approach to interpolate an unlimited number of curves meeting at an extraordinary point on a subdivision surface. At that point, the curves can all meet with either C ⁰ or C ¹ continuity, yet still have common tangent plane. A straight forward application is the generation of subdivision surfaces through 3‐regular meshes of curves for which an easy interface can be used.     

Interference detection through rasterization

Interference detection is a useful technique, but it is also generally time-consuming. In this paper, a new type of interference detection algorithm is proposed for real-time interference detection. The algorithm first rasterizes the projection of the target objects and calculates the z-values, just as done by the z-buffer visible surface algorithm. For interference detection, all z-values and pointers to the corresponding faces of objects are saved in a z-list for each pixel. Sorting the z-list against the z-values allows the detection of overlapping objects in the z-direction at each pixel position and, thus, finds interfering faces by referring to the face pointers in the z-list. The algorithm is simple and easy to implement. Its computational complexity is directly proportional to the number of polygons, and, in addition, standard graphics hardware can be used to greatly accelerate execution. Another advantage is that the algorithm can be applied to all ‘ray-traceable’ objects, including algebraic surfaces, and procedurally defined objects; traditionally these were not suitable subjects for interference detection. The algorithm is implemented on a graphics workstation using a standard graphics library. Interference detection at a practical interaction speed is achieved for complicated objects such as polyhedra with thousands of polygons. The algorithm can be used in two ways: for inexpensive interference detection, and as an efficient culling method for more precise collision/interference detection algorithms.     

多分辨率三维建筑群建模与实时绘制技术

将纹理特征分析技术引入到多边形网格建模中,提出一种基于高程特征值进行曲面细分的算法以构建多分辨率虚拟建筑群模型。该算法给出一种三角边与纹理特征曲线相交的三角面分裂方法构造自适应细分三角网格。通过设计细分三角网格的二叉树数据结构和开发测试程序进行测试,表明该算法具有自适应网格速度快和保持几何特征较好的特点,可以满足在PC机上实现三维建筑群的大范围建模和实时交互显示要求。     

Efficient Depth of Field Rasterization Using a Tile Test Based on Half‐Space Culling

For depth of field (DOF) rasterization, it is often desired to have an efficient tile versus triangle test, which can conservatively compute which samples on the lens that need to execute the sample‐in‐triangle test. We present a novel test for this, which is optimal in the sense that the region on the lens cannot be further reduced. Our test is based on removing half‐space regions of the (u, v) ‐space on the lens, from where the triangle definitely cannot be seen through a tile of pixels. We find the intersection of all such regions exactly, and the resulting region can be used to reduce the number of sample‐in‐triangle tests that need to be performed. Our main contribution is that the theory we develop provides a limit for how efficient a practical tile versus defocused triangle test ever can become. To verify our work, we also develop a conceptual implementation for DOF rasterization based on our new theory. We show that the number of arithmetic operations involved in the rasterization process can be reduced. More importantly, with a tile test, multi‐sampling anti‐aliasing can be used which may reduce shader executions and the related memory bandwidth usage substantially. In general, this can be translated to a performance increase and/or power savings.     

An adjustable subdivision surface modeling scheme

Based on triangle and quadrilateral meshes, this paper presents an adjustable subdivision surface scheme. The scheme can produce subdivision surface of Cl continuity of limit surface Since an adjustable parameter is introduced to the scheme, the surface modeling is flexible. Depended on given initial data, the limited surface shape can be adjusted and controlled through selecting appropriate parameters. The method is effective in generating smooth surfaces.     

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three‐dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle‐triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.     

基于二次误差的三角网格自适应细分算法研究

提出一种基于二次误差的三角网格自适应细分算法,该算法采用二次误差描述三角网格的曲率变化情况,只对二次误差大于阈值的三角面片进行细分,避免了在较平坦区域再进行细分,以较少的三角面片表达了模型的特征,实现三角网格的自适应细分.与全局细分相比,自适应细分既可增加模型光顺性,又可减少模型的数据处理量,提高细分效率.     

Modeling smooth shape using subdivision on differential coordinates

Traditional subdivision schemes are applied on Euclidean coordinates (the spatial geometry of the control mesh). Although the subdivision limit surfaces are almost everywhere C² continuous, their mean-curvature normals are only C⁰. In order to generate higher quality surfaces with better-distributed mean-curvature normals, we propose a novel framework to apply subdivision for shape modeling, which combines subdivision with differential shape processing. Our framework contains two parts: subdivision on differential coordinates (a kind of differential geometry of the control mesh), and mutual conversions between Euclidean coordinates and differential coordinates. Further discussions about various strategies in both parts include a special subdivision method for mean-curvature normals, additional surface editing options, and a version of our framework for curve design. Finally, we demonstrate the improvement on surface quality by comparing the results between our framework and traditional subdivision methods.     

DigitalSculpture: a subdivision-based approach to interactive implicit surface modeling

This paper presents DigitalSculpture, an interactive sculpting framework founded upon iso-surfaces extracted from recursively subdivided, 3D irregular grids. Our unique implicit surface model arises from an interpolatory, volumetric subdivision scheme that is C¹ continuous across the domains defined by arbitrary 3D irregular grids. We assign scalar coefficients and color to each control vertex and allow these quantities to participate in the volumetric subdivision of irregular grids. In the subdivision limit, a virtual sculpture is obtained by extracting the zero-level from the volumetric, scalar field defined over the irregular grid. This novel shape geometry extends concepts from solid modeling, recursive subdivision, and implicit surfaces; facilitates many techniques for interactive sculpting; permits rapid, local evaluation of iso-surfaces; and affords level-of-detail control of the sculpted surfaces.     

Adaptive mesh subdivision for efficient light baking

Light baking has long been a popular technique for real-time rendering. It usually precomputes and bakes the global lighting effects as vertex attributes or textures. Vertex baking requires less memory but can cause artifacts for large triangles. Texture baking can avoid this and generate a high-quality visual effect in real-time rendering. However, it requires significant memory consumption, which may limit the real-time performance and usage. To address this problem, we propose an adaptive mesh subdivision algorithm for memory-efficient light baking, including a fast triangle subdivision level determination method and an optimized solution to calculate vertex colors. Only the subdivided mesh is required during the real-time rendering. Therefore, memory requirements can be significantly reduced while keeping the visual effect. Besides, the subdivision level is allowed to be intuitively controlled by users with a specified parameter. Our algorithm can be easily implemented on commodity graphics hardware and integrated in existing real-time applications such as online preview systems.     

Ternary Sparse Matrix Representation for Volumetric Mesh Subdivision and Processing on GPUs

In this paper, we present a novel volumetric mesh representation suited for parallel computing on modern GPU architectures. The data structure is based on a compact, ternary sparse matrix storage of boundary operators. Boundary operators correspond to the first‐order top‐down relations of k‐faces to their (k ? 1)‐face facets. The compact, ternary matrix storage format is based on compressed sparse row matrices with signed indices and allows for efficient parallel computation of indirect and bottom‐up relations. This representation is then used in the implementation of several parallel volumetric mesh algorithms including Laplacian smoothing and volumetric Catmull‐Clark subdivision. We compare these algorithms with their counterparts based on OpenVolumeMesh and achieve speedups from 3× to 531×, for sufficiently large meshes, while reducing memory consumption by up to 36%.