Texture mapping with hard constraints using warping scheme

Texture mapping with positional constraints is an important and challenging problem in computer graphics. In this paper, we first present a theoretically robust, foldover-free 2D mesh warping algorithm. Then we apply this warping algorithm to handle mapping texture onto 3D meshes with hard constraints. The proposed algorithm is experimentally evaluated and compared with the state-of-the-art method for examples with more challenging constraints. These challenging constraints may lead to large distortions and foldovers. Experimental results show that the proposed scheme can generate more pleasing results and add fewer Steiner vertices on the 3D mesh embedding.     

曲线插值约束的LOOP细分曲面

提出基于Loop细分方法的曲线插值方法,不需要修改细分规则,只需以插值曲线的控制多边形为中心多边形,向其两侧构造对称三角网格带,该对称三角网格带将收敛于插值曲线。因此,包含有该三角网格带的多面体网格的极限曲面将经过插值曲线。若要插值多条相交曲线只需在交点处构造全对称三角网格。运用该方法可在三角网格生成的细分曲面中插值多达六条的相交曲线。     

Taxonomy of interpolation constraints on recursive subdivision surfaces

Published online: 14 May 2002     

Reverse engineering with subdivision surfaces

Reverse engineering is concerned with the reconstruction of surfaces from three-dimensional point clouds originating from laser-scanned objects. We present an adaptive surface reconstruction method providing a hierarchy of quadrilateral meshes adapting surface topology when a mesh is refined. This way, a user can choose a model with proper resolution and topology from the hierarchy without having to run the algorithm multiple times with different parameters. The multiresolution mesh representation can be used subsequently for view-dependent rendering and wavelet compression.     

Blending parametric patches with subdivision surfaces

In this paper the problem of blending parametric surfaces using subdivision patches is discussed.A new approach,named removing-boundary,is presented to generate piecewise-smooth subdivision surfaces through discarding the outmost quadrilaterals of the open meshes derived by each subdivision step.Then the approach is employed both to blend parametric bicubic B-spline surfaces and to fill n-sided holes.It is easy to produce piecewise-smooth subdivision surfaces with both convex and concave corners on the boundary,and limit surfaces are guaranteed to be C^2 continuous on the boundaries except for a few singular points by the removing-boundary approach Thus the blending method is very efficient and the blending surface generated is of good effect.     

Algorithmic shape modeling with subdivision surfaces

We present methods for synthesizing 3D shape features on subdivision surfaces using multi-scale procedural techniques. Multi-scale synthesis is a powerful approach for creating surfaces with different levels of detail. Our methods can also blend multiple example multi-resolution surfaces, including procedurally defined surfaces as well as captured models.     

Dynamic Catmull-Clark subdivision surfaces

Recursive subdivision schemes have been extensively used in computer graphics, computer-aided geometric design, and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. We present a new dynamic surface model based on the Catmull-Clark subdivision scheme, a popular technique for modeling complicated objects of arbitrary genus. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme, as well as those of the physics-based models. This new model provides a direct and intuitive means of manipulating geometric shapes, and an efficient hierarchical approach for recovering complex shapes from large range and volume data sets using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the “physical” quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces. The governing dynamic differential equation is derived using Lagrangian mechanics and the finite element method. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design, and scientific visualization     

Virtual control with hard constraints

Practical control problems are always subject to plant state and/or input constraints, which make designing an effective controller a challenging task. This paper introduces a novel virtual control approach to handling the presence of hard constraints in control systems by utilizing virtual mechanisms in the form of nonlinear springs and dampers. The augmented virtual mechanisms are to assist in better shaping the closed‐loop responses, especially when operating near the constrained boundary. A linear quadratic regulator based model predictive control method is utilized to develop stabilizing controllers that not only achieve desired system performance, but also meet the imposed hard constraints. The basic idea is to dramatically increase control penalty by way of tuning the spring and damper effect when the constrained state/input response is close to its hard constraint. The proposed method is applied to a balancing ball problem to demonstrate its applicability and effectiveness, and the simulation results validate the proposed concept.     

Texture mapping on surfaces of arbitrary topology using norm preserving-based optimization

A simple and yet highly efficient, high-quality texture mapping method for surfaces of arbitrary topology is presented. The new method projects the given surface from the 3D object space into the 2D texture space to identify the 2D texture structure that will be used to texture the surface. The object space to texture space projection is optimized to ensure minimum distortion of the texture mapping process. The optimization is achieved through a commonly used norm preserving minimization process on edges of the surface. The main difference here is, by using an initial value approach, the optimization problem can be set up as a quadratic programming problem and, consequently, solved by a linear least squares method. Three methods to choose a good initial value are presented. Test cases show that the new method works well on surfaces of arbitrary topology, with the exception of surfaces with exceptionally abnormal curvature distribution. Other advantages of the new method include uniformity and seamlessness of the texture mapping process. The new method is suitable for applications that do not require precise texture mapping results but demand highly efficient mapping process such as computer animation or video games.     

Optimal LPV control with hard constraints

This paper considers the optimal control of polytopic, discrete-time linear parameter varying (LPV) systems with a guaranteed ?₂ to ?_∞ gain. Additionally, to guarantee robust stability of the closed-loop system under parameter variations, H _∞ performance criterion is also considered as well. Controllers with a guaranteed ?₂ to ?_∞ gain and a guaranteed H _∞ performance (?₂ to ?₂ gain) are a special family of mixed H ₂=H _∞ controllers. Normally, H₂ controllers are obtained by considering a quadratic cost function that balances the output performance with the control input needed to achieve that performance. However, to obtain an optimal controller with a guaranteed ?₂ to ?_∞ gain (closely related to the physical performance constraint), the cost function used in the H₂ control synthesis minimizes the control input subject to maximal singular-value performance constraints on the output. This problem can be efficiently solved by a convex optimization with linear matrix inequality (LMI) constraints. The main contribution of this paper is the characterization of the control synthesis LMIs used to obtain an LPV controller with a guaranteed ?₂ to ?_∞ gain and >H _∞ performance. A numerical example is presented to demonstrate the effectiveness of the convex optimization.     

Computing surfaces invariant under subdivision

In this paper, we propose a general subdivision algorithm for generating surfaces. The algorithm has as motivation our earlier work on the design of free form curves where similar ideas were investigated. Here we describe some properties of uniform refinement algorithms for surface generation. A detail analysis of their properties will be given later by one of us.     

Smooth blending of subdivision surfaces

A technique for merging freeform objects modeled with subdivision surfaces is proposed. The method takes into consideration the possible distortion of the surfaces while the smoothness of the blending surface connecting the objects is maintained. A blend region between the given object surfaces is located. The topology of the surfaces in the vicinity of the blend region is refined and the boundary curves of the blend region are smoothed to reduce possible distortion in the blending surface. A blend curve between the two surfaces is used for connecting the given surfaces. Vertices on the blend curve are regular whereas extraordinary vertices are restricted to locate along the boundary of the blend region. Locations of the blend vertices are adjusted such that the curvature of the surfaces along the boundaries of the blend region is minimized. This ensures extraordinary vertices on the boundaries to lie on a relatively flat region, and hence, minimizes the distortion of the surfaces in the merging process.     

Efficient skeleton-guided displaced subdivision surfaces

Displacement mapping is a computer graphics technique that uses scalar offsets along normals on a base surface to represent and render a model with highly geometric details. The technique natively compresses the model and saves memory I/O. A subdivision surface is the ideal base surface, due to its good geometric properties, such as arbitrary topology, global smoothness, and multi-resolution via hardware tessellation, among others. Two of the main challenges in displacement mapping representation are constructing the base surface faithfully and generating displacement maps efficiently. In this paper, we propose an efficient skeleton-guided displaced subdivision surfaces method. The construction of the base mesh is guided by a sketched skeleton. To make the shape of the base surface fit the input model well, we develop an efficient progressive GPU-based subdivision fitting method. Finally, a GPU-based raycasting method is proposed to sample the input model and generate the displacement maps. The experimental results demonstrate that the proposed method can efficiently generate a high-quality displacement mapping representation. Compared with the traditional displaced subdivision surface method, the proposed method is more suitable for the modern rendering pipeline and has higher efficiency.     

Boolean surfaces with shape constraints

In this paper, we present a new method for the construction of parametric surfaces reproducing an object from a set of spatial data. We adopt a hybrid scheme, based on the Boolean sum of variable degree spline operators, which both interpolate a set of grid lines and approximate the data. As usual the variable degrees can be chosen to satisfy proper shape constraints.     

Optimization methods for scattered data approximation with subdivision surfaces

We present a method for scattered data approximation with subdivision surfaces which actually uses the true representation of the limit surface as a linear combination of smooth basis functions associated with the control vertices. A robust and fast algorithm for exact closest point search on Loop surfaces which combines Newton iteration and non-linear minimization is used for parameterizing the samples. Based on this we perform unconditionally convergent parameter correction to optimize the approximation with respect to the L² metric, and thus we make a well-established scattered data fitting technique which has been available before only for B-spline surfaces, applicable to subdivision surfaces. We also adapt the recently discovered local second order squared distance function approximant to the parameter correction setup. Further we exploit the fact that the control mesh of a subdivision surface can have arbitrary connectivity to reduce the L^∞ error up to a certain user-defined tolerance by adaptively restructuring the control mesh. Combining the presented algorithms we describe a complete procedure which is able to produce high-quality approximations of complex, detailed models.     

Variational reconstruction using subdivision surfaces with continuous sharpness control

Computational Visual Media - We present a variational method for subdivision surface reconstruction from a noisy dense mesh. A new set of subdivision rules with continuous sharpness control is...     

Generating subdivision surfaces from profile curves

The construction of freeform models has always been a challenging task. A popular approach is to edit a primitive object such that its projections conform to a set of given planar curves. This process is tedious and relies very much on the skill and experience of the designer in editing 3D shapes. This paper describes an intuitive approach for the modeling of freeform objects based on planar profile curves. A freeform surface defined by a set of orthogonal planar curves is created by blending a corresponding set of sweep surfaces. Each of the sweep surfaces is obtained by sweeping a planar curve about a computed axis. A Catmull-Clark subdivision surface interpolating a set of data points on the object surface is then constructed. Since the curve points lying on the computed axis of the sweep will become extraordinary vertices of the subdivision surface, a mesh refinement process is applied to adjust the mesh topology of the surface around the axis points. In order to maintain characteristic features of the surface defined with the planar curves, sharp features on the surface are located and are retained in the mesh refinement process. This provides an intuitive approach for constructing freeform objects with regular mesh topology using planar profile curves.     

Estimating subdivision depth of Catmull-Clark surfaces

In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection,mesh generation, numerical control machining and surface rendering.     

Lens-shaped surfaces and C 2 subdivision

Lens-shaped surfaces (with vertices of valence 2) arise for example in automatic quad-remeshing. Applying standard Catmull–Clark subdivision rules to a vertex of valence 2, however, does not yield a C ¹ surface in the limit. When correcting this flaw by adjusting the vertex rule, we discover a variant whose characteristic ring is z → z ². Since this conformal ring is of degree bi-2 rather than bi-3, it allows constructing a subdivision algorithm that works directly on the control net and generates C ² limit surfaces of degree bi-4 for lens-shaped surfaces. To further improve shape, a number of re-meshing and re-construction options are discussed indicating that a careful approach pays off. Finally, we point out the analogy between characteristic configurations and the conformal maps z ^4/n , cos z and e ^z .