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刘艳, 寿华好, 季康松(浙江工业大学理学院, 杭州 310023)

摘 要
目的 对采样设备获取的测量数据进行拟合,可实现原模型的重建及功能恢复。但有些情况下,获取的数据点不仅包含位置信息,还包含法向量信息。针对这一问题,本文提出了基于圆平均的双参数4点binary非线性细分法与单参数3点ternary插值非线性细分法。方法 首先将线性细分法改写为点的重复binary线性平均,然后用圆平均代替相应的线性平均,最后用加权测地线平均计算的法向量作为新插入顶点的法向量。基于圆平均的双参数4点binary细分法的每一次细分过程可分为偏移步与张力步。基于圆平均的单参数3点ternary细分法的每一次细分过程可分为左插步、插值步与右插步。结果 对于本文方法的收敛性与C1连续性条件给出了理论证明;数值实验表明,与相应的线性细分相比,本文方法生成的曲线更光滑且具有圆的再生力,可以较好地实现3个封闭曲线重建。结论 本文方法可以在带法向量的初始控制顶点较少的情况下,较好地实现带法向约束的离散点集的曲线重建问题。
Circle average nonlinear subdivision curve design with normal constraints

Liu Yan, Shou Huahao, Ji Kangsong(College of Science, Zhejiang University of Technology, Hangzhou 310023, China)

Objective The subdivision technique has been developing in relevant to the design of efficient, hierarchical, local, and adaptive algorithms for modeling, plotting, and manipulating arbitrary topology-related free-shaped objects beyond the non-uniform rational B-splines (NURBS). The initial subdivision step is oriented at a control polygon or mesh. First, novel vertices can be involved in and the existed vertices can be optimized. Next, a new control polygon or control mesh can be obtained. Finally, the target curves of surfaces can be replicated to produce. Subdivision schemes can be segmented into two categories:1) linear-based and 2) nonlinear-based because the issue of new points can change the linear combinations of old points in the iterative process. Generally speaking, linear subdivision schemes are easier to be implemented, but there are inflection points plotting on the limit curves and it is challenged to represent precise circle, while nonlinear subdivision schemes can eliminate inflection points and reproduce a circle accurately. The smooth curve-fit point clouds problem is concerned more in the context of computer-aided geometric design (CAGD) and computer graphics (CG). Measurement data can be obtained on real objects via such techniques like laser scanning, structure light source converter, and X-ray tomography. To perform a commonly-used model reconstruction and functional recovery for the original model or product, these discrete data points are scanned and used for data fitting. But, data points are often linked to position information and normal vector information like optical reflector design. To resolve this problem, we develop two schemes of nonlinear subdivision in related to a parameters-dual 4-point binary and a parameter-solo 3-point ternary interpolation in terms of circle average. Method First, a two points and its normal vector-related binary nonlinear circle average is introduced. This task is targeted on circle average because the new point is on the circle-constructed derived from the original two points and corresponding normal vectors. Next, linear subdivision method is rewritten into a replicated binary average of points. In order to optimize linear subdivision schemes, linear average is replaced by the circle average. Third, the weighted geodesic average is used to calculate the newly vertex-inserted normal vector. The two kinds of circle average-based nonlinear subdivision schemes are obtained through melting the operations mentioned above into linear parameters-dual 4-point binary subdivision and parameter-solo 3-point ternary interpolation subdivision. For circle average-related two-parameter four-point binary subdivision scheme, each subdivision process is composed of two steps of displacement and tension both. For circle average-based single-parameter three-point ternary subdivision method, each subdivision step is reconstructed by left interpolation, interpolation, and right interpolation. In addition, the feasibility of these methods is tested theoretically and numerically. Some theorems of convergence and consistency of two proposed methods are illustrated because normal vectors-proved have factor-contracted, and the data points have factor-contracted and backup-displaced in the subdivision process. Result These methods are implemented in terms of MATLAB overall. The issue of parameters is studied on the two proposed subdivision schemes. First, for the circle average-based two-parameter 4-point binary subdivision scheme, the smaller of the tension parameter is, the limit curve is closer to the initial control polygon. The smaller of the displacement parameter is the limit curve is near to the initial control vertex. When the parameter-displaced is zero, the subdivision method is transferred to interpolation subdivision. For the circle average-based solo-parameter 3-point ternary subdivision method, the circle average is first to be applied to the linear ternary interpolation subdivision, which makes the vertices-controlling more fast. Then, for the same initial control vertices, the normal vector of one fixed control vertex is changed to produce different limit curves freely. Test results show that the selection of parameters and initial normal vectors can be used to control the shape of limit curves effectively. Finally, our nonlinear subdivision schemes proposed are compared to the corresponding linear subdivision schemes. When the initial control vertices are sampled from the circle, the corresponding normal vectors will be pointed between the center of the circle and the vertex. Test results show that our nonlinear subdivision schemes proposed can reconstruct the circle through the proposed nonlinear subdivision schemes and the corresponding linear subdivision schemes-reconstructed, but the corresponding linear subdivision schemes cannot be used to reconstruct the circle. Furthermore, three sorts of case studies for curve models are selected in comparison with curve reconstruction from multiple subdivision methods. The initial control vertices and their normal vectors are sampled based on curves-consistent, and they are subdivided for 8 times totally. Our nonlinear subdivision-schemed limit curve is much smoother, while the corresponding linear subdivision schemes have their sharp points. Conclusion Theoretically, it shows that our circle average-based two nonlinear subdivisions proposed are convergent and consistent with C1. Experimental results indicate that our nonlinear subdivision schemes can optimize linear subdivision-schemed modeling ability, and it has its circular regenerative potentials. Normal vectors-selected is beneficial to the shapes of limit curves to some extent.