目的 随着几何造型、计算机动画等领域的快速发展，曲线的自由变形技术在近年来受到了广泛的关注。为了获得更多有趣、逼真的变形效果，提出基于渐进迭代逼近与主顶点方法的曲线局部变形算法。方法 给定数据点集，首先采用渐进迭代逼近方法或是基于最小二乘的渐进迭代逼近方法产生待变形曲线;其次对待变形区域使用延拓准则，基于主顶点方法与待变形曲线的形状信息选取控制顶点进行调整;最后对调整后的控制顶点运用局部渐进迭代逼近方法生成逼近曲线,得到期望的变形效果。结果 此变形操作借助于局部渐进迭代逼近方法，具有较好的灵活性。文末展示的茶壶、面部轮廓、手等数值实例，表明了该方法可以得到良好的变形效果。进一步地，借助于叠加变形还可以得到整体的、周期的、伸缩的等各类更加丰富的变形效果。结论 本文研究渐进迭代逼近在曲线变形上的应用，将主顶点方法引入曲线的变形之中，把两者相结合提出了基于渐进迭代逼近与主顶点方法的曲线局部变形算法。该算法不仅具备渐进迭代逼近方法的收敛稳定性，且借助于主顶点方法，可以得到较好的变形效果。该方法适用于曲线的局部变形，丰富了曲线的变形效果。
Application of progressive iterative approximation method to curve deformations
Zhang Shijie,zhangli,Yu Liping,Liu Wenzheng()
Objective With the rapid development of geometric modeling and computer animation, freeform deformation techniques of curves and surfaces have received widespread concentration recently. In order to acquire interesting and lifelike deformation effects, a new local deformation algorithm for curves based on progressive iterative approximation (PIA) and dominant point method is proposed in this paper. The new deformation method not only produces various deformation effects, but also possesses properties such as flexibility and convergence by virtue of PIA method.Method The initial curves aimed for deformation are first obtained through using PIA or least squares progressive iterative approximation(LSPIA) method. Secondly, by calculating and comparing the curvatures corresponding to the control points, the dominant points which involve in maximum local curvature points and two end points are selected among the initial control points. In this phase, we detect the maximum local curvature points using the rule that if the curvature of a point is bigger than the curvatures of its neighboring points. Thus, maximum local curvature points have been selected as dominant points. Then, an extension rule is constructed on the region expected to be deformed. That is to say, after we select the region aimed for deformation according to real needs, we extend this region along the curve until it meets two closest dominant points. In this way, we can obtain a segment which is bounded by two dominant points. With this extension rule, we classify the situation into two categories based on the number of dominant points in the segment we obtain. The control points that will be adjusted later are now picked out according to the dominant points and the shape information of the curves. If the region ready for deformation contains one dominant point, then after using extension rule, there will be three dominant points in the segment we obtain, the dominant point in the middle is picked out to adjust later. If the region ready for deformation does not contain any dominant points, then after extension, there will be only two dominant points in the segment we obtain, we select a control point according to the shape information of the curve, which is useful in dealing with some complex deformation problems. In this situation, we first calculate a shape parameter for each control point in the segment we obtain. The shape parameter represents how complex the curve is and indicates the difference between two adjacent segments. Then, after comparing these parameters, the control point which has the smallest shape parameter is picked out to adjust later. We call the procedure stated above the dominant point method. Moreover, if there are more than one dominant point in the region expected to be deformed, we can split the segment according to the distribution of dominant points to make sure that each segment contains less than one dominant point. Then, we can use the dominant point method mentioned above to select control points. Finally, the local progressive iterative approximation (LPIA) method is carried out to generate the final curves after local deformation.Result The proposed deformation method selects control points based on how complex the curves’ shapes are. Since we use LPIA method to fit the data set after adjustment, the deformation method is convergent and can be excuted flexibly. It can also highlight the features of the deformed regions through iterating locally. By using B-spline basis, which is the most commonly used basis in geometric design and processes good local properties, the numerical examples such as teapot, face contour and hand show that the proposed method can obtain good deformation effects. More specifically, the teapot mouth is stretched by adjusting the chosen control point. By using the deformation algorithm, the lips, eyebrows and earlobes of the face contour are deformed to generate a more fascinating face. The fingers are also stretched to make the hand more nature. We also demonstrate the distortion situation which occurs when we do not use the deformation method proposed in this paper, which is illustrated in the teapot example. We can clearly see that if we do not adjust the control points generated by the algorithm we give, the curve after deformation will distort and lack reality. Furthermore, this algorithm can be used repeatedly to generate the global, local, periodic and elastic deformation effects.Conclusion The main work of the paper focuses on the application of progressive iterative approximation method to the local curve deformations. We first talk about PIA method, which presents an intuitive and straightforward way to fit data points and brings more flexibility for shape controlling in data fitting. Then we introduce the notion of dominant points into the curve deformations. By combining these two means, we finally propose a new deformation method based on PIA and dominant point method. The algorithm not only possesses the property of convergence and stability which is owned by PIA method, but also produces various good deformation effects through the selection of dominant points. During the implementation of the deformation method, the user just need to select the regions expected to be deformed and determine the deformation scales according to his needs, which guarantee the interactivity of this algorithm. In sum, this proposed method greatly enriches the deformation effects of curves.