基于加速多维尺度变换算法的三维模型形变规范型计算方法

针对如何建立可有效表示三维模型，使模型在大尺度形变过程中保持等距不变特性，并能保留原模型几何细节特征的形变规范型这一问题，提出基于加速多维尺度变换算法的三维模型形变规范型计算方法。生成包含有三维几何模型体积信息的四面体网格模型，避免形变过程中由体积变化引起的模型失真；使用最远点采样算法对原模型上的顶点进行采样，获取更具代表性的顶点；使用加速多维尺度变换算法得到原始模型的初始规范型；利用三维模型注册的思想，通过建立能量函数并计算最优解的方法将输入的四面体网格模型按照初始规范型的姿态进行形变，从而得到保留输入模型几何特征的形变模型。实验结果表明，以四面体网格模型作为输入，经加速多维尺度变换算法计算得到的初始规范型为基础，通过三维模型注册能量函数的最小化求解过程，可以构建出保留更多原模型几何细节特征的形变规范型。和已有算法相比，加速多维尺度变换算法计算三维模型形变规范型的方法更适用于处理具有复杂几何结构的三维模型，可以得到姿态更为自然的模型形变结果，普适性较高。

Calculation of deformation canonical forms of 3D shape based on accelerate multidimensional scaling algorithm

Aiming at how to create a deformation canonical form which can effectively represent the 3D shape,maintain the isometric invariance during the large-scale deformation process, and preserve the geometric details of the original 3D shape, a calculation of 3D shape deformation canonical form based on the accelerated multidimensional scaling algorithm (Accelerated MDS) is proposed in this paper.First,a tetrahedral mesh containing volume information of 3D shape is generated to avoid distortion caused by volume changes during deformation. Second, the farthest point sampling (FPS) algorithm is used to sample the vertices on the original shape, and then the Accelerated MDS algorithm is used to calculate the square geodesic distance matrix, and obtain the spectral embedding coordinates of the original shape through matrix decomposition. Finally, based on the idea of 3D shape registration, the input tetrahedral mesh is deformed according to the initial canonical form by establishing the energy function to calculate the optimal solution. Thus, the deformed shape which retains the geometric details of the input shape is obtained. The experimental results show that taking tetrahedral mesh shape as input, based on the initial canonical form calculated by the Accelerate MDS algorithm, through the minimization of the 3D shape registration energy function, a deformation canonical form which retains more geometric details of the original shape can be obtained. Compared with the existing algorithms, our algorithm is more suitable for calculating deformation canonical form of 3D shape with complex geometric structures and gets more natural shape deformation results with higher universality.