校准三维模型基矩阵的函数映射的对应关系计算

在利用函数映射计算模型间对应关系时，提出了一种校准三维几何模型之间基矩阵的新方法，将模型间对应关系的构建转化为由模型特征函数构建的基矩阵之间的校准运算。首先计算三维模型的Laplace算子，获得模型的特征值和特征向量，并利用所得到的特征向量构建基矩阵；其次，提出了协方差的最小值校准算法，以计算模型基矩阵之间的校准矩阵 S ，并用矩阵 S 对两个模型的函数基进行校准；最后，计算校准模型所有点的高斯曲率来采样源模型尖端特征点，并在校准后的目标模型上遍历所有点，以寻求最优对应点来构建等距变换（或近似等距变换）的三维模型间的对应关系。通过计算采样点与最优对应点的测地错误，以衡量所提算法的匹配准确率。实验结果表明，与已有算法相比，本算法可以较为准确地构建两个或多个模型间的对应关系，同时也克服了模型自身对称性影响对应关系计算的问题。

An Algorithm for Calculating Shape Correspondences Using Functional Maps by Calibrating Base Matrix of 3D Shapes

A new algorithm is proposed to calibrate the base matrix between 3D geometric shapes for calculating correspondences using functional maps, in which shape correspondences can be represented as the calibration operation between the base matrices constructed by the shape eigenfunctions. First, the Laplace operators of 3D shapes are calculated to obtain eigenvectors and eigenvalues, and the basis matrix is constructed using the eigenvectors. Second, a calibration algorithm based on covariance mini-mum is proposed to calculate a calibration matrix S between shapes, and used to calibrate the basis matrices of function space of the two given shapes. Third, the Gauss curvature of all points of the source shape is calculated to sample some feature points, and traverses all points on the calibrated target model in order to find the optimal corresponding points to construct the correspondence between 3D shapes with isometric transformation (or approximate isometric transformation). Finally, the matching accuracy of the proposed algorithm is measured by calculating the geodesic error between the sampling points and the optimal points. Experiment results show that our algorithm is better than existing methods for establishing an accurate correspondence between two or more shapes, moreover, it significantly solves symmetry ambiguities problem which influence calculation of shape correspondence.