Department of Computer Engineering, Koç University, Istanbul, Turkey.
IEEE Trans Pattern Anal Mach Intell. 2012 Nov;34(11):2203-15. doi: 10.1109/TPAMI.2012.26.
We present a purely isometric method that establishes 3D correspondence between two (nearly) isometric shapes. Our method evenly samples high-curvature vertices from the given mesh representations, and then seeks an injective mapping from one vertex set to the other that minimizes the isometric distortion. We formulate the problem of shape correspondence as combinatorial optimization over the domain of all possible mappings, which then reduces in a probabilistic setting to a log-likelihood maximization problem that we solve via the Expectation-Maximization (EM) algorithm. The EM algorithm is initialized in the spectral domain by transforming the sampled vertices via classical Multidimensional Scaling (MDS). Minimization of the isometric distortion, and hence maximization of the log-likelihood function, is then achieved in the original 3D euclidean space, for each iteration of the EM algorithm, in two steps: by first using bipartite perfect matching, and then a greedy optimization algorithm. The optimal mapping obtained at convergence can be one-to-one or many-to-one upon choice. We demonstrate the performance of our method on various isometric (or nearly isometric) pairs of shapes for some of which the ground-truth correspondence is available.
我们提出了一种纯粹等距的方法，用于建立两个（几乎）等距形状之间的 3D 对应关系。我们的方法从给定的网格表示中均匀地采样高曲率顶点，然后寻找从一个顶点集到另一个顶点集的单射映射，该映射最小化等距变形。我们将形状对应问题表述为所有可能映射的组合优化问题，然后在概率设置中将其简化为对数似然最大化问题，我们通过期望最大化（EM）算法来解决该问题。EM 算法通过经典多维尺度（MDS）对采样顶点进行变换，在谱域中初始化。然后，在 EM 算法的每次迭代中，在原始 3D 欧几里得空间中，通过首先使用二分完美匹配，然后使用贪婪优化算法，实现等距变形的最小化，从而实现对数似然函数的最大化。在收敛时获得的最优映射可以是一对一或一对多，具体取决于选择。我们在一些具有真实对应关系的等距（或几乎等距）形状对上展示了我们方法的性能。
