Consistent Shape Maps via Semidefinite Programming
Consistent Shape Maps via Semidefinite Programming
复制标题
DOI：
10.1111/cgf.12184
复制
发表时间：
2013-08-01
影响因子：
2.5
通讯作者：
Guibas, Leonidas
中科院分区：
文献类型：
--
作者：
Huang, Qi-Xing;Guibas, Leonidas
形状匹配的最新进展表明，与孤立地估计形状对之间的映射相比，联合优化集合中的形状之间的映射可以导致显著的改进。这些方法通常调用一个循环一致性标准，即沿着沿着一个形状循环的映射组合应该近似于恒等映射。这个条件使网络规则化，并允许校正个别地图中的错误和缺陷。特别是，它鼓励通过沿着沿着更相似形状的路径的地图的合成来估计不同形状之间的地图。在本文中，我们介绍了一种新的方法来获得一致的形状映射在一个集合，制定的周期一致性约束作为解决方案的半定规划（SDP）。所提出的方法是基于这样的观察，即如果形状之间的地面真值映射是循环一致的，则将所有成对映射存储在块中的矩阵是低秩和半正定的。最近的进展，通过半定规划的低秩矩阵恢复技术的启发，我们制定的问题，估计周期一致的地图，找到最接近的半正定矩阵的输入矩阵，存储所有的初始地图。通过分析该方案的Karush-Kuhn-Tucker（KKT）最优性条件，得到了该算法的理论保证，保证了当输入映射的误差不超过一定阈值时恢复的正确性.除了这个理论保证，在基准数据集上的实验结果表明，所提出的方法优于国家的最先进的多形状匹配方法。
Recent advances in shape matching have shown that jointly optimizing the maps among the shapes in a collection can lead to significant improvements when compared to estimating maps between pairs of shapes in isolation. These methods typically invoke a cycle-consistency criterion the fact that compositions of maps along a cycle of shapes should approximate the identity map. This condition regularizes the network and allows for the correction of errors and imperfections in individual maps. In particular, it encourages the estimation of maps between dissimilar shapes by compositions of maps along a path of more similar shapes. In this paper, we introduce a novel approach for obtaining consistent shape maps in a collection that formulates the cycle-consistency constraint as the solution to a semidefinite program (SDP). The proposed approach is based on the observation that, if the ground truth maps between the shapes are cycle-consistent, then the matrix that stores all pair-wise maps in blocks is low-rank and positive semidefinite. Motivated by recent advances in techniques for low-rank matrix recovery via semidefinite programming, we formulate the problem of estimating cycle-consistent maps as finding the closest positive semidefinite matrix to an input matrix that stores all the initial maps. By analyzing the Karush-Kuhn-Tucker (KKT) optimality condition of this program, we derive theoretical guarantees for the proposed algorithm, ensuring the correctness of the recovery when the errors in the inputs maps do not exceed certain thresholds. Besides this theoretical guarantee, experimental results on benchmark datasets show that the proposed approach outperforms state-of-the-art multiple shape matching methods.