Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Consequently, we focus on Riemannian gradient descent for minimization of the error and obtain a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that the non-degeneracy of a perfect alignment is equivalent to the local rigidity of the resulting realization. By specializing the characterization of a non-degenerate alignment to the noiseless setting, we obtain necessary and sufficient conditions on the overlapping structure of the views for a locally rigid realization. Similar results are also obtained in the context of global rigidity.

翻译：给定一个数据集的一组重叠局部视图（补丁），我们考虑寻找最小化基于$2$-范数对齐误差的视图刚性对齐问题。通常，视图存在噪声，完美对齐可能不存在。在这项工作中，我们基于某个矩阵的核与正定性，刻画了有噪声情形下对齐的非退化性。这引出了一个用于测试给定对齐非退化性的多项式时间算法。因此，我们聚焦于黎曼梯度下降法以最小化误差，并得到了算法（局部）线性收敛到该对齐的充分条件。在无噪声视图情况下，存在完美对齐，从而得到尊重视图几何的点实现。在视图的温和条件下，我们证明完美对齐的非退化性等价于所得实现的局部刚性。通过将非退化对齐的刻画专门化到无噪声情形，我们得到了实现局部刚性所需视图重叠结构的充分必要条件。类似结果也在全局刚性的背景下获得。