We propose a general convex optimization problem for computing regularized geodesic distances. We show that under mild conditions on the regularizer the problem is well posed. We propose three different regularizers and provide analytical solutions in special cases, as well as corresponding efficient optimization algorithms. Additionally, we show how to generalize the approach to the all pairs case by formulating the problem on the product manifold, which leads to symmetric distances. Our regularized distances compare favorably to existing methods, in terms of robustness and ease of calibration.

翻译：我们提出一个通用的凸优化问题，用于计算正则化测地线距离。我们证明，在正则化项的温和条件下，该问题是适定的。我们提出了三种不同的正则化项，并在特殊情况下给出解析解，以及相应的高效优化算法。此外，我们展示了如何通过将问题表述在乘积流形上，将这种方法推广到所有点对的情况，从而得到对称距离。我们的正则化距离在鲁棒性和易校准性方面优于现有方法。