论文标题

一个统治所有这些的戒指:与异常值证明具有强大的几何感知

One Ring to Rule Them All: Certifiably Robust Geometric Perception with Outliers

论文作者

Yang, Heng, Carlone, Luca

论文摘要

我们提出了第一个一般和实用的框架,用于在存在大量异常值的情况下设计可认证的算法,以实现可靠的几何感知。我们研究了截短的最小二乘(TLS)成本函数的使用,这对离群值是可靠的,但是导致硬,非convex和非平滑优化问题。我们的首要贡献是表明,对于一系列几何感知问题,可以对多项式的优化进行重新估算,而Lasserre的凸力矩放松的层次结构在经验上以最小的放松顺序在经验上紧张(即,认证,确认的全球最低限度最低限度的最小值)。我们的第二个贡献是利用客观和约束多项式的结构稀疏性,并利用基础减少,以显着降低由于瞬间放松而导致的半决赛计划(SDP)的大小,而不会损害其紧密度。我们的第三个贡献是从平方之和的镜头(SOS)放松镜头开发可扩展的双重最优认证符,这些镜头可以计算次要差距,并可能证明任何候选解决方案的全球最优性(例如,由诸如RANSAC或诸如RANSAC之类的快速启发式方法返回,例如RANSAC或毕业于毕业的非统一性)。我们的双重认证者利用道格拉斯·拉赫福德(Douglas-Rachford)分裂来解决凸的可行性SDP。跨不同感知问题的数值实验,包括单旋平均,形状比对,3D点云和网格注册以及高融合性卫星姿势估计,证明了我们放松的紧密性,认证的正确性以及所提出的双重认证的可伸缩性,超出了当前SDP溶剂的大型问题。

We propose the first general and practical framework to design certifiable algorithms for robust geometric perception in the presence of a large amount of outliers. We investigate the use of a truncated least squares (TLS) cost function, which is known to be robust to outliers, but leads to hard, nonconvex, and nonsmooth optimization problems. Our first contribution is to show that -for a broad class of geometric perception problems- TLS estimation can be reformulated as an optimization over the ring of polynomials and Lasserre's hierarchy of convex moment relaxations is empirically tight at the minimum relaxation order (i.e., certifiably obtains the global minimum of the nonconvex TLS problem). Our second contribution is to exploit the structural sparsity of the objective and constraint polynomials and leverage basis reduction to significantly reduce the size of the semidefinite program (SDP) resulting from the moment relaxation, without compromising its tightness. Our third contribution is to develop scalable dual optimality certifiers from the lens of sums-of-squares (SOS) relaxation, that can compute the suboptimality gap and possibly certify global optimality of any candidate solution (e.g., returned by fast heuristics such as RANSAC or graduated non-convexity). Our dual certifiers leverage Douglas-Rachford Splitting to solve a convex feasibility SDP. Numerical experiments across different perception problems, including single rotation averaging, shape alignment, 3D point cloud and mesh registration, and high-integrity satellite pose estimation, demonstrate the tightness of our relaxations, the correctness of the certification, and the scalability of the proposed dual certifiers to large problems, beyond the reach of current SDP solvers.

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