We provide a dynamical perspective on the classical problem of 3D point cloud registration with correspondences. A point cloud is considered as a rigid body consisting of particles. The problem of registering two point clouds is formulated as a dynamical system, where the dynamic model point cloud translates and rotates in a viscous environment towards the static scene point cloud, under forces and torques induced by virtual springs placed between each pair of corresponding points. We first show that the potential energy of the system recovers the objective function of the maximum likelihood estimation. We then adopt Lyapunov analysis, particularly the invariant set theorem, to analyze the rigid body dynamics and show that the system globally asymptotically tends towards the set of equilibrium points, where the globally optimal registration solution lies in. We conjecture that, besides the globally optimal equilibrium point, the system has either three or infinite "spurious" equilibrium points, and these spurious equilibria are all locally unstable. The case of three spurious equilibria corresponds to generic shape of the point cloud, while the case of infinite spurious equilibria happens when the point cloud exhibits symmetry. Therefore, simulating the dynamics with random perturbations guarantees to obtain the globally optimal registration solution. Numerical experiments support our analysis and conjecture.
翻译:我们对3D点云的典型问题提供了动态视角。 点云被视为由粒子组成的僵硬体体。 记录两点云的问题被表述为一个动态系统, 动态模型云在固定场点云的粘稠环境中, 在每对对应点之间放置的虚拟弹簧所引发的力量和托盘下, 向静态场点云转化和旋转。 我们首先显示, 系统的潜在能量恢复了最大概率估计的客观功能。 然后我们采用 Lyapunov 分析, 特别是变量设置定定定的定理, 来分析硬体体体动态, 并显示全球系统在静态点的设置上, 而全球最佳登记解决方案就处于静态点。 我们推测, 除了全球最佳平衡点之外, 系统还有三个或无限的“ 纯度” 平衡点, 而这些刺激的平衡点都是局部不稳定的。 三个虚假的平衡性案例与点云的普通形状相对对应, 而当点的精确度定调度则发生于无限的断断断点点点的模型时, 当我们获得最佳的图像定位时, 我们的模拟的模拟的模拟的模拟的模拟的模拟模型分析时, 。