Non-rigid 3D registration, which deforms a source 3D shape in a non-rigid way to align with a target 3D shape, is a classical problem in computer vision. Such problems can be challenging because of imperfect data (noise, outliers and partial overlap) and high degrees of freedom. Existing methods typically adopt the $\ell_p$ type robust norm to measure the alignment error and regularize the smoothness of deformation, and use a proximal algorithm to solve the resulting non-smooth optimization problem. However, the slow convergence of such algorithms limits their wide applications. In this paper, we propose a formulation for robust non-rigid registration based on a globally smooth robust norm for alignment and regularization, which can effectively handle outliers and partial overlaps. The problem is solved using the majorization-minimization algorithm, which reduces each iteration to a convex quadratic problem with a closed-form solution. We further apply Anderson acceleration to speed up the convergence of the solver, enabling the solver to run efficiently on devices with limited compute capability. Extensive experiments demonstrate the effectiveness of our method for non-rigid alignment between two shapes with outliers and partial overlaps, with quantitative evaluation showing that it outperforms state-of-the-art methods in terms of registration accuracy and computational speed. The source code is available at https://github.com/yaoyx689/AMM_NRR.

翻译：非刚性三维配准，即通过非刚性方式变形源三维形状以对齐目标三维形状，是计算机视觉领域的一个经典问题。由于数据不完美（噪声、离群点和部分重叠）以及高自由度，此类问题极具挑战性。现有方法通常采用$\ell_p$型鲁棒范数来度量对齐误差并正则化变形的平滑性，同时使用近端算法求解由此产生的非光滑优化问题。然而，这类算法收敛速度慢，限制了其广泛应用。本文提出一种基于全局光滑鲁棒范数进行对齐与正则化的鲁棒非刚性配准公式，该公式能有效处理离群点和部分重叠。问题通过最大化-最小化算法求解，每次迭代可简化为具有闭式解的凸二次问题。我们进一步应用Anderson加速技术加快求解器的收敛速度，使其能在计算能力有限的设备上高效运行。大量实验证明了该方法在两具有离群点和部分重叠的形状间进行非刚性对齐的有效性，定量评估表明其在配准精度和计算速度上均优于现有最先进方法。源代码见https://github.com/yaoyx689/AMM_NRR。