An important part of the well-known iterative closest point algorithm (ICP) is the variational problem. Several variants of the variational problem are known, such as point-to-point, point-to-plane, generalized ICP, and normal ICP (NICP). This paper proposes a closed-form exact solution for orthogonal registration of point clouds based on the generalized point-to-point ICP algorithm. We use points and normal vectors to align 3D point clouds, while the common point-to-point approach uses only the coordinates of points. The paper also presents a closed-form approximate solution to the variational problem of the NICP. In addition, the paper introduces a regularization approach and proposes reliable algorithms for solving variational problems using closed-form solutions. The performance of the algorithms is compared with that of common algorithms for solving variational problems of the ICP algorithm. The proposed paper is significantly extended version of Makovetskii et al. (CCIS 1090, 217–231, 2019).

众所周知的迭代最近点算法（ICP）的重要部分是变分问题。已知变分问题的几种变体，例如点对点，点对平面，广义ICP和常规ICP（NICP）。本文提出了一种基于广义点对点ICP算法的点云正交配准的闭式精确解。我们使用点和法向矢量来对齐3D点云，而常见的点对点方法仅使用点的坐标。本文还为NICP的变分问题提供了一种封闭形式的近似解。此外，本文介绍了一种正则化方法，并提出了使用闭式解来解决变分问题的可靠算法。将算法的性能与普通算法的性能进行比较，以解决ICP算法的变型问题。拟议论文是Makovetskii等人的显着扩展版本。（CCIS 1090，217–231，2019）。