A non‐iterative method for approximation of the exact solution to the point‐to‐plane variational problem for orthogonal transformations

Abstract

The most popular algorithm for aligning of three-dimensional point data is the iterative closest point (ICP). In this paper, a new algorithm for orthogonal registration of point clouds based on the point-to-plane ICP algorithm is proposed.The algorithm consists of three steps: first, a matrix of affine transformation between two given point clouds are calculated; second, the affine transformation matrix is projected onto the manifold SO(3) of orthonormal matrices; finally, a translation vector is reestimated. The proposed algorithm does not require an approximate initial estimate. At each iterative step of the ICP algorithm, an approximated closed-form solution for the orthogonal transformation is derived.The performance of the proposed algorithm is compared with that of common algorithms for the geometrical transformations estimation. KEYWORDS exact solution, ICP, point-to-plane, registration, variational problem 9218The variational problem point-to-point for orthogonal transformations is mathematically equivalent to the absolute orientation problem in photogrammetry. 4 In this case, the closed-form solution was obtained by Horn. 5,6 The solution in the earlier work of Horn 5 is expressed with quaternions, whereas the solution in his other work 6 uses orthogonal matrices. Note that the computational complexity of the solutions is linear with respect to the number of point pairs.The original ICP algorithm is widely used for registration of rigid objects, but it often fails for the case of nonrigid objects. Recently, an extension of the ICP algorithm for scaling has been proposed. 7 An efficient algorithm for arbitrary affine transformations was proposed. 8,9 A closed-form solution to the point-to-point problem was also derived. 10,11 The aforementioned approaches for the variational ICP problem are based on the point-to-point metric. It has been shown that the point-to-plane metric is better than the point-point metric in terms of accuracy and convergence rate. 12 Search for the closed-form solution to the point-to-plane case for orthogonal transformations is an open problem. Instead, iterative methods based on the Levenberg-Marquardt 13 linear least squares optimization or closed-form methods for small angles only are often used. 12 Note that all iterative solutions require an initial approximation of the geometrical transformation, and they usually possess poor convergence, sometimes converge to a local minimum, and sometimes do not converge.A closed-form solution to the point-to-plane problem for an arbitrary affine transformation was proposed. [14][15][16] The affine approach works well when the correspondence between point clouds is good. In this case, the affine point-to-plane algorithm precisely reconstructs the original geometric transformation for arbitrary affine transformations, particularly for orthogonal transformations. 15,16 When the correspondence between clouds is poor, the affine approach is unable to recover the original orthogonal transformation. Note that the variational problem for orthogona...

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“…Finally, the translation vector is computed. The variational problem λ-ICP [35], NICP [7] and point-to-plane ICP [36] is solved similarly.…”