Convergence analysis and validation of low cost distance metrics for computational cost reduction of the Iterative Closest Point algorithm

Shape registration is a key stage in the process of reconstruction or acquisition of three-dimensional surfaces so that it is considered a cornerstone in the fields of computer vision and graphics. It is also a task of major importance in many other fields like biometric applications [1], 3D model reconstruction from multiple range images [2], medical diagnostic support tools and computer aided interventions [3], quality control of manufactured pieces [4] and robotic applications ranging from full environment reconstruction for navigation to particular objects 3D model creation and tracking [5]. A remarkable study of the different registration techniques both rigid and non-rigid is the work by Tam et al. [6] which includes an in depth review of the registration problem; other important surveys are the ones by Van Kaik et al. [7] (focused on shape correspondence techniques) and Audette et al. [3] (centered on shape registration applied to medical images). In addition, registration can be classified according to its granularity, distinguishing coarse and fine grained methods. The objective of coarse grained registration is to obtain a quick estimate of the transformation to roughly align both shapes while fine grained techniques use that initial estimate to refine it iteratively in order to find the best alignment in terms of precision under a set of restrictions. Reviews of multiple techniques and methods that can be applied to solve both coarse and fine problems have been carried out by Salvi et al. [8] paying special attention to precision, robustness and efficiency assessment. In this work we will deal with the ICP algorithm which is a fine grained registration method which is currently the most popular algorithm for 3D rigid registration for the Robotics community; the cause of that popularity is mainly its simplicity and effectiveness as well as the many variants that have been developed, adapting the ICP to different scenarios to improve its efficiency and precision [9]. It is a fact that any general improvement of the algorithm which is able to accelerate its execution, without affecting the quality nor reducing its possible application scenarios, is a step forward for all those applications with needs of precision and quality in their rigid registrations. In that sense, this work proposes a general improvement for the algorithm, carried out by an interdisciplinary research based on the fusion of mathematical and geometric concepts, such as distance metrics, with its computational component by taking into account their associated operative cost and their impact on the algorithm’s execution time. Our research group has experience and successful examples of this kind of research about accelerating mathematical methods at a low level [10, 11]. The hypothesis of the improvement is the following one: distance metrics whose computational cost is reduced respect to the Euclidean’s one, like Manhattan or Chebyshev distances, may replace it and effectively reduce the computational cost while preserving the algorithm’s convergence properties as well as the registration quality in terms of convergence domain and final registration error.