Abstract
In geodetic surveying, input data from two coordinates are needed to compute rigid transformations. A common solution is a least-squares algorithm based on a Gauss–Markov model, called iterative closest point (ICP). However, the error in the ICP algorithm only exists in target coordinates, and the algorithm does not consider the source model’s error. A total least-squares (TLS) algorithm based on an errors-in-variables (EIV) model is proposed to solve this problem. Previous total least-squares ICP algorithms used a Euler angle parameterization method, which is easily affected by a gimbal lock problem. Lie algebra is more suitable than the Euler angle for interpolation during an iterative optimization process. In this paper, Lie algebra is used to parameterize the rotation matrix, and we re-derive the TLS algorithm based on a GHM (Gauss–Helmert model) using Lie algebra. We present two TLS-ICP models based on Lie algebra. Our method is more robust than previous TLS algorithms, and it suits all kinds of transformation matrices.
Highlights
An iterative closest point (ICP) algorithm [1] estimates a pose transformation matrix using coordinates from a source model and target model
We use a variant of the Gauss–Helmert [16] model, which is a general method to solve nonlinear total least-squares (TLS) problems without assumptions, and Lie algebra is used to parameterize the rotation matrix without the gimbal lock problem
ICP Algorithm Based on Lie Algebra In Section 2.1, we describe the fundamental ICP problem and introduce the TLS model to solve it
Summary
An iterative closest point (ICP) algorithm [1] estimates a pose transformation matrix using coordinates from a source model and target model. In [11], Ohta constructs a cost function using elements of the rotation matrix directly This method is convenient for obtaining results using existing methods such as [7,12]. It minimizes the cost function in Euclidean space, which does not conform to reality because the rotation matrix belongs to a special orthogonal group. We use a variant of the Gauss–Helmert [16] model, which is a general method to solve nonlinear TLS problems without assumptions, and Lie algebra is used to parameterize the rotation matrix without the gimbal lock problem.
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