Bias-Corrected Weighted Total Least-Squares Adjustment of Condition Equations

Abstract

The total least-squares (TLS) method and its variations have recently received increasing research attention. However, little attention has been given to the weighted TLS adjustment method with condition equations. In this paper, a weighted TLS method designed for condition equations (WTLSC) is presented with the assumption that both the observation vector and design matrix contain errors. The covariance matrices are estimated for both the observation vector and design matrix after the adjustment, and the biases are corrected for the adjusted observation vector, design matrix, and corresponding covariance matrices in the WTLSC method. The proposed approach was used in an adjustment problem of an object point photographed by three terrestrial cameras. The results show that the proposed method resolves the condition equations with errors in the design matrix without linearization in the case study. The proposed WTLSC method generates stable error vector and matrix for the observation vector and design matrix, which satisfy the condition equation in the repeated simulation experiments. The results also show that there are biases in the adjusted observation vector, design matrix, and corresponding covariance matrices, although the biases are small in the case study.