L-1 Norm Minimization in Partial Errors-In-Variables Model

The weighted total least-squares (WTLS) estimate is sensitive to outliers and will be strongly disturbed if there are outliers in the observations and coefficient matrix of the partial errors-in-variables (EIV) model. The L 1 norm minimization method is a robust technique to resist the bad effect of outliers. Therefore, the computational formula of the L 1 norm minimization for the partial EIV model is developed by employing the linear programming theory. However, the closed-form solution cannot be directly obtained since there are some unknown parameters in constrained condition equation of the presented optimization problem. The iterated procedure is recommended and the proper condition for stopping iteration is suggested. At the same time, by treating the partial EIV model as the special case of the non-linear Gauss–Helmert (G–H) model, another iterated method for the L 1 norm minimization problem is also developed. At last, two simulated examples and a real data of 2D affine transformation are conducted. It is illustrated that the results derived by the proposed L 1 norm minimization methods are more accurate than those by the WTLS method while the observations and elements of the coefficient matrix are contaminated with outliers. And the two methods for the L 1 norm minimization problem are identical in the sense of robustness. By comparing with the data-snooping method, the L 1 norm minimization method may be more reliable for detecting multiple outliers due to masking. But it leads to great computation burden.