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For visually aided industrial robots, the transformation between the mounted visual information system and the end-effector must be calibrated prior to the practical use. However, during the data acquisition stage, measurement uncertainties will lead to inevitable mismatched data. This article proposes a novel correspondence matching method for the hand-eye calibration system of the type $\boldsymbol {AX} = \boldsymbol {XB}$. The correspondence matching refers to the problem of finding out the well-matched $\boldsymbol {A}$ and $\boldsymbol {B}$ and the relative time delay between them, from a series of measurements. A neat Lie algebra formulation has been obtained to cast the original problem into a high-dimensional point-cloud registration problem. This solution allows for the simultaneous solution of the hand-eye calibration, correspondence matching, and time-delay estimation as well as the uncertainty description of the obtained results. Synthetic simulations verify the correctness of the proposed method. Experimental studies have also confirmed its practical feasibility on an automatic welding platform. [ABSTRACT FROM AUTHOR]