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The iterative closest point (ICP) algorithm is an accurate approach for the registration between two point sets on the same scale. However, it can not handle the case with different scales. This paper proposes a fast and robust ICP algorithm for isotropic scaling point sets registration (FRISICP). In order to accurately and directly estimate the scale factor without any constraints, we introduce a bidirection distance measurement method into the least square (LS) problem. Then to keep computational efficiency when the number of points in the set increasing, we further introduce a sparse-to-dense hierarchical model in ICP algorithm to speed up the isotropic scaling point set matching process. Experimental results demonstrate that the proposed FRISICP method outperforms other algorithms on both 2D and 3D point sets.