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Inspired by reversible data hiding methods that applied the PE (predicted error) strategy will have the advantage of good performances. It means that the closer the PE approaches its climax, the higher embedding capacity and image quality will have. This paper provides four stages of two-phase reversible data hiding method to produce prediction images with Lagrange interpolation on odd and even pixels, and to hide confidential information into the differences between the predicted image and the original image. Lagrange interpolation will be crossly applied on odd and even pixels to produce predicted images, and then histogram shifting modification will be applied to embed secret data into. The experimental results demonstrate the quality of Lena image from this scheme has a PSNR value of 43.97 dB and an embedding capacity of 0.38 bpp, for one level. For enhancing the embedding capacity to 0.98 bpp, our scheme can still maintain a PSNR value of 31.14 dB. From the demonstrated results, it can be obvious to know that our proposed algorithm has superior performances in terms of image quality and embedding capacity than that the algorithms of Ni et al. and Tai et al. [ABSTRACT FROM AUTHOR]