In this paper, we propose a new reversible watermarking scheme. Onefirst contribution is a histogram shifting modulation which adaptively takes care of the local specficities of the image content. By applying it to the image prediction-errors and by considering their immediate neighborhood, the scheme we propose inserts data in textured areas where other methods fail to do so. Furthermore, our scheme makes use of a classification process for identifying parts of the image that can be watermarked with the most suited reversible modulation. This classfication is based on a reference image derived from the image itself, a prediction of it, which has the property of being invariant to the watermark insertion. In that way, the watermark embedder and extractor remain synchronized for message extraction and image reconstruction. The experiments conducted so far, on some natural images and on medical images from different modalities, show that for capacities smaller than 0.4 bpp, our method can insert more data with lower distortion than any existing schemes. For the same capacity, we achieve a peak signal-to-noise ratio (PSNR) of about 1-2 dB greater than with the scheme of Hwang et al., the most efficient approach actually.