Reversible watermarking methods enable auxiliary data to be embedded in data sets, such as images, audio, video and software. The reversible nature of the method enables the original data set to be perfectly restored. Control systems with feedback loops are used to optimize embedding based on distortion or auxiliary data capacity constraints. The watermarking may be applied recursively to embed several layers, where subsequent layers are embedded into a previously watermarked data set. To recover the original data, each layer is extracted and the data restored in reverse order of the embedding. Sets of elements that are expanded to carry auxiliary data in each layer overlap or are interleaved to maximize embedding capacity or quality of the host data.