A dual watermarking scheme based on Sobolev type orthogonal moments, Charlier and Meixner, is proposed based on different discrete measures. The existing relation through the connection formulas allows to provide with structure and recurrence relations, together with two difference equations satisfied by such families. Weighted polynomials derived from them are being applied in an embedding and extraction watermarking algorithm, comparing the results obtained in imperceptibly and robustness tests with other families of polynomials.

翻译：基于不同离散测度，提出了一种以Sobolev型正交矩（Charlier矩与Meixner矩）为基础的双重水印方案。通过连接公式所建立的现有关系，可提供结构与递推关系，以及此类正交多项式族满足的两个差分方程。由这些多项式导出的加权多项式被应用于水印嵌入与提取算法中，并在不可感知性与鲁棒性测试中与其他多项式族所获得的结果进行了比较。