A robust confirmable watermarking algorithm for 3D mesh based on manifold harmonics analysis

Owing to the manifold harmonics analysis, a robust non-blind spectral watermarking algorithm for a two-manifold mesh is presented, which can be confirmed by a trusted third party. Derived from the Laplace–Beltrami operator, a set of orthogonal manifold harmonics basis functions is first adopted to span the spectral space of the underlying three-dimensional (3D) mesh. The minimal number of the basis functions required in the proposed algorithm is also determined, which can effectively accelerate the spectrum computations. Then, to assert ownership and resist 3D mesh forging, a digital signature algorithm is adopted to sign the watermark in the embedding phase and to verify the signature in the extraction phase, which could optimize the robust non-blind spectral watermarking algorithm framework. To improve the robustness of the embedded watermark signature, the input 3D mesh will be segmented into patches. The watermark signature bits are embedded into the low-frequency spectral coefficients of all patches repeatedly and extracted with regard to the corresponding variations of their coefficients. Extensive experimental results demonstrate the efficiency, invisibility, and robustness of the proposed algorithm. Compared with existing watermarking algorithms, our algorithm exhibits better visual quality and is more robust to resist various geometric and connectivity attacks.