Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the “true” geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem.
Discrete Laplacian Operator and Its Applications in Signal Processing
