Manifold Graph Signal Restoration using Gradient Graph Laplacian Regularizer

In the graph signal processing (GSP) literature, graph Laplacian regularizer (GLR) was used for signal restoration to promote smooth reconstructions with respect to the underlying graph -- typically signals that are (piecewise) constant. However, for graph signals that are (piecewise) planar, GLR may suffer from the well-known "staircase" effect. In this paper, focusing on manifold graphs -- sets of uniform discrete samples on low-dimensional continuous manifolds -- we generalize GLR to gradient graph Laplacian regularizer (GGLR) that provably promotes piecewise planar (PWP) signal reconstruction. Specifically, for a graph endowed with latent space coordinates (e.g., 2D images, 3D point clouds), we first define a gradient operator, using which we construct a higher-order gradient graph for the computed gradients in each latent dimension. This maps to a gradient-induced nodal graph (GNG) and a Laplacian matrix for a signed graph that is provably positive semi-definite (PSD), thus suitable for quadratic regularization. For manifold graphs without explicit latent coordinates, we propose a fast parameter-free spectral method to first compute latent space coordinates for graph nodes based on generalized eigenvectors. We derive the means-square-error minimizing weight parameter for GGLR efficiently, trading off bias and variance of the signal estimate. Experimental results show that GGLR outperformed previous graph signal priors like GLR and graph total variation (GTV) in a range of graph signal restoration tasks.