Manifold Graph Signal Restoration using Gradient Graph Laplacian Regularizer

TL;DR

This paper introduces a gradient graph Laplacian regularizer (GGLR) for manifold graph signal restoration, addressing limitations of traditional graph Laplacian regularizers by promoting piecewise planar signals and reducing staircase effects.

Contribution

The paper generalizes the graph Laplacian regularizer to a gradient-based version for manifold graphs, including a method for embedding graphs without explicit coordinates and deriving optimal weight parameters.

Findings

GGLR outperforms GLR and GTV in restoration tasks

Effective for signals on low-dimensional manifolds

Handles graphs without explicit sampling coordinates

Abstract

In the graph signal processing (GSP) literature, graph Laplacian regularizer (GLR) was used for signal restoration to promote piecewise smooth / constant reconstruction with respect to an underlying graph. However, for signals slowly varying across graph kernels, GLR suffers from an undesirable "staircase" effect. In this paper, focusing on manifold graphs -- collections of uniform discrete samples on low-dimensional continuous manifolds -- we generalize GLR to gradient graph Laplacian regularizer (GGLR) that promotes planar / piecewise planar (PWP) signal reconstruction. Specifically, for a graph endowed with sampling coordinates (e.g., 2D images, 3D point clouds), we first define a gradient operator, using which we construct a gradient graph for nodes' gradients in sampling manifold space. This maps to a gradient-induced nodal graph (GNG) and a positive semi-definite (PSD) Laplacian matrix with planar signals as the 0 frequencies. For manifold graphs without explicit sampling coordinates, we propose a graph embedding method to obtain node coordinates via fast eigenvector computation. 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.