Graph Fourier Transform with Negative Edges for Depth Image Coding

TL;DR

This paper introduces the signed graph Fourier transform (SGFT), a novel graph-based transform that incorporates negative edges to model anti-correlations, improving depth image coding performance.

Contribution

The paper develops SGFT, a new transform using negative edges in graphs to better capture anti-correlations in depth images, outperforming previous positive-edge-only methods.

Findings

SGFT approximates the optimal Karhunen-Loève Transform.

SGFT outperforms previous graph transforms in depth image coding.

Self-loops are crucial for ensuring positive semi-definiteness of the graph Laplacian.

Abstract

Recent advent in graph signal processing (GSP) has led to the development of new graph-based transforms and wavelets for image / video coding, where the underlying graph describes inter-pixel correlations. In this paper, we develop a new transform called signed graph Fourier transform (SGFT), where the underlying graph G contains negative edges that describe anti-correlations between pixel pairs. Specifically, we first construct a one-state Markov process that models both inter-pixel correlations and anti-correlations. We then derive the corresponding precision matrix, and show that the loopy graph Laplacian matrix Q of a graph G with a negative edge and two self-loops at its end nodes is approximately equivalent. This proves that the eigenvectors of Q - called SGFT - approximates the optimal Karhunen-Lo`eve Transform (KLT). We show the importance of the self-loops in G to ensure Q is positive semi-definite. We prove that the first eigenvector of Q is piecewise constant (PWC), and thus can well approximate a piecewise smooth (PWS) signal like a depth image. Experimental results show that a block-based coding scheme based on SGFT outperforms a previous scheme using graph transforms with only positive edges for several depth images.