Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Graph-Based Transform for 2D Piecewise Smooth Signals With Random Discontinuity Locations

Abstract: The graph-based block transform recently emerged as an effective tool for compressing some special signals such as depth images in 3D videos. However, in existing methods, overheads are required to describe the graph of the block, from which the decoder has to calculate the transform via time-consuming eigendecomposition. To address these problems, in this paper, we aim to develop a single graph-based transform for a class of 2D piecewise smooth signals with similar discontinuity patterns. We first consider the deterministic case with a known discontinuity location in each row. We propose a 2D first-order autoregression (2D AR1) model and a 2D graph for this type of signals. We show that the closed-form expression of the inverse of a biased Laplacian matrix of the proposed 2D graph is exactly the covariance matrix of the proposed 2D AR1 model. Therefore, the optimal transform for the signal are the eigenvectors of the proposed graph Laplacian. Next, we show that similar results hold in the random case, where the locations of the discontinuities in different rows are randomly distributed within a confined region, and we derive the closed-form expression of the corresponding optimal 2D graph Laplacian. The theory developed in this paper can be used to design both pre-computed transforms and signal-dependent transforms with low complexities. Finally, depth image coding experiments demonstrate that our methods can achieve similar performance to the state-of-the-art method, but our complexity is much lower.

Nonparametric High-dimensional K-sample Comparison

Abstract: High-dimensional k-sample comparison is a common applied problem. We construct a class of easy-to-implement nonparametric distribution-free tests based on new tools and unexplored connections with spectral graph theory. The test is shown to possess various desirable properties along with a characteristic exploratory flavor that has practical consequences. The numerical examples show that our method works surprisingly well under a broad range of realistic situations.

ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem

Abstract: We review a recent algorithm for localization of points in Euclidean space from a sparse and noisy subset of their pairwise distances. Our approach starts by extracting and embedding uniquely realizable subsets of neighboring sensors called patches. In the noise-free case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation, and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. In other words, to every patch, there corresponds an element of the Euclidean group Euc(3) of rigid transformations in \({\mathbb{R}}^{3}\), and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The algorithm is scalable as the number of nodes increases, and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity and running time.

A Spectral Graph Theoretical Approach to Oriented Energy Features

Abstract: In this work, we propose a novel method for determining oriented energy features of an image. Oriented energy features, useful for many machine vision applications like contour detection, texture segmentation and motion analysis, are determined from the filters whose outputs are enhanced at the edges of the image at a given orientation. We use the eigenvectors and eigenvalues of graph Laplacian for determining the oriented energy features of an image. Our method is based on spectral graph theoretical approach in which a graph is assigned complex-valued edge weights whose phases encode orientation information. These edge weights give rise to a complex-valued Hermitian Laplacian whose spectrum enables us to extract oriented energy features of the image. We perform a set of numerical experiments to determine the efficiency and characteristics of the proposed method. In addition, we apply our feature extraction method to texture segmentation problem. We do this in comparison with other known methods, and show that our method performs better for various test textures.

On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

Abstract: Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy ( energy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from its energy.