图信号的采样与重构理论研究

发布时间：2018-04-30 18:26

本文选题：图信号 + 图信号处理　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：随着信息技术的飞速发展,人们迈入了海量信息时代,产生了来自不同场景数据的处理需求,比如大数据、信息网络、传感器网络、社交网络等数据类型。在这些问题中,需要处理的往往是网状的高维度数据,与传统的时域或空域信号相比具有不规则的拓扑结构,为传统的信号处理方法带来了挑战。为了刻画和处理这类结构复杂的数据,“图信号”的概念便应运而生。这种新的信号形式定义于加权图上:数据间的拓扑结构被抽象为加权图,将信号值分别映射在加权图的各顶点上,即形成图信号。图信号处理主要研究图信号的表示、分析和变换等概念与方法,通过加权图揭示信号之间的相互作用和联系,将传统数字信号处理理论扩展到不规则的图信号上,为处理结构复杂的数据提供了有效手段,在生物医学、计算机视觉、机器学习等领域广泛应用。经典采样理论在传统数字信号处理中发挥着重要的作用。同样,图信号的采样与重构理论在图信号处理中也扮演着重要的角色,然而其采样却比传统采样更为复杂。这是因为图信号的底层结构是随机的不规则加权图,其顶点序号可随机排列,因此无法按顶点的序号均匀获取采样值;图信号在其变换域也无法明确定义频谱混叠效应。鉴于此,图信号采样与重构理论的研究是十分必要的。本文重点研究了图信号的采样与重构问题。在现有的图信号处理相关理论的基础上,本文首先综述了图信号和图傅里叶变换的基本概念,分析归纳了图信号的基本性质、运算及定理,揭示了图信号变换的机理及其信号处理的基本原理;其次给出了传统离散时间信号的图信号描述,证明了离散时间信号的DFT变换是图傅里叶变换的简单特例,并从信号空间的角度阐述了经典香农采样定理的机理,从而将其推广至离散时间信号,构建了环形图信号的采样定理,为研究一般图信号的采样提供了切入点;最后,以有限维离散信号的采样和插值理论为基础,本文进一步从信号空间的角度,根据图信号采样空间和插值空间的关系,建立了一般的图信号采样定理,讨论了该定理的基本性质,并归纳总结了实现图信号无损失恢复的条件和步骤,通过必要的数值举例和仿真分析具体说明了实现过程,验证了理论结果的正确性。
[Abstract]:With the rapid development of information technology, people have entered a mass of information era, resulting in the processing needs from different scenarios, such as big data, information networks, sensor networks, social networks and other data types. Among these problems, the high-dimensional data often needs to be processed, which has irregular topology compared with the traditional signal in time domain or spatial domain, which brings challenges to the traditional signal processing methods. In order to describe and deal with this kind of complicated data, the concept of "graph signal" came into being. The new signal form is defined on the weighted graph: the topological structure between the data is abstracted into the weighted graph, and the signal values are mapped to each vertex of the weighted graph, that is, the graph signal is formed. Image signal processing mainly studies the concepts and methods of the representation, analysis and transformation of graph signals. The traditional digital signal processing theory is extended to irregular graph signals through the weighted graph to reveal the interaction and relationship between the signals. It is widely used in biomedicine, computer vision, machine learning and so on. Classical sampling theory plays an important role in traditional digital signal processing. Similarly, the theory of graph signal sampling and reconstruction also plays an important role in graph signal processing, but its sampling is more complex than traditional sampling. This is because the underlying structure of the graph signal is a random irregular weighted graph, the vertex ordinal number can be arranged randomly, so the sampling value can not be obtained uniformly by the ordinal number of the vertex, and the spectrum aliasing effect can not be clearly defined in the transformation domain of the graph signal. In view of this, it is necessary to study the theory of graph signal sampling and reconstruction. This paper focuses on the sampling and reconstruction of graph signals. Based on the existing theories of graph signal processing, this paper first summarizes the basic concepts of graph signal and graph Fourier transform, and analyzes and summarizes the basic properties, operations and theorems of graph signal. The mechanism of graph signal transformation and the basic principle of signal processing are revealed. Secondly, the graph signal description of traditional discrete time signal is given, and the DFT transform of discrete time signal is proved to be a simple special case of graph Fourier transform. From the point of view of signal space, the mechanism of classical Shannon sampling theorem is expounded, which is extended to discrete time signal, and the sampling theorem of ring graph signal is constructed, which provides a breakthrough point for studying the sampling of general graph signal. Based on the theory of sampling and interpolation of finite dimensional discrete signals, a general sampling theorem of graph signals is established from the point of view of signal space, according to the relationship between sampling space and interpolation space of graph signals. The basic properties of the theorem are discussed, and the conditions and steps to realize the lossless recovery of the graph signal are summarized. The realization process is illustrated by the necessary numerical examples and simulation analysis, and the correctness of the theoretical results is verified.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TN911.73

【参考文献】