本文選題：圖信號 + 圖信號處理��； 參考：《哈爾濱工業(yè)大學》2017年碩士論文

【摘要】：隨著信息技術的飛速發(fā)展,人們邁入了海量信息時代,產(chǎn)生了來自不同場景數(shù)據(jù)的處理需求,比如大數(shù)據(jù)、信息網(wǎng)絡、傳感器網(wǎng)絡、社交網(wǎng)絡等數(shù)據(jù)類型。在這些問題中,需要處理的往往是網(wǎng)狀的高維度數(shù)據(jù),與傳統(tǒng)的時域或空域信號相比具有不規(guī)則的拓撲結構,為傳統(tǒng)的信號處理方法帶來了挑戰(zhàn)。為了刻畫和處理這類結構復雜的數(shù)據(jù),“圖信號”的概念便應運而生。這種新的信號形式定義于加權圖上:數(shù)據(jù)間的拓撲結構被抽象為加權圖,將信號值分別映射在加權圖的各頂點上,即形成圖信號。圖信號處理主要研究圖信號的表示、分析和變換等概念與方法,通過加權圖揭示信號之間的相互作用和聯(lián)系,將傳統(tǒng)數(shù)字信號處理理論擴展到不規(guī)則的圖信號上,為處理結構復雜的數(shù)據(jù)提供了有效手段,在生物醫(yī)學、計算機視覺、機器學習等領域廣泛應用。經(jīng)典采樣理論在傳統(tǒng)數(shù)字信號處理中發(fā)揮著重要的作用。同樣,圖信號的采樣與重構理論在圖信號處理中也扮演著重要的角色,然而其采樣卻比傳統(tǒng)采樣更為復雜。這是因為圖信號的底層結構是隨機的不規(guī)則加權圖,其頂點序號可隨機排列,因此無法按頂點的序號均勻獲取采樣值;圖信號在其變換域也無法明確定義頻譜混疊效應。鑒于此,圖信號采樣與重構理論的研究是十分必要的。本文重點研究了圖信號的采樣與重構問題。在現(xiàn)有的圖信號處理相關理論的基礎上,本文首先綜述了圖信號和圖傅里葉變換的基本概念,分析歸納了圖信號的基本性質、運算及定理,揭示了圖信號變換的機理及其信號處理的基本原理;其次給出了傳統(tǒng)離散時間信號的圖信號描述,證明了離散時間信號的DFT變換是圖傅里葉變換的簡單特例,并從信號空間的角度闡述了經(jīng)典香農(nóng)采樣定理的機理,從而將其推廣至離散時間信號,構建了環(huán)形圖信號的采樣定理,為研究一般圖信號的采樣提供了切入點;最后,以有限維離散信號的采樣和插值理論為基礎,本文進一步從信號空間的角度,根據(jù)圖信號采樣空間和插值空間的關系,建立了一般的圖信號采樣定理,討論了該定理的基本性質,并歸納總結了實現(xiàn)圖信號無損失恢復的條件和步驟,通過必要的數(shù)值舉例和仿真分析具體說明了實現(xiàn)過程,驗證了理論結果的正確性。
[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.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：TN911.73

【參考文獻】

相關期刊論文 前1條

1 史雪松;馮輝;楊濤;胡波;;圖上低頻信號譜域變換中的邊權重優(yōu)化設計[J];復旦學報(自然科學版);2015年06期

相關博士學位論文 前2條

1 歐璐;圖譜理論在齒輪箱故障診斷中的應用研究[D];湖南大學;2016年

2 劉鵬飛;圖上信號的降維與重建方法研究[D];清華大學;2015年

相關碩士學位論文 前1條

1 楊漢鍵;基于圖信號處理的滾動軸承故障特征提取方法研究[D];湖南大學;2016年

，

本文編號：1825673