图形信号处理：顶点乘法
Graph Signal Processing: Vertex Multiplication
摘要：
在经典信号处理的欧几里德域上，将信号样本链接到底层坐标结构是直接的。虽然图邻接矩阵完全定义了底层图顶点之间的定量关联，但图信号处理中的主要问题是缺乏顶点与底层定量坐标结构的显式关联。为了使这个链接，我们提出了一个操作，称为顶点乘法，这是为图形定义的，可以对图形信号进行操作。顶点乘法是对时间序列信号中坐标乘法运算的推广，它可以被解释为一种为图指定坐标结构的运算符。利用微分的图域扩张和图傅里叶变换（GFT），定义了顶点乘，使其具有傅里叶对偶性，即微分和坐标乘运算在傅里叶变换下是相互独立的.所提出的定义减少到坐标乘法对应的时间序列的图形。最后给出了数值算例。
英文摘要：
On the Euclidean domains of classical signal processing, linking of signal samples to the underlying coordinate structure is straightforward. While graph adjacency matrices totally define the quantitative associations among the underlying graph vertices, a major problem in graph signal processing is the lack of explicit association of vertices with an underlying quantitative coordinate structure. To make this link, we propose an operation, called the vertex multiplication, which is defined for graphs and can operate on graph signals. Vertex multiplication, which generalizes the coordinate multiplication operation in time series signals, can be interpreted as an operator which assigns a coordinate structure to a graph. By using the graph domain extension of differentiation and graph Fourier transform (GFT), vertex multiplication is defined such that it shows Fourier duality, which states that differentiation and coordinate multiplication operations are duals of each other under Fourier transformation (FT). The proposed definition is shown to reduce to coordinate multiplication for graphs corresponding to time series. Numerical examples are also presented.
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