论文标题
用于光谱处理的基于傅立叶和理性的图形过滤器
Fourier-based and Rational Graph Filters for Spectral Processing
论文作者
论文摘要
数据表示为多种应用中的图形，例如计算机视觉（例如，图像）和图形（例如3D网格），网络分析（例如社交网络）和生物信息学（例如分子）。在这种情况下，我们的总体目标是由理性多项式为图处理引起的新型基于傅立叶和图形过滤器的定义，这些滤波器概括了多项式过滤器，而傅立叶变换为非欧几里得域。为了有效评估离散傅立叶基于傅立叶和小波的算子，我们引入了一种无光谱的方法，该方法需要解决一小部分稀疏，对称，条件良好的线性系统的解决方案，并忽略了Laplacian或Kernel Spectrum的评估。使用有理多项式近似任意的图形过滤器为多项式提供了更准确和数值稳定的替代方案。为了实现这些目标，我们还研究了光谱运算符，小波和通过光谱内核引起的积分运算符过滤的卷积之间的联系。根据我们的测试，提出的方法的主要优势是（i）在输入数据（例如，图形，3D形状）方面的一般性，应用程序（例如，信号重建和形状对应关系，形状对应关系）和过滤器（例如，多项式，理性多项式）和（ii）和（ii）和（ii）和（ii）consectry conpercation和spectry comperation Apectry Computation and Spectry Computation and Spectry Computation and Spectry Computation and Spectry Computation and Spectry Compation和（ii）complational Comperation和（ii）均等计算。
Data are represented as graphs in a wide range of applications, such as Computer Vision (e.g., images) and Graphics (e.g., 3D meshes), network analysis (e.g., social networks), and bio-informatics (e.g., molecules). In this context, our overall goal is the definition of novel Fourier-based and graph filters induced by rational polynomials for graph processing, which generalise polynomial filters and the Fourier transform to non-Euclidean domains. For the efficient evaluation of discrete spectral Fourier-based and wavelet operators, we introduce a spectrum-free approach, which requires the solution of a small set of sparse, symmetric, well-conditioned linear systems and is oblivious of the evaluation of the Laplacian or kernel spectrum. Approximating arbitrary graph filters with rational polynomials provides a more accurate and numerically stable alternative with respect to polynomials. To achieve these goals, we also study the link between spectral operators, wavelets, and filtered convolution with integral operators induced by spectral kernels. According to our tests, main advantages of the proposed approach are (i) its generality with respect to the input data (e.g., graphs, 3D shapes), applications (e.g., signal reconstruction and smoothing, shape correspondence), and filters (e.g., polynomial, rational polynomial), and (ii) a spectrum-free computation with a generally low computational cost and storage overhead.
