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.

翻译：数据以图解形式呈现在广泛的应用中,例如计算机视野(例如图像)和图形(例如3D模版)、网络分析(例如3D模版)、网络分析(例如社交网络)和生物信息学(例如分子)。在这方面,我们的总体目标是界定由理性多元图处理法引出的基于Fourier和图形的新过滤器,这些过滤器一般地将多边光学过滤器和Fourier转换为非欧洲大陆域。为了有效地评价离散光谱、基于Fourier和波盘的对流操作者,我们采用了无频分析(例如社交网络)、网络分析(例如社交网络)和生物信息信息学(例如分子)。在这方面,我们的总目标是界定由理性的多元线谱处理器或内核谱处理引出的新的Fourier和图形过滤过滤器,这些过滤器提供了与多元度有关的更准确和数字稳定的替代器。为了实现这些目标,我们还研究光谱光谱光谱光谱光谱光谱光谱、波段、波段计算法和透视器的图像应用之间的链接(例如光谱操作者、电路路路段、电路测算、电路段的计算和过滤器),它们与总体的精测测算。