We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.

通过引入图拉普拉斯特征向量之间的“自然”距离，而不是简单地使用特征值排序，我们为利用图的“双”域的图上的信号引入了一套新颖的多尺度基变换。这些基本字典可以看作是经典Shannon小波包字典对任意图的推广，并且不依赖于Laplacian特征值的频率解释。我们描述了构建这些基本字典的算法（涉及矢量旋转或正交化），使用它们通过最佳的基本搜索有效地逼近图信号，并展示了这些基本字典对于在向日葵图和街道网络上测量的图信号的优势。