CGD-Based Inpainting Algorithm for Time-Varying Signals on Strong Product Graph

A multi-dimensional signal on graph has different features along different directions/dimensions. For example, image signal is isotropic along horizontal and vertical directions, and time-varying signal presents different correlation characters in the time and vertex domains. Nevertheless, the current graph signal processing concentrates on the single shift, which makes it difficult to differentiate the correlation features along different directions. In this paper, we define a multi-shift notion for the time-varying signals on graphs enabling us to separately analyze the multi-dimensional signal along different directions captured by diverse shifts. Furthermore, the multi-shift notion is leveraged to formulate the inpainting problem for time-varying signals on the strong product graph, which can be exploited to characterize three different kinds of elements interaction of time-varying signals by using three different kinds of shifts. The conjugate gradient descent algorithm is further deployed to solve the inpainting problem. Numerical experiments conducted on the synthetic signal and real-world data show the potentiality of the multi-shift representation and the effectiveness of the inpainting algorithm.