Fusion frame, relay fusion frame and signal reconstruction

In this paper, we demonstrate that any signal exhibiting a certain sparse pattern can be recovered in a stable and resilient manner through the utilization of the fusion frame approach. The theoretical analysis highlights that the deviation of the approximate solution is effectively controlled. Furthermore, the adoption of different norms contributes to further reinforcing the guarantees of robustness and stability. Driven by the ideas of compressed sensing and fusion frames, we extend the setting to relay fusion frames. With the help of operator theory, we provide several recovery guarantee conditions based on the relay fusion frames. Finally, the relationship between relay fusion frames and compressed sensing is elucidated.