Three-dimensional signal processing: a new approach in dynamical sampling via tensor products
This work addresses signal recovery in 3D dynamical systems, offering a novel theoretical framework for applications like imaging or sensing, but it appears incremental as it extends known 1D and 2D methods to 3D with tensor products.
The paper tackles the dynamical sampling problem for three-dimensional signals by characterizing evolution via tensor products, providing a necessary condition for sampling sets and reformulating reconstruction as an efficient optimization task, with numerical simulations demonstrating performance.
The dynamical sampling problem is centered around reconstructing signals that evolve over time according to a dynamical process, from spatial-temporal samples that may be noisy. This topic has been thoroughly explored for one-dimensional signals. Multidimensional signal recovery has also been studied, but primarily in scenarios where the driving operator is a convolution operator. In this work, we shift our focus to the dynamical sampling problem in the context of three-dimensional signal recovery, where the evolution system can be characterized by tensor products. Specifically, we provide a necessary condition for the sampling set that ensures successful recovery of the three-dimensional signal. Furthermore, we reformulate the reconstruction problem as an optimization task, which can be solved efficiently. To demonstrate the effectiveness of our approach, we include some straightforward numerical simulations that showcase the reconstruction performance.