用张量积方法解决三维信号动态采样重建问题
Three-dimensional signal processing: a new approach in dynamical sampling via tensor products
- 基于张量积构建三维信号演化模型
- 给出确保信号可重建的采样集必要条件
- 适用于需高效重建三维动态信号的研究者
动态采样问题聚焦于从可能含噪的时空样本中重构随时间演化的信号。该领域在一维信号中已有深入研究，多维信号恢复也有所探索，但主要局限于驱动算子为卷积算子的情形。本文转向三维信号恢复场景，其中演化系统可用张量积表征。我们给出了确保三维信号成功重建的采样集必要条件，并将重构问题重构为可高效求解的优化任务。通过若干直观数值模拟，验证了所提方法在信号重建性能上的有效性。
原文摘要 · Abstract (English)
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.
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