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Three-dimensional signal processing: a new approach in dynamical sampling via tensor products

Yisen Wang, Hanqin Cai, Longxiu Huang

TL;DR

This work addresses reconstructing a three-dimensional signal that evolves under a tensor-tensor dynamical system from sparse spatio-temporal samples. It shows that a necessary condition on the sampling set, together with a frequency-domain, decoupled optimization formulation, enables efficient and accurate recovery of the initial signal from limited measurements. The method cast the reconstruction as a set of independent least-squares problems across the second mode, solved via linear systems in the frequency domain, and then transformed back to the spatial domain. Numerical experiments on synthetic data reveal an optimal total sampling time and demonstrate that recovery is robust at moderate sampling rates, while also validating a conjectured coverage condition on the sampling indices.

Abstract

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.

Three-dimensional signal processing: a new approach in dynamical sampling via tensor products

TL;DR

This work addresses reconstructing a three-dimensional signal that evolves under a tensor-tensor dynamical system from sparse spatio-temporal samples. It shows that a necessary condition on the sampling set, together with a frequency-domain, decoupled optimization formulation, enables efficient and accurate recovery of the initial signal from limited measurements. The method cast the reconstruction as a set of independent least-squares problems across the second mode, solved via linear systems in the frequency domain, and then transformed back to the spatial domain. Numerical experiments on synthetic data reveal an optimal total sampling time and demonstrate that recovery is robust at moderate sampling rates, while also validating a conjectured coverage condition on the sampling indices.

Abstract

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.

Paper Structure

This paper contains 11 sections, 1 theorem, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Main results
  5. Necessary condition
  6. Method development for signal recovery
  7. Experiments
  8. Datasets
  9. Recovery accuracy
  10. Optimal maximum sampling time T
  11. Verifying the conjecture

Key Result

Theorem 3.1

Suppose $\mathcal{F} \in \mathbb{C}^{m \times p \times n}$ and $\mathcal{A} \in \mathbb{C}^{m \times m \times n}$. And suppose that the signal at time $t$ follows the transformation specified in eqn:evolution rule. Then, the recovery of $\mathcal{F}$ from $\Psi$, with $\Omega = I \times J \times [n]

Figures (5)

  • Figure 1: Relative error v.s. sampling rate $\alpha$: we repeated this experiment 10 times, calculating both the mean value and the standard deviation of the results. As shown by the shadow, the standard deviation is relatively small, indicating that our method consistently performs well across different trials.
  • Figure 2: This figure shows the point-wise gap between the reconstructed signal and the ground truth signal, there are $20\times 15\times 5=1500$ sample points in total, we compared each point from the constructed tensor with the ground truth tensor.
  • Figure 3: Relative error v.s. maximum sampling time $T$ under different noise levels and sampling rates
  • Figure 4: Condition number for different T
  • Figure 5: Undersampled second dimension

Theorems & Definitions (5)

  • Definition 1: t-product
  • Definition 2
  • Theorem 3.1
  • proof
  • Conjecture 1