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

2025-02-04

AI Summary

This paper addresses dynamic sampling and reconstruction of three-dimensional time-varying signals under noise. We consider systems whose evolution is governed by tensor-product operators, where conventional methods struggle with high-dimensional spatiotemporal coupling. First, we extend dynamic sampling theory to 3D tensor-evolving systems—establishing, for the first time, necessary sampling conditions guaranteeing stable signal reconstruction. Second, we formulate a convex optimization model incorporating tensor-structured priors, enabling efficient and robust recovery. Theoretical analysis derives a minimal sampling density requirement for the measurement set, ensuring identifiability. Numerical experiments demonstrate that the proposed method achieves high reconstruction accuracy and strong robustness against noise, significantly improving both efficiency and reliability in reconstructing 3D dynamic signals compared to existing approaches.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: 3D Computer Vision
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

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.

Problem

• Reconstructing 3D dynamical signals
• Using tensor product characterization
• Optimizing for efficient signal recovery

Innovation

• Tensor product dynamical sampling
• Optimization-based signal reconstruction
• Three-dimensional signal recovery