Computational time-reversal imaging with a small number of random and noisy measurements

Computational time reversal imaging can be used to locate the position of multiple scatterers in a known background medium. The current methods for computational time reversal imaging are based on the null subspace projection operator, obtained through the singular value decomposition of the frequency response matrix. Here, we discuss the image recovery problem from a small number of random and noisy measurements, and we show that this problem is equivalent to a randomized approximation of the null subspace of the frequency response matrix.

💡 Research Summary

Computational time‑reversal imaging (TRI) exploits the physical principle that a wave field, once recorded and time‑reversed, refocuses at the locations of scatterers within a known background medium. Classical implementations rely on the full frequency‑response matrix (H(\omega)) obtained from an array of transmitters and receivers. By performing a singular‑value decomposition (SVD) of (H) and extracting the subspace associated with near‑zero singular values, one constructs the null‑space projection operator (P_{\mathcal N}=V_{\text{null}}V_{\text{null}}^{\dagger}). Applying (P_{\mathcal N}) to the measured data yields an image in which energy concentrates only at true scatterer positions, delivering high‑resolution localization.

The major practical bottleneck is the need to acquire the entire matrix (H). In realistic scenarios—high‑frequency electromagnetic or ultrasonic imaging, large‑scale sensor arrays, or portable devices—the number of required measurements can be prohibitive in terms of acquisition time, hardware cost, and data‑processing load. Moreover, each measurement is inevitably corrupted by additive noise, calibration errors, and environmental fluctuations. The paper addresses these challenges by proposing a randomized, noise‑robust approximation of the null space that works with a dramatically reduced set of measurements.

Key methodological steps

1. Random sub‑sampling – From the full matrix (H\in\mathbb{C}^{M\times N}) (with (M) transmit and (N) receive channels) a subset of columns (or rows) indexed by a random set (\Omega) of size (m\ll N) is selected. The sampling operator (P_{\Omega}) zero‑pads all unselected columns, yielding the reduced matrix (H_{\Omega}=P_{\Omega}H). The expectation satisfies (\mathbb{E}