Innovation Rate Sampling of Pulse Streams with Application to Ultrasound Imaging (1003.2822v4)

Abstract: Signals comprised of a stream of short pulses appear in many applications including bio-imaging and radar. The recent finite rate of innovation framework, has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals. Unfortunately, for high rates of innovation, existing sampling schemes are numerically unstable. In this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses. We begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples. We then design a compactly supported class of filters, satisfying this condition. The periodic solution is extended to finite and infinite streams, and is shown to be numerically stable even for a large number of pulses. High noise robustness is also demonstrated when the delays are sufficiently separated. Finally, we process ultrasound imaging data using our techniques, and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved.

• The paper extends the Finite Rate of Innovation framework and introduces novel Sum of Sincs filters for stable, low-rate sampling and reconstruction of pulse streams, addressing instability issues.
• The proposed method is theoretically validated for periodic and non-periodic pulse streams, demonstrating numerical stability even with noise when pulses are well-separated.
• Applied to ultrasound imaging, this framework enables data rate reductions of up to two orders of magnitude, paving the way for more efficient and portable devices.

Innovation Rate Sampling of Pulse Streams: A Methodological Advancement in Ultrasound Imaging

The paper authored by Ronen Tur, Yonina C. Eldar, and Zvi Friedman presents a comprehensive paper on innovation rate sampling of pulse streams with particular emphasis on ultrasound imaging. This research identifies the challenges associated with traditional high-rate sampling methods and proposes a novel framework for achieving efficient low-rate sampling without compromising accuracy or stability.

Summary of Core Contributions

The central contribution of this paper lies in extending the Finite Rate of Innovation (FRI) framework to effectively sample and reconstruct signals composed of streams of short pulses. The authors tackle the inherent instability issues of previous methods when applied to high innovation rates. Their innovative approach involves designing compactly supported filters, termed Sum of Sincs (SoS), which meet specific conditions to ensure perfect reconstruction. The paper demonstrates that this class of filters is both efficient and robust, even in the presence of large numbers of closely spaced pulses.

Stable Recovery and Filter Design

A significant portion of the paper is dedicated to the development and theoretical validation of the SoS filter design, which satisfies a defined mathematical condition to enable accurate retrieval of periodic pulse streams from a minimum number of samples. The authors further broaden the applicability of their results by extending the periodic solution to both finite and infinite pulse streams. This extension proves numerically stable even in scenarios with considerable noise, particularly when time delays are well-separated.

Implications and Practical Application

One of the most impactful applications of this research is in the domain of medical ultrasound imaging. The traditional ultrasound sampling methods are data-intensive, leading to high computational costs and power consumption. Tur, Eldar, and Friedman demonstrate that their proposed sampling scheme can achieve substantial reductions in data rate by two orders of magnitude compared to conventional methods. These improvements imply significant potential benefits for portable and power-efficient ultrasound devices, enhancing accessibility and point-of-care diagnostics.

Speculation on Future Directions

Given the promising results in the ultrasound context, it is plausible that this framework could be adapted for other fields characterized by pulse stream signals such as radar and telecommunication systems. Future research could explore the adaptability of these compactly supported filters under varying physical constraints or develop optimized filter design techniques tailored to specific industrial applications.

Concluding Remarks

The methodologies proposed by Tur, Eldar, and Friedman stand out for their balance of theoretical rigor and practical applicability. By addressing the instability of high innovation rate sampling and demonstrating robust performance under realistic conditions, this paper paves the way for more efficient signal processing frameworks. As systems continue to demand high accuracy with lower data rates, such advancements are crucial in driving the evolution of signal processing technologies.

Overall, this paper provides a valuable contribution to the field of applied signal processing, with significant implications for future research and development across various domains reliant on pulse stream data.