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Innovation Rate Sampling of Pulse Streams with Application to Ultrasound Imaging
Ronen Tur, Yonina C. Eldar, Zvi Friedman
TL;DR
Existing pulse-stream sampling schemes can become numerically unstable as innovation rates and pulse counts increase. This paper derives a general kernel condition, designs compactly supported filters, extends the approach across stream settings, and demonstrates stable recovery and substantial sample-rate reduction in ultrasound data.
Problem
Existing sampling schemes for finite-rate-of-innovation pulse streams are numerically unstable at high innovation rates, motivating stable recovery with many pulses.
Method
The paper derives a sampling-kernel condition for periodic streams, designs compactly supported filters satisfying it, and extends the resulting scheme to finite and infinite streams.
Results
The method is numerically stable for large pulse counts and achieves high-accuracy ultrasound estimation while reducing samples by two orders of magnitude.
Takeaways & Limitations
Compactly supported filters provide a common sampling and reconstruction approach across periodic, finite, and infinite pulse-stream settings.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper develops minimal-rate sampling and reconstruction schemes for periodic, finite, and infinite pulse streams, targeting numerical stability with many pulses and application to ultrasound imaging.
- Motivation: Pulse-stream signals are parameterized by pulse time-delays and amplitudes, which motivates recovering these parameters from low-rate samples.In ultrasound, delays indicate scatterer positions and amplitudes indicate scatterer strength.
- Limitations of prior work: Existing finite-stream approaches become numerically unstable at high innovation rates, while prior high-order problems lacked a reported stable sampling and reconstruction scheme.Gaussian-tail methods are unstable because the tails take small values.
- Contributions: The paper studies periodic, finite, and infinite pulse streams using minimal sampling rate, numerical stability for sufficiently separated delays, and minimal restrictions on pulses per sampling period.The stated goal is a stable minimal-rate single-channel scheme even with many pulses.
- Periodic streams: For periodic streams, the authors derive a general sampling-kernel condition and introduce compactly supported Sum of Sincs filters that satisfy it.The paper presents these as the first finite-support filters solving the periodic case.
- Finite streams: Compact support enables extension to finite streams with perfect reconstruction from a minimal number of samples when the pulse shape has compact support.The resulting reconstruction is reported as numerically stable for both small L and large pulse counts, including L = 100.
- Results and application: The method remains stable for closely spaced pulses, keeps signal-structure constraints independent of L, requires lower sampling rate for L ≥3, and achieves high-accuracy ultrasound estimation with two orders of magnitude fewer samples.The ultrasound application uses real data and reports a substantial reduction relative to current imaging techniques.
II. PERIODIC STREAM OF PULSES
The periodic pulse stream is modeled by unknown delays and amplitudes, then recovered from Fourier coefficients using spectral-analysis tools at the critical sample count.
- The signal contains L pulses with known pulse shape and period, but unknown delays t_l and amplitudes a_l.
- The target is minimal-rate reconstruction, with 2L samples expected because the periodic stream has 2L degrees of freedom.
- Directly sampling short-support pulses can produce many zero samples because uniform sample locations rarely hit a pulse.
- After suitable preprocessing, Fourier coefficients form a sum of L complex exponentials whose frequencies and amplitudes can be recovered spectrally.
- The Vandermonde system has full column rank when the number of coefficients satisfies M ≥ L and the delays are distinct.
- The annihilating-filter method recovers the frequencies using the critical number M = 2L, unlike MUSIC and ESPRIT, which require oversampling.
B. Obtaining The Fourier Series Coefficients
The paper designs a sampling kernel that extracts the needed Fourier coefficients from time-domain samples, enabling minimal-rate recovery of periodic streams.
- In the special case N = M and T = τ/N, recovery applies a DFT to the sample vector followed by a sampling-filter correction matrix.
- The sampling kernel is designed to pass selected Fourier coefficients X[k] for k ∈ K while suppressing all others.
- The resulting sample combinations are linearly independent, giving the recovery system full column rank and allowing the coefficient vector x to be solved.
- Periodic signals are uniquely determined by samples c[n] when N ≥ |K| ≥ 2L and the pulse transform is nonzero on K.
- The framework extends to nonuniform sampling by substituting nonuniform sampling times into the recovery formulation.
- The prior ideal low-pass special case needs N ≥ M ≥ 2L + 1 samples because its symmetric coefficient set must have odd cardinality.
- Unlike the ideal low-pass filter, the proposed non-bandlimited kernels have compact time support, allowing extension to finite and infinite pulse streams.
C. Compactly Supported Sampling Kernels
The paper develops compactly supported sampling filters whose coefficients can be selected for implementation and noise-performance objectives. The construction supports exact recovery while exposing a tradeoff between filter approximation quality and required sample count.
- Filter construction: The proposed filter class is a sum-of-sincs construction with coefficients {b_k} that determine the sampling kernel.The class can be generalized using a function φ(ω) satisfying the stated conditions, enabling smooth analog-filter versions.
- Filter construction: The filter is real-valued when K is symmetric and coefficients satisfy b_k = b*_−k.
- Design tradeoff: Using more coefficients improves approximation of the analog filter but requires more samples because N must exceed |K|.
- Noise-aware design: Theorem 2 selects coefficients minimizing the mean-squared error of a linear estimator under the paper’s noise assumptions.For equal pulse-spectrum magnitudes, the optimal coefficients have equal squared magnitudes, |b_i|^2 = 1/|K|.
- Finite-stream extension: The compactly supported construction extends the periodic scheme to finite streams while preserving its samples and recovery procedure.The resulting finite-stream recovery is exact to numerical precision, and the paper reports numerical stability and high noise robustness.
D. Simulations
The simulations use high-rate digital signals to mimic analog filtering and sampling operations.
- Simulation setup: The simulations approximate analog filtering and sampling with high-rate digital signals.
1) Demonstration of Our Sampling Scheme:
The sampling scheme is demonstrated on Gaussian and impulse streams, recovering delays and amplitudes from low-rate samples. Noise experiments examine SNR effects and the accuracy gains from oversampling.
- Demonstration: A stream of five delayed and weighted Gaussian pulses is used to demonstrate the proposed sampling scheme.
- Demonstration: With five pulses, near-critical sampling uses 11 uniformly spaced samples over one period.The filter uses K = {−L, …, L}, with |K| = 11 and N = 11.
- Demonstration: The annihilating-filter reconstruction estimates delays and amplitudes, and both estimation and signal reconstruction are exact to numerical precision.
- Noisy case: The noisy experiments evaluate delay and amplitude estimation error as a function of SNR, with delay estimation receiving primary attention.
- Noisy case: Oversampling factors of 1, 2, 4, and 8 are used to improve reconstruction accuracy, followed by total least-squares estimation and Cadzow denoising.
- Finite streams: For finite streams, compactly supported filtering produces the same samples as the periodic case, so the same delay-and-amplitude recovery applies.
B. Simulations
A second simulation tests exact recovery for streams of Dirac pulses using the proposed filter at near-critical sampling rates.
- Exact recovery: The signal is filtered with a three-period kernel and sampled uniformly N = M = 5 times.
- Exact recovery: Perfect reconstruction is achieved, with the estimation reported as exact.
2) High Order Problems:
The SoS filter remains stable as the number of pulses increases, unlike competing methods whose errors grow or become unstable. Its compact support also enables sequential local reconstruction of separated infinite pulse bursts under stated spacing and location assumptions.
- Noisy finite streams: The E-spline simulation is limited to L = 5 because computing high-order time-domain expressions is computationally complex.The E-splines were simulated only up to order 9, which permits L = 5 pulses.
- Noisy finite streams: For L = 2, all methods are stable, with Gaussian and SoS approaches demonstrating the lowest errors.E-splines outperform B-splines in this low-order case.
- Noisy finite streams: For L = 20, the SoS filter retains nearly unchanged performance while B-spline and Gaussian methods are unstable.For L ≥5, Gaussian and spline errors approach the order of τ.
- Infinite streams: The infinite-stream construction assumes bursts have maximal duration τ, at most L pulses, quiet phases, and known burst locations.Samples are taken during the burst duration because samples outside it can be contaminated by adjacent bursts.
- Infinite streams: The compactly supported SoS filter allows an infinite stream to be solved sequentially as local finite-order problems.For the g3p(t) filter, burst spacing greater than 3τ/2 prevents one burst from influencing samples taken during another.
V. RELATED WORK
The paper relates its SoS filters to prior low-rate pulse-stream methods and identifies stability, compact support, and sampling efficiency advantages. These advantages become especially clear for high pulse counts and infinite streams.
- Comparison with prior kernels: The SoS class satisfies the general exponential reproduction property while introducing a new sampling scheme with advantages over E-splines.With suitable spanning coefficients, SoS filters reproduce exponentials at frequencies {2πk/τ}k∈K.
- Infinite streams: Compact support keeps the SoS filter’s support independent of L, whereas E-spline support grows with filter order and makes infinite-stream constraints more stringent.The E-spline constraint tightens quadratically with L, while the SoS constraint is independent of L.
- High-order stability: For L ≥5, simulations show SoS reconstruction remains stable at high order, while B-spline and E-spline errors become very large.The SoS method remains stable even for L = 100 and outperforms both spline approaches.
C. Infinite Streams
The paper extends its pulse-stream framework to infinite bursts and applies it to one-dimensional ultrasound data. The method achieves accurate delay estimation with roughly two orders of magnitude fewer samples than the high-rate recording.
- Infinite-stream reconstruction: Infinite pulse streams can be divided into finite problems when bursts contain at most L pulses and occupy intervals of duration LPT.Quiet phases of 1.5τ between bursts of length τ enable the finite-case reduction.
- Infinite-stream reconstruction: The infinite-stream method improves stability and noise robustness in high-order problems but requires quiet phases between bursts.This quiet-phase requirement is an additional condition of the approach.
- Ultrasound model: The ultrasound echo signal is modeled as a finite pulse stream whose delays represent scatterer locations and amplitudes represent reflection coefficients.The received pulse shape is assumed known and modeled as a Gaussian.
- Ultrasound experiment: The experiment processes a single probe element from GE Healthcare’s Vivid-i system and digitally emulates the analog sampling scheme.The original data contains 4160 high-rate samples sampled at fs = 20 MHz.
- Ultrasound results: Around 30 samples replace 4160 while estimating scatterer locations with high accuracy.The sampling rate is reduced by 2 orders of magnitude, and pulse-location error is around 0.1 mm.
VII. CONCLUSIONS
The paper develops compactly supported sampling filters for pulse streams, extending minimal-rate periodic recovery to finite and infinite settings. Simulations and ultrasound experiments show improved stability, noise robustness, and substantial sampling-rate reduction.
- VII. CONCLUSIONS: The paper derives a sampling-kernel condition for minimal-rate recovery of periodic pulse streams and proposes compactly supported filters satisfying it.Previous work is identified as a special case of the general periodic result.
- VII. CONCLUSIONS: The resulting finite-stream method outperforms previous techniques in high-order stability and noise robustness.The paper also extends the approach to infinite pulse streams.
- VII. CONCLUSIONS: Compact support enables local reconstruction and lowers the complexity of the infinite-stream problem.The method is demonstrated on real ultrasound data.
APPENDIX
The appendix derives an optimal diagonal estimator design by reducing MSE minimization to a constrained optimization over diagonal coefficients. KKT conditions characterize the solution, and monotonicity establishes a unique parameter satisfying the constraint.
- Optimization formulation: The estimator’s MSE objective is reduced to maximizing its B-dependent term after substituting the relevant covariance and matrix expressions.The derivation uses white Gaussian noise with covariance Rww = σ2I and diagonal matrices B and ˜H.
- Optimization formulation: Writing βi = |bi|2 converts the optimization into a convex problem with constraints.The diagonal structure of the matrices supports this scalar coefficient formulation.
- KKT solution: KKT conditions determine which coefficients vanish: βi = 0 when λ exceeds the corresponding threshold, and µi = 0 otherwise.The threshold is expressed using |˜hi|4N/σ2, while λ > 0 is selected to satisfy the constraint.
- KKT solution: The nonzero coefficients form an ordered tail: if βi ≠ 0 for i < j, then βj ≠ 0 as well.This follows from the increasing ordering of the magnitudes |˜hi|.
- Uniqueness: A unique λ satisfies the constraint because G(λ) decreases monotonically over the relevant interval, remains −1 above the largest threshold, and is positive as λ approaches zero.The appendix concludes uniqueness before substituting the result into the constraint equation.