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Compressed Beamforming in Ultrasound Imaging

Noam Wagner, Yonina C. Eldar, Zvi Friedman

arXiv:1202.6037v2cs.ITcs.CV

TL;DR

Applying low-rate FRI/Xampling schemes to individual ultrasound transducer signals is hindered by low SNR and parameter-interpretation challenges. The paper integrates beamforming into compressed sampling, enabling two-dimensional cardiac ultrasound images that depict strong tissue perturbations with up to seven-fold lower sample rates.

  • Problem

    Applying FRI Xampling to biological-tissue signals faces low SNR and difficulties interpreting estimated parameters given the transmitted-beam profile.

  • Method

    The paper generalizes FRI Xampling to multiple receiving elements by integrating beamforming into the low-rate sampling process, termed compressed beamforming.

  • Results

    Up to seven-fold sample-rate reduction was achieved while constructing two-dimensional ultrasound images that depict strong tissue perturbations.

  • Takeaways & Limitations

    Compressed beamforming combines beamforming and compressed-domain sampling to support ultrasound imaging with substantially fewer samples than standard imaging techniques.

Abstract

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Emerging sonography techniques often require increasing the number of transducer elements involved in the imaging process. Consequently, larger amounts of data must be acquired and processed. The significant growth in the amounts of data affects both machinery size and power consumption. Within the classical sampling framework, state of the art systems reduce processing rates by exploiting the bandpass bandwidth of the detected signals. It has been recently shown, that a much more significant sample-rate reduction may be obtained, by treating ultrasound signals within the Finite Rate of Innovation framework. These ideas follow the spirit of Xampling, which combines classic methods from sampling theory with recent developments in Compressed Sensing. Applying such low-rate sampling schemes to individual transducer elements, which detect energy reflected from biological tissues, is limited by the noisy nature of the signals. This often results in erroneous parameter extraction, bringing forward the need to enhance the SNR of the low-rate samples. In our work, we achieve SNR enhancement, by beamforming the sub-Nyquist samples obtained from multiple elements. We refer to this process as "compressed beamforming". Applying it to cardiac ultrasound data, we successfully image macroscopic perturbations, while achieving a nearly eight-fold reduction in sample-rate, compared to standard techniques.

I. INTRODUCTION

Ultrasound imaging must process growing data volumes as more transducer elements are used. The paper combines FRI Xampling with beamforming to reduce sampling demands while addressing low-SNR tissue signals.

  • Increasing array size raises acquisition and processing rates, affecting machinery size and power consumption.
  • Classical systems reduce rates by exploiting the limited bandpass bandwidth of detected signals.
  • FRI modeling represents ultrasound echoes as a relatively small number of known-shape pulses defined by 2L unknown delays and amplitudes.
  • Applying FRI Xampling independently to receiver elements is impractical because poor SNR causes erroneous parameter extraction.
  • Compressed beamforming integrates beamforming into low-rate sampling, producing a signal equivalent to Xampling the beamformed output.
  • The imaging setup transmits a pulse along direction θ and combines delayed receiver signals to form scanline-related echoes.

III. SAMPLE RATE REDUCTION USING THE FRI MODEL

The FRI model enables ultrasound signals to be reconstructed from a small set of Fourier samples rather than Nyquist-rate samples. Recovery requires estimating pulse delays and amplitudes from these coefficients.

  • Each receiver signal is modeled as a sum of L replicas of a known pulse shape, with unknown arrival times and amplitudes.
  • The FRI representation contains 2L real-valued degrees of freedom corresponding to the pulse delays and amplitudes.
  • FRI sampling projects the signal onto a 2L-dimensional subspace represented by selected Fourier series coefficients.
  • When selected frequency indices are consecutive, the resulting Vandermonde system has full column rank if delays are distinct and K_m ≥ L.
  • Spectral methods can recover the unknown parameters when K_m ≥ 2L frequency samples are available.
  • A single-channel Xampling scheme estimates K_m frequency coefficients from p filtered-signal samples using a linear transformation, with p ≥ K_m.

IV. WHY COMPRESSED BEAMFORMING?

Independent low-rate processing is hindered by noisy cardiac ultrasound traces and ambiguous cross-element pulse matching. The paper therefore Xamples the beamformed signal through compressed beamforming.

  • Applying FRI Xampling separately to receiver elements faces low SNR and difficulty interpreting parameters under the transmitted beam profile.
  • Cardiac receiver traces contain noise from constructive and destructive interference involving dense sub-wavelength tissue scatterers.
  • Compressed beamforming Xamples the beamformed signal instead of individual signals and obtains an equivalent result from low-rate receiver samples.
  • The beamformed signal retains strong, typical-shaped pulses that may overlap, supporting its treatment within the FRI framework.
  • Beamforming produces a higher-SNR signal and avoids matching pulses across separately detected receiver signals.
  • The target pipeline estimates necessary beamformed-signal samples from low-rate samples of filtered individual receiver signals.

V. COMPRESSED BEAMFORMING

Compressed beamforming assumes that appropriately distorted and averaged receiver signals remain approximately finite-rate-of-innovation signals. Empirical results support this approximation, while showing that projection distortion decreases with increasing pulse delay and depends on receiver position.

  • FRI modeling: Beamforming attenuates off-axis reflections through destructive interference, leaving a signal focused on reflections near the central beam axis.This produces a narrower effective sector and focuses analysis on a single scanline.
  • FRI modeling: Under the stated support and delay assumptions, the beamformed signal is approximately a sum of L shifted replicas of the two-way pulse.The approximation follows from bounding the distorted pulse support and using ∆≪tl.
  • Distortion analysis: As γm approaches tl, σm,l decreases and beamforming distortion increases, making the shifted-replica approximation less accurate.The resulting projection error is examined as a function of γm, tl, and θ.
  • Distortion analysis: For δ31 = 0.29mm, the SNR exceeds 25dB beyond 1/50 of imaging depth, whereas for δ1 = 8.99mm it exceeds 10dB beyond 1/5.Receivers closer to the reference origin improve more quickly than receivers farther away.
  • Validation: Empirical results justify the approximation, and appropriate apodization may further improve it before applying Xampling reconstruction.The farthest receivers are combined for imaging depths greater than 9.1mm under the proposed apodization.

B. Compressed Beamforming with Distorted Analog Kernels

The distorted-kernel scheme extracts low-rate Fourier information about the beamformed signal directly from filtered measurements of individual receiver signals. It uses multichannel kernels whose outputs are combined to obtain improved-SNR coefficients for FRI parameter recovery.

  • Scheme: The modified scheme extracts the beamformed signal’s necessary low-rate samples by sampling filtered receiver signals at sub-Nyquist rates.This avoids directly sampling Φ(t; θ), which does not exist in the analog domain.
  • Scheme: The beamformed signal’s Fourier series is defined over [0,T), with cj denoting the kjth Fourier series coefficient.The support bound TB(θ)≤T permits this representation.
  • Scheme: Indicator functions zero intervals assumed signal-free in the model, preventing practical noise in those intervals from entering the coefficient computation.The resulting operator is applied directly to the receiver signals.
  • Multichannel implementation: Each receiver signal is multiplied by a bank of distorted kernels gj,m(t; θ) and integrated over [0,T) to form coefficient vectors.The vectors from all M channels are then combined for beamformed coefficient extraction.
  • Parameter recovery: The combined coefficients provide improved SNR and support extraction of the 2L parameters defining the beamformed signal.The coefficient relation uses a diagonal matrix H and a Vandermonde-type matrix V, with spectral analysis followed by least squares recovery of b.
  • Kernel examples: Figure 5 plots real parts of gj,m(t; θ=0) for kj=3 and kj=5 across seven receiver elements from a 64-element array.The elements are spaced 0.49mm apart and sampled at indices m0 through m0+30 in steps of five.

VI. SIMPLIFIED XAMPLING MECHANISM

The simplified Xampling mechanism approximates Fourier coefficients from low-rate samples using a linear transformation, avoiding complex analog kernels while retaining controlled accuracy.

  • Approximation: The scheme approximates each desired coefficient c_j,m by truncating the Fourier-series summation to a finite index range.For any ε > 0, suitable bounds N1 and N2 ensure |c_j,m − ĉ_j,m|^2 < ε.
  • Approximation: The approximation error can be made arbitrarily small by selecting truncation bounds that capture a sufficiently large fraction of the coefficient energy.The construction uses the ratio ρ^2 = ∥b_t∥_2^2/∥b∥_2^2 and chooses bounds offline.
  • Linear transformation: The required Fourier coefficients are collected into a vector and mapped to the approximated coefficients through a K × K_m matrix A_m(θ).The matrix is built from selected Fourier-series indices and the truncated coefficients Q_j,m;θ[n].
  • Sampling implementation: Fourier coefficients can be obtained from point-wise sub-Nyquist samples after filtering with an appropriate kernel and applying a linear transformation W_m.The Sum of Sincs is given as an example of a suitable filtering kernel.
  • Sampling implementation: The simplified scheme uses more output samples per element than desired coefficients, but avoids complicated analog kernels and incurs only a small sampling overhead in imaging.Here K_m > K, while the approximation is reported to remain good in an actual imaging scenario.
  • Sampling implementation: The resulting Xampling scheme is illustrated in Fig. 6 and uses Fourier samples of ϕ_m(t).

VII. SIGNAL RECONSTRUCTION

The paper reconstructs beamformed ultrasound signals from estimated finite-rate parameters and then forms two-dimensional images using standard envelope extraction and post-processing.

  • Reconstruction pipeline: The estimated parameters {t_l, b_l} determine the beamformed signal Φ(t; θ) within the FRI framework.
  • Image formation: For multiple steering angles θ, reconstructed signals are converted into scanlines by Hilbert-transform envelope extraction followed by logarithmic compression.The resulting scanlines are combined to form a two-dimensional image.
  • Phase modeling: The reconstruction model is generalized to include unknown carrier phases of reflected pulses, which the Xampling approach can estimate.
  • Alternative reconstruction: An alternative reconstruction of Φ(t; θ) is formulated using Compressed Sensing methodology.

A. Signal Reconstruction Assuming Unknown Carrier Phase

The unknown-phase reconstruction model explains complex coefficients arising from reflected-pulse phase shifts and provides a closed-form signal reconstruction that improves the recovered envelope.

  • Phase model: The reconstructed spectral coefficients become complex because reflections introduce unknown phase shifts into the detected pulses.The phase of each coefficient corresponds to the phase shift β_l − β of a reflected pulse.
  • Signal reconstruction: A significant improvement is observed when comparing the envelope of the reconstructed signal with that of the original signal.
  • Phase model: A reflected ultrasonic pulse is modeled as a shifted replica of a carrier-modulated baseband waveform rather than an exact replica with fixed phase.The phase shift is attributed to the relative complex impedances involved in reflection.
  • Coefficient representation: The jth Fourier coefficient is reformulated using the carrier spectrum G(ω) and the unknown reflected-pulse phases.
  • Signal reconstruction: The reconstructed signal can be formed from the complex coefficients and then processed using standard ultrasound post-processing techniques.

B. CS Approach for Signal Reconstruction

The CS reconstruction approach casts Fourier-domain recovery as a sparse-vector problem, using randomly selected Fourier measurements under an RIP-based sampling condition.

  • Sparse formulation: Quantized time delays allow the Fourier coefficients to be written as a matrix projection of an N-dimensional L-sparse vector.The vector contains reflected-pulse amplitudes at delay-grid locations and zeros elsewhere.
  • Sparse formulation: The reconstruction is a classic Compressed Sensing problem: recover x from K projections onto selected Fourier basis vectors.
  • Sampling condition: Randomly selecting Fourier rows can provide the Restricted Isometry Property with high probability when the stated sampling bound is satisfied.
  • Sampling condition: The delay-grid resolution directly affects the Restricted Isometry Property of the sensing matrix.
  • Comparison with spectral methods: Unlike spectral methods requiring carefully chosen consecutive samples, the CS formulation permits randomly selected sensing vectors and accommodates oversampling for noisy data.

VIII. COMPARISON BETWEEN RECOVERY METHODS

The study compares four recovery methods for beamformed ultrasound signals reconstructed from Fourier-series measurements under varying SNR and oversampling. OMP methods outperform spectral-analysis methods, with random OMP showing the strongest performance advantage.

  • Simulation setup: The evaluation uses Field II simulations of beamformed signals from phantoms containing six strong axial reflectors and distributed speckle reflectors.The aperture contains 64 transducer elements, and recovery is assessed from estimated reflector time delays.
  • Evaluation: Recovery probability is evaluated over multiple combinations of SNR and oversampling factor using four reconstruction methods.The methods are total least-squares with Cadzow enhancement, matrix pencil, consecutive-coefficient OMP, and randomly distributed-coefficient OMP.
  • Results: Matrix pencil achieves high recovery probabilities over a wider range of SNR and oversampling than total least-squares with Cadzow enhancement.The comparison identifies matrix pencil as preferable among the two spectral-analysis techniques.
  • Results: Both OMP methods outperform the spectral-analysis methods, with an obvious advantage for OMP using randomly distributed Fourier coefficients.The random-coefficient selection is evaluated alongside consecutive Fourier coefficients.
  • Hardware trade-off: Random Fourier-coefficient selection can increase Xampling hardware complexity because the sampling kernel must be specifically designed for that selection.Consecutive coefficient selection permits a relatively simple kernel, while the alternative Xampling scheme is practically invariant to coefficient selection.

IX. EXPERIMENTS ON CARDIAC ULTRASOUND DATA

The cardiac-data experiments apply two Xampling schemes to multichannel raw RF data and compare their images and reconstructed beamformed signals with standard imaging. The approximated scheme achieves a seven-fold sample-rate reduction while preserving the strong perturbations seen in the reference image.

  • Data acquisition: The cardiac experiment uses raw RF data from a 64-channel scanner and a 64-element phased-array probe acquired along 120 beams.The data cover a 60° sector to a depth of 16cm.
  • Reference imaging: The standard reference acquires 10389 real-valued samples per element for one scanline at 50MHz, then downsamples them to 1662 samples for beamforming.The reference image is used to reproduce macroscopic reflectors with Xampling.
  • Non-approximated Xampling: The non-approximated Xampling scheme reproduces the strong perturbations in the standard image and keeps isolated reflectors near the array in focus.The scheme uses 100 consecutive frequency indices with two-fold oversampling for L = 25 reflectors.
  • Approximated Xampling: Seven-fold sample-rate reduction is obtained with the approximated scheme, requiring an average of 116 complex samples per element for one scanline.The maximum reaches 133 samples for certain elements and beam angles; a common sensor rate still provides a six-fold reduction.
  • Evaluation scope: SNR comparisons do not fully measure overall Xampling performance because the reference standard image includes speckle whereas Xampling targets strong macroscopic reflections.The paper also evaluates recovery by matching Xampling pulses to the strongest local maxima in standard beamformed signals.

X. CONCLUSION

The paper generalizes Xampling to beamformed signals, combining low-rate acquisition, compressed sensing recovery, and phase-shift estimation for two-dimensional ultrasound imaging. Cardiac data depict strong tissue perturbations with up to seven-fold sample-rate reduction.

  • Compressed beamforming: The proposed array-based Xampling scheme reconstructs two-dimensional ultrasound images by applying the one-dimensional method to beamformed signals.Beamformed signals have enhanced SNR and represent reflections from a narrower sector than individual element signals.
  • Compressed beamforming: The first scheme uses multiple modulation and integration channels with analog kernels, while the second approximates beamformed-signal parameters from selected Fourier-series projections.The second approach also reduces front-end data transmission through a relatively simple linear transformation, including when preliminary sampling is Nyquist-rate.
  • Signal recovery: CS-based recovery is generally comparable to spectral analysis at equal sample-set cardinality and typically performs better in noise when frequency samples are widely spread.With few samples, CS permits wide frequency distribution without requiring unique sample configurations.
  • Signal model: Unknown phase shifts are incorporated into the signal model and estimated by interpreting extracted coefficients without changing the recovery method.This extends the assumed FRI structure beyond pulses with known phase behavior.
  • Experimental outcome: Up to seven-fold sample-rate reduction was achieved while constructing cardiac ultrasound images that depict strong tissue perturbations.The comparison is against standard imaging techniques.

APPENDIX A

The appendix bounds the beamformed signal’s temporal support using pulse duration, propagation-delay mappings, and the assumption that the pulse duration is much shorter than the observation interval.

  • Support assumptions: The pulse h(t) is supported on [0, ∆), while each received signal ϕ_m(t) is supported within [0, T).The interval [0, T) reflects transmission at t = 0 and a penetration-depth cutoff beyond which reflections fall below the noise level.
  • Delay mapping: The delay relation t_l,m = τ_m(t_l; θ) and monotonicity of τ_m(t; θ) allow inverse-delay expressions to be used in the support analysis.The appendix explicitly invokes the inverse of τ_m(t; θ).
  • Support bound: Under ∆ ≪ T, the beamformed signal support is bounded by T_B(θ) = min 1≤m≤M τ_m^-1(T; θ), with T_B(θ) ≤ T.The bound is constructed so that τ_m(T_B(θ); θ) ≤ T for every receiver element.
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