PaperlayerView exact revision on arXiv

Source-linked AI summary

Fourier Domain Beamforming: The Path to Compressed Ultrasound Imaging

Tanya Chernyakova, Yonina C. Eldar

arXiv:1307.6345v1cs.IT

TL;DR

Digital ultrasound beamforming requires sampling rates well above the signal’s Nyquist rate, creating substantial data and processing demands. The paper extends compressed beamforming to general frequency-domain processing and combines partial-bandwidth sampling with compressed sensing. The methods preserve image quality, achieve up to 28-fold sampling-rate reduction, and are demonstrated on real-time cardiac ultrasound hardware.

  • Problem

    Digital beamforming requires sampling rates above the signal’s Nyquist rate, while prior compressed beamforming cannot recover weak reflectors and therefore loses medically important speckle.

  • Method

    The paper performs beamforming through frequency-domain relationships and combines partial-bandwidth sampling with compressed sensing when signal structure is available.

  • Results

    Up to 28 fold reduction in sampling rate is achieved, with frequency-domain beamforming preserving image integrity and compressed sensing recovering speckle.

  • Takeaways & Limitations

    Sub-Nyquist processing is feasible for medical ultrasound and could reduce the size, power consumption, and cost of future ultrasound machines.

Abstract

from arXiv · show

Sonography techniques use multiple transducer elements for tissue visualization. Signals detected at each element are sampled prior to digital beamforming. The sampling rates required to perform high resolution digital beamforming are significantly higher than the Nyquist rate of the signal and result in considerable amount of data, that needs to be stored and processed. A recently developed technique, compressed beamforming, based on the finite rate of innovation model, compressed sensing (CS) and Xampling ideas, allows to reduce the number of samples needed to reconstruct an image comprised of strong reflectors. A drawback of this method is its inability to treat speckle, which is of significant importance in medical imaging. Here we build on previous work and extend it to a general concept of beamforming in frequency. This allows to exploit the low bandwidth of the ultrasound signal and bypass the oversampling dictated by digital implementation of beamforming in time. Using beamforming in frequency, the same image quality is obtained from far fewer samples. We next present a CS-technique that allows for further rate reduction, using only a portion of the beamformed signal's bandwidth. We demonstrate our methods on in vivo cardiac data and show that reductions up to 1/28 over standard beamforming rates are possible. Finally, we present an implementation on an ultrasound machine using sub-Nyquist sampling and processing. Our results prove that the concept of sub-Nyquist processing is feasible for medical ultrasound, leading to the potential of considerable reduction in future ultrasound machines size, power consumption and cost.

I. INTRODUCTION

Ultrasound beamforming requires substantial sampling and processing because digital time-domain delays demand rates above the signal's Nyquist rate. The paper extends compressed beamforming to frequency-domain processing, combining it with compressed sensing for further reduction while addressing speckle recovery.

  • Motivation: 4-10 times the transmitted pulse’s central frequency may be required to avoid artifacts from digital time-domain beamforming.The resulting data volume grows with the number of transducer elements and image lines.
  • Prior work: Compressed beamforming models detected ultrasound signals with finite rate of innovation and reconstructs images from reduced-rate measurements.Prior work combines FRI modeling, Xampling, and compressed sensing to recover images containing strong reflectors.
  • Contributions: Frequency-domain beamforming applies to signals without assuming a structured model, while compressed sensing can provide further reduction when structure exists.The frequency-domain relationship is equivalent to time-domain beamforming and uses weighted Fourier-coefficient averaging.
  • Results: 4-10 fold fewer samples preserve image integrity on in vivo cardiac data, while partial-bandwidth processing and CS capture speckle with up to 28 fold reduction.The method was also implemented on a stand-alone ultrasound machine using real-time data from a 64-element cardiac probe.

II. CONVENTIONAL PROCESSING IN ULTRASOUND IMAGING

Conventional ultrasound imaging transmits and receives acoustic pulses with arrays of transducer elements, then aligns and averages received signals to form focused image lines. This process improves localization and SNR but requires analog-to-digital sampling before digital beamforming.

  • System operation: Multiple transducer elements transmit and receive pulses to steer and focus ultrasound energy for tissue visualization.Beamforming is used during transmission and reception in modern imaging systems.
  • System operation: Reception beamforming applies dynamically changing delays to element signals before averaging them.The changing delays move the focal point with depth and improve angular resolution.
  • Beamforming in Time: The array geometry determines echo detection times, and beamforming compensates those differences before averaging receiver signals.The resulting beam represents reflected energy along the central transmission axis.
  • Beamforming in Time: The aligned signals are averaged to produce a beam that is optimally focused at each depth.This improves angular localization and enhances SNR.
  • Digital implementation: Digital beamforming samples amplified transducer signals with an ADC before applying the alignment and averaging operations.The analog signals are preceded by anti-aliasing filtering.

B. Rate Requirements

Digital time-domain beamforming imposes high sampling and processing rates because delays require fine temporal resolution. The paper motivates frequency-domain and sub-Nyquist approaches to obtain equivalent image quality with fewer samples.

  • Rate Requirements: Hundreds of MHz can be required for digital beamforming because delay accuracy demands sampling far above the signal’s Nyquist rate.Typical sampling intervals are on the order of nanoseconds.
  • Rate Requirements: Interpolation lowers the physical sampling rate but leaves the beamforming rate high because interpolation operates at the high digital rate.The reduced sampling rate is therefore offset by additional computational load.
  • Rate Requirements: 4-10 times the transducer central frequency can be needed in phase-rotation-based beamforming to avoid degradation in array gain and sidelobe level.The rule follows from the relationship between signal bandwidth and sampling rate.
  • Rate Requirements: Growing element counts increase real-time data transmission and processing burdens on ultrasound DSP hardware.This motivates reducing data as close to the system front-end as possible.
  • Frequency-domain alternative: Frequency-domain beamforming forms beam DFT coefficients from linear combinations of the individual signals’ DFT coefficients.This provides a path to substantial sample reduction while obtaining the same image quality.

A. Implementation and Properties

Frequency-domain beamforming derives beam coefficients from Fourier/DFT coefficients of the detected signals, enabling finite-bandwidth computation. The coefficient decay allows accurate approximation using a small set of dominant terms.

  • Frequency-domain formulation: Beam Fourier coefficients are derived from the Fourier coefficients of the detected signals through a finite approximation based on the distortion-function coefficients.The approximation truncates an infinite summation according to the decay properties of Qk,m;θ[n].
  • Coefficient properties: Most of the energy in {Qk,m;θ[n]} is concentrated around the DC component across choices of k, m, and θ.The reported example uses k = 100, m = 14, and θ = 0.421 [rad].
  • Coefficient properties: 20 most significant elements of {Qk,m;θ[n]} contain, on average, more than 95% of the entire energy.The method therefore uses 20 elements throughout the work.
  • Bandwidth: The beam bandwidth contains at most B + N1 + N2 nonzero frequency components, which is approximately B when B is much larger than N1 and N2.In typical imaging setups, B is of order hundreds of coefficients while N1 and N2 are no larger than 10.
  • DFT implementation: The DFT relationship preserves the beam-to-signal frequency-domain correspondence, and the zero-frequency beam coefficient yields the beamformed signal in time.Standard image-generation steps then include log-compression and interpolation.

B. Simulations and Validation

The proposed frequency-domain and standard time-domain beamforming methods were compared on in vivo cardiac data. Their outputs were visually and quantitatively highly similar at both signal and image levels.

  • Visual comparison: In vivo cardiac images produced by time-domain and frequency-domain beamforming look identical.The comparison used an imaging setup with fs = 16 MHz.
  • Quantitative validation: Validation compares one-dimensional beamformed signals and the resulting two-dimensional image after envelope detection with a Hilbert transform.The image-line evaluation uses J = 120 image lines and NRMSE for signal comparison.
  • Quantitative validation: SSIM-based image comparison and signal NRMSE verify that the two methods produce extremely similar signals and images.SSIM is used to compare the resulting images.

IV. RATE REDUCTION BY BEAMFORMING IN FREQUENCY

The paper reduces sampling and processing requirements by translating beamforming into the frequency domain. It first exploits the signal’s effective bandwidth, then applies compressed-sensing recovery when the beamformed signal has exploitable structure.

  • Rate reduction by beamforming in frequency: Frequency-domain beamforming can operate at the Nyquist rate defined by the ultrasound signal’s effective bandwidth, avoiding time-domain oversampling.This reduction does not initially require using the structure of the beamformed signal.
  • Rate reduction by beamforming in frequency: Compressed-sensing recovery provides further rate reduction by exploiting the finite-rate-of-innovation structure of the beamformed signal.The approach uses only a portion of the beamformed signal’s bandwidth.

A. Exploiting Frequency Domain Relationship

Frequency-domain beamforming computes the beam from low-rate Fourier-domain measurements, exploiting the detected signals’ limited effective bandwidth. This bypasses time-domain oversampling while preserving comparable image quality.

  • A. Exploiting Frequency Domain Relationship: The beam’s bandwidth contains approximately B nonzero frequency components, so beamforming can use only those components rather than all N samples.The resulting bandwidth ratio is B/N = 1/4-1/10 under the stated oversampling definition.
  • A. Exploiting Frequency Domain Relationship: 4-10 fold reduction follows when B low-rate samples replace the N samples required by standard beamforming.This ratio is determined by the oversampling factor used for time-domain digital beamforming.
  • A. Exploiting Frequency Domain Relationship: B low-rate samples per detected signal can provide the required DFT coefficients through Xampling with an appropriate pulse-dependent kernel.The outputs’ DFT equals the desired coefficients, so the number of samples matches the number of coefficients computed.
  • A. Exploiting Frequency Domain Relationship: The beamformed frequency coefficients are converted back to time through an inverse DFT, with zero-padding used to match the standard processing grid.Padding with N − B zeros improves time resolution and enables comparison on the same sampling grid.
  • A. Exploiting Frequency Domain Relationship: 416 real-valued samples per image line produced frequency-domain images comparable to standard beamforming using 3360 samples.The reported comparison uses cardiac images and evaluates similarity with NRMSE and SSIM.
  • A. Exploiting Frequency Domain Relationship: Frequency-domain processing can reduce similarity to standard beamforming because it filters out noise outside the retained signal bandwidth.Time-domain beamforming retains full-spectrum noise, whereas bandwidth-limited frequency processing excludes it.

V. FURTHER REDUCTION VIA COMPRESSED SENSING

Further rate reduction uses only a subset of the beam’s nonzero Fourier coefficients and reconstructs the beam from partial frequency data. The formulation represents recovery as a structured inverse problem.

  • V. FURTHER REDUCTION VIA COMPRESSED SENSING: A subset μ of the beam’s nonzero coefficients reduces each channel’s required samples to M + N1 + N2.Here M = |μ| is smaller than the full beam bandwidth BBF.
  • V. FURTHER REDUCTION VIA COMPRESSED SENSING: The partial-frequency recovery problem is formulated using a parametric FRI representation of the beam.The beam is modeled as replicas of the known transmitted pulse with unknown amplitudes and delays.
  • V. FURTHER REDUCTION VIA COMPRESSED SENSING: Sampling the model at the beamforming rate and taking its DFT yields coefficients that depend on the unknown beam parameters.Recovering the beam is therefore recast as determining the coefficient vector from selected frequency measurements.
  • V. FURTHER REDUCTION VIA COMPRESSED SENSING: The measurement system uses an M-length vector of selected coefficients, an M × M diagonal pulse matrix, an M × N partial DFT matrix, and an N-length coefficient vector.This vector-matrix form explicitly connects partial frequency observations to the full beam representation.

B. Prior Work

Prior compressed beamforming exploits sparsity to recover strong reflectors from few Fourier measurements. Its central limitation is that treating weak echoes as noise removes speckle information.

  • B. Prior Work: When M ≥ 2L, matrix pencil or annihilating-filter methods can recover the unknown delays and amplitudes from selected coefficients.Rate reduction is obtained when 2L << N.
  • B. Prior Work: Compressed sensing narrows the infinitely many solutions of the underdetermined system by exploiting sparsity in the unknown vector.The approach is especially useful at moderate to high noise levels.
  • B. Prior Work: Randomly selected DFT rows satisfy the RIP with high probability when K ≥ CL(log N)^4, assuming L << N.Because random frequency sampling is impractical in hardware, randomly distributed frequency bands provide an alternative.
  • B. Prior Work: The prior model treats strong reflections as sparse signal components and weaker scattered echoes as noise.This matches the assumed L-sparse representation used for recovery.
  • B. Prior Work: The method cannot restore weak reflectors, so the resulting images lose speckle that supports cardiac tissue motion and deformation tracking.This loss substantially reduces the medical-imaging value of the reconstructed images.

C. Alternative Approach

An alternative model treats the complete beam as approximately sparse rather than exactly sparse, allowing recovery of both strong reflectors and weak scattered echoes. The resulting l1 formulation is essential for capturing this signal structure.

  • C. Alternative Approach: The revised model recovers the entire signal, including both strong reflectors and weak scattered echoes.This changes the target from strong-reflector-only reconstruction to full-signal recovery.
  • C. Alternative Approach: Because weak echoes are roughly two orders of magnitude weaker on average, the coefficient vector is modeled as compressible rather than exactly sparse.The l1 norm is used to represent this approximate sparsity.
  • C. Alternative Approach: The l1 optimization problem can be solved with second-order interior-point methods or first-order iterative-shrinkage methods.These are alternative numerical approaches to the same convex formulation.
  • C. Alternative Approach: Replacing the sparse formulation with the l1 formulation is crucial because it captures the signal structure and boosts sub-Nyquist processing performance.The paper presents this substitution as more than a generic convex relaxation in this setting.

VI. SIMULATIONS AND RESULTS

The evaluation compares low-rate frequency-domain beamforming with standard time-domain beamforming and prior l0 optimization on cardiac data and hardware. The l1 approach preserves strong reflectors, speckle, and clinically relevant structures while substantially reducing sampling or processing rates.

  • In Vivo Cardiac Data: 100 DFT coefficients obtained from 120 real-valued samples provide a 28-fold sampling reduction and 14-fold processing reduction versus 3360 standard samples.The difference arises because DFT coefficients are complex-valued.
  • In Vivo Cardiac Data: The l1 frequency-domain reconstruction preserves both strong reflectors and speckle, although its images are not identical to standard beamforming.The preserved image content includes heart-wall thickness, valves, and the speckle pattern used by tracking tools.
  • In Vivo Cardiac Data: The l0 optimization comparison depicts strong reflectors but completely loses speckle, degrading the overall image.The comparison assumes L = 25 strong reflectors in each direction.
  • Hardware Implementation: The implementation used a 64-channel ultrasound setup with a 15.7 cm imaging depth, 204 µsec signal duration, and 1.77 MHz bandwidth centered near 3.4 MHz.The initial hardware implementation sampled each channel at a high rate and reduced data and processing rates after DFT computation.
  • Hardware Implementation: Frequency-domain beamforming retains sufficient cardiac image quality despite a significant processing-rate reduction on an ultrasound system.The comparison is shown between low-rate frequency-domain and standard time-domain beamforming images.

VII. CONCLUSION

The paper extends compressed beamforming to frequency-domain beamforming, avoiding time-domain oversampling without additional signal assumptions. Combining partial-bandwidth sampling with a compressible-signal model enables larger reductions, and real-time implementation demonstrates feasibility for medical ultrasound.

  • VII. CONCLUSION: Frequency-domain beamforming achieves a 4-10-fold sampling-rate reduction without compromising image quality or adding signal assumptions.The approach avoids oversampling required for digital beamforming in time.
  • VII. CONCLUSION: Up to 28-fold sampling-rate reduction is obtained by using only part of the beam's bandwidth and reconstructing the beamformed signal with FRI and compressed-sensing methods.A compressible coefficient vector is assumed to preserve both strong reflections and weaker scattered echoes.
  • VII. CONCLUSION: Real-time processing on a standalone ultrasound machine with a 64-element probe achieves a 10-fold rate reduction relative to the lowest processing rates currently achievable.The implementation processes data obtained while scanning a heart.
  • VII. CONCLUSION: The results establish sub-Nyquist processing as feasible for medical ultrasound, with potential reductions in future machine size, power consumption, and cost.
Loading 1307.6345v1…