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Double-Stage Delay Multiply and Sum Beamforming Algorithm: Application to Linear-Array Photoacoustic Imaging

Moein Mozaffarzadeh, Ali Mahloojifar, Mahdi Orooji, Saba Adabi, Mohammadreza Nasiriavanaki

arXiv:1801.07446v1eess.SPcs.IT

TL;DR

PAI beamforming with DAS suffers from low resolution and off-axis contributions, while DMAS remains sensitive to high noise and offers insufficient resolution improvement. The paper introduces DS-DMAS by inserting DMAS correlation into the expanded DMAS formulation, and reports better image quality than DAS and DMAS at higher computational cost.

  • Problem

    DAS produces low-resolution PAI images with substantial off-axis contributions, while DMAS remains sensitive to high noise and provides unsatisfying resolution improvement.

  • Method

    DS-DMAS replaces DAS terms in the expanded DMAS algebra with DMAS beamforming, creating an additional correlation stage.

  • Results

    DS-DMAS outperforms DAS and DMAS, reducing sidelobe levels by about 15 dB and improving SNR, FWHM, and CR by about 13%, 30%, and 35% versus DMAS.

  • Takeaways & Limitations

    DS-DMAS improves sidelobe suppression, resolution, SNR, FWHM, and CR across the reported numerical and experimental evaluations.

  • Takeaways & Limitations

    DS-DMAS and DMAS have O(M^2) computational complexity versus O(M) for DAS, with DS-DMAS adding linearly increasing overhead relative to DMAS as array size grows.

Abstract

from arXiv · show

Photoacoustic imaging (PAI) is an emerging medical imaging modality capable of providing high spatial resolution of Ultrasound (US) imaging and high contrast of optical imaging. Delay-and-Sum (DAS) is the most common beamforming algorithm in PAI. However, using DAS beamformer leads to low resolution images and considerable contribution of off-axis signals. A new paradigm namely Delay-Multiply-and-Sum (DMAS), which was originally used as a reconstruction algorithm in confocal microwave imaging, was introduced to overcome the challenges in DAS. DMAS was used in PAI systems and it was shown that this algorithm results in resolution improvement and sidelobe degrading. However, DMAS is still sensitive to high levels of noise, and resolution improvement is not satisfying. Here, we propose a novel algorithm based on DAS algebra inside DMAS formula expansion, Double Stage DMAS (DS-DMAS), which improves the image resolution and levels of sidelobe, and is much less sensitive to high level of noise compared to DMAS. The performance of DS-DMAS algorithm is evaluated numerically and experimentally. The resulted images are evaluated qualitatively and quantitatively using established quality metrics including signal-to-noise ratio (SNR), full-width-half-maximum (FWHM) and contrast ratio (CR). It is shown that DS-DMAS outperforms DAS and DMAS at the expense of higher computational load. DS-DMAS reduces the lateral valley for about 15 dB and improves the SNR and FWHM better than 13% and 30%, respectively. Moreover, the levels of sidelobe are reduced for about 10 dB in comparison with those in DMAS.

I. INTRODUCTION

PAI combines ultrasound-like spatial resolution with optical contrast, but reconstruction artifacts and DAS limitations motivate improved beamforming. DMAS addresses some DAS weaknesses through correlation-based sample multiplication, while retaining computational and noise-related challenges.

  • Motivation: PAI reconstructs optical absorption maps from ultrasound waves generated by short laser pulses.Its appeal is combining high spatial resolution associated with ultrasound imaging and high contrast associated with optical imaging.
  • Motivation: Reconstruction artifacts remain a crucial challenge across transducer configurations and imaging media.PA images share formation characteristics with ultrasound images, enabling adapted ultrasound beamforming algorithms.
  • DMAS: DMAS requires computationally expensive pairwise operations, and its correlation procedure can remain vulnerable to high noise and off-axis contributions.The supplied introduction describes procedural modifications intended to reduce sign, absolute-value, and square-root operation costs.
  • Limitations of DAS: DAS is widely used because of simple implementation and real-time capability, but it produces high sidelobes, low resolution, and weak off-axis rejection.DAS delays and samples signals from array elements before summation.
  • DMAS: DMAS multiplies delayed samples from array elements before summation, using correlation to improve sidelobe levels and resolution relative to DAS.The method can be viewed as a nonlinear spatial-coherence or aperture-autocorrelation process.

III. PROPOSED METHOD

DS-DMAS replaces DAS terms within the expanded DMAS formulation with another DMAS correlation stage. This design targets blurring, noise, and off-axis contributions by applying adaptive correlation to the expansion terms.

  • Proposed method: Each expansion term combines delayed detected signals from array elements before the second-stage beamforming operation.The delayed signals are denoted x_id(k) and x_jd(k) for elements i and j.
  • Expansion: The expanded DS-DMAS expression organizes pairwise products into multiple terms, including first, second, intermediate, and final terms.The displayed expansion explicitly labels the first, second, (M−2)th, and (M−1)th terms.
  • Proposed method: DS-DMAS uses DMAS instead of DAS between terms in the expanded DMAS algebra, adding a second correlation procedure.The expansion contains summations with DAS-like structure; DS-DMAS substitutes DMAS beamforming within those terms.
  • Rationale: Using DMAS within the expansion is intended to prevent blurring caused by treating all calculated samples identically in non-adaptive DAS.The paper also states that the same computational shortcut used for DMAS is applied to DS-DMAS.
  • Rationale: The proposed procedure is reported to improve resolution while reducing reconstructed-image noise and sidelobe levels.These outcomes are presented as the expected performance of DS-DMAS in the numerical-results section.

IV. NUMERICAL RESULTS AND PERFORMANCE ASSESSMENT

The numerical-results section evaluates DS-DMAS against DMAS and DAS using simulated imaging experiments. The stated assessment focuses on comparative beamformer performance.

  • Numerical evaluation: Numerical experiments compare the performance of DS-DMAS with DMAS and DAS.The section presents numerical results to illustrate comparative algorithm performance.
  • Numerical evaluation: The numerical study assesses the proposed beamformer through simulated imaging results.The supplied passage identifies numerical results as the basis of this evaluation.
  • Numerical evaluation: DS-DMAS is evaluated as the proposed algorithm against the two established beamformers.The comparison includes both DMAS and DAS as reference methods.

A. Simulated Point Target

Numerical point-target simulations compare DAS, DMAS, and DS-DMAS across imaging depths and noise conditions using sidelobe, SNR, and FWHM measures. DS-DMAS provides the strongest sidelobe suppression, SNR, and resolution performance among the three beamformers.

  • Simulation setup: The simulations used a 128-element, 7 MHz linear array to image targets across depths from 25 to 60 mm.The imaging region was 40 mm laterally by 60 mm vertically, with 0.1 mm spherical absorbers positioned along the vertical axis.
  • Sidelobe performance: At 55 mm depth, sidelobe levels were about -42 dB for DAS, -58 dB for DMAS, and -72 dB for DS-DMAS.DS-DMAS therefore had the lowest sidelobe level in the compared depth profiles.
  • Resolution and noise: At 50 mm depth, lateral-variation valleys were reduced by about 30 dB, 51 dB, and 63 dB for DAS, DMAS, and DS-DMAS, respectively.The three-point target simulation used noisy detected signals to assess resolution and noise reduction.
  • Resolution and noise: With 10 dB Gaussian noise, DS-DMAS produced higher-quality reconstructed images than DAS and DMAS, while DAS targets became barely detectable beyond 45 mm.The noisy point-target images used a 60 dB dynamic range.
  • Quantitative metrics: At 50 mm, SNR was about 30.1526 dB for DAS, 38.5532 dB for DMAS, and 43.7768 dB for DS-DMAS.DS-DMAS exceeded DMAS by about 5.2236 dB and DAS by about 13.6242 dB at this depth.
  • Quantitative metrics: At 55 mm, FWHM was 2.2668 mm for DAS, 1.6922 mm for DMAS, and 1.2376 mm for DS-DMAS.Across depth, FWHM degraded more slowly with DS-DMAS than with DAS or DMAS.

B. Simulated Circular Cyst

The simulated two-cyst phantom evaluates DAS, DMAS, and DS-DMAS using contrast ratio, showing stronger contrast enhancement for DS-DMAS.

  • B. Simulated Circular Cyst: The phantom contains two 4 mm-radius cysts located at depths of 15 mm and 24 mm.The images compare DAS, DMAS, and DS-DMAS under this two-cyst configuration.
  • B. Simulated Circular Cyst: 12.5727 dB and 21.591 dB CR enhancement were obtained by DMAS and DS-DMAS, respectively, compared with DAS at 15 mm depth.DS-DMAS provided the larger enhancement over DAS.
  • B. Simulated Circular Cyst: 9.0183 dB CR enhancement was obtained by DS-DMAS compared with DMAS for the cyst at 15 mm depth.
  • B. Simulated Circular Cyst: 7.388 dB CR improvement was obtained by DS-DMAS compared with DMAS for the cyst at 24 mm depth.DMAS-based algorithms outperformed DAS for the 24 mm cyst.

C. Low Contrast Target

The low-contrast simulation examines point-target visibility and sidelobe behavior under substantial added noise, with DS-DMAS retaining the background and reducing sidelobes.

  • C. Low Contrast Target: DS-DMAS makes the low-contrast point targets more detectable while retaining the imaging-medium background.
  • C. Low Contrast Target: 50 dB noise was added to the detected signals in the three-point target simulation.Images use a 60 dB dynamic range.
  • C. Low Contrast Target: 23 dB and 11 dB sidelobe reductions were obtained by DS-DMAS compared with DAS and DMAS, respectively.

D. Sensitivity to Sound Velocity Inhomogeneities

The robustness simulation introduces a 5% sound-velocity overestimation to represent medium inhomogeneity, and DS-DMAS produces narrower mainlobes and lower sidelobes than DAS and DMAS.

  • D. Sensitivity to Sound Velocity Inhomogeneities: A 5% sound-velocity overestimation was used to evaluate robustness against errors from medium inhomogeneities.The paper describes this error as covering and possibly exceeding typical estimation error.
  • D. Sensitivity to Sound Velocity Inhomogeneities: DS-DMAS produces narrower mainlobes and lower sidelobe levels than DAS and DMAS at both presented depths.

E. Processing Complexity

DS-DMAS and DMAS require substantially greater computational complexity than DAS, while DS-DMAS adds a linearly increasing overhead relative to DMAS as array size grows.

  • E. Processing Complexity: O(M^2) computational complexity is required by DS-DMAS and DMAS, compared with O(M) for DAS.The higher complexity is identified as the cost of their higher performance relative to DAS.
  • E. Processing Complexity: DS-DMAS has a linearly increasing computational overhead compared with DMAS as the number of transducer elements increases.
  • E. Processing Complexity: Processing complexity is compared across the different beamformers using the operation counts reported in Table IV.

V. EXPERIMENTAL RESULTS

Experimental linear-array imaging shows that DS-DMAS produces lower sidelobes and higher SNR than DAS and DMAS, while retaining detectable point targets.

  • Experimental setup: The experiments used a 128-element linear-array PAI system with a Verasonics data-acquisition system and a 532 nm, 10 ns Q-switched Nd:YAG laser.
  • Experimental imaging: DS-DMAS produced the lowest lateral-variation valley among the three beamformers at both experimental imaging depths.At 20 mm, the valleys were -40 dB for DAS, -52 dB for DMAS, and -63 dB for DS-DMAS.
  • Experimental imaging: 23.1804 dB and 11.9131 dB: DS-DMAS improved SNR over DAS and DMAS, respectively, at 20 mm depth.
  • Experimental imaging: DS-DMAS retained detectable point targets while reducing sidelobes and artifacts in both presented lateral variations.

VI. DISCUSSION

The discussion attributes DS-DMAS improvements to two-stage correlation and weighting that suppress noise, artifacts, off-axis contributions, and sidelobes more effectively than DAS and DMAS. Experimental and simulated comparisons show improved image quality and resolution, with greater computational cost.

  • Beamformer behavior: DAS suffers from high sidelobes and artifacts because of non-adaptiveness, blindness, and strong off-axis signal contributions.
  • Beamformer behavior: DMAS improves contrast resolution and noise reduction through correlation-based weighting, but its image quality worsens with increasing depth or high noise.
  • DS-DMAS mechanism: DS-DMAS uses two DMAS stages to modify calculated weights, improving detectability and image quality especially at high imaging depths and high medium noise.
  • Computational trade-off: The performance gains require more operations than DAS and DMAS.
  • Robustness and contrast: DS-DMAS retained strong signals while degrading weak signals, producing lower sidelobes and artifacts than DAS and DMAS for low-contrast targets and sound-velocity overestimation.
  • Quantitative evaluation: DS-DMAS improved experimental SNR and resolution relative to DAS and DMAS, as reported in Tables V and VI.

VII. CONCLUSION

The paper introduces DS-DMAS, which replaces DAS terms inside the expanded DMAS formulation with DMAS-based processing. Across numerical and experimental evaluations, DS-DMAS reduces sidelobes and improves SNR, FWHM, and CR relative to DMAS, with higher computational cost.

  • Conclusion: DS-DMAS replaces existing DAS operations inside the expanded DMAS algorithm with DMAS processing.
  • Conclusion: Numerical simulations and experimental results evaluated DS-DMAS against DAS and DMAS using point targets, noise levels, cyst targets, and experimental data.
  • Conclusion: DS-DMAS outperformed DMAS, while DMAS-based algorithms reduced sidelobes and improved resolution compared with DAS.
  • Conclusion: 15 dB: DS-DMAS reduced sidelobe levels, while improving SNR, FWHM, and CR by about 13%, 30%, and 35%, respectively, compared with DMAS.
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