3D Null Subtraction Imaging (3D NSI)

• 3D NSI is a nonlinear beamforming framework for volumetric ultrasound imaging that uses null-subtraction and multiplexing-aware sparse aperture designs to enhance resolution and contrast.
• It incorporates multiple apodization masks—zero-mean and DC-biased—to produce synthetic beampatterns with narrowed main lobes and suppressed sidelobes across nine diverging-wave angles.
• Experimental results demonstrate significant improvements in beam area, contrast, and volumetric rate, albeit with a modest reduction in speckle contrast-to-noise ratio.

Three-dimensional Null Subtraction Imaging (3D NSI) is a nonlinear beamforming framework for volumetric ultrasound imaging that enables enhanced spatial resolution and contrast while addressing acquisition speed and hardware limitations. By combining a computationally efficient null-subtraction process with multiplexing-aware sparse aperture designs, 3D NSI offers a practical pathway toward high-quality real-time 4D imaging using matrix arrays, especially in systems constrained by low-channel-count data acquisition (Dai et al., 15 Nov 2025).

1. Beamforming Model and Null-Subtraction Principle

3D NSI is formulated within the context of diverging-wave volumetric ultrasound imaging. Each array element $i$ receives an RF signal: $r_i(t) = s\left( t - \tau_i(\mathbf{r}) \right)$ where $\tau_i(\mathbf{r})$ denotes the two-way delay between the virtual point source and voxel location $\mathbf{r}$.

In conventional delay-and-sum (DAS) beamforming, the voxel value is: $$B_\text{DAS}(\mathbf{r}) = \sum_{i=1}^{N_\text{el}} w_i\, r_i \left( t_0 + \tau_i(\mathbf{r}) \right)$$ with apodization weights $w_i$. The spatial beampattern, for direction $\theta$, is: $$B_\text{DAS}(\theta) \propto \sum_{i=1}^{N_\text{el}} w_i\, e^{j k\,\mathbf{d}_i \cdot \hat{\mathbf{k}}(\theta)}$$

3D NSI introduces three receive apodizations:

• Zero-mean (ZM) mask:

$$A_{\text{ZM},i} = \begin{cases} -1, & 0 \leq r_i \leq r_{\rm in} \ +1, & r_{\rm in} < r_i \leq r_{\rm out} \end{cases}$$

with $r_i = \|\mathbf{d}_i\|$ and inner/outer regions containing approximately equal element counts. This ensures an on-axis null.

• Two DC-biased versions:

$$A_{\text{DC1},i} = A_{\text{ZM},i} + \mathrm{dc}, \qquad A_{\text{DC2},i} = -A_{\text{ZM},i} + \mathrm{dc}$$

where $\mathrm{dc}=+1$ in typical practice.

Each apodization is used in a separate DAS pass, yielding images $E_{\rm ZM}$, $E_{\rm DC1}$, and $E_{\rm DC2}$. The final nonlinear combination is: $$E_{\rm NSI} = \frac{E_{\rm DC1} + E_{\rm DC2}}{2} - E_{\rm ZM}$$ This operation produces a synthetic beampattern with a narrowed main lobe and suppressed sidelobes, leading to improved spatial resolution and contrast.

2. Aperture and Apodization Configurations

3D NSI has been implemented on a 1024-element (32×32) matrix array with 300 µm pitch, operated under a 4:1 multiplexing scheme for 256 physical channels. Three aperture designs were evaluated:

• Fully Addressed Circular Aperture:
  • All elements within a specified radial boundary centered on the array (radius $r_{\rm out}$).
  • 812 active elements, requiring 16 transmit/receive (TX/RX) events per steering angle due to multiplexing.
• Fermat’s Spiral Sparse Aperture (with element reuse):
  • 256 ideal spiral points mapped to nearest physical elements.
  • Element indices are reused across multiplexed banks.
  • 256 active elements, still requiring 16 events per angle.
• Spiral “No-Reuse” Multiplexing-Aware Sparse Aperture:
  • 256 ideal spiral points, with unique element selection per bank to avoid conflicts.
  • Candidate elements are scored:

  $$S_{\rm final}(i,j) = \exp\left(-\frac{d_{\min}(i,j)^2}{2\sigma_d^2}\right)$$

  ($\sigma_d=0.7$). For each spiral point, one best-matching element per bank is chosen. - 240 active elements, needing only a single TX/RX event per angle.

Table: Hardware and Acquisition Features per Configuration

Configuration	Active Elements	Events/Volume	Max Volume Rate
Circular	812	144	76 vol/s
Fermat Spiral	256	144	76 vol/s
Spiral No-Reuse	240	9	1222 vol/s

3. 3D NSI Beamforming Algorithm

For each aperture design, 3D NSI is executed along nine diverging-wave steering angles. The key procedural steps are:

1. Define apodization masks $A_{\text{ZM}}, A_{\text{DC1}}, A_{\text{DC2}}$.

2. Acquire RF data per steering angle, respecting multiplexing:

   • For circular and spiral-reuse, each angle requires 4 TX and 4 RX events (one per bank).
   • For spiral no-reuse, all active elements are used in a single event.
3. Beamform and envelope-detect three volumes (one per mask).
4. Nonlinear combination via null-subtraction:

$$E_{\rm NSI}(\mathbf{r}) = \frac{E^{\rm DC1}(\mathbf{r}) + E^{\rm DC2}(\mathbf{r})}{2} - E^{\rm ZM}(\mathbf{r})$$

1. Postprocessing via log compression for visualization.

The computational cost is approximately threefold that of standard DAS, though the dominant time factor remains the underlying RF acquisition and bank scheduling.

4. Performance and Experimental Validation

Benchmarks demonstrate substantial improvements in spatial resolution and contrast:

• Resolution (point targets, 40 mm depth):
  • Circular: Lateral FWHM reduced from 3.35→2.68 mm (−20%) and elevational from 3.04→2.41 mm (−21%); combined beam area −36%.
  • Spiral: 2.74→2.47 mm (lateral), 2.47→2.41 mm (elevational).
  • Spiral No-Reuse: 2.66→2.66 mm (lateral), 2.41→2.41 mm (elevational).
• Contrast metrics (anechoic cysts):
  • Circular: CR 0.47→0.61 (+29%)
  • Spiral: 0.37→0.47 (+27%)
  • No-Reuse: 0.41→0.49 (+19%)
• Phantom results: Resolution and contrast gains confirmed for point, wire, and cyst targets; wire sidelobes improved by 1–3 dB.
• Volumetric rate:
  • Circular and spiral-reuse: 76 volumes/s (256 channels; 9 angles × 16 events).
  • Spiral no-reuse: 1222 volumes/s (9 events).
  • RF data per volume reduced eightfold (71 MB→21 MB), with a 16-fold volumetric rate gain.

A limitation is the modest reduction in speckle contrast-to-noise ratio (CNR), a trade-off inherent in nonlinear subtraction. The practical achievable rate is also determined by front-end bandwidth, memory, computational throughput, and thermal design.

5. Trade-offs, Practical Deployment, and Guidelines

3D NSI’s structure allows flexible deployment:

• Applicability: Any 2D matrix array with balanced azimuth/elevation layout can apply 3D NSI, provided the inner/outer regions have nearly equal element counts.
• Aperture strategy: The spiral no-reuse design is uniquely suited for multiplexed arrays, eliminating channel conflicts and maximizing acquisition speed.
• Parameter tuning: The DC bias parameter ($\mathrm{dc}$) can be varied to balance SNR against resolution, though $\mathrm{dc}=1$ is effective in most scenarios.
• Integration: As a software post-processing step, 3D NSI does not require changes to TX sequences and can be retrofitted into existing volumetric systems.
• Extensions: Coded excitation or dual-polarity waveforms may help recover SNR loss due to subtraction, especially for deep imaging.

The technology achieves improved volumetric ultrasound resolution (~36% beam-area reduction), 20–30% contrast ratio gain, and >1000 volumes/s with only a moderate increase in computational load over DAS, positioning it effectively for dynamic 4D imaging applications such as cardiac, vessel flow, and interventional guidance (Dai et al., 15 Nov 2025).

6. Experimental Highlights and Application Domains

Experiments conducted in simulation (Field II, MATLAB) and on tissue-mimicking phantoms demonstrate:

• Main-lobe narrowing and sidelobe suppression (SMER improved by 2–3 dB).
• Direct validation of spatial and contrast improvements across multiple volumetric targets.
• Recovery of near-matching image quality using the spiral no-reuse aperture, with a 16× volumetric rate boost.
• A trade-off between enhanced resolution/contrast and reduced speckle CNR, characteristic of the nonlinear subtraction approach.

This suggests that, despite specific texture compromises, 3D NSI achieves a favorable balance between imaging quality, acquisition speed, and hardware resource use, enabling a broad range of real-time volumetric applications in both research and clinical contexts (Dai et al., 15 Nov 2025).