Fermat’s Spiral Sparse Apertures in Ultrasound Imaging

• Fermat’s spiral sparse apertures are a density-tapered geometric design for volumetric ultrasound arrays that optimizes element distribution for even angular coverage.
• They integrate with advanced 3D Null Subtraction Imaging to balance trade-offs among spatial resolution, contrast, and acquisition speed using tailored apodization schemes.
• The spiral no-reuse method significantly increases imaging speed, achieving up to 1,222 volumes/s on a 256-channel system while reducing hardware conflicts.

Fermat’s spiral sparse apertures are a geometric and algorithmic approach to sparse array design, enabling efficient and high-performance volumetric ultrasound imaging. By leveraging the mathematically optimal distribution properties of the Fermat (density-tapered) spiral on circular apertures, these layouts provide both hardware-tractable and computationally efficient sampling for modern matrix arrays, especially under constraints imposed by multiplexed hardware. Recent advances focus on integrating Fermat spiral apertures with nonlinear beamforming frameworks, such as three-dimensional Null Subtraction Imaging (3D NSI), to resolve longstanding trade-offs among spatial resolution, contrast, volume acquisition rate, and system complexity (Dai et al., 15 Nov 2025).

1. Fermat Spiral Geometry and Discretization

Fermat spiral sparse aperture designs employ a density-tapered Fermat spiral to generate $N$ ideal element positions in a circular aperture of outer radius $R_o$. The $n$th position ($n=1...N$) is defined in polar coordinates:

• $r_n = R_o \cdot \sqrt{n/N}$
• $\theta_n = n \cdot \gamma$, with $\gamma = 2\pi(1-1/\phi) \approx 137.5^\circ$, where $\phi = (1+\sqrt{5})/2$ is the golden ratio.

This construction ensures near-uniform angular distribution and a smooth radial density taper. In practical array implementations (e.g., a 32×32 element Vermon probe), these ideal coordinates are mapped to the physical element grid using nearest-neighbor quantization. For the standard spiral aperture, $N = 256$, resulting in 256 active elements after quantization (Dai et al., 15 Nov 2025).

2. Spiral Sparse Aperture and Apodization Schemes

Two principal spiral-based sparse aperture strategies are utilized:

• Standard Fermat Spiral Sparse Aperture: Here, 256 ideal spiral positions are mapped to the device’s physical grid, with channel reuse occurring where multiple idealized points correspond to the same hardware element across banked sub-arrays. Apodization for Delay-and-Sum (DAS) beamforming is a rectangular mask (weight 1 on active, 0 elsewhere). For 3D NSI, three receive apodization masks are used, each balancing inner versus outer element populations:
  • Zero-mean (ZM): outer half +1, inner half –1.
  • DC1: $\mathbf{A}_{DC1} = \mathbf{A}_{ZM} + dc$
  • DC2: $\mathbf{A}_{DC2} = -\mathbf{A}_{ZM} + dc$
  • With $dc = 1$.
• Spiral "No-Reuse" Aperture: To eliminate multiplexing conflicts (i.e., redundant element use across array banks), for each ideal spiral position $i$ and candidate physical element $j$, a score $S_{final}(i, j)$ is computed:

$$S_{final}(i,j) = \exp\left[-\frac{d_{min}(i,j)^2}{2\sigma_d^2}\right], \quad \sigma_d = 0.7 \cdot \text{pitch}$$

The best match across all banks is selected for each $i$, yielding a set of 240 unique, non-overlapping active elements. Both DAS and NSI apodizations apply as above; now all selected elements can transmit and receive without inter-bank conflict (Dai et al., 15 Nov 2025).

3. Multiplexing Constraints and Volume Rate

Aperture designs impact scan efficiency in hardware-limited (multiplexed) matrix arrays. Standard spiral and circular apertures require time-multiplexing due to overlapping channel assignments: each steering angle involves $4$ transmit × $4$ receive banks = $16$ TX/RX events. For $9$ compounding angles, this totals $144$ events per volumetric frame ($\approx 76$ volumes/s).

By contrast, the spiral no-reuse aperture requires only $1$ TX/RX event per angle (no channel sharing), totaling $9$ events for all $9$ angles and enabling a maximum acquisition rate of $\approx 1\,222$ volumes/s, a $16$-fold increase over the multiplexed design when using the same 256-channel system.

Configuration	Active Elements	Events/Angle	Events/Volume	Max Volume Rate
Circular	812	16	144	76 Hz
Spiral (256 el)	256	16	144	76 Hz
Spiral no-reuse	240	1	9	1,222 Hz

4. Integration with Null Subtraction Imaging (3D NSI)

Both Fermat spiral and spiral no-reuse apertures integrate directly with 3D NSI—an extension of null-subtraction beamforming to volumetric imaging. For each of 9 diverging-wave angles, raw RF data are beamformed three times using the ZM, DC1, and DC2 apodizations. Envelope detection yields $E_{ZM}$, $E_{DC1}$, $E_{DC2}$, with nonlinear combination:

$E_{NSI} = \frac{1}{2}(E_{DC1} + E_{DC2}) - E_{ZM}$

Subsequent normalization and log-compression are performed. This workflow applies identically to circular, standard spiral, and spiral no-reuse apertures. The computational burden is under 3× that of standard DAS beamforming, making real-time kilohertz-rate imaging feasible (Dai et al., 15 Nov 2025).

5. Quantitative Performance Analysis

Performance metrics quantify improvements in spatial resolution, sidelobe suppression, contrast, and throughput, with the following representative results:

• Resolution (FWHM) at 40 mm Depth (Simulations):

Metric	Circ DAS	Circ NSI	Spiral NSI	Spiral no-reuse NSI
Azimuth FWHM (mm)	3.35	2.68	2.74	2.66
Elevation FWHM (mm)	3.04	2.41	2.47	2.41
Azimuth SMER (dB)	–16.14	–18.64	–18.35	–18.71
Elevation SMER (dB)	–16.42	–19.26	–19.03	–19.24

• Contrast Ratio (CR) in Cyst Phantom (40 mm, X–Z plane):

Aperture	DAS	NSI	ΔCR (%)
Circular	0.47	0.61	+29
Spiral	0.37	0.47	+27
Spiral no-reuse	0.41	0.49	+20

• Phantom Wire Experiment (54 mm): NSI yields azimuthal FWHM (circular) 4.08→2.97 mm ($-27$\%), elevation 3.27→2.43 mm ($-26$\%), beam area reduction $-47$\%; spiral and spiral no-reuse show matching improvements.
• CIRS Cyst (25 mm): Circular: CR 0.88→0.92 (+4.5%), Spiral: 0.66→0.72 (+9%), Spiral no-reuse: 0.61→0.69 (+13%).

The NSI approach, while enhancing contrast, does lower the contrast-to-noise ratio (CNR) relative to DAS, a known effect of non-linear beamforming operations.

6. Practical Implications and Hardware Efficiency

Fermat’s spiral sparse apertures, particularly the no-reuse variant, provide a solution for maximizing spatiotemporal resolution and throughput in resource-constrained volumetric imaging systems. They achieve:

• Beam narrowing by 20–25\% (≈36\% lower area)
• Sidelobe suppression by 2–3 dB
• 20–30% higher contrast
• Ultrafast rates (up to 1,200 volumes/s with spiral no-reuse on a 256-channel system)
• Computational efficiency, with overall workload $<3\times$ DAS

These properties enable practical real-time 4D imaging, addressing core limitations of prior high-density, time-multiplexed array approaches without significantly increasing hardware or computational complexity (Dai et al., 15 Nov 2025).

7. Context within Advanced Ultrasound Imaging

The integration of Fermat spiral sparse apertures with 3D NSI exemplifies a trend toward computationally aware array design. The density-tapered spiral geometry provides improved incoherence and apodization profiles compared to random or grid-based sparsification, facilitating both enhanced image fidelity and hardware compatibility. The spiral no-reuse approach is specifically motivated by channel-sharing limitations in existing 2D matrix transducer hardware, highlighting the role of physical and electronic constraints in dictating optimal sampling strategies.

A plausible implication is that future systems may further blend algorithmic, geometric, and hardware-aware design to maximize information extraction per acquisition event, leveraging similar mathematically grounded sparse patterns in other modalities and beamforming frameworks.