Delay-Multiply-And-Sum Beamforming for Real-Time In-Air Acoustic Imaging

TL;DR

This work tackles the limited dynamic range and contrast of conventional Delay-and-Sum beamforming in in-air acoustic imaging by introducing higher-order non-linear Delay-Multiply-and-Sum (DMAS) beamforming with Coherence Factor (CF) weighting. The authors derive an efficient, generalizable DMAS framework using Newton–Girard identities to compute $E_n$ from power sums, enabling $O(N)$ per-pixel computation and real-time GPU acceleration on embedded platforms. They provide explicit expansions for orders $n=2$..$5$ and demonstrate substantial improvements in image contrast and dynamic range over DAS in both simulations and real-world data, while preserving spatial resolution dictated by array geometry. The approach, validated on embedded GPUs and demonstrated with a 32-element MEMS array in broadband 25–50 kHz operation, offers a practical, high-performance solution for real-time in-air acoustic imaging with potential impact on leak detection, machinery diagnostics, and autonomous sensing.

Abstract

In-air acoustic imaging systems demand beamforming techniques that offer a high dynamic range and spatial resolution while also remaining robust. Conventional Delay-and-Sum (DAS) beamforming fails to meet these quality demands due to high sidelobes, a wide main lobe and the resulting low contrast, whereas advanced adaptive methods are typically precluded by the computational cost and the single-snapshot constraint of real-time field operation. To overcome this trade-off, we propose and detail the implementation of higher-order non-linear beamforming methods using the Delay-Multiply-and-Sum technique, coupled with Coherence Factor weighting, specifically adapted for ultrasonic in-air microphone arrays. Our efficient implementation allows for enabling GPU-accelerated, real-time performance on embedded computing platforms. Through validation against the DAS baseline using simulated and real-world acoustic data, we demonstrate that the proposed method provides significant improvements in image contrast, establishing higher-order non-linear beamforming as a practical, high-performance solution for in-air acoustic imaging.

Figures (8)

• Figure 1: On the left side of the figure we depict a 3D in-air sonar device (eRTISlaurijssen2025ruggedizedultrasoundsensingharsh) as used in the experiments with the typical situation of an active pulse-echo measurement. The signal from the emitter reflects on an object and is received by all microphones, which then goes through the several signal processing steps to result in the final acoustic image. The steps used in this processing pipeline are matched filtering (pulse compression), beamforming and a final envelope extraction phase. In this paper, we will focus on improvements of the beamforming stage, which converts the signals of the microphone array into a spatialized set of signals. The figure furthermore shows the coordinate system with the azimuth angle $\theta$ and elevation angle $\varphi$, used to define directions of interest $\psi$ or sound locations. In the right sketch of the acoustic image, the vertical axis uses the range dimension $r$, converted from the time domain $t$.
• Figure 2: Directional component of the point-spread function in both azimuth and elevation, plotted using a Lambert Equal Area Projection. The resulting PSF shows a higher dynamic range for increasing orders of DMAS processing. Furthermore, the addition of the CF post-processing step further increases the dynamic range of the PSF. Gridlines are spaced by 30°.
• Figure 3: Horizontal slice of the Point Spread Function (PSF) of the acoustic imaging system, shown in a range/azimuth plot (for elevation equal to 0 degrees). Similar to the directional slice through the PSF, an increase in dynamic range can be observed for increasing orders of DMAS processing, with an additional gain to be made when applying the CF post-processing step. No discernible increase in the range resolution can be observed, as is expected from the theory underpinning the far-field imaging system.
• Figure 4: Image SNR results for varying input noise levels. The image SNR quantities were calculated for input microphone noise levels $\mathrm{mic}$ ranging from -40 to 10. The left panel shows the SNR results obtained without CF post-processing, while the right panel shows the results with CF post-processing applied. The plots illustrate the superior noise suppression of the DMAS-class algorithms, as evidenced by increasing image SNR values with higher DMAS orders. The inclusion of the CF post-processing step further enhances the image SNR across all algorithms.
• Figure 5: Spatial resolution analysis of DMAS-class beamformers. Two reflectors were simulated at a fixed range of 1.5, symmetrically positioned around the X-axis with a varying half-angle $a$. The acoustic image $I$ was computed in the horizontal plane, and the angular responses at the reflector range were aggregated over all inner-angle configurations. The results show that while DMAS methods improve the dynamic range, the spatial resolution remains largely unchanged, confirming that it is primarily determined by the microphone array geometry rather than the beamforming algorithm.