Delay-Multiply-and-Sum Beamforming

• Delay-Multiply-and-Sum (DMAS) is a nonlinear beamforming method that replaces traditional DAS summation with pairwise signal multiplications to improve spatial coherence and image quality.
• It achieves significant enhancements by reducing sidelobe levels and noise, yielding up to 25 dB higher SNR and narrower FWHM compared to DAS.
• Variants such as DS-DMAS and MVB-DMAS incorporate adaptive weighting techniques to further suppress clutter, although they increase computational complexity.

The Delay-Multiply-and-Sum (DMAS) technique is a nonlinear beamforming algorithm that enhances image contrast and suppresses sidelobes and noise in array-based imaging systems. Originally motivated by the limitations of the conventional Delay-and-Sum (DAS) beamformer in modalities such as photoacoustic, ultrasound, and acoustic imaging, DMAS substitutes the linear sum of delayed sensor signals with pairwise multiplications before summation. This pairwise correlation process exploits spatial coherence within the array, leading to marked improvements in image quality, particularly in high-clutter and low signal-to-noise environments. DMAS forms the computational and conceptual core for a family of algorithms, including its higher-order generalizations and hybrid variants with adaptive or minimum variance weighting.

1. Mathematical Formulation and Algorithmic Structure

For an array of $N$ sensors, with delayed signals $x_i$ per channel at the focal time or location, the classical Delay-and-Sum beamformer output is

$S_\mathrm{DAS} = \sum_{i=1}^{N} x_i.$

DMAS enhances this approach by performing a summation over the products of all unique channel pairs: $$S_\mathrm{DMAS} = \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \mathrm{sgn}(x_i x_j) \sqrt{|x_i x_j|},$$ where $\mathrm{sgn}$ denotes the sign function and the square root companding controls the dynamic range of the output (Jansen et al., 12 Nov 2025). Alternatively, in standard baseband processing or with real-valued signals,

$S_\mathrm{DMAS} = \sum_{i<j} x_i x_j$

is often used (Paridar et al., 2018, Mozaffarzadeh et al., 2017).

The DMAS algebra permits reformulation via power sums,

$$S_\mathrm{DMAS} = \frac{1}{2}\left[(\sum_{i=1}^N x_i)^2 - \sum_{i=1}^N x_i^2\right],$$

demonstrating that DMAS is a nonlinear transformation of the DAS output, with the subtraction term effectively removing autocorrelation components (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).

For higher-order DMAS, products over $n$-tuples of channels are used and expressed efficiently using symmetric polynomials and the Newton–Girard formulas, allowing $O(N)$ implementation even for $n>2$ (Jansen et al., 12 Nov 2025).

2. Motivation and Performance Relative to DAS

DAS suffers from high sidelobe levels, low contrast, and limited resolution due to its non-adaptive and strictly linear construction. DMAS was introduced to address these deficits by:

• Enhancing pixel-wise coherence via nonlinear (second or higher-order) channel correlation
• Suppressing incoherent off-axis energy and noise, resulting in lower sidelobes
• Increasing image dynamic range and contrast

Empirical results demonstrate that, compared to DAS, DMAS yields up to 25 dB higher SNR, 0.2 mm narrower FWHM in transverse profile (photoacoustic tomography), and sidelobe levels reduced by 10–25 dB in typical array setups (Paridar et al., 2018, Mozaffarzadeh et al., 2018, Jansen et al., 12 Nov 2025). In in-air acoustic imaging, the dynamic range advantage exceeds 20 dB for DMAS2 and reaches over 80 dB for fifth-order DMAS (Jansen et al., 12 Nov 2025).

3. Extensions: Filtered, Double-Stage, and Modified DMAS

Several DMAS variants have been developed to further exploit its nonlinear filtering capability:

• Filtered DMAS (F-DMAS): Band-pass filtering (typically at $2f_c$, with $f_c$ system center frequency) is applied to the DMAS output to remove DC and preserve frequency-specific coherence. This leads to further enhancements in local contrast and lateral spread, yielding up to 5% better vessel contrast and 10–25% narrower spread in ultrasound localization microscopy (Madhavanunni et al., 27 Feb 2024).
• Double-Stage DMAS (DS-DMAS): A DMAS stage is applied recursively to the partial-coherence sums of the first DMAS stage. This hierarchical multiplication leads to an additional 25% sidelobe reduction and SNR improvements of 22–23% over DMAS, at commensurately higher computational cost (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).
• Modified DMAS (M-DMAS): Reformulates DMAS so that the conventional DAS sum and its square are combined to suppress undesired terms. Empirically, this achieves up to 19 dB better lateral valley suppression and 25 dB lower sidelobes in high-noise photoacoustic imaging (Mozaffarzadeh et al., 2018).

4. Integration with Adaptive Beamformers

Recognizing that DMAS's algebra embeds DAS operations, recent work replaces DAS terms within the DMAS expansion with outputs from adaptive beamformers:

• Minimum Variance-Based DMAS (MVB-DMAS): Uses Minimum Variance weights instead of uniform summation within the DMAS algebra, improving both mainlobe width and sidelobe suppression. MVB-DMAS achieves about 31 dB, 18 dB, and 8 dB sidelobe reduction compared to DAS, MV, and DMAS, respectively; FWHM improves by 96%, 94%, and 45% over DAS, DMAS, and MV, while SNR increases by 89%, 15%, and 35% (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2018).
• Eigenspace-Based MV (EIBMV-DMAS): Projects minimum variance weights into the signal subspace for improved covariance estimation. EIBMV-DMAS demonstrates up to 113 dB sidelobe reduction and 75% SNR improvement versus DMAS in experimental linear-array PAI (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017).

Adaptive DMAS hybrids thus combine nonlinear spatial coherence filtering with statistical adaptivity, yielding the highest SNR, mainlobe narrowing, and sidelobe suppression seen in array imaging beamformers.

5. Computational Complexity and Efficient Implementation

The principal drawback of DMAS and its higher-order or adaptive extensions is computational. DMAS’s naive implementation incurs $O(N^2)$ complexity per focal sample, with DS-DMAS requiring potentially $O(N^4)$ (mitigated by exploiting symmetry). Efficient $O(N)$ formulations have been developed using Newton–Girard identities for DMAS-n, facilitating GPU-based real-time acceleration (Jansen et al., 12 Nov 2025). In embedded GPU benchmarks, second-order DMAS executes in approximately twice the time of DAS, with fifth-order DMAS overhead at 24–150% depending on platform and with minimal impact from post-hoc coherence weighting.

6. Empirical Results and Quantitative Metrics

Published results across photoacoustic, ultrasound, and acoustic testbeds converge on several key metrics:

Beamformer	Sidelobe Reduction	SNR Gain	FWHM Improvement	Reference
DMAS	10–25 dB	3–25 dB	15–40%	(Paridar et al., 2018, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2017)
DS-DMAS	~25% over DMAS	~23% over DMAS	~22% over DMAS	(Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018)
MVB-DMAS	31 dB (vs DAS)	89% (vs DAS)	96% (vs DAS)	(Mozaffarzadeh et al., 2017)
EIBMV-DMAS	365% (vs DAS)	158% (vs DAS)	~94% (FWHM)	(Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017)

DMAS and its variants consistently outperform DAS in contrast, lateral and axial resolution, and sidelobe floor, facilitating visualization of high-resolution and low-contrast structures in challenging noise environments.

7. Applications and Limitations

DMAS is broadly applicable to any array-based imaging system where spatial coherence is diagnostically relevant, including photoacoustic imaging, clinical and preclinical ultrasound, and acoustic field mapping (Paridar et al., 2018, Jansen et al., 12 Nov 2025). Filtered and double-stage variants further specialize DMAS for robust microvascular imaging and clutter-suppressed power Doppler mapping in ultrafast ultrasound.

The main limitations are computational: even with $O(N)$ optimizations, higher-order and adaptive DMAS variants pose challenges for real-time volumetric (3D/4D) imaging on large arrays. Furthermore, DMAS spatial resolution is ultimately limited by physical array aperture unless combined with appropriate adaptive or subspace beamformers, and performance can degrade in the presence of unmodeled propagation delays or strong aberrations. Nonlinear operations can yield modest reductions in contrast-to-noise ratio for very weak targets (Mozaffarzadeh et al., 2018).

Further developments include real-time FPGA/GPU implementations, extension to curvilinear and 3D arrays, and hybrid adaptive–nonlinear strategies that balance computational load with image quality objectives.