Delay-Multiply-and-Sum (DMAS) Beamforming

Updated 17 March 2026

Delay-Multiply-and-Sum (DMAS) is a nonlinear beamforming method that forms pairwise multiplicative correlations of delayed signals to improve contrast and suppress sidelobes.
It employs sign-root normalization and higher-order correlations to achieve significant SNR enhancements and resolution improvements over conventional DAS.
Recent closed-form O(N) implementations and GPU acceleration have made DMAS viable for real-time medical imaging and other high-performance applications.

Delay-Multiply-and-Sum (DMAS) is a non-linear beamforming algorithm originally introduced in confocal microwave imaging for breast cancer detection and now extensively used in medical ultrasound (US), photoacoustic imaging (PAI), and in-air acoustic imaging. DMAS enhances image quality by suppressing sidelobes and increasing contrast relative to linear Delay-and-Sum (DAS), leveraging pairwise or higher-order multiplicative correlations of delayed channel signals. The method is characterized by its nonlinear summation structure, which extracts spatial coherence across array elements and thus attenuates noise and off-axis clutter more effectively than DAS. Although its computational load historically limited its use in real-time systems, recent closed-form $O(N)$ implementations and GPU-based strategies have made DMAS and its extensions viable even in embedded settings.

1. Mathematical Formulation and Algorithmic Structure

The canonical DMAS beamformer, for an $M$ -element array, forms all pairwise products of the delay-corrected radio-frequency (RF) signals: $y_{\mathrm{DMAS}}(\mathbf{r}) = \sum_{i=1}^{M-1} \sum_{j=i+1}^{M} s_i\bigl(\tau_i(\mathbf{r})\bigr) \, s_j\bigl(\tau_j(\mathbf{r})\bigr)$ where $s_i(\cdot)$ is the time-delayed signal for element $i$ and $\tau_i(\mathbf{r})$ is the time-of-flight to the reconstruction point $\mathbf{r}$ (Ettefagh et al., 2017, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).

Variants introduce a sign-root normalization to moderate dynamic range: $y_{\mathrm{DMAS}}(\mathbf{r}) = \sum_{i=1}^{M-1} \sum_{j=i+1}^{M} \mathrm{sign}\left[u_i(\mathbf{r})\,u_j(\mathbf{r})\right] \sqrt{|u_i(\mathbf{r})\,u_j(\mathbf{r})|}$ This configuration accentuates coherent signal contributions and suppresses uncorrelated noise.

Algebraic expansion reveals an embedded DAS term, enabling combinations with adaptive beamformers, and paves the way for higher-order (K-wise) multiplicative generalizations (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

2. Theoretical Underpinnings and Nonlinear Properties

DMAS functions as a spatial autocorrelator, reinforcing coherent in-phase signals while averaging out incoherent noise and off-axis contributions:

True scatterers yield constructive products because delayed signals align with similar phase and amplitude.
Noise and off-axis echoes, being less correlated, yield products with near-zero mean contributions.

Mathematically, the point-spread-function (PSF) of DMAS is the autoconvolution of the DAS PSF, resulting in a narrowed mainlobe and suppressed sidelobe structure (Paridar et al., 2018).

By extension, higher-order DMAS ( $\text{DMAS}_K$ ) leverages products of $K$ signals. For example: $S_{\mathrm{DMAS}-3} = \sum_{i<j<k} \sqrt[3]{s_i\,s_j\,s_k}$ Closed-form Newton-Girard expansions allow these higher-order correlations to be computed in $O(M)$ per pixel (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

3. Practical Implementation and Computational Strategies

The core computational burden of DMAS is $O(M^2)$ multiplies and sums per pixel, compared to $O(M)$ for DAS. This cost is dominated by the combinatorics of all pairs (or higher-order tuples) of delayed signals. Modern implementation strategies include:

Sign-root pre-processing for numeric stability.
Band-pass filtering of DMAS output, particularly in filtered DMAS (F-DMAS), to isolate second-harmonic components and suppress DC leakage (Madhavanunni et al., 2024, Malamal et al., 2021).
GPU acceleration: Recent closed-form reductions and massively parallel implementations achieve real-time DMAS and high-order DMAS on standard GPUs. For $2048^2$ photoacoustic frames, up to 12 fps is reported for $\text{DMAS}_5$ (Mulani et al., 2022).
Adaptive frameworks: DMAS can be fused with adaptive or minimum-variance weights (MVB-DMAS, EIBMV-DMAS) for further mainlobe sharpening and sidelobe rejection (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).
Local segmentation: To reduce total computational load, region-of-interest segmentation schemes process only promising sub-regions at full resolution (Ettefagh et al., 2017).

4. Quantitative Performance Gains

Experimental and simulation results across multiple domains consistently demonstrate that DMAS provides:

Sidelobe suppression: Reductions of 8–20 dB relative to DAS are typical (Ettefagh et al., 2017, Paridar et al., 2018, Mozaffarzadeh et al., 2018). In adaptive variants (MVB-DMAS, EIBMV-DMAS), total reductions exceed 30–100 dB depending on depth and imaging modality (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2018).
Narrower Mainlobe (Resolution improvement): FWHM shrinks by 10–40%, from $\sim$ 1.2 mm to $\sim$ 1.0 mm for point targets, and further improvement is achieved in higher-order DMAS and adaptive combinations (Paridar et al., 2018, Mulani et al., 2022).
SNR Increase: SNR gains of 3–30 dB, with the largest improvements seen in higher-order DMAS and minimum-variance variants (Mulani et al., 2022, Mozaffarzadeh et al., 2017).
Contrast Ratio (CR): Up to $\sim$ 14 dB improvement in CR for reflected or specular targets versus DAS (Malamal et al., 2021, Madhavanunni et al., 2024).
Artifact Suppression: Experimental phantoms, B-mode, and power Doppler maps show improved vessel delineation and reduced speckle and clutter (Madhavanunni et al., 2024).

Table: Representative Quantitative Gains (Selected Depths and Modalities)

Method	Sidelobe (dB)	FWHM (mm)	SNR Gain (dB)	CR (dB)
DAS	–10 to –31	1.05–2.41	—	14–28
DMAS	–14 to –42	0.75–1.8	+3 to +25	24–52
MVB-DMAS	–22 to –25	0.55–0.6	+11 to +12	—
EIBMV-DMAS	–140	0.10	+45.6	—
F-DMAS	–	0.35λ	SNR↑6–8%	CR↑6–8%
DMAS-5	–109	1.6	+25	CR/gCNR↑

(Mozaffarzadeh et al., 2017, Mulani et al., 2022, Madhavanunni et al., 2024, Malamal et al., 2021, Paridar et al., 2018, Mozaffarzadeh et al., 2017, Jansen et al., 12 Nov 2025, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018)

5. Extensions: Higher-Order DMAS, Adaptive and Hybrid Approaches

Beyond pairwise DMAS, higher-order generalizations (DMAS- $K$ for $K>2$ ) further amplify coherence, yielding substantial SNR and contrast improvements—up to 81% SNR gain over DAS and 39% over DMAS (FWHM also narrows by 51% over DAS), with $O(M)$ complexity via polynomial expansions (Mulani et al., 2022, Jansen et al., 12 Nov 2025). Adaptive extensions—such as MVB-DMAS (minimum-variance weights) and EIBMV-DMAS (eigenspace-based weights)—integrate DMAS’s coherence sensitivity with statistical spatial filtering, achieving extreme sidelobe and mainlobe performance at higher computational cost, suitable for deep tissue or high-contrast applications (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2018). Double-Stage DMAS (DS-DMAS) recursively structures the multiplication and summation, combining first, second, and third moments for additional robustness to noise and further SNR, FWHM, and CR gains (Mozaffarzadeh et al., 2018).

Coherence Factor (CF) weighting, often post-applied to DMAS outputs, enhances contrast by penalizing incoherent signal sums and aiding in artifact suppression—especially valuable in speckle-dominated or in-air acoustic imaging (Jansen et al., 12 Nov 2025).

Filtered DMAS (F-DMAS) applies band-pass filtering to the DMAS output, extracting second-harmonic content for improved lateral resolution, particularly in ultrafast ultrasound localization microscopy and specular tissue characterization where harmonic content improves target detectability (Madhavanunni et al., 2024, Malamal et al., 2021).

6. Application Domains and Empirical Validation

DMAS and its extensions are widely validated in:

Medical ultrasound imaging: Improved spatial resolution and clutter rejection in phantom and in vivo studies, especially in dense tissue, vascular, and needle visualization scenarios (Paridar et al., 2018, Malamal et al., 2021).
Photoacoustic tomography: Major SNR, sidelobe, FWHM and CR enhancements in both 2D and 3D configurations, for clinical targets such as breast tumors, sentinel lymph nodes, and deep tissue mapping (Ettefagh et al., 2017, Paridar et al., 2018, Mozaffarzadeh et al., 2018).
Ultra-wideband and microwave imaging: Enhanced tumor localization accuracy in breast cancer detection, with adaptive spatial-resolution frameworks for computational efficiency (Ettefagh et al., 2017).
In-air acoustic imaging and sonar: Robust performance under single-snapshot constraints for real-time 3D acoustic imaging, including mobile robotics and non-contact applications (Jansen et al., 12 Nov 2025).

Key empirical findings confirm that, while mainlobe width is fixed by aperture and bandwidth, DMAS dramatically lowers sidelobe energy and background noise, with higher-order variants pushing dynamic range to 75–80 dB and enabling clear separation of closely spaced sources (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

7. Limitations, Trade-Offs, and Implementation Considerations

The principal trade-off in DMAS is computational complexity: $O(M^2)$ for standard DMAS and $O(M^K)$ for naive K-wise extensions, although explicit expansions reduce this to $O(M)$ for practical $K$ . The following limitations and considerations are reported:

Hardware acceleration is typically necessary (GPU/FPGA/ASIC), especially for real-time or volumetric scenarios (Mulani et al., 2022, Jansen et al., 12 Nov 2025).
Resolution gains plateau with increasing DMAS order beyond $K=3$ –5, with diminishing returns and potential instability due to phase errors (Mulani et al., 2022).
No significant improvement in spatial resolution beyond the aperture- and bandwidth-limited mainlobe—the main gain resides in sidelobe and noise suppression (Jansen et al., 12 Nov 2025, Madhavanunni et al., 2024).
Parameter tuning (especially in F-DMAS and CF variants) can affect artifact suppression and contrast; deep targets and specular geometries benefit most when sufficient aperture coverage is ensured (Malamal et al., 2021).
In shallow or angular-limited configurations, DMAS can underperform due to insufficient correlation statistics among elements (Malamal et al., 2021).
Adaptive and hybrid schemes (e.g., MVB-DMAS, EIBMV-DMAS) achieve best-in-class performance for deep or high-contrast imaging, but at $O(M^3)$ cost due to covariance estimation and inversion operations (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017).

Recent work proposes hierarchical or locally adaptive frameworks to reduce per-frame computational costs, and closed-form $O(M)$ expansions for real-time feasibility even in embedded and portable imaging scenarios (Ettefagh et al., 2017, Mulani et al., 2022, Jansen et al., 12 Nov 2025).