---
title: Delay-Multiply-and-Sum (DMAS) Beamforming
url: https://www.emergentmind.com/topics/delay-multiply-and-sum-dmas
type: topic
---

# Delay-Multiply-and-Sum (DMAS) Beamforming

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 $\mathbf{r}$ [1709.02549][1805.09913][1801.07446].

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 [2203.14906][2511.09165].

## 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 [1802.09310].

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 [2203.14906][2511.09165].

## 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 [2402.17643][2107.08069].
- **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 $\text{DMAS}_5$ [2203.14906].
- **Adaptive frameworks**: DMAS can be fused with adaptive or minimum-variance weights (MVB-DMAS, EIBMV-DMAS) for further mainlobe sharpening and sidelobe rejection [1709.06523][1709.07965][1801.06421][1807.07405].
- **Local segmentation**: To reduce total computational load, region-of-interest segmentation schemes process only promising sub-regions at full resolution [1709.02549].

## 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 [1709.02549][1802.09310][1801.07446]. In adaptive variants (MVB-DMAS, EIBMV-DMAS), total reductions exceed 30–100 dB depending on depth and imaging modality [1709.07965][1709.06523][1807.07405].
- **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 [1802.09310][2203.14906].
- **SNR Increase**: SNR gains of 3–30 dB, with the largest improvements seen in higher-order DMAS and minimum-variance variants [2203.14906][1709.07965].
- **Contrast Ratio (CR)**: Up to $\sim$14 dB improvement in CR for reflected or specular targets versus DAS [2107.08069][2402.17643].
- **Artifact Suppression**: Experimental phantoms, B-mode, and power Doppler maps show improved vessel delineation and reduced speckle and clutter [2402.17643].

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↑  |

[1709.06523][2203.14906][2402.17643][2107.08069][1802.09310][1709.07965][2511.09165][1801.07446][1805.09913]

## 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 [2203.14906][2511.09165]. 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 [1709.06523][1801.06421][1709.07965][1807.07405]. 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 [1801.07446].

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 [2511.09165].

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 [2402.17643][2107.08069].

## 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 [1802.09310][2107.08069].
- **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 [1709.02549][1802.09310][1805.09913].
- **Ultra-wideband and microwave imaging**: Enhanced tumor localization accuracy in breast cancer detection, with adaptive spatial-resolution frameworks for computational efficiency [1709.02549].
- **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 [2511.09165].

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 [2203.14906][2511.09165].

## 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 [2203.14906][2511.09165].
- **Resolution gains plateau with increasing DMAS order beyond $K=3$–5**, with diminishing returns and potential instability due to phase errors [2203.14906].
- **No significant improvement in spatial resolution** beyond the aperture- and bandwidth-limited mainlobe—the main gain resides in sidelobe and noise suppression [2511.09165][2402.17643].
- **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 [2107.08069].
- **In shallow or angular-limited configurations, DMAS can underperform due to insufficient correlation statistics among elements** [2107.08069].
- **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 [1709.06523][1709.07965].

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 [1709.02549][2203.14906][2511.09165].

---

**Key Citations**:  
[1709.02549]  
[1709.06523]  
[1709.07965]  
[1801.07446]  
[1802.09310]  
[1805.09913]  
[1807.07405]  
[2107.08069]  
[2203.14906]  
[2402.17643]  
[2511.09165]

Source: https://www.emergentmind.com/topics/delay-multiply-and-sum-dmas