Filtered DMAS: Advanced ULM Beamforming

• Filtered DMAS is a beamforming algorithm that extends DAS and DMAS by emphasizing coherent microbubble echoes and rejecting clutter through pairwise channel correlation.
• It applies frequency doubling and narrow bandpass filtering centered at 2f₍c₎ to improve image contrast and lateral resolution in ultrafast ultrasound localization microscopy.
• Experimental results in both in-silico and in-vitro studies demonstrate enhanced local contrast, reduced lateral spread, and improved microbubble tracking.

Filtered Delay Multiply and Sum (F-DMAS) is a beamforming algorithm designed for high-resolution image reconstruction in Ultrafast Ultrasound Localization Microscopy (ULM). F-DMAS extends the Delay-and-Sum (DAS) and Delay-Multiply-and-Sum (DMAS) beamformers by emphasizing coherent microbubble echoes while rejecting clutter, exploiting pairwise channel correlations and spectral filtering. This approach produces microvascular maps with enhanced contrast and lateral resolution, facilitating more accurate microbubble localization and velocity tracking compared to conventional DAS-based methods (Madhavanunni et al., 2024).

1. Mathematical Foundation of F-DMAS

Let an $N_c$-element receive array acquire baseband signals $u_i[n]$ at element $i$. For pixel $p$ at $(x_p, z_p)$, the sample delay to element $i$ is

$$\Delta n_{p,i} = \mathrm{round}\left(f_s \cdot \frac{\sqrt{(x_i - x_p)^2 + z_p^2}}{c}\right)$$

where $c$ denotes sound speed and $f_s$ the sample rate. For non-steered (plane-wave) transmissions, the classical DAS beamformer is: $$y_{\mathrm{DAS}}[p] = \sum_{i=1}^{N_c} w_i[p] \, u_i[n - \Delta n_{p,i}]$$ where $w_i[p]$ applies apodization.

DMAS generalizes this by forming pairwise products of delayed, signed square-root processed channels: $$v_i[n] = \mathrm{sign}\left(u_i[n]\right) \, \sqrt{|u_i[n]|}$$

$$\hat{y}_{\mathrm{DMAS}}[p] = \sum_{i=1}^{N_c-1} \sum_{j=i+1}^{N_c} v_i[n - \Delta n_{p,i}] \cdot v_j[n - \Delta n_{p,j}]$$

This process multiplies two bandpass signals at $f_c$, producing spectral components at DC and $2f_c$.

F-DMAS introduces a narrow bandpass filter $H(\omega)$ centered on $2f_c$, yielding the final beamformed output: $$y_{\mathrm{F-DMAS}}[p] = H * \hat{y}_{\mathrm{DMAS}}[p]$$ where $*$ denotes convolution. The filter passband is selected to match $[2f_c-\Delta f, 2f_c+\Delta f]$, suppressing near-DC (clutter) and isolating the coherent $2f_c$ content (Madhavanunni et al., 2024).

2. Signal Processing Principles and Image Quality Improvement

F-DMAS enhances local contrast and lateral resolution by three primary mechanisms:

• Synthetic aperture widening: Pairwise channel multiplication mimics the effect of combining channel apertures, which increases the effective aperture and tightens the main lobe.
• Coherence weighting: Cross-multiplication down-weights incoherent noise and off-axis (side lobe) artifacts, yielding improved suppression of clutter and side lobes.
• Frequency-doubling plus filtering: Focusing on the $2f_c$ spectral component through bandpass filtering removes low-frequency reverberation and stationary echoes, further elevating image contrast.

These characteristics collectively lead to more faithful and contrasted depiction of small vessels and microbubble tracks, as verified by both qualitative visualization and quantitative metrics (Madhavanunni et al., 2024).

3. Experimental Protocols and Pipeline Integration

Experiments assessed F-DMAS performance in both in-silico and in-vitro settings:

In-silico Phantom (PALA Dataset):

• Eleven-tube microvascular structure.
• Plane-wave transmit at 15.625 MHz, PRF 500 Hz.
• 128-element linear array, simulated in Field II (pitch 0.11 mm, $f_s = 100$ MHz, $c = 1540$ m/s).

In-vitro Phantom:

• PVA-based flow phantom (110 μm channel diameters).
• Verasonics Vantage 128 with L11-5v probe (7.6 MHz, 1.5 $\lambda$ pitch), plane waves at 2 kHz PRF, $f_s = 31.25$ MHz.
• Blood mimic: 3% potato-starch in water/glycerol; SVD-based clutter removal.

The ULM processing pipeline is as follows:

1. Raw RF data —> beamforming (DAS or F-DMAS) —> IQ demodulation.
2. Region cropping with $\lambda \times \lambda$ pixel sizes.
3. Microbubble localization via the LOTUS toolbox, applying four sub-pixel localization methods: spline interpolation (Sp-Interp), Gaussian fitting, weighted average (WA), and radial symmetry (RS).
4. Microbubble tracking, velocity calculation, and rendering of B-mode, power Doppler, and super-resolution maps (Madhavanunni et al., 2024).

4. Novel Image Quality Metrics

Two bespoke metrics were introduced to address limitations in conventional quality measures that require knowledge of ground truth targets:

Local Contrast Score ($\mu_{LC} \pm \sigma_{LC}$)

• Compute a local standard deviation map $S(x,y)$ over the localization density image $R(x, y)$ using a $2 \times 2$ kernel ($0.2\lambda \times 0.2\lambda$ window):

$$S(x, y) = \sqrt{\frac{1}{4}\sum_{(u,v) \in 2\times2_{\text{patch}}}[R(x+u, y+v) - \overline{R}_{\text{patch}}]^2}$$

• Report field-wide mean $\mu_{LC}$ and standard deviation $\sigma_{LC}$. Greater $\sigma$ signifies better microvascular contrast because small vessels drive local intensity variations.

Lateral Spread Score ($L_{FWHM}$)

• Select a straight vessel (e.g., vertical channel), normalize the axial cross-section to $[0, 1]$, extract the full width at half maximum (FWHM) laterally, and average over slices.
• Express $L_{FWHM}$ in units of $\lambda$. Smaller $L_{FWHM}$ indicates finer lateral resolution of vessel walls and tracks (Madhavanunni et al., 2024).

5. Empirical Performance and Comparative Assessment

F-DMAS yielded improvements over DAS in both simulated and physical phantom scenarios:

Metric / Method	DAS + Sp-Interp	F-DMAS + Sp-Interp	Ground Truth
Local Contrast $\mu_{LC}$	0.905	0.932	0.952
Local Contrast $\sigma_{LC}$	0.213	0.199	0.173
Lateral Spread $L_{FWHM}$ ($\lambda$)	0.263	0.257	0.238

• F-DMAS increased $\mu_{LC}$ by 3.0%, reduced $\sigma_{LC}$ by 6.6%, and reduced $L_{FWHM}$ by 2.3% relative to DAS with Sp-Interp, with consistent improvements for RS, WA, and Gaussian Fit methods.
• In in-silico localization maps, F-DMAS produced thinner and higher-contrast vessels, suppressed side-lobe artifacts, and recovered channels not visible in DAS.
• Velocity maps generated under F-DMAS recovered fine vessel tracks that DAS could not resolve.
• In-vitro B-mode and power Doppler images exhibited sharper vessel boundaries and improved contrast/continuity under F-DMAS (Madhavanunni et al., 2024).

6. Implementation Considerations and Prospects

• F-DMAS computational cost scales as $O(N_c^2)$ due to exhaustive pairwise channel products. Real-time or high-throughput scenarios may require GPU or FPGA acceleration.
• Filter design for $2f_c$ should maintain signal bandwidth while minimizing group delay.
• Unlike adaptive beamformers, F-DMAS does not require per-scan parameter optimization, supporting clinical integration without recovery-impairing manual tuning.
• Future directions include in vivo validation (e.g., small-animal studies), plane-wave compounding or steered DMAS to enhance coverage, integration with advanced tissue-bubble separation filters (e.g., SVD-based methods), and 2D/3D acquisition for volumetric ULM (Madhavanunni et al., 2024).

F-DMAS is a low–tuning-overhead approach that leverages synthetic aperture widening and coherence-based weighting to achieve superior vessel contrast and resolution, as substantiated by new localized metrics and multi-modal phantom experiments in the context of ULM.