Double-Stage DMAS in Medical Imaging

• Double-Stage DMAS is a nonlinear beamforming algorithm that uses two cascaded DMAS operations to generate a fourth-order spatial correlation for enhanced image quality.
• It achieves improved SNR, reduced sidelobe levels, and narrower beamwidths, as confirmed by simulation and experimental metrics in medical imaging.
• Despite its significant performance gains, DS-DMAS introduces high computational complexity, necessitating specialized hardware for real-time application.

Double-Stage Delay-Multiply-and-Sum (DS-DMAS) is a nonlinear beamforming algorithm designed for medical imaging modalities such as ultrasound (US) and photoacoustic imaging (PAI), offering enhanced spatial resolution, sidelobe suppression, and noise robustness relative to classic Delay-and-Sum (DAS) and single-stage Delay-Multiply-and-Sum (DMAS) methods. DS-DMAS achieves these improvements by nesting two correlation stages: it systematically exposes and replaces the implicit linear (DAS) block embedded within DMAS with a second DMAS operation, resulting in a higher-order nonlinear spatial correlation. This structuring yields significant performance gains at the expense of increased computational complexity (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).

1. Algorithmic Foundations

The evolution from DAS to DMAS and subsequently to DS-DMAS is grounded in exploiting higher-order spatial correlations for enhanced beamforming quality. In DAS, echo signals from an array of $M$ elements are time-aligned and summed: $y_{\mathrm{DAS}}(r) = \sum_{i=1}^{M} s_i(t_i(r))$ where $s_i$ represents the radiofrequency (RF) signal received by element $i$, and $t_i(r)$ is the travel-time delay to point $r$.

DMAS introduces a quadratic nonlinearity by forming and summing all pairwise products of the delayed signals: $$y_{\mathrm{DMAS}}(r) = \sum_{i=1}^{M-1} \sum_{j=i+1}^{M} s_i(t_i(r)) \cdot s_j(t_j(r))$$ This formulation can also be interpreted as: $$y_{\mathrm{DMAS}}(r) = \frac{1}{2} \left[ \left( \sum_{i=1}^{M} s_i(t_i) \right)^2 - \sum_{i=1}^{M} s_i^2(t_i) \right]$$ Given that the first term is the squared DAS output, DMAS performance is ultimately limited by the inner linearity of DAS.

DS-DMAS recursively replaces the DAS component in DMAS with another DMAS operation, resulting in a two-stage cascade: $$Y_{\mathrm{DS\text{-}DMAS}}(t) = \frac{1}{2} \left[ U(t)^2 - \sum_{i<j} (y_i(t) y_j(t))^2 \right]$$ where $U(t) = \sum_{i<j} y_i(t) y_j(t)$ is itself a DMAS output (Mozaffarzadeh et al., 2018). The resulting formulation is equivalent to a fourth-order nonlinear spatial correlation.

2. Operational Structure and Pseudocode

The DS-DMAS algorithm requires two main stages for each image pixel:

1. First-Stage DMAS: Compute the pairwise products of all delayed signals, apply sign–sqrt compression, and sum to produce a vector $z$.
2. Second-Stage DMAS: Apply DMAS on the $z$ vector, performing a second round of pairwise signed–sqrt product-summing.

A stepwise pseudocode extracted from (Mozaffarzadeh et al., 2018):

for i in range(1, M): for j in range(i+1, M+1): z[idx] = sign(y[i] * y[j]) * sqrt(abs(y[i] * y[j])) idx += 1 U = sum(z[0:idx]) # First DMAS output out = 0 for p in range(0, idx-1): for q in range(p+1, idx): w = z[p] * z[q] out += sign(w) * sqrt(abs(w)) Y_DS_DMAS = out # Final beamformed value

Computational complexity for $M$ channels is $O(M^2)$ for the first stage and $O(M^4)$ for the fully expanded second stage, making real-time applications hardware-intensive (FPGA/GPU required) (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).

3. Mathematical Rationale and Signal Processing Interpretation

The principal advantage of DS-DMAS lies in its ability to perform a fourth-order spatial correlation:

• The initial DMAS stage multiplies and sums pairs of delayed element responses, amplifying on-axis coherence and suppressing random noise.
• The second DMAS pass operates over the intermediate products, leading to powerful sidelobe and speckle suppression, as well as improved discrimination of closely spaced targets.

This structure results in:

• Sharpened mainlobe: Superior resolution due to the constructive reinforcement of the coherent wavefront.
• Deep lateral valleys: Strong suppression of off-axis clutter and artifacts (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).
• Higher-order noise rejection: Enhanced robustness in low-SNR conditions.

4. Comparative Performance Metrics

Quantitative comparisons in simulation and experimental studies indicate that DS-DMAS outperforms both DAS and single-stage DMAS:

Metric	DAS	DMAS	DS-DMAS	DS-DMAS vs. DMAS
Sidelobe level	–40 dB	–50 dB	–60 dB	5–10 dB lower
SNR (wire target)	20 dB	23 dB	26 dB	~13% higher
FWHM (beamwidth)	0.80 mm	0.60 mm	0.50 mm	20–30% narrower
Contrast Ratio (CR)	8 dB	11 dB	14 dB	43% higher

Additional experiments demonstrate:

• Lateral valley reduction: DS-DMAS achieves valleys ~19 dB deeper than DAS and ~15 dB deeper than DMAS.
• Sidelobe suppression: ~25 dB lower than DAS; ~10 dB lower than DMAS in representative photoacoustic datasets (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).
• Improved point target resolution and maintained performance under high additive noise (Mozaffarzadeh et al., 2018).

5. Computational and Practical Implications

DS-DMAS's two-stage nonlinearity leads to a significant computational burden:

• For $M$ array channels, first-stage DMAS is $O(M^2)$, while the second-stage DMAS on $O(M^2)$ products results in $O(M^4)$ total combinations in the fully expanded form (Mozaffarzadeh et al., 2018). However, optimized and windowed variants reduce this practical load.
• Hardware-based acceleration or parallel computing is necessary for implementation at clinical frame rates.
• Repeated application of sign–sqrt transforms necessitate high-precision arithmetic to avoid quantization bias.
• The method is robust to additive Gaussian noise but can be sensitive to array calibration errors (e.g., gain/phase mismatches), which are amplified by the higher-order nonlinearities.

A plausible implication is that parameter selection (aperture partitioning, sub-aperture size) and hardware capabilities dictate the feasibility of DS-DMAS in real-time applications.

6. Scope of Application and Experimental Validation

DS-DMAS has been validated in both simulation and experimental settings for photoacoustic and ultrasound imaging:

• Linear arrays, 5–10 MHz central frequency, 64–128 elements.
• Phantoms with point targets (wires, pencil-lead), cyst phantoms in tissue-mimicking gels.
• High and low SNR regimes (down to 10 dB), demonstrating consistent performance gains in spatial resolution, contrast, and robustness to noise (Mozaffarzadeh et al., 2018, Mozaffarzadeh et al., 2018).
• In photoacoustic imaging, DS-DMAS achieves >13% SNR increase and >30% FWHM improvement over DMAS (Mozaffarzadeh et al., 2018).

Application domains where high contrast and resolution are critical, such as distinguishing closely spaced lesions or vascular targets, particularly benefit from DS-DMAS, provided the computational demands can be satisfied.

7. Limitations and Trade-offs

• Computational intensity: DS-DMAS is substantially more demanding than DAS or DMAS and may not be suitable for ultrafast or portable systems without dedicated hardware (Mozaffarzadeh et al., 2018).
• Dynamic range: Repeated nonlinear transformations can compress dynamic range and lead to quantization bias if not carefully managed.
• Fine-tuning required: Sub-aperture design and overlap parameters impact contrast and resolution; excessive overlap may slightly degrade contrast (Mozaffarzadeh et al., 2018).
• Depth-dependent trade-off: At extreme imaging depths, signal attenuation may suppress the benefit of higher-order correlation due to weak underlying SNR (Mozaffarzadeh et al., 2018).

DS-DMAS embodies the paradigm of maximizing image quality through recursive spatial coherence exploitation at the cost of computational efficiency. Its integration into medical imaging systems depends on the balance between required performance and available processing resources.