So you think you can DAS? A viewpoint on delay-and-sum beamforming

DAS position in ultrafast ultrasound imaging

• The paper examines DAS for diverging and plane waves, which insonify a large region and support high-frame-rate image acquisition.Multiple transmissions are commonly combined to obtain higher-quality images.Source: §1 · p. 1, A. Pulse-echo with a virtual point… · p. 2

Motivation

• DAS is widely used in high-frame-rate ultrasound because it is simple, fast, numerically robust, parallelizable, and compatible with real-time applications.Its data-independent operation also preserves temporal coherence and the statistical properties of real envelopes.Source: §1 · pp. 1–2

Fundamentals

• DAS forms an image by estimating round-trip traveltimes and summing signal amplitudes along the hyperbolic signatures associated with scatterers.The formulation uses transmit and receive distances under a uniform sound-speed assumption.Source: A. Pulse-echo with a virtual point… · p. 2

Transmit modeling

• Circular-wave transmission is parameterized by a virtual point source, with plane-wave imaging obtained as the angular width β tends toward zero.The transmit distance depends on the virtual source and the array geometry.Source: A. Pulse-echo with a virtual point… · p. 3

Model assumptions

• The transmit-distance expression is constrained to points in the sensor-array shadow and assumes that the minimum transmission delay is zero.The Heaviside term selects the shortest distance from the virtual source to the transducer.Source: A. Pulse-echo with a virtual point… · p. 3

Hyperbolic signatures

• Scatterers appear as hyperbolic signatures when element signals are arranged in the x–fast-time plane.The hyperbola is derived from the scatterer location and propagation delays.Source: B. Hyperbolic signatures and DAS · p. 3

Beamformed outputs

• Beamformed I/Q data can produce B-mode images from log-compressed moduli or Doppler images from temporal phase shifts.Only a subset of correctly selected I/Q signals may need to be summed along the hyperbolas.Source: B. Hyperbolic signatures and DAS · p. 4

DAS operation

• DAS forms an amplitude image by adding signal amplitudes along the hyperbolas associated with point scatterers.This treats each reflector’s contribution as if it were emitted simultaneously.Source: B. Hyperbolic signatures and DAS · p. 4

Receive aperture

• A subset of array-element signals can be used in Eq. (9) to account for element directivity through an f-number.This corresponds to discarding some signals during receive beamforming.Source: B. Hyperbolic signatures and DAS · p. 4

Numerical implementation

• Sampled signals at the required delays are estimated from nearby discrete values using q-point interpolation.Examples range from nearest-neighbor and linear interpolation to 5-lobe Lanczos interpolation.Source: B. Hyperbolic signatures and DAS · p. 4

I/Q beamforming

• I/Q signals are low-frequency representations that contain amplitude and phase information for B-mode and Doppler imaging.They can be beamformed directly onto an image grid such as 256×256.Source: C. Beamforming I/Q signals · p. 4

Phase preservation

• Phase rotators are necessary when delayed-and-summed I/Q signals are beamformed after demodulation.Summation and demodulation are non-commuting operations.Source: C. Beamforming I/Q signals · p. 4

Element directivity

• The directivity curve and its greater-than−3 dB conical sector visualize the angular sensitivity of one array element.The sector depends on element width relative to wavelength.Source: Figure 4 · p. 4

Parameter selection

• The proposed f-number is linked to the element directivity and the corresponding angle-of-view.The angle-of-view is represented as 2α.Source: Figure 5 · p. 4

Element directivity

• Element directivity depends on the element-width-to-wavelength ratio and determines how uniformly signals are received across directions.Larger W/λ produces higher directivity, while smaller ratios are more omnidirectional.Source: C. Beamforming I/Q signals · p. 4

Directivity

• Element directivity depends on propagation angle relative to the z-axis and decreases as the angle approaches ±π/2.The received sound pressure is maximal at zero angle.Source: D. Receive f-number · p. 5

Definition

• The receive f-number is the scatterer depth divided by the width of the receive aperture and is related to angle-of-view 2α.Using the top part of the hyperbolas is recommended because signal-to-noise ratio decreases away from the vertex.Source: D. Receive f-number · p. 5

Threshold choice

• A −3 dB threshold corresponds to D_thresh = 0.71 as a rule-of-thumb compromise between image quality and signal-to-noise ratio.This threshold defines which directivity amplitudes are discarded.Source: D. Receive f-number · p. 5
• When W/λ = 1, the directivity-based expressions yield f# = 1.2.The relevant wavelength should be the smallest significant wavelength of the signal.Source: D. Receive f-number · p. 5

DAS implementation

• Applying the f-number modifies the DAS equation by truncating the summed aperture.The receive operation therefore uses only the permitted portion of each hyperbolic signature.Source: D. Receive f-number · p. 5

Sound-speed sensitivity

• The speed of sound governs hyperbola shape, and a mismatch causes DAS to sum amplitudes along incorrect paths.This mismatch can significantly affect image quality and distort the output point-spread function.Source: E. Speed of sound · p. 5

Estimation principle

• The expected speed of sound is defined as the value that uniformizes phase along the delayed hyperbolas.The phase-based criterion is evaluated over beamformed points.Source: E. Speed of sound · p. 5
• The Qp metric weights phase-variance information by intensity, prioritizing contributions from bright speckles.Earlier work minimized only the sum of phase variances.Source: E. Speed of sound · pp. 5–6

Scope of estimation

• The proposed procedure estimates a constant average speed of sound rather than a spatial map.The region of interest and its average speed are treated as more or less time-invariant during organ scanning.Source: E. Speed of sound · p. 6

Numerical implementation

• DAS beamforming can be represented as multiplication by a large sparse matrix containing interpolation weights, phase rotators, and f-number truncation.The matrix maps raw signal samples to beamformed data points.Source: Figure 7 · p. 6

• DAS is a linear operator that transforms element-recorded I/Q signals into beamformed I/Q data.Source: F. DAS as a matrix product · p. 6
• The DAS matrix contains interpolation weights, phase rotators, and sum truncation induced by a positive f-number.Source: F. DAS as a matrix product · p. 6
• For M beamforming pixels, the DAS matrix has size M × n_sN_e and produces a beamformed vector of length M.Source: F. DAS as a matrix product · p. 6
• The matrix is large but sparse, with limited receive apertures reducing its number of nonzero elements.Source: F. DAS as a matrix product · p. 6
• With unchanged imaging geometry, the DAS matrix needs to be calculated only once, after which sparse matrix-vector multiplication enables fast real-time visualization.Source: F. DAS as a matrix product · p. 6
• The experimental evaluation used PICMUS CIRS-phantom data and carotid acquisitions to examine parameter effects in vitro and in vivo.Source: F. DAS as a matrix product · p. 7, §3 · p. 7, Figure 8 · p. 7

In vitro results

• 𝑓# ≈1.4 was obtained from element directivity, while maximizing 𝒬𝒑 with one plane wave yielded an optimal speed of sound of 1570 m/s.Source: A. In vitro results · p. 7
• Both speed of sound and f-number significantly affect CNR and lateral resolution in the CIRS phantom.Source: A. In vitro results · p. 7
• Physically estimated speed-of-sound and f-number values were consistent with brute-force search results.Source: A. In vitro results · p. 8

In vivo results

• In eight carotids, maximizing 𝒬𝒑 produced estimated speeds of sound of 1510 ± 41 m/s.Source: Figure 10 · p. 7, B. In vivo results · p. 8
• One carotid produced an estimated speed of sound of 1400 m/s, which improved the image metric despite probably not representing the artery’s average speed of sound.Source: B. In vivo results · p. 8, Figure 11 · p. 8
• The directivity-derived f-number had an obvious positive impact on image quality, whereas speed of sound mainly changed the z-direction scale in the example.Source: B. In vivo results · p. 8

• The f-number is related to transducer-element directivity and depends on element width, center frequency, and pulse-echo bandwidth.Source: A. The 𝑓𝑓-number and speed of sound… · p. 8
• An excessively large f-number creates bias by summing too few elements, whereas an excessively small f-number increases variance through low-directivity, low-SNR contributions.Source: A. The 𝑓𝑓-number and speed of sound… · p. 8
• A -3dB directivity threshold was heuristically identified as a good f-number compromise, with D_thr=0.71 in Eq. (13).Source: A. The 𝑓𝑓-number and speed of sound… · p. 8
• The smallest vector-velocity errors in a rotating-disk example coincided with the directivity-derived f-number of approximately 2.6.Source: A. The 𝑓𝑓-number and speed of sound… · p. 8, Figure 12 · p. 8
• The speed of sound determines which diffraction hyperbola is selected, while the f-number determines which portion of that hyperbola is included.Source: A. The 𝑓𝑓-number and speed of sound… · pp. 8–9, Figure 13 · p. 8
• In vitro, speed-of-sound choice significantly affected CNR and FWHM, whereas in vivo appearance was largely unchanged except for axial dimensions.Source: A. The 𝑓𝑓-number and speed of sound… · p. 9
• Adaptive and deep-learning methods may depend on properly tuned DAS, because DAS can serve as a reference or provide training images.Source: B. DAS vs. advanced beamformers · p. 9
• The conclusion recommends optimizing DAS parameters because neglecting the f-number or choosing the speed of sound poorly can impair measured image quality.Source: §5 · p. 9

• A focus-point derivation analogous to Eq. (5) is used to obtain an appendix distance relation.Source: §6 · p. 9
• Equations (5) and (20) combine into a generalized transmit distance covering focusing and diverging wavefronts.Source: A. Generalized transmit distance… · p. 9
• Taking the virtual-source limit β→0+ produces the plane-wave limiting case.Source: B. Plane wave as a limit of diverging… · p. 9
• The derivation expands square terms, factors the result, and applies a first-order Taylor series to simplify the limiting expression.Source: B. Plane wave as a limit of diverging… · pp. 9–10
• The resulting limit expresses d_TX(x_s,z_s) using the functions f(θ)=L cos(θ)sin(θ) and g(θ)=L cos^2(θ).Source: B. Plane wave as a limit of diverging… · pp. 9–10

• The f-number script finds the angle whose element directivity reaches 0.71, then computes fnumber=1/(2 tan(alpha)).Source: C. Determine the f-number in Matlab · p. 10
• EZDAS beamforms RF or I/Q signals at coordinates specified by x and z using a virtual source and parameter structure.Source: D. A simple Matlab code for DAS… · p. 10
• The implementation represents each column of SIG as a fast-time RF or I/Q signal from one array element.Source: D. A simple Matlab code for DAS… · p. 10
• For a uniform linear array, the code computes element coordinates from pitch and derives the array length L.Source: D. A simple Matlab code for DAS… · p. 10
• DAS delays combine transmit distance from the virtual source with receive distances from each element to each image point.Source: D. A simple Matlab code for DAS… · p. 10
• The receive aperture retains elements satisfying abs(x-xe)<=z/(2*param.fnumber), thereby enforcing the selected f-number.Source: D. A simple Matlab code for DAS… · p. 10
• The code uses linear interpolation and, for complex I/Q data, applies phase rotations before constructing the sparse DAS matrix.Source: D. A simple Matlab code for DAS… · p. 10
• The DAS matrix maps raw signals into interpolated I/Q values along the M hyperbolas associated with the beamformed pixels.Source: Figure 14 · p. 10, D. A simple Matlab code for DAS… · p. 10

• EZSOS estimates the speed of sound that yields an “optimal” real-envelope image after DAS beamforming.Source: E. A simple Matlab code for estimating… · p. 11
• The estimator analyzes phase behavior along diffraction hyperbolas whose vertices are located at the supplied image coordinates.Source: E. A simple Matlab code for estimating… · p. 11
• For each candidate speed c, the code updates the beamforming speed parameter and recomputes DAS using coordinates scaled by c/c0.Source: E. A simple Matlab code for estimating… · p. 11
• The matrix product M*IQ produces the diffraction-hyperbola signals used for the phase analysis.Source: E. A simple Matlab code for estimating… · p. 11
• The criterion combines mean hyperbola-signal magnitude with unwrapped phase dispersion, using a negative averaged score for optimization.Source: E. A simple Matlab code for estimating… · p. 11

• EZRF2IQ converts RF bandpass signals into complex I/Q data, with real and imaginary parts representing the in-phase and quadrature components.The routine accepts sampling frequency and center frequency as inputs.Source: F. A simple Matlab code for I/Q… · p. 11
• The routine demodulates 2-D and 3-D RF data along columns, treating each column as one RF signal.Source: F. A simple Matlab code for I/Q… · p. 11
• The code converts row-vector input to a column vector before processing and restores the original row-vector shape afterward.Source: F. A simple Matlab code for I/Q… · p. 11
• Downmixing multiplies the RF data by exp(-1i*2*pi*Fc*t) to shift the signals using the center frequency.Source: F. A simple Matlab code for I/Q… · p. 11
• A fifth-order Butterworth low-pass filter is applied with filtfilt, and the result is scaled by 2.Source: F. A simple Matlab code for I/Q… · p. 11