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So you think you can DAS? A viewpoint on delay-and-sum beamforming
Vincent Perrot, Maxime Polichetti, François Varray, Damien Garcia
TL;DR
DAS is widely used in high-frame-rate ultrasound, but its image quality depends on correctly setting the receive aperture and speed of sound. This viewpoint explains DAS theory and proposes physically motivated ways to select these parameters. The authors show that optimizing them can substantially improve DAS image quality in vitro and in vivo.
Problem
DAS is widespread in high-frame-rate ultrasound, yet it is often used as a substandard reference without sufficient attention to receive aperture and speed of sound.
Method
The paper provides a theoretical and pedagogical account of DAS and proposes selecting the f-number from transducer-element directivity and sound speed by maximizing phase uniformity.
Results
Optimizing the f-number and speed of sound substantially improved DAS-derived image quality in the reported in vitro and in vivo investigations.
Takeaways & Limitations
DAS should be implemented and compared with alternative beamformers only after its receive aperture and sound speed have been properly optimized.
Abstract
from arXiv · showhide
Delay-and-sum (DAS) is the most widespread digital beamformer in high-frame-rate ultrasound imaging. Its implementation is simple and compatible with real-time applications. In this viewpoint article, we describe the fundamentals of DAS beamforming. The underlying theory and numerical approach are detailed so that users can be aware of its functioning and limitations. In particular, we discuss the importance of the f-number and speed of sound on image quality, and propose one solution to set their values from a physical viewpoint. We suggest determining the f-number from the directivity of the transducer elements and the speed of sound from the phase dispersion of the delayed signals. Simplified Matlab codes are provided for the sake of clarity and openness. The effect of the f-number and speed of sound on the lateral resolution and contrast-to-noise ratio was investigated in vitro and in vivo. If not properly preset, these two factors had a substantial negative impact on standard metrics of image quality (namely CNR and FWHM). When beamforming with DAS in vitro or in vivo, it is recommended to optimize these parameters in order to use it wisely and prevent image degradation.
1. Introduction
DAS is a simple, efficient beamformer widely used in high-frame-rate ultrasound, but correct implementation requires careful control of receive aperture and sound speed. The paper develops its theory and practical parameter-selection approaches for circular and plane-wave imaging.
- 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.
- 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.
- 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.
- 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.
- 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.
B. Hyperbolic signatures and DAS
DAS interprets scatterer echoes as hyperbolic signatures across element position and fast time, then sums appropriately delayed signal samples along those hyperbolas. Sampling requires interpolation, and selected element subsets may limit the receive aperture.
- 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.
- 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.
- 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.
- 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.
- 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.
C. Beamforming I/Q signals
I/Q signals can be beamformed directly on the image grid, but demodulation and summation do not commute. Phase rotators are therefore required to preserve relative phases during delayed summation.
- 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.
- Phase preservation: Phase rotators are necessary when delayed-and-summed I/Q signals are beamformed after demodulation.Summation and demodulation are non-commuting operations.
- 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.
- 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.
- 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.
D. Receive f-number
The receive f-number restricts DAS to the top portion of each hyperbolic signature, where element directivity provides stronger received signals. It can be selected from the directivity-based angle-of-view using a −3 dB compromise.
- 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.
- 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.
- 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.
- Threshold choice: When W/λ = 1, the directivity-based expressions yield f# = 1.2.The relevant wavelength should be the smallest significant wavelength of the signal.
- 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.
E. Speed of sound
The speed of sound controls the shape of DAS hyperbolas, so mismatch with the medium can distort the point-spread function. The paper estimates an average speed by maximizing phase uniformity weighted by signal intensity.
- 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.
- 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.
- Estimation principle: The Qp metric weights phase-variance information by intensity, prioritizing contributions from bright speckles.Earlier work minimized only the sum of phase variances.
- 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.
- 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.
F. DAS as a matrix product
DAS beamforming can be represented as a sparse matrix-vector product that maps recorded element I/Q signals to beamformed data. This formulation supports interpolation, phase rotation, aperture truncation, and efficient real-time computation.
- DAS is a linear operator that transforms element-recorded I/Q signals into beamformed I/Q data.
- The DAS matrix contains interpolation weights, phase rotators, and sum truncation induced by a positive f-number.
- For M beamforming pixels, the DAS matrix has size M × n_sN_e and produces a beamformed vector of length M.
- The matrix is large but sparse, with limited receive apertures reducing its number of nonzero elements.
- With unchanged imaging geometry, the DAS matrix needs to be calculated only once, after which sparse matrix-vector multiplication enables fast real-time visualization.
- The experimental evaluation used PICMUS CIRS-phantom data and carotid acquisitions to examine parameter effects in vitro and in vivo.
A. In vitro results
In vitro and in vivo analyses show that speed of sound and f-number substantially affect DAS image quality. Physically estimated parameters were consistent with brute-force optimization and improved relevant image metrics.
- 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.
- In vitro results: Both speed of sound and f-number significantly affect CNR and lateral resolution in the CIRS phantom.
- In vitro results: Physically estimated speed-of-sound and f-number values were consistent with brute-force search results.
- In vivo results: In eight carotids, maximizing 𝒬𝒑 produced estimated speeds of sound of 1510 ± 41 m/s.
- In vivo results: 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.
- In vivo results: 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.
A. The 𝑓𝑓-number and speed of sound in DAS
The f-number and speed of sound determine how DAS selects and aligns received signals, and both can materially affect image-quality metrics. The paper proposes physically grounded parameter choices and recommends optimization to avoid degradation.
- The f-number is related to transducer-element directivity and depends on element width, center frequency, and pulse-echo bandwidth.
- 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.
- A -3dB directivity threshold was heuristically identified as a good f-number compromise, with D_thr=0.71 in Eq. (13).
- The smallest vector-velocity errors in a rotating-disk example coincided with the directivity-derived f-number of approximately 2.6.
- The speed of sound determines which diffraction hyperbola is selected, while the f-number determines which portion of that hyperbola is included.
- In vitro, speed-of-sound choice significantly affected CNR and FWHM, whereas in vivo appearance was largely unchanged except for axial dimensions.
- Adaptive and deep-learning methods may depend on properly tuned DAS, because DAS can serve as a reference or provide training images.
- The conclusion recommends optimizing DAS parameters because neglecting the f-number or choosing the speed of sound poorly can impair measured image quality.
6. Appendix
The appendix extends DAS distance calculations across focusing, diverging, and plane-wave cases. It derives a limiting transmit-distance expression as the virtual-source angle approaches zero.
- A focus-point derivation analogous to Eq. (5) is used to obtain an appendix distance relation.
- Equations (5) and (20) combine into a generalized transmit distance covering focusing and diverging wavefronts.
- Taking the virtual-source limit β→0+ produces the plane-wave limiting case.
- The derivation expands square terms, factors the result, and applies a first-order Taylor series to simplify the limiting expression.
- 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).
C. Determine the f-number in Matlab
The Matlab appendix provides a compact DAS implementation and a directivity-based f-number calculation. The code forms transmit and receive delays, applies the receive aperture, interpolates signals, and assembles a sparse DAS matrix.
- The f-number script finds the angle whose element directivity reaches 0.71, then computes fnumber=1/(2 tan(alpha)).
- EZDAS beamforms RF or I/Q signals at coordinates specified by x and z using a virtual source and parameter structure.
- The implementation represents each column of SIG as a fast-time RF or I/Q signal from one array element.
- For a uniform linear array, the code computes element coordinates from pitch and derives the array length L.
- DAS delays combine transmit distance from the virtual source with receive distances from each element to each image point.
- The receive aperture retains elements satisfying abs(x-xe)<=z/(2*param.fnumber), thereby enforcing the selected f-number.
- The code uses linear interpolation and, for complex I/Q data, applies phase rotations before constructing the sparse DAS matrix.
- The DAS matrix maps raw signals into interpolated I/Q values along the M hyperbolas associated with the beamformed pixels.
E. A simple Matlab code for estimating the speed of sound
The appendix code estimates speed of sound by evaluating DAS-beamformed I/Q phase behavior along diffraction hyperbolas. It selects the speed producing an optimal real-envelope image criterion.
- EZSOS estimates the speed of sound that yields an “optimal” real-envelope image after DAS beamforming.
- The estimator analyzes phase behavior along diffraction hyperbolas whose vertices are located at the supplied image coordinates.
- For each candidate speed c, the code updates the beamforming speed parameter and recomputes DAS using coordinates scaled by c/c0.
- The matrix product M*IQ produces the diffraction-hyperbola signals used for the phase analysis.
- The criterion combines mean hyperbola-signal magnitude with unwrapped phase dispersion, using a negative averaged score for optimization.
F. A simple Matlab code for I/Q demodulation
The Matlab routine demodulates RF bandpass signals into complex I/Q components by downmixing and low-pass filtering, while handling vector orientation and multidimensional data.
- 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.
- The routine demodulates 2-D and 3-D RF data along columns, treating each column as one RF signal.
- The code converts row-vector input to a column vector before processing and restores the original row-vector shape afterward.
- Downmixing multiplies the RF data by exp(-1i*2*pi*Fc*t) to shift the signals using the center frequency.
- A fifth-order Butterworth low-pass filter is applied with filtfilt, and the result is scaled by 2.