Retrospective Transmit Beamforming (RTB)

• Retrospective Transmit Beamforming (RTB) is a post-acquisition transmit modeling method that synthesizes dynamic elevational focus from multiple acquisitions using a virtual-source framework.
• It extends conventional 2D imaging by enabling synthetic transmit aperture compounding in elevation, thereby improving resolution and contrast over a volumetric imaging slab.
• RTB enhances image quality in modalities like FORCES and uFORCES while introducing trade-offs in acquisition rate and computational load, with parallels in wireless beamforming strategies.

Searching arXiv for the cited RTB-related papers to ground the article in current records. {"queries":[{"query":"id:(Caulfield et al., 10 Sep 2025)"},{"query":"id:(0909.4983)"},{"query":"id:(Krikidis, 2018)"}]} I found the relevant arXiv records for the three cited papers, including "Improving the Elevational Focusing of Fast Orthogonal Row-Column Electronic Scanning (FORCES) Ultrasound Imaging using Retrospective Transmit Beamforming (RTB)" (Caulfield et al., 10 Sep 2025), "Event-Driven Optimal Feedback Control for Multi-Antenna Beamforming" (0909.4983), and "Retrodirective Large Antenna Energy Beamforming in Backscatter Multi-User Networks" (Krikidis, 2018). Retrospective Transmit Beamforming (RTB) denotes a post-acquisition transmit modeling strategy in which the effective transmit focus is synthesized after data collection rather than fixed entirely at firing time. In the context of Fast Orthogonal Row-Column Electronic Scanning (FORCES) and reduced-transmit uFORCES on bias-programmable Top Orthogonal to Bottom Electrode (TOBE) arrays, RTB is introduced as a method to recover dynamic transmit focusing in elevation from acquisitions made with a fixed elevational focus by treating the fixed elevational focal arc as a virtual source and retrospectively re-timing multiple acquisitions acquired at different elevational positions (Caulfield et al., 10 Sep 2025). The term also appears in wireless beamforming literature in different senses, including reuse of outdated CSI and retrodirective phase-conjugating transmission, so its technical meaning is domain-dependent (0909.4983, Krikidis, 2018).

1. Definition and conceptual scope

In conventional 2D phased-array imaging, transmit beamforming chooses a 3D focal point in advance, computes transmit delays and apodizations across elements, fires once, and focuses the transmitted wave at that point or along a line. Receive beamforming is retrospective in the usual sense because RF data are recorded first and then delayed and summed in software. RTB, as introduced for FORCES and uFORCES, is different: it is a post-acquisition reinterpretation of the transmit paths that realizes dynamic transmit focusing in elevation from data acquired with a fixed elevational focus (Caulfield et al., 10 Sep 2025).

The core idea is that, after Hadamard decoding, each FORCES transmit event can be treated as if a long column element emitted an elevationally focused beam. The focal zone of that transmit forms an arc-shaped virtual source in the azimuthal slice. By walking the elevational focal arc through space, acquiring multiple FORCES image planes at different elevational positions, and combining the resultant data offline with transmit path modeling, the method synthesizes an effective dynamic elevational transmit focus at arbitrary 3D points. This makes RTB in this setting neither conventional receive beamforming nor a change to the physical transmit event itself; it is a retrospective transmit-path construction applied after acquisition (Caulfield et al., 10 Sep 2025).

A common misconception is to equate RTB here with standard receive-side synthetic focusing. The paper explicitly distinguishes the two. Receive beamforming still happens as usual, whereas RTB modifies how the transmit path is modeled, using a virtual source associated with the fixed elevational focal arc. Another misconception is that RTB provides unrestricted real-time transmit focusing from a single acquisition. The reported implementation instead exploits a set of predefined acquisitions made while walking the focal zone in elevation and then beamforms later as if a transmit focus had been placed at each voxel (Caulfield et al., 10 Sep 2025).

2. TOBE arrays, FORCES, and the source of the elevational focusing problem

The method is formulated for a 128×128 element Top-Orthogonal-to-Bottom Electrode (TOBE) row-column array in which each physical element corresponds to the intersection of a top electrode and a bottom electrode. The piezoelectric response is controlled by the DC bias polarity and magnitude across these electrodes, so elements can be biased positive, negative, or left at $0\ \mathrm{V}$. With bias-switching electronics, this architecture supports Hadamard-like aperture encoding and fast reconfiguration of active intersections (Caulfield et al., 10 Sep 2025).

In Fast Orthogonal Row-Column Electronic Scanning (FORCES), the Verasonics system drives the rows with an AC waveform that is elevationally focused in the $y$ direction, while the bias electronics apply a pattern along the columns following rows of a Hadamard matrix. On receive, RF signals are recorded from orthogonal row-column combinations and then decoded by multiplying by the transpose of the Hadamard matrix, yielding an equivalent set of single-column synthetic aperture transmits. After decoding, FORCES effectively provides synthetic aperture transmits where each transmit is equivalent to a single long column element that is elevationally focused on transmit and laterally focused on receive (Caulfield et al., 10 Sep 2025).

This architecture produces full transmit and receive focusing in the image plane, with coordinates given as $x$ for azimuth, $y$ for elevation, and $z$ for depth. Its limitation is specifically elevational: the transmit focus in elevation is fixed for the entire acquisition. At the nominal elevational focal depth the resolution is good, but away from that depth the elevational beam becomes large, out-of-plane clutter increases, and the elevational point-spread function broadens substantially. Each FORCES acquisition therefore yields essentially one B-scan at a given elevational plane, and volumetric coverage requires mechanically or electronically walking the focal zone and repeating acquisitions (Caulfield et al., 10 Sep 2025).

The reduced-transmit variant uFORCES uses fewer bias patterns. It requires 16 transmits per FORCES image and yields 15 usable SA transmits after decoding. The paper describes this as providing much higher frame rate, at the cost of fewer coherent transmits and some loss of image quality relative to full FORCES (Caulfield et al., 10 Sep 2025).

3. Virtual-source formulation and delay model

RTB is implemented by acquiring multiple FORCES images at different elevational focal positions and then beamforming them jointly as if they were produced by a dense set of elevational virtual sources. The array is operated in walking FORCES, where the hardware elevational focal zone is shifted in steps of $2 \times$ pitch across the array. This yields 64 FORCES image slices, each spaced 500 $\mu\mathrm{m}$ in elevation. All reported acquisitions are made on a Sun Nuclear ATS-539 phantom at 4.3 MHz to image wires and tubular cysts, with the phantom oriented so that the targets are parallel to the imaging plane and therefore emphasize the elevational response (Caulfield et al., 10 Sep 2025).

For RTB emulation, subsets of the decoded FORCES data are grouped across imaging planes. For uFORCES RTB, the method uses 15 SA transmits from each of 16 different imaging planes, giving $15 \times 16 = 240$ effective SA transmits. For FORCES RTB, it uses 128 SA transmits from each of 16 imaging planes, giving $128 \times 16 = 2048$ effective SA transmits. The data are decoded first by standard Hadamard multiplication, and subsets of the decoded SA data are then used to emulate uFORCES and RTB variants under identical probe positioning and noise conditions (Caulfield et al., 10 Sep 2025).

The virtual-source interpretation is central. After Hadamard decoding, each transmit is equivalent to one long column emitter, while the elevational focus of that emitter at depth $f_{\text{dist}}$ is approximated by an arc of radius $y$0 in the azimuthal slice. The paper treats that focal arc as a virtual source. The transmit path is therefore modeled as a path from the physical transmitter to the virtual source, followed by a path from the virtual source to the image point (Caulfield et al., 10 Sep 2025).

Without RTB, standard FORCES uses the two-way path-length model

$y$1

In RTB, the path is decomposed into $y$2 from transmitter to virtual source and $y$3 from virtual source to pixel, so that

$y$4

The focal-arc radius is taken as the focal distance, so $y$5. The transmit segment is then separated into a component in the $y$6-$y$7 plane and a purely elevational component: $y$8

$y$9

and the 3D transmit path is taken as

$x$0

The sign in the RTB path model depends on whether the pixel is above or below the focal zone along elevation. Once the total path length is available, the beamformer applies standard delay-and-sum summation with a constant F-number of 1 on both transmit and receive (Caulfield et al., 10 Sep 2025).

This formulation amounts to synthetic transmit aperture compounding in elevation. A plausible implication is that the method extends the synthetic-aperture logic already present laterally in FORCES into the out-of-plane direction by replacing a single fixed elevational focus with a discrete set of retrospectively addressable virtual sources.

4. Experimental evaluation and reported performance

The experimental system uses a 128×128 TOBE array (CliniSonix), $x$1-pitch, center frequency 6.2 MHz, driven here at 4.3 MHz, together with a Verasonics Vantage platform for AC transmit and receive and a CliniSonix adapter plate and high-voltage bias unit for DC bias switching. Evaluation is performed on a Sun Nuclear ATS-539 QA phantom containing wire targets and anechoic tubular cyst targets (Caulfield et al., 10 Sep 2025).

Elevational resolution is quantified by Full Width at Half Maximum (FWHM) measured from the wire response. The expected theoretical FWHM in ultrasound is reported as

$x$2

For contrast evaluation on the tubular cysts, the paper uses generalized Contrast-to-Noise Ratio (gCNR) following Rodriguez-Molares et al.: $x$3 where $x$4 and $x$5 are the gray-level probability density functions in the background and inside the cyst, respectively (Caulfield et al., 10 Sep 2025).

On the wire phantom, the comparison emphasized Walking FORCES versus uFORCES RTB. Near the focal zone, both methods achieve ~1.5 mm FWHM at ~2.2 mm from the focal zone, indicating that RTB does not degrade performance at focus. Far from the focal zone, at 50 mm from focus, the reported values are FWHM $x$6 for Walking FORCES and FWHM $x$7 for uFORCES RTB. The paper states that the uFORCES RTB FWHM-versus-depth curve closely follows the theoretical $x$8 trend over a large depth range (Caulfield et al., 10 Sep 2025).

On the tubular cyst phantom, images are formed with FORCES, uFORCES, FORCES RTB, and uFORCES RTB, with the elevational focus placed at 85 mm. Near the focal zone, all methods yield relatively good contrast, uFORCES RTB is comparable to FORCES at focus, and FORCES RTB is slightly better or similar. Away from the focal zone, both RTB methods outperform their non-RTB counterparts. The reported improvement is particularly pronounced for shallow cysts closest to the array, where gCNR approximately doubles when RTB is used instead of non-RTB FORCES or uFORCES. The paper further states that FORCES RTB consistently outperforms standard FORCES at all tested depths, whereas uFORCES RTB performs better than, or similar to, standard FORCES at all depths except exactly at the focal zone, where FORCES may have a slight edge because of less grating-lobe contamination (Caulfield et al., 10 Sep 2025).

The volumetric wire-phantom rendering is used as a qualitative demonstration. Walking FORCES yields wires that become indistinct away from the focal slab, whereas uFORCES RTB maintains localization across a thicker slab. The paper summarizes this as extending FORCES and uFORCES from principally “2D with a thin slab” toward practically useful 3D imaging with more uniform elevational resolution across the slab (Caulfield et al., 10 Sep 2025).

5. Acquisition burden, limitations, and future directions

The improvement in elevational focusing is accompanied by a substantial acquisition-rate trade-off. For a 128×128 TOBE array at 4 kHz PRF, the paper reports the following operating points (Caulfield et al., 10 Sep 2025):

Method	Transmits / imaging planes	Frame rate
FORCES	128 transmits / 1 imaging plane	$x$9 fps
uFORCES	16 transmits / 1 imaging plane	$y$0 fps
FORCES RTB	2048 transmits / 16 imaging planes beamformed	$y$1 fps
uFORCES RTB	256 transmits / 16 imaging planes beamformed	$y$2 fps

The reported consequence is that RTB increases the number of transmits per “RTB volume” by a factor of 16 for both FORCES and uFORCES while providing simultaneous 16-plane volumetric coverage with improved elevational focus. Computational load also increases because many more transmit events must be beamformed per volume (Caulfield et al., 10 Sep 2025).

Several limitations are identified explicitly. First, RTB is motion-sensitive because it combines data from many transmits and multiple elevational planes; motion during acquisition can produce decorrelation and artifacts. Second, the effective pitch between virtual sources in elevation is 500 $y$3; at 4.3 MHz and speed of sound 1452 m/s, the paper gives $y$4, so the effective pitch is approximately $y$5 and grating lobes are expected. The study reports that, experimentally, grating lobes become more prominent far from the focal zone for uFORCES RTB. Third, the virtual-source model assumes an arc-shaped focal zone with ideal focusing and no lateral transmit directivity, whereas real beams deviate from this ideal. Fourth, the approach requires TOBE arrays with high-voltage bias-switching electronics and rapid DC pattern changes (Caulfield et al., 10 Sep 2025).

Future directions stated in the paper include combining RTB with HERCULES, implementing real-time RTB, investigating trade-offs among virtual-source spacing, grating-lobe level, resolution, and frame rate, and applying RTB to tasks such as 3D carotid imaging and carotid plaque volume estimation. The paper also notes that an RTB sequence can beamform an ~8 mm slab per acquisition, which it identifies as attractive for 3D carotid imaging (Caulfield et al., 10 Sep 2025).

6. Domain-specific meanings beyond ultrasound

The phrase “Retrospective Transmit Beamforming” is not unique to ultrasound. In wireless communications, closely related usage appears in work on transmit beamforming with feedback control, where the operational issue is whether a transmitter should continue using stale CSI or refresh it. In that setting, the state variable

$y$6

measures the alignment between the current channel direction and the current beamformer, and the optimal event-driven feedback policy is proved to be of threshold type: feedback is triggered whenever the CSI-quality variable falls below a threshold that depends on channel power (0909.4983). The paper does not use the ultrasound virtual-source model; instead, “retrospective” refers to continued use of a beamformer based on past CSI. This suggests that in communications RTB is fundamentally a state-control problem, not a post-acquisition imaging reconstruction.

A second distinct usage appears in energy beamforming for backscatter multi-user networks, where the transmitter forms beams back toward the direction from which it has just received a backscattered signal. The energy transmitter receives a composite waveform from passive reflectors, applies matched filtering and phase conjugation, and retransmits using the normalized conjugated signal as the beamforming vector. The harvested energy at receiver $y$7 is decomposed into an omnidirectional term and a retrodirective term,

$y$8

and the method is characterized as low-complexity and CSI-free because it relies on reciprocity and passive backscatter rather than explicit channel estimation or feedback (Krikidis, 2018).

Across these domains, the common element is retrospective determination of the effective transmit beam from information not fixed solely at the original transmit-design step. The underlying mechanisms, however, differ sharply. In ultrasound RTB for FORCES, the retrospective step is virtual-source-based transmit-path modeling over multiple elevational acquisitions (Caulfield et al., 10 Sep 2025). In wireless feedback control, it is the decision to retain or refresh a beamformer based on the staleness of CSI (0909.4983). In retrodirective energy beamforming, it is phase-conjugating retransmission along the direction encoded in a just-received waveform (Krikidis, 2018).