FORCES Ultrasound Imaging

• FORCES is an ultrasound imaging method using bias-switchable row‐column TOBE arrays and Hadamard encoding to achieve full transmit/receive focusing while reducing wiring complexity.
• It employs Hadamard-derived bias sequences for aperture encoding and synthetic-aperture decoding, improving lateral resolution by approximately 25% and enhancing contrast metrics compared to TPW and VLS methods.
• Advancements such as retrospective transmit beamforming and recursive decoding mitigate fixed elevational focus and motion sensitivity issues, with realtime GPU-based implementations supporting volumetric imaging.

Searching arXiv for recent FORCES / TOBE ultrasound papers to ground the article. arxiv_search(query="Fast Orthogonal Row-Column Electronic Scanning FORCES TOBE ultrasound", max_results=10, sort_by="relevance") arxiv_search(query="row-column array FORCES TOBE ultrasound Hadamard", max_results=10, sort_by="relevance") Fast Orthogonal Row-Column Electronic Scanning (FORCES) is an ultrasound imaging scheme for bias-switchable row-column arrays, particularly Top Orthogonal to Bottom Electrode (TOBE) arrays, in which orthogonal Hadamard aperture encoding and synthetic-aperture decoding are used to obtain full transmit and receive focusing in an electronically chosen scan plane while retaining the reduced wiring burden of row-column architectures (Palamar et al., 12 Jun 2025). In the cited implementations, a row-column array consists of $N$ row electrodes on one face and $N$ orthogonal column electrodes on the opposite face of a 2-D electrostrictive slab, so only $2N$ channels are required rather than $N^2$; later developments extend FORCES with retrospective elevational focusing, motion-robust decoding, and realtime GPU beamforming (Caulfield et al., 10 Sep 2025, Henry et al., 10 Sep 2025, Palamar et al., 11 Dec 2025).

1. Array architecture and element control

FORCES is defined by the conjunction of a row-column array topology and a bias-switchable transducer material. In the TOBE implementations, the array geometry is a $128\times128$ $\lambda$-pitch arrangement of 128 rows and 128 columns, where each effective element is the intersection of one row electrode and one column electrode (Palamar et al., 12 Jun 2025). The electrostrictive relaxor material is PMN-PT-based; it is not piezoelectric until a DC bias field is applied, so a bias voltage $V_{\text{bias}}$ polarizes only those row-column intersections with the desired polarity (Palamar et al., 12 Jun 2025).

Two sets of 128 bias-tee lines feed DC biases to rows and columns, and transmit/receive pulses are AC-coupled through the same lines. By choosing a bias pattern on rows and columns, one can turn any sub-subset of the $128\times128$ intersections on or off. This element-addressing capability is the hardware prerequisite for FORCES, because the method depends on switching bias patterns between transmit events rather than permanently wiring each matrix element independently (Palamar et al., 12 Jun 2025).

A central distinction from conventional RCAs follows directly from this control model. Conventional RCAs are described as facing challenges related to their long elements, including an inability to image beyond the shadow of the aperture and an inability to focus in both transmit and receive for desired scan planes. TOBE arrays are reported to provide novel opportunities to read out from every element of the array and achieve high-quality images, and FORCES is the imaging scheme that exploits that capability (Palamar et al., 12 Jun 2025). A common misconception is therefore that reduced-channel row-column hardware necessarily precludes full two-way focusing; the TOBE/FORCES combination is presented specifically as a counterexample within the cited work (Palamar et al., 12 Jun 2025).

2. Encoding sequence and synthetic-aperture decoding

FORCES uses Hadamard-derived bias patterns to encode the aperture over a sequence of transmits. In one formulation, the columns are biased with a Hadamard sequence of $\pm1$ weights while one row or a set of rows is fired; after $H$ transmit events, with $N$0 and often $N$1, the receive echoes are decoded by multiplication with the transpose of the $N$2 Hadamard matrix, yielding $N$3 effective synthetic-aperture transmit events (Caulfield et al., 10 Sep 2025). In the 128-order TOBE implementation, the columns are biased with the $N$4-th codeword of a 128-order Hadamard matrix while the rows are held at $N$5, thus spatially encoding the receive aperture (Palamar et al., 12 Jun 2025).

The sequence can be expressed in a standard Hadamard framework. If

$N$6

then

$N$7

Using the notation of the motion paper, the encoded data satisfy

$N$8

where $N$9 is the synthetic transmit aperture multistatic dataset recovered after decoding (Henry et al., 10 Sep 2025).

The scan topology is bidirectional. In Mode A, transmission occurs via an elevationally focused aperture along rows and reception occurs on all columns with Hadamard bias encoding; in Mode B, the row and column roles are swapped (Palamar et al., 12 Jun 2025). To form one 2D plane requires 128 transmit events in Mode A, and an orthogonal B-scan can be formed with another 128 transmits after electronically swapping the row and column roles (Palamar et al., 12 Jun 2025).

The timing model is equally explicit. For each FORCES transmission, the sequence consists of sending an electronic trigger from the ultrasound platform to the bias controller, switching to a new bias pattern, allowing bias settling with $2N$0–$2N$1, firing the transmit pulse, and recording the receive data. The acoustic transmit-receive window is reported as $2N$2–$2N$3 (Palamar et al., 12 Jun 2025). This sequence structure is important for later discussions of motion sensitivity, because the method depends on an ensemble of coded transmits rather than a single unfocused emission (Henry et al., 10 Sep 2025).

3. Delay laws, focusing strategy, and image formation

The defining beamforming claim for FORCES is that the column coding is orthogonal to the acoustic focusing, giving full transmit/receive focusing everywhere in the $2N$4 plane (Caulfield et al., 10 Sep 2025). In the standard fixed-elevational-focus formulation, let $2N$5 be the pixel position, $2N$6 the center of the transmitting row element, and $2N$7 the center of the receiving column element. The total two-way path length is

$2N$8

with corresponding delay

$2N$9

On receive, a matched-filter or dynamic-receive delay is applied so that signals from $N^2$0 sum coherently, and the receive weight is

$N^2$1

where $N^2$2 is an elevational apodization, for example a constant F-number apodization (Caulfield et al., 10 Sep 2025).

The transmit side is likewise dynamically controlled. Transmit focusing in the elevational direction is achieved by applying time delays across the excited row elements so that their wavefronts converge at a user-specified depth $N^2$3 in the imaging plane. The transmit apodization across rows is chosen to maintain a constant F-number, commonly $N^2$4 (Caulfield et al., 10 Sep 2025). In the GPU implementation, the more general delay-and-sum expression over a 3D point $N^2$5 is

$N^2$6

and the beamformed signal is

$N^2$7

with $N^2$8 incorporating any transmit and receive apodization (Palamar et al., 11 Dec 2025).

Within the comparison study, the synthetic-aperture delay law for a 2D scan plane is given as

$N^2$9

where $128\times128$0 is the transmit sub-aperture center and $128\times128$1 (Palamar et al., 12 Jun 2025). Axial resolution is described as approximately

$128\times128$2

while lateral resolution for a focused aperture of width $128\times128$3 at depth $128\times128$4 is

$128\times128$5

The cited interpretation is that axial resolution is set by bandwidth, whereas FORCES improves lateral beamwidth because both transmit and receive use the full aperture with dynamic delays (Palamar et al., 12 Jun 2025).

4. Comparative performance against conventional RCA imaging

The principal experimental comparison places FORCES against Tilted Plane Wave (TPW) compounding and Virtual Line Source (VLS) imaging on two electrostrictive relaxor TOBE arrays (Palamar et al., 12 Jun 2025). The study states that conventional RCA schemes can be run on the same hardware by fixing a constant bias, but only FORCES requires and exploits bias-pattern switching to gain two-way focusing and extended coverage (Palamar et al., 12 Jun 2025).

At 30 mm depth in the ATS-539 phantom, the quantitative metrics reported for axial resolution $128\times128$6, lateral resolution $128\times128$7, and generalized Contrast-to-Noise Ratio (gCNR) are as follows (Palamar et al., 12 Jun 2025):

Configuration	FORCES	TPW / VLS
3.3 MHz TOBE array (60% BW)	$128\times128$8 mm; $128\times128$9 mm; gCNR $\lambda$0	TPW: $\lambda$1 mm, $\lambda$2 mm, $\lambda$3; VLS: $\lambda$4 mm, $\lambda$5 mm, $\lambda$6
7.8 MHz TOBE array (70% BW)	$\lambda$7 mm; $\lambda$8 mm; gCNR $\lambda$9	TPW: $V_{\text{bias}}$0 mm, $V_{\text{bias}}$1 mm, $V_{\text{bias}}$2; VLS: $V_{\text{bias}}$3 mm, $V_{\text{bias}}$4 mm, $V_{\text{bias}}$5

The paper’s own summary is that axial resolution is similar for all methods, whereas lateral resolution improves by approximately 25% with FORCES and gCNR for small cysts nearly doubles (Palamar et al., 12 Jun 2025). Additional qualitative findings are also concrete. The 3.3 MHz FORCES configuration resolves targets as deep as 14 cm outside the aperture shadow. The 7.8 MHz configuration clearly resolves 4 mm, 3 mm, and 2 mm cysts with gCNR $V_{\text{bias}}$6, whereas TPW and VLS fail at the edges. In a wire-target phantom, FORCES shows finer lateral definition and reveals 100 $V_{\text{bias}}$7m wires close to the array face that are obscured in TPW and VLS (Palamar et al., 12 Jun 2025).

Volumetric acquisition is achieved by walking individual B-scan planes in elevation and stitching the resulting images. One reported volume uses 64 FORCES planes spaced by 300 $V_{\text{bias}}$8m with focal depth 35 mm, yielding a volume of approximately $V_{\text{bias}}$9 (Palamar et al., 12 Jun 2025). The same study states that these volumes exhibit uniform contrast and detail beyond the usual aperture shadow, and an ex vivo mouse flank experiment with an 8.3 MHz TOBE array demonstrates a high-resolution cross-plane B-scan through a subcutaneous melanoma tumor with crisp boundary delineation and speckle texture throughout (Palamar et al., 12 Jun 2025).

5. Fixed elevational focus and retrospective transmit beamforming

Although FORCES provides full transmit and receive focusing in the image plane, its original formulation has a fixed elevational transmit focus. Because the transmit lens on the rows is focused at a single plane $128\times128$0, the elevational beamwidth broadens substantially above or below that focal depth. For a fixed transmit aperture $128\times128$1 and wavelength $128\times128$2, the approximate one-dimensional beamwidth is

$128\times128$3

and the Full Width at Half Maximum grows roughly linearly with $128\times128$4 (Caulfield et al., 10 Sep 2025). The reported practical consequence is poor elevational resolution and large out-of-plane artifacts away from the focal zone, which makes volumetric acquisition of thick slabs impractical (Caulfield et al., 10 Sep 2025).

Retrospective Transmit Beamforming (RTB) is introduced as a modification of FORCES and uFORCES that remedies this fixed-focus limitation without hardware changes (Caulfield et al., 10 Sep 2025). The method treats each elevational focus position as a virtual source on an arc of radius $128\times128$5 centered at $128\times128$6, $128\times128$7. By walking the focus across $128\times128$8 elevational planes spaced $128\times128$9 apart, one acquires $\pm1$0 FORCES or uFORCES datasets. After Hadamard decoding, each decoded transmit behaves as if it originated from one point on the virtual-source arc, which allows retrospective focusing at any $\pm1$1 within the volume (Caulfield et al., 10 Sep 2025).

For the simplified 2D cross-plane $\pm1$2 slice, the retrospective delay is given as

$\pm1$3

The full two-way FORCES-RTB path is expressed in the paper by splitting the transmit path into a row-to-virtual-source leg of length $\pm1$4 and a virtual-source-to-pixel leg (Caulfield et al., 10 Sep 2025).

The experimental setup uses a CliniSonix $\pm1$5 TOBE array with $\pm1$6-pitch and nominal $\pm1$7 MHz, driven at 4.3 MHz for phantom tests. Wire targets consist of parallel 20 $\pm1$8m-diameter wires embedded in a CIRS ATS-539 phantom; tubular cysts are anechoic cylinders of known diameter with wall parallel to the image plane (Caulfield et al., 10 Sep 2025). The acquisition counts reported are specific: standard FORCES uses 128 transmits to yield 128 synthetic-aperture transmits for one imaging plane; uFORCES uses 16 transmits to yield 15 synthetic-aperture transmits; FORCES RTB uses 16 elevational planes $\pm1$9 128 transmits for 2048 total transmits and 2048 synthetic-aperture transmits to beamform one 8 mm slab; uFORCES RTB uses $H$0 total transmits and yields 240 synthetic-aperture transmits (Caulfield et al., 10 Sep 2025).

The reported gains are substantial away from focus (Caulfield et al., 10 Sep 2025):

Metric	Standard	RTB
FWHM at $H$1 mm	Matched at focus	$H$2 mm for both FORCES and uFORCES RTB
FWHM at $H$3 mm	FORCES $H$4 mm	uFORCES RTB $H$5 mm
gCNR, shallow cyst	FORCES $H$6; uFORCES $H$7	FORCES RTB $H$8; uFORCES RTB $H$9

The same study states that near focus all methods yield gCNR of approximately 0.8–0.9, whereas at $N$00–40 mm from focus FORCES RTB yields approximately 0.6–0.7 versus standard FORCES at approximately 0.3, and uFORCES RTB yields approximately 0.5–0.6 versus uFORCES at approximately 0.25–0.3 (Caulfield et al., 10 Sep 2025). In the two shallow cysts closest to the probe, gCNR roughly doubles with RTB. A volumetric demonstration shows that FORCES RTB can beamform an approximately 8 mm-thick elevational slab per acquisition by retrospective focusing across 16 planes, and that reducing the plane spacing $N$01 can suppress grating lobes, with effective pitch reported as $N$02 for $N$03 mm at 4.3 MHz, at the cost of extra transmits (Caulfield et al., 10 Sep 2025).

6. Motion sensitivity, recursive decoding, and realtime implementations

FORCES is also characterized as a Hadamard-encoded Synthetic Transmit Aperture sequence that produces higher Signal-to-Noise Ratio and improved penetration depth compared with traditional STA techniques, but suffers from motion sensitivity due to ensemble size and aperture encoding (Henry et al., 10 Sep 2025). In the signal model given there, each encoded transmit uses the full aperture, so under thermal noise assumptions the SNR scales as approximately $N$04 relative to firing one element at a time, and the penetration depth extension follows from the same gain. The trade-off is that the sequence duration grows as $N$05, which increases motion sensitivity and reduces effective frame rate (Henry et al., 10 Sep 2025).

The motion paper makes the mechanism explicit. Because FORCES decodes using the entire $N$06-pulse sequence, any motion of scatterers during $N$07 produces a time-averaged multistatic dataset rather than the true static one, yielding ghosting and blurring. If a target translates at velocity $N$08, then over the sequence it shifts by $N$09, so spatial blurring is approximately $N$10, and preserving sharpness requires $N$11 pixel size (Henry et al., 10 Sep 2025).

Recursive Aperture Decoded Ultrasound Imaging (READI) addresses this by splitting $N$12 into $N$13, forming multiple low-resolution images from subsets of the FORCES sequence that are less susceptible to motion but sum to the complete FORCES image. The low-resolution images are then aligned by Estimated Motion-Compensated Compounding (EMC$N$14), in which a spatial warp is estimated by block matching with normalized cross-correlation on a coarse grid and sub-pixel paraboloid fitting, and the warped images are coherently compounded (Henry et al., 10 Sep 2025). The experimental claims are specific: point spread function simulation in Field II confirmed that $N$15 to numerical precision for $N$16; in lateral probe motion at approximately 15 cm/s, EMC$N$17-corrected images recover native gCNR versus the static case for depths below 70 mm; in a beating-heart phantom at 120 bpm, 6 MHz, and PRF 4 kHz, EMC$N$18 restores tissue speckle and sharp chamber walls; and in a flow phantom with a 6 mm vessel, 8 MHz imaging, and 42 cm/s flow, the method recovers blood speckle and a parabolic profile (Henry et al., 10 Sep 2025).

A separate line of work places FORCES in a realtime reconstruction stack. An open-source GPU-accelerated reconstruction and rendering suite integrated with programmable ultrasound hardware and TOBE arrays supports cross-plane aperture-encoded synthetic-aperture imaging and aperture-encoded volumetric scanning (Palamar et al., 11 Dec 2025). The reconstruction pipeline is reported as: demodulate and matched filter to obtain complex IQ, Hadamard decode, delay-and-sum beamforming on a user-specified 2D or 3D grid, and envelope detection with log compression for display (Palamar et al., 11 Dec 2025).

The implementation uses OpenGL compute shaders, ring-buffered uploads, separate upload and compute queues, register- and LDS-caching, and just-in-time shader specialization (Palamar et al., 11 Dec 2025). Benchmark figures are given in ns/point for beamforming one million pixels in a B-scan and 16 million voxels in a volume. For FORCES B-scan, the reported values are 61.9 on an RTX4090, 29.7 on an AMD 9070 XT, and 682 on an Adreno X1. The paper also states that at 30 ns/point on a modern AMD GPU, one can deliver 60 frames/s for a $N$19 B-scan, while the live setup is limited to approximately 24 Hz by the Verasonics-to-PC PCIe3$N$20 link; a sustained GPU upload rate above 18 GB/s on PCIe Gen4$N$21 is reported to guarantee zero stalls if the acquisition link is at least 16 GB/s (Palamar et al., 11 Dec 2025).

Taken together, these developments define the current technical profile of FORCES in the cited literature: a reduced-channel, bias-switchable, Hadamard-encoded row-column imaging scheme with full two-way focusing in electronically selected planes, superior lateral resolution and cyst contrast relative to TPW and VLS on the same TOBE hardware, a known elevational fixed-focus limitation addressed by retrospective transmit beamforming, and a known motion sensitivity addressed by recursive decoding and motion-compensated compounding, with contemporary implementations extending to realtime GPU-based reconstruction (Palamar et al., 12 Jun 2025, Caulfield et al., 10 Sep 2025, Henry et al., 10 Sep 2025, Palamar et al., 11 Dec 2025).