Hadamard-encoded synthetic beamforming for high frame rate acoustoelectric current source imaging

Abstract

Acoustoelectric (AE) imaging can potentially image the current source density (CSD) in biological systems using ultrasound with high spatial and temporal resolution. However, existing approaches are limited by poor signal-to-noise ratio (SNR) and low frame rate, as conventional focused-beam scanning requires many scan lines to cover a large field of view (FOV). Here, we present spatial encoding using Hadamard synthetic beamforming (HSB) for ultrafast AE imaging. HSB was implemented using 16 transmit events for each image frame. While this number of transmissions severely restricts the FOV in conventional focused-beam scanning, the same number of HSB transmissions sufficiently samples the aperture, enabling reconstructions limited only by the transducer geometry. We validate this approach using a 1.5D transducer array operating in real time at a center frequency of 0.6 MHz to image the CSD of single and multiple dipole current sources varying at 200 Hz in a bio-mimicking phantom at a frame rate of 500 Hz, demonstrating the potential for ultrafast AE imaging. Measurements demonstrate an expected ~5 dB improvement in SNR compared to conventional focused-beam scanning after pressure equalization. The spatial resolution of HSB was comparable to that of the focused ultrasound beam, with fewer waveform artifacts observed at the same frame rate. Finally, imaging distributed current sources at various depths further demonstrated improved image quality using dynamic beamforming with HSB compared with conventional beam scanning at a fixed focal depth. These results demonstrate that HSB provides a practical and robust solution for ultrafast AE imaging of bioelectric currents.

1. Introduction

High-frame-rate (HFR) ultrasound (US) imaging enables visualization of mechanical and physiological processes evolving on millisecond time scales (>100 frames/s), far beyond the limits of conventional focused US imaging (30–100 frames/s) [1]. By transmitting plane or diverging waves from full or synthetic apertures, HFR imaging circumvents line-by-line scanning and allows an entire image to be reconstructed from a small number of transmit events [1]. Frame rates exceeding 10,000 frames/s have been demonstrated [1-7], supporting applications that require ultrafast temporal resolution, including cardiac and hemodynamic imaging [2,7,8], shear-wave imaging, and quantitative elastography [1,3]. In these approaches, spatial image formation is achieved primarily on receive through dynamic beamforming of multi-channel radio-frequency (RF) data, enabling efficient image reconstruction with minimal transmit events.

Extending HFR imaging paradigms to electrophysiological imaging would enable direct, noninvasive visualization of bioelectric current sources in electrically active organs such as the heart and brain, where rapid spatiotemporal dynamics are critical for understanding normal and pathological functions. Acoustoelectric (AE) imaging uniquely addresses this need by exploiting ultrasound-induced modulation of electrical conductivity to localize internal current sources using externally measured electrical potentials. Unlike conventional electrophysiological techniques that rely on sparse surface measurements or invasive electrodes, AE imaging offers the potential for deep, noninvasive current source density (CSD) imaging with high spatial and temporal resolution determined by the ultrasound field [9-22].

However, the fundamental imaging paradigm of AE differs markedly from that of typical HFR ultrasound. AE imaging is a one-way modality in which ultrasound pulses are transmitted into tissue, but imaging signals arise from conductivity-mediated voltage perturbations rather than from pulse-echo reception. Consequently, AE imaging relies entirely on transmit beamforming, and high-quality image formation typically requires many transmissions. This requirement poses a fundamental challenge to achieving high frame rates and large fields of view (FOV) while maintaining adequate signal-to-noise ratio (SNR). Although ultrafast AE imaging using multi–plane-wave transmission has been demonstrated [23,24], such approaches suffer from large near-field dead zones and limited sensitivity. Synthetic-aperture AE imaging provides an alternative strategy [25], but its intrinsically low SNR further constrains practical frame rates.

Prior work has addressed the frame rate–image quality trade-off in ultrasound imaging by reconstructing images from reduced data [26-30]. Compressed sensing leverages sparsity to recover RF signals from undersampled measurements [26], while tensor completion exploits low-rank structure to reconstruct missing scan lines, transmit events, or compounding angles [27-30]. Applied to synthetic aperture and plane-wave imaging, including echocardiography, these methods reduce data acquisition while preserving image quality metrics such as resolution and contrast.

Currently, focused-beam scanning is the dominant approach in AE imaging, as it provides sufficient SNR at the expense of frame rate and spatial coverage [9-22]. To mitigate the poor SNR associated with unfocused multi–plane-wave and synthetic-aperture transmission, several studies have proposed encoding strategies that more efficiently utilize transmitted acoustic energy, including encoding of plane-wave transmission sequences [31-33], synthetic-aperture excitation sequences [34-37], and temporal pulse coding [38-40]. However, increasing encoding complexity—particularly with large numbers of plane waves or pulses—exacerbates near-field dead-zone artifacts and limits scalability.

In this study, we introduce a Hadamard-encoded ultrasound transmission framework for AE imaging that enables ultrafast acquisition without sacrificing SNR or FOV. Like prior synthetic-aperture encoding approaches in conventional ultrasound imaging [34-37], the proposed method combines synthetic beamforming with full-aperture transmission, overcoming the low-SNR limitation of conventional synthetic-aperture imaging [41]. As described previously, the proposed framework employs one-way AE imaging with recording electrodes that are much sparser than conventional ultrasound receive arrays. Consequently, spatial encoding is achieved through transmit-side beam synthesis. Because it does not rely on multi–plane-wave transmission, the method avoids large near-field dead zones and achieves AE image reconstruction over a broader FOV while maintaining the same number of transmissions and frame rate as conventional focused-beam scanning. Experimental validation using real-time AE imaging in a biomimetic phantom demonstrates improved SNR after pressure equalization relative to focused-beam scanning, along with reduced distortion in reconstructed CSD waveforms. In addition, decoded signals enable dynamic beamforming to image current sources at varying depths, as demonstrated with a distributed multi-electrode current source.

2. Hadamard-encoded synthetic aperture

2.1. Acoustoelectric Imaging

Acoustoelectric (AE) imaging relies on the acoustoelectric effect and is inherently unidirectional. US pulses are transmitted from the transducer, and the propagating pressure field locally and linearly perturbs the electrical conductivity of a conductive medium. These small perturbations are detected as changes in surface voltage potential measured using surface electrodes [42-44]. The resulting AE voltage measured between an electrode pair is

$$\Delta V\left(t\right)=-{\rho }_{0}K\int \int \int \left({\mathit{J}}^L•{\mathit{J}}^s\right)p(\mathit{r},t)d\mathit{r},$$ (1)

where ${\rho }_0$ is the electrical resistivity of the medium before modulation and K is the interaction constant, which are $0.6\text{-}0.7\Omega •m$ and 10⁻⁹Pa⁻¹ in 0.9% NaCl solution, the saline bath condition used in this work. In (1), $p(\mathit{r},t)$ is the US pressure field, at position $\mathit{r}=(r_x,r_y,r_z)$ and time $t$, ${\mathit{J}}^L$ is the lead field of a pair of electrodes and ${\mathit{J}}^s$ is the current field from the current source.

Because AE signals are measured using electrodes rather than acoustic receivers, the transducer's receive elements (which typically share hardware with the transmitter) are not used and receive beamforming does not apply to AE imaging. This presents a critical challenge, as techniques such as multi–plane-wave transmission (MPW) and synthetic-aperture transmission (SAT) are traditionally combined with receive beamforming to enhance SNR and spatial resolution in conventional ultrasound imaging. In the next section, we describe three transmit schemes that can be used for AE imaging, their associated limitations, and how our proposed method addresses these limitations.

2.2. Ultrasound Transmission Schemes

Here we review three ultrasound transmission schemes (Fig. 1) for AE imaging and discuss their respective advantages and inherent limitations, which collectively motivate the proposed Hadamard-encoded transmission framework. The results presented in Fig. 1 are acoustic fields from k-Wave [45] simulations of a custom 1.5D transducer array (18 × 7 elements) operating at a center frequency of 0.6 MHz, with an azimuthal aperture of 48.6 mm (18 elements, pitch = 2.7 mm) and an elevational aperture of 29.3 mm (7 elements, pitch = 4.18 mm) [19, 22]. This configuration matches the geometry of the transducer used in the experiments presented below. All k-Wave simulations were performed in full 3D over a computational domain spanning −30 mm to 30 mm along the x-axis (azimuth), −30 mm to 30 mm along the y-axis (elevation), and 0 mm to 80 mm along the z-axis (depth). Just like all experiments, the simulations approximated 2-D conditions by applying zero relative delays to elements in a column (i.e., 7 elements in elevation) so that transmitted US pulses were focused elevationally at the transducer’s natural focus of 35 mm [19,22]. The acoustic fields in Fig. 1 correspond to 2D cross-sectional images at the central elevational plane (y = 0).

Fig. 1. Comparison of three ultrasound transmission schemes.

(a) Focused-beam scanning, in which a focused US beam is sequentially steered across the angular FOV. (b) Multi–plane-wave (MPW) transmission, in which plane waves are transmitted at multiple steering angles and subsequently compounded coherently. (c) Synthetic-aperture transmission (SAT), in which US pulses are transmitted from individual synthetic sub-apertures. For all cases, the transducer was driven with the same excitation voltage, and the resulting pressure field amplitudes illustrate that focused-beam scanning produces the largest peak pressure, followed by MPW and then SAT. For both (b) and (c), spatial image formation is performed synthetically using multiple transmissions.

First, focused-beam scanning (Fig. 1A) represents the conventional approach and is described by

$$p_i(\mathit{r},t)={\sum }_{m=1}^M{a}_{i,m}s\left(t-{\tau }_{i,m}^{foc}\left(\mathit{r}\right)\right)$$ (2)

where $p_i(\mathit{r},t)$ is the pressure as a function of position and time for the i^th beam in an N-beam sequence (i.e., i=1,2, …, N), and $a_{i,m}$ and ${\tau }_{i,m}^{foc}\left(\mathit{r}\right)$ denote the apodization and focusing delay applied to the m^th transducer element, respectively, for the i^th beam line, and $s\left(t\right)$ denotes the transmitted US pulse waveform. Note that the beam is formed prospectively, which means that the elemental delays do not change as a function of position and the beam is focused to a specific depth. By concentrating energy at a localized focal region, this approach provides high SNR; however, it suffers from a limited FOV and low frame rate since it requires sequential line-by-line scanning (i.e., N transmissions over the FOV), making it inadequate for HFR-AE imaging.

An alternative strategy is MPW transmission (Fig. 1B), given by

$$p(\mathit{r},t)=\sum _{k=1}^K\sum _{m=1}^{M}s\left(t-{\tau }_{m,k}^{pw}\left(\mathit{r}\right)\right),$$ (3)

where k denotes the k^th plane wave and ${\tau }_{m,k}^{pw}\left(\mathit{r}\right)$ denotes the time delay to steer the k^th plane wave from the m^th transducer element to field point $\mathit{r}=(r_x,r_y,r_z)$. If K is much less than the typical value of N for focused-beam scanning, MPW enables high frame rates and a large FOV. Note that the image is produced retrospectively such that the delay can be changed for every image point to effectively form a beam focused to each point. However, it introduces a significant near-field dead zone. Additionally, compared to focused-beam scanning, MPW transmits energy across multiple emissions, resulting in a reduced effective SNR per transmission.

Equation (3) corresponds to K sequential transmissions, one per plane wave. In practical MPW imaging, multiple plane waves are often transmitted using coding schemes (e.g., Hadamard encoding [26-28]) to improve SNR without increasing the total number of transmissions; however, the achievable frame rate is still determined by the K transmissions, and near-field dead-zone limitations persist. As discussed below, the large dead zone for MPW is a significant issue in AE imaging.

A third approach is SAT (Fig. 1C), expressed as

$$p(\mathit{r},t)=\sum _{l=1}^L\sum _{m\in A_l}s\left(t-{\tau }_{m,l}^{sa}\left(\mathit{r}\right)\right),$$ (4)

where $A_l$ denotes the l^th sub-aperture, and the m^th transducer element is within the defined sub-aperture $A_l$. Spatial information is synthesized by compounding pressure fields generated from L sub-aperture transmissions. Note that the imaging field is produced retrospectively such that the delay can be changed for every point in the reconstructed image. While SAT enables synthetic focusing without the dead-zone limitation associated with MPW, it is intrinsically SNR-limited because only a fraction of the full aperture contributes to transmitting energy during each emission.

Finally, for these three transmission schemes the resulting image frame rates are identical when the total number of focused beams N, the number of transmitted plane waves K, and the number of sub-apertures L are equal (i.e., N = K = L).

2.3. Multi-Plane Wave Transmission and Challenges in AE Imaging

Before introducing the proposed Hadamard-encoded synthetic-aperture transmission framework, we first examine a fundamental limitation of MPW transmission for AE imaging—namely, the near-field dead-zone. Previous studies on ultrafast MPW imaging have reported the presence of substantial near-field dead zones arising from the sequential emission of multiple plane waves and the associated transmit–receive overlap and pulse interference, which fundamentally restricts the imaging depth at high frame rates [26-28]. Although MPW compounding is widely used for HFR imaging, it presents a significant challenge for AE imaging, particularly when combined with complex encoding schemes. Note that AE signals cannot be recorded on electrodes when the transducer array is excited because of strong EM interference.

Fig. 2(a) illustrates MPW transmission using eight (K=8) plane waves steered from −15° to +15° for a 48.6-mm aperture at 0.6 MHz, for which angular sampling is Nyquist-adequate only up to a depth of about 17 mm; beyond this, the angular spectrum becomes increasingly undersampled, leading to degraded lateral resolution and compounding performance. Fig. 2(a) illustrates a Hadamard-encoded excitation with the first sequence (all plane waves have the same polarity). For a length 8 code, seven additional transmissions are required, resulting in eight transmissions in total. In addition, as shown in Fig. 2(a), a substantial near-field dead zone is created by the sequential transmission of plane waves. Within this region, transducer elements are actively transmitting and, consequently, imaging cannot be performed.

Fig. 2. Near-field dead zone in multi–plane-wave (MPW) transmission for AE imaging.

(a) Example of eight plane-wave transmissions steered from −15° to +15° using a 48.6-mm aperture transducer at 0.6 MHz, illustrating the sequential emission of plane waves. For this configuration, the angular sampling is Nyquist-adequate only up to a depth of approximately 17 mm, beyond which the angular spectrum becomes under-sampled. (b) Near-field dead-zone depth computed as a function of the number of plane waves. Because AE imaging is a one-way modality, the dead-zone depth corresponds to a one-way propagation distance (d = ct), resulting in a substantially larger non-imaging region compared with conventional pulse–echo ultrasound.

In applying Hadamard encoding to plane-wave–based ultrafast imaging to improve SNR, the number of firings equals the number of transmitted plane waves, so the achievable frame rate decreases as more plane waves are used to improve beam quality through coherent compounding. Consequently, conventional MPW compounding cannot simultaneously achieve high image quality, high frame rate, and a minimal dead zone—particularly for near-field and rapidly varying targets.

This limitation is further exacerbated in AE imaging, which is an inherently one-way (unidirectional) modality. Unlike pulse–echo US, the imaging depth corresponds to one-way propagation, such that d = ct rather than d = ct/2. Fig. 2(b) demonstrates how the dead zone increases as the number of plane waves increases. For this aperture and frequency, MPW imaging with eight plane waves is not possible up to a depth of about 70 mm. If the number of plane waves increases to 16 (K=16), the dead zone expands further, preventing imaging up to about 110 mm. This fundamental limitation motivated our search for alternative transmission strategies that can support HFR imaging without significant near-field dead zones.

2.4. Hadamard-Encoded Synthetic Apertures for AE Imaging

While SAT does not have a significant near-field dead zone, it is typically SNR-limited, as only a subset of transducer elements transmits ultrasound pulses in each emission. In this work, we overcome this intrinsic SNR limitation by encoding all transducer elements using a Hadamard sequence, thereby enabling full-aperture energy transmission while preserving the ability for synthetic decoding and beamforming.

Fig. 3 illustrates the pressure gain achieved through Hadamard encoding and decoding using a simulation setup identical to the experimental configuration described in the following section. Fig. 3(a) shows the encoded Hadamard matrix of order 16 (i.e., L=16, top) and the corresponding decoded sequence (bottom). The boxes in Fig. 3(a) highlight encoded and decoded sequences, as presented in Fig. 3(b) and Fig. 3(c), respectively. From the perspective of a single transducer element, decoding yields a theoretical pressure gain of 16, as indicated by the color scale; however, system gain must use the pressure fields synthesized across all elements in the encoded and decoded cases.

Fig. 3. Pressure gain achieved through Hadamard encoding and decoding with pressure equalization.

(a) Encoded Hadamard transmission sequence (top) and the corresponding decoded sequence (bottom) for a length 16 code. (b) Example pressure field generated by the fourth encoded transmission. (c) Corresponding decoded acoustic field synthesized at the ninth (center) element of the synthetic aperture. (d) Comparison of the maximum pressures for the encoded and decoded sequences and their mean values, illustrating the pressure gain achieved after decoding. (e) Focused scan after pressure equalization to (b), showing that Hadamard decoding yields a 3.08-fold pressure gain.

Fig. 3(b) shows the simulated acoustic field for the fourth Hadamard-encoded transmission, while Fig. 3(c) shows the corresponding decoded pressure field synthesized at the ninth (center) element. Clear pressure amplification is observed after decoding, as evidenced by the change in dynamic range. Fig. 3(d) quantitatively summarizes this effect, showing that the average pressure gain from the encoded to the decoded fields is 3.08-fold. With pressure equalization—i.e., scaling the Hadamard transmission voltage such that the maximum encoded pressure matches that of the focused scan—this gain corresponds to an SNR improvement of 20log10 $\sqrt{3.08}=4.9\text{dB}$.

3. Experimental investigation

We investigate the feasibility and performance of Hadamard-encoded AE imaging using both M-mode and real-time scanning in a biomimetic phantom. It consists of a cube filled with 0.9% NaCl with stimulating and recording electrodes embedded inside. The stimulating electrodes consisted of pairs of platinum electrodes (Natus Medical Inc., Middleton, WI, USA) driven by a 200 Hz sinusoidal current with opposite polarity and a peak-to-peak voltage of 6 V, delivered by a multi-channel stimulator (STG 4008, Multichannel Systems, Reutlingen, Germany), serving as the current sources to be imaged. The recording electrodes—gold cup electrodes (Natus Medical Inc., Middleton, WI, USA)—were positioned at x = 0 and at the same depth as the current electrodes, and were offset by 40 mm in the y-direction (elevation direction) to measure the resulting AE signals. We first analyzed a single dipole current source centered at x = 0 and a depth of z = 45 mm, formed by two stimulating electrodes separated by approximately 10 mm, to evaluate SNR gain, spatial resolution, and the reconstructed current-source waveform for Hadamard encoding. Subsequently, using the same experimental setup, we imaged multiple dipole current sources formed by four electrodes distributed over depths ranging from 25 mm to 40 mm to demonstrate the full advantage of dynamic beamforming, which cannot be achieved with conventional scanning at a fixed focal depth.

Fig. 4(a) (bottom left) shows the custom 1.5D ultrasound transducer (H235, Sonic Concepts, Seattle, WA, USA) used in the experiments, which was driven by a Vantage 256 system (Verasonics, Kirland, WA, USA). It contains 126 elements arranged in an 18 × 7 configuration. Consistent with the simulation framework described earlier, Hadamard encoding was applied only along the azimuthal (x) direction, while all elements along the elevational (y) direction were driven with the same code. Zero relative delays were applied to all elements along the y-axis so that transmitted US pulses focused energy elevationally at the 35-mm natural focus of the H235 transducer [19,22]. Because the order-16 Hadamard scheme uses 16 active channels, the inner 16 of the 18 azimuthal elements were used during encoded transmissions. This constraint was explicitly incorporated into the pressure-equalization simulations: the focused scan employed all 18 azimuthal elements, whereas the Hadamard scan used 16.

Fig. 4. Experimental setup and AE measurements using focused and Hadamard-encoded transmissions.

(a) Schematic of the experimental setup showing the biomimetic cube phantom filled with 0.9% NaCl, the ultrasound transducer, and the stimulating and recording electrodes used for AE measurements. (b) Representative M–mode measurements acquired using focused-beam scanning and Hadamard-encoded transmission at a single slow-time frame, illustrating the encoded AE signals prior to decoding. (c) Real-time focused-scan AE measurements acquired at a frame rate of 500 Hz for a 200-Hz sinusoidal current source. (d) Corresponding real-time Hadamard-encoded AE measurements under identical acquisition conditions. Real-time data in (c) and (d) were further processed to reconstruct AE images of two current sources, as described in the text.

3.1. Data Acquisition

Hadamard-encoded AE imaging was investigated using two acquisition modes: M-mode and real-time scanning. In M-mode, repeated US transmissions were performed at a fixed lateral position until the entire slow-time current waveform was acquired. A total of 41 (i.e., N=41) focused beams were transmitted at a frame rate of 4000 Hz, whereas the Hadamard M-mode acquisition employed 16 (i.e., L=16) encoded transmissions, corresponding to a Hadamard matrix of order 16, at the same 4000-Hz frame rate. Although M-mode is not practical for HFR imaging, it can quantitatively validate the performance of Hadamard-encoded imaging relative to conventional focused scanning under well-controlled conditions. For both M-mode and real-time scanning, a 40-ms slow-time data record was acquired using an NI PXI-5105 data acquisition card (National Instruments, Austin, TX, USA) to sufficiently capture three cycles of the 200 Hz current source waveform between 5 and 20 ms.

For real-time scanning, both focused and Hadamard acquisitions were performed using 16 transmit events per image frame to ensure an equivalent frame rate (i.e., L=N=16). The real-time frame rate of 500 Hz was sufficiently high to sample the 200-Hz sinusoidal current-source waveform used throughout the experiments. The corresponding pulse-repetition rate (PRR) was 8000 Hz (500 Hz × 16 transmits per frame). Thus, while the frame rate differs between M-mode and real-time imaging (4000 Hz versus 500 Hz), the frame rate for focused and Hadamard acquisitions was preserved for each mode. For a fair comparison between M-mode and real-time acquisitions, SNR values must account for the factor-of-8 frame-rate difference, corresponding to an expected 9-dB difference, which was consistently observed in the experiments discussed below.

A critical step for AE image reconstruction and other reconstructive imaging approaches is to correctly register the transmitted ultrasound pulse in slow time for each coded transmission. This is especially critical for imaging fast-moving objects; when slow-time registration is ignored, reconstructed images have significant artifacts. In our study, we enabled time registration by upsampling real-time data acquired at a 500 Hz frame rate by a factor of 16, which equals the PRR of 8000 Hz. Because the frame rate properly samples the current waveform, upsampled waveforms are equivalent to direct sampling at the PRR. Time differences between transmissions relative to the first transmission (e.g., 1/8000 Hz, 2/8000 Hz, …, 15/8000 Hz, …) were adjusted so that all slow-time frames after the first are aligned, ensuring correct time registration. Proper time alignment is critical for all synthetic beamforming approaches.

Fig. 4(a) shows the experimental setup. Fig. 4(b) presents representative M-mode measurements using focused scanning and Hadamard encoding at a single slow-time frame. Figs. 4(c) and 4(d) show corresponding real-time measurements for focused and Hadamard scans, respectively. Real-time data were processed to reconstruct AE images of two current sources, equivalent to the left image in Fig. 4(b).

To ensure pressure-equalized comparison between transmission schemes, the drive voltage for focused scanning was adjusted based on the pressure normalization procedure described in Fig. 3. Specifically, the maximum focal pressure at 45 mm obtained using focused transmission was compared with the mean of the maximum pressures across the 16 unfocused Hadamard transmissions, yielding a ratio of 2.27. Accordingly, the transducer drive voltage for focused scanning was set 2.27 times lower than that used for Hadamard transmission (11 V for focused scanning and 25 V for Hadamard-encoded transmissions) for both M-mode and real-time acquisitions to equalize the effective acoustic output.

3.2. Dynamic Beamforming

Because the proposed AE imaging framework is based on Hadamard-encoded synthetic-aperture transmission, full synthetic beamforming is performed in post-processing after time alignment and decoding. This enables two key beamforming capabilities not available in conventional focused-beam scanning: dynamic focusing and dynamic apodization.

In Hadamard-encoded AE imaging, decoded measurements retain element-wise spatial information, enabling synthetic dynamic focusing on all image depths during post-processing. Consequently, AE images can be formed with depth-dependent focusing across the entire FOV. In contrast, conventional focused-beam scanning employs a fixed transmit focal depth, which cannot be modified after transmission and therefore limits optimal resolution to a narrow depth region. In all Hadamard-based AE reconstructions presented here, dynamic focusing was applied for image formation at all depths. Dynamic focusing is captured in Eq. (4), where the delay term ${\tau }_{m,l}^{sa}\left(\mathit{r}\right)$ varies with spatial location (depth), effectively acting as a dynamic focal point implemented in post-processing [25]—a capability not available in conventional focused-beam scanning.

Decoded Hadamard measurements also enable dynamic apodization during synthetic beamforming. Element-wise dynamic apodization, which is naturally applicable in synthetic-aperture imaging, also provides a key advantage over MPW imaging, where plane-wave acquisition generally requires more complex angular apodization strategies [46-51]. Indeed, with a limited number of Fourier coefficients (i.e., plane waves), it is inherently difficult to design spatial-filter coefficients yielding a robust and well-behaved apodization function. In this study, cosine apodization with a pedestal was applied in post-processing to adjust the effective aperture as a function of depth. Dynamic apodization suppresses noise and sidelobes while preserving spatial resolution, consistent with established beamforming principles. The effect of dynamic apodization is presented in Fig. 8. All other results (Figs. 5-7) were reconstructed without dynamic apodization to isolate the effects of Hadamard encoding and dynamic focusing alone.

Fig. 8. AE imaging of spatially distributed current sources using dynamic synthetic beamforming with Hadamard-encoded transmission in comparison with conventional focused scanning.

(a) AE image of the first dipole current source reconstructed using focused scanning with a fixed transmit focal depth of 25 mm under real-time imaging conditions. (b) Corresponding AE image reconstructed using Hadamard-encoded transmission with dynamic focusing and dynamic apodization, showing improved image sharpness across depth. (c) AE image of the second dipole current source reconstructed using focused scanning, illustrating degraded image quality away from the fixed transmit focus; the apparent enhancement in S2⁺ is a sampling and interpolation artifact caused by spatial undersampling. (d) Corresponding AE image reconstructed using Hadamard-encoded transmission, in which dynamic focusing enables depth-dependent focusing for both dipoles, and dynamic apodization suppresses near-field noise, yielding consistent image quality for current sources distributed from 25 mm to 40 mm depth.

Fig. 5. Decoded Hadamard signals and beamformed AE images compared with focused-scan results in M–mode imaging.

(a) Decoded M–mode AE signals obtained from Hadamard-encoded measurements. (b) Envelope-detected AE image reconstructed from decoded Hadamard signals after dynamic focusing, showing two clearly resolved current sources. (c) Corresponding AE image reconstructed using conventional focused-beam scanning in M–mode. (d) Signal-to-noise ratio (SNR) computed using spatial windows defined over the AE signal region [from (b)] and background noise region [from (c)], plotted as a function of slow time. The signal region fully encompassed the AE signal, while the noise region (2–30 mm depth over all angles) was selected outside the AE signal to estimate the background noise. The SNR follows the 200-Hz injected current waveform, with a peak at approximately 13.5 ms; the SNR gain, defined as the difference between the maximum SNR values of Hadamard and focused scans (solid red horizontal line), is close to the expected theoretical gain of 4.9 dB. (e) One-dimensional lateral projection through the right current source for Hadamard and focused scans, demonstrating comparable spatial resolution.

Fig. 7. Comparison of slow-time AE waveform fidelity in real-time imaging using Hadamard-encoded and focused scans.

Slow-time AE waveforms extracted from the Hadamard-encoded reconstruction and the conventional focused scan for both the S⁺ and S⁻ signals. The Hadamard-encoded reconstruction preserves temporal waveform fidelity of the 200-Hz current source, whereas the focused scan exhibits pronounced waveform distortion due to severe spatial under-sampling in real-time imaging.

3.3. AE Images in M-Mode and Analysis

Fig. 5(a) shows the decoded M-mode AE signals at the peak current time (13.5 ms, slow time) obtained from Hadamard-encoded measurements and Fig. 5(b) presents the envelope-detected AE image formed from the decoded Hadamard signals after dynamic focusing. Note that the x-axis corresponds to the transducer aperture axis, which, after decoding, aligns with the elements along the azimuthal axis. Fig. 5(c) shows the corresponding AE image obtained using conventional focused-beam scanning. Figures. 5(b) and 5(c) clearly demonstrate that two current sources are clearly resolved in both cases, confirming successful AE image reconstruction using the Hadamard approach.

Quantitatively, Fig. 5(d) shows the SNR computed using spatial windows encompassing the AE signal region [Fig. 5(b)] and background noise region [Fig. 5(c)], plotted as a function of slow time. The SNR variation follows the 200-Hz current-source waveform, with a peak at approximately 13.5 ms. The SNR gain, defined as the difference between the maximum SNR values of Hadamard and focused scans (solid red horizontal line), is close to the expected theoretical gain of 4.9 dB.

Finally, Fig. 5(e) shows the one-dimensional lateral projection through the right current source for both Hadamard and focused scans, demonstrating comparable spatial resolution with the same full-width half maximum (FWHM) of 6 mm. Together, these results confirm that Hadamard encoding can produce AE images with enhanced SNR while preserving spatial resolution.

3.4. AE Images in Real-Time Scan and Analysis

In real-time imaging, an SNR reduction of approximately 9 dB is expected relative to M-mode due to the eightfold reduction in the number of acquired samples. The results below confirm that this behavior is observed for both focused and Hadamard-encoded scans, while the key conclusions drawn from the M-mode analysis are preserved.

Fig. 6(a) shows decoded real-time AE signals obtained from Hadamard-encoded measurements at the peak current time (13.5 ms, slow time). Fig. 6(b) presents the envelope-detected AE image formed from the decoded Hadamard signals after dynamic focusing, while Fig. 6 (c) shows the corresponding AE image using conventional focused-beam scanning. The AE image in Fig. 6(c) shows an interpolated reconstruction from the original 16 beam lines spanning −20° to 20°, aligned with the same scan-lines in Fig. 6(b). Despite interpolation, the image exhibits noticeably weaker signals on the left (S+) compared to the Hadamard reconstruction, attributable to lateral undersampling. This undersampling effect is further evidenced in the slow-time waveforms extracted at the two primary locations, as discussed in the following section. Comparison of Figs. 6(b) and 6(c) demonstrates that two current sources are clearly resolved in both cases, confirming successful AE image reconstruction in real time.

Fig. 6. Decoded Hadamard signals and beamformed AE images compared with focused-scan results in real-time imaging.

(a) Decoded real-time AE signals obtained from Hadamard-encoded measurements. (b) Envelope-detected AE image reconstructed from the decoded Hadamard signals after dynamic focusing in real-time mode, showing two clearly resolved current sources. (c) Corresponding AE image reconstructed using conventional focused-beam scanning in real-time mode using 16 beam lines, which were then interpolated to match the number of lines in (b). (d) Signal-to-noise ratio (SNR) computed using spatial windows defined over the AE signal region [from (b)] and background noise region [from (c)], plotted as a function of slow time. The signal region fully encompassed the AE signal, while the noise region (2–30 mm depth over all angles) was selected outside the AE signal to estimate the background noise. The SNR follows the 200-Hz injected current waveform, with a peak at approximately 13.5 ms; the SNR gain, defined as the difference between the maximum SNR values of Hadamard and focused scans (solid red horizontal line), remains close to the expected theoretical gain of 4.9 dB, while the absolute SNR levels are reduced by approximately 9 dB relative to M–mode due to the reduced sampling rate. (e) One-dimensional lateral projection through the right current source for the Hadamard and focused scans, demonstrating comparable spatial resolution under real-time imaging conditions.

Like Figs. 5(d), Fig. 6(d) shows the SNR computed using spatial windows containing the AE signal and background noise, plotted as a function of slow time. The SNR follows the 200-Hz injected current waveform, with a peak at approximately 13.5 ms. The SNR gain (solid red horizontal line) is close to the expected theoretical gain of 4.9 dB. Comparison between Figs. 5(d) and 6(d) further confirm that the relative SNR gain is preserved, while absolute SNR levels are reduced by about the expected 9 dB in real time (measured reductions of approximately 10.5 dB for focused scanning and 10.8 dB for Hadamard scanning).

Next, Fig. 6(e) shows the one-dimensional lateral projection through the right current source for both the Hadamard and focused scans, demonstrating comparable spatial resolution, with the FWHM of 6 mm. These results confirm that the Hadamard-encoded approach consistently improves SNR without degrading spatial resolution under real-time imaging constraints, in agreement with the M-mode findings.

The goal of real-time AE scanning is to image current changes at the slow-time frame rate. The first supplemental movie compares AE images of the current sources in Fig. 6 as a function of slow time. Here, AE images are shown using a color scale from −0.5 to 0.5, with amplitudes normalized to the maximum value of the slow-time signal. This movie can be used to extract the slow-time waveform at any image point. Fig. 7 compares slow-time waveform artifacts obtained from focused scanning and Hadamard beamforming. Here, the S− site time waveforms have been inverted for direct comparison with the S+ site. The Hadamard-encoded reconstruction preserves the S⁺ and S⁻ waveform characteristics, whereas the focused scan exhibits pronounced waveform distortion evident by the discrepancy between S+ and S− waveforms. Specifically, the normalized cross-correlation between the S− and S+ waveforms yielded a coefficient of 0.999 for the Hadamard reconstruction in Fig. 7(a), indicating near-identical waveforms, whereas the corresponding coefficient for the focused-scan waveforms was 0.933, capturing the pronounced discrepancy observed in Fig. 7(b). These artifacts arise primarily from severe spatial undersampling in the focused real-time scan, where only 16 beams (i.e., N=16 for real-time scanning compared to N=41 for M-Mode) are used to cover a 40° lateral FOV. In contrast, the Hadamard-encoded approach adequately samples the transmit aperture, enabling reconstruction over a broader FOV and yielding substantially improved slow-time waveform fidelity under real-time constraints.

3.5. AE Image of Distributed Current Sources

One of the key advantages of synthetic beamforming—dynamic focusing and apodization across depth—is demonstrated using spatially distributed current sources, highlighting its superiority over conventional focused scanning with a fixed focal depth. To this end, an experimental configuration with two dipole current sources formed by four electrodes was implemented, with the sources distributed over depths ranging from 25 mm to 40 mm.

Figs. 8(a) and 8(b) show AE images of the first current dipole at the peak current time (13.5 ms, slow time) obtained using focused scanning and Hadamard-encoded scanning, respectively, under real-time imaging conditions. Figs. 8(c) and 8(d) present the corresponding results for the second current dipole at the peak current time (27 ms, slow time). For the focused scan (Figs. 8(a) and (c)), the transmit focal depth was fixed at 25 mm, resulting in optimal image quality only near this depth. In contrast, the Hadamard-encoded reconstruction with dynamic focusing and apodization, consistently produces sharper AE images for all distributed current sources.

A quantitative analysis compared the RMS levels of the two primary AE signals at different depths (Fig. 8). The ratio of the larger signal (S+) to the smaller (S−) was computed in dB for all panels (a–d). For conventional focused-beam scanning, the ratios were higher—3.4 dB in (a) and 2.7 dB in (c)—indicating a weaker deeper signal. In contrast, Hadamard-based dynamic beamforming yielded lower ratios—1.7 dB in (b) and 1.0 dB in (d)—showing more comparable amplitudes and effective focusing across depths.

In addition, dynamic apodization significantly suppresses near-field noise, further improving overall image quality. The second supplemental movie illustrates AE images in Fig. 8 over slow-time, highlighting the consistently improved image quality achieved by the Hadamard-encoded reconstruction across all slow-time frames.

It should be noted that the apparent enhancement of S2⁺ observed in Fig. 8(c) is a sampling and interpolation artifact arising from spatial undersampling, rather than a true improvement in spatial resolution.

4. Discussion

The proposed Hadamard-encoded synthetic-aperture transmission framework simultaneously addresses two fundamental limitations of existing ultrafast AE imaging strategies: the significant near-field dead-zone inherent to MPW transmission and the intrinsically low SNR associated with conventional SAT. By encoding all transducer elements in each transmission and performing synthetic beamforming after decoding, this approach preserves full-aperture acoustic energy while retaining the flexibility of synthetic focusing and apodization, avoiding both the MPW dead zone and the poor SNR of SAT. The decoding process is linear and can be implemented via simple matrix multiplication of RF signals based on the Hadamard encoding sequence. The computational complexity of subsequent beamforming is comparable to that of conventional synthetic aperture beamforming. Namely, the computational complexity scales with the product of the number of transmissions and image pixels during synthetic beamforming. The advantages are demonstrated not only under well-controlled M-mode conditions but also validated in real-time AE imaging at a frame rate of 500 Hz, highlighting its potential for ultrafast AE imaging (typically exceeding the 30–100 Hz range of conventional imaging [1]). Real-time results confirm that the expected SNR gain relative to focused scanning is preserved, while maintaining spatial resolution and waveform fidelity.

A preliminary comparison of conventional synthetic aperture acquisition and Hadamard encoding showed SNR improvements consistent with theoretical $\sqrt{N}$ scaling. Together with the SNR gains over focused-beam scanning observed here, this supports the effectiveness of the encoding/decoding process. However, performance may be affected by non-idealities such as element-wise sensitivity variations and system-level imperfections, especially as encoding complexity and sequence length increase. Together, these findings show the potential of HSB in AE imaging as a high-frame-rate, high-SNR AE imaging modality in real-time, overcoming previous limitations of ultrafast AE imaging. Experimental validation was conducted in a homogeneous conductive medium to isolate the performance of the proposed framework; the effects of heterogeneity, acoustic aberration, and spatially varying conductivity in a more complex environment (e.g., ex vivo transcranial scenarios involving ultrasound propagation through the human skull) require further study. The aforementioned factors, together with encoding/decoding imperfections and phase errors, may degrade HSB performance by reducing reconstruction fidelity and image quality. For example, significant encoding or decoding errors could prevent realization of the theoretical $\surd \mathrm{N}$ SNR gain over conventional synthetic aperture imaging, resulting in a smaller practical improvement. Quantifying these effects under realistic conditions will require further investigation in future ex vivo and in vivo studies.

An important step in all synthetic, real-time imaging is accurate temporal registration of transmitted ultrasound pulses in slow time. Here we resampled the acquired slow-time sequence after temporal upsampling and aligned decoded signals (i.e., individual elemental signals) within a frame. This step is critical for preserving coherence across encoded firings and directly impacts image quality and reconstruction fidelity. Similar time-registration challenges arise broadly in HFR synthetic and coded imaging systems and become more important as frame rates increase.

As with all ultrasound-based imaging modalities, AE imaging involves tradeoffs among FOV, spatial resolution, temporal sampling, acquisition time, and SNR. For a fixed PRR and acquisition time, the achievable frame rate is set by the number of transmissions and directly limits lateral image sampling. In conventional focused-beam scanning, the lateral FOV is constrained by the number of scan lines; increasing frame rate therefore requires fewer lines, leading to spatial undersampling. In contrast, HSB decouples lateral FOV from the number of focused transmissions, with FOV determined solely by the transducer’s physical aperture. In this study, 16 transmissions achieved a real-time frame rate of 500 Hz while reconstructing a lateral FOV from −20° to 20° that satisfies the Nyquist sampling criterion. Under identical conditions, a conventional focused scan with 16 beam lines over the same angular range showed clear lateral undersampling and Nyquist violation (Fig. 6(b) vs. Fig. 6(c)). These results highlight HSB as a robust and efficient approach for ultrafast AE imaging of biocurrents, enabling wide-FOV imaging without compromising temporal resolution.

While this work demonstrates that Hadamard-encoded synthetic-aperture transmission can substantially improve SNR in AE imaging while preserving spatial resolution and enabling dynamic synthetic beamforming, several challenges remain before this approach can be generalized to 3D AE imaging and to more complex applications such as transcranial brain imaging.

The first limitation is scalability to 3D. In the present 2D configuration, 16 Hadamard-encoded transmissions were sufficient to reconstruct 2D AE images with the desired SNR and resolution. Extending the same scheme to a fully sampled 2D transmit aperture (imaging in 3D) would require, for example, on the order of 16×16=256 firings, implying roughly an order-of-magnitude reduction in frame rate for volumetric imaging. Similar constraints have been reported for 3D ultrafast and synthetic-aperture imaging more generally, where high volumetric frame rates are challenging due to the required number of transmissions and the associated data throughput and processing load [52,53]. Another potential issue in extended 3D transmission sequences is decoding error, which can affect reconstruction accuracy. Although not observed in 2D imaging, such imperfections may become more pronounced as encoding order increases, potentially amplifying errors.

Recent work on coded 3D synthetic aperture—e.g., Hadamard Aperiodic Interval (HAPI) codes and related schemes—has explored long coded transmissions, sparse 2D arrays, and separable transmit–receive strategies to reduce the number of required firings while preserving the point spread function (PSF) and SNR. Similar ideas, such as partial Hadamard encoding, overlapping virtual sources, interleaved transmission, and/or compressed-sensing reconstructions that exploit spatial or temporal sparsity, may provide a path toward 3D AE imaging with fewer firings than direct 3D extension. System-level constraints—including hardware channel count, memory bandwidth, and on-probe or on-board processing—will also become increasingly important as volumetric AE imaging is performed.

A second challenge is the translation of Hadamard-encoded AE imaging to more noisy and complex propagation environments, such as transcranial brain imaging. For these applications, the US field is strongly affected by the skull, which introduces aberration, phase distortion, and frequency-dependent attenuation. Existing work on coded and synthetic-aperture imaging in conventional US has shown that while encoding schemes (i.e., Hadamard, Golay, and other spatial or temporal codes) can yield substantial SNR and contrast gains, they are also sensitive to propagation distortions, motion, and nonlinearities, which can break the assumed linear encoding–decoding model [54-56].

Here, SNR was improved relative to conventional focused scanning under pressure-equalized conditions, but further SNR gains will likely be required for realistic transcranial applications, where skull attenuation and inhomogeneity can significantly reduce acoustic pressure at depth. Clearly, a combination of strategies is needed: (1) multi-channel AE acquisition using multiple recording electrodes to increase measurement redundancy and improve conditioning of the inverse problem [57,58]; (2) adaptive or model-based beamforming that explicitly accounts for skull-induced aberration; and (3) joint optimization of code design and reconstruction, for example, by integrating Hadamard or related spatial/temporal codes with regularized or model-based inversion frameworks.

Future work will therefore focus on (1) reducing the number of required transmissions for 3D AE imaging using optimized or partial coding strategies, possibly in combination with compressed or adaptive sampling; (2) extending the present framework to heterogeneous media, including skull phantoms and in vivo–relevant configurations; and (3) exploring multi-channel AE acquisition and advanced reconstruction methods to fully exploit the information content of Hadamard-encoded datasets in realistic neuroimaging scenarios.

5. Conclusion

In this work, we demonstrated a Hadamard-encoded synthetic-aperture transmission framework for high-frame-rate acoustoelectric (HFR-AE) imaging that overcomes fundamental limitations associated with conventional focused and multi–plane-wave transmission schemes. By encoding all transducer elements in each transmission and using synthetic beamforming after decoding with proper time alignment in slow-time, the proposed method preserves full-aperture acoustic energy while enabling dynamic focusing and dynamic apodization in post-processing—capabilities not available in conventional focused-beam AE imaging.

Experimental validation using a biomimetic 0.9% NaCl solution-filled cube phantom with both M-mode and real-time acquisitions shows that the Hadamard-encoded approach achieves an SNR gain of approximately 4.9 dB for the simple case considered here, in close agreement with theoretical predictions, while maintaining spatial resolution comparable to focused scanning. Real-time imaging results further confirm that the relative SNR advantage is preserved despite the expected ~9 dB absolute SNR reduction due to reduced temporal sampling. In addition, slow-time waveform fidelity is significantly improved for Hadamard encoding compared with focused real-time scanning.

Using spatially distributed current sources over depths from 25 to 40 mm, we further demonstrate that dynamic synthetic beamforming enables depth-independent focusing and effective near-field noise suppression, producing consistently sharp AE images across depth in contrast to the depth-limited performance of fixed-focus scanning.

Together, these results establish Hadamard-encoded AE imaging as a powerful strategy for high-SNR, and depth-flexible HFR-AE imaging, with strong potential for advancing real-time functional and bioelectrical imaging applications.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ultras.2026.108229.

Data availability

Data will be made available on request.