EMC²: Motion-Compensated Ultrasound Imaging

• The paper demonstrates that EMC² restores phase-coherent compounding by estimating inter-acquisition motion in both READI/FORCES and ultrafast echocardiography settings.
• EMC² employs motion estimation methods like NCC block matching and Doppler-based autocorrelation to realign low-resolution sub-images before coherent summation.
• EMC² enhances image quality by restoring contrast, speckle, and sharpness while enabling advanced modalities such as second-harmonic imaging.

Searching arXiv for the specified EMC² papers and closely related context. Searching arXiv for the specified EMC² papers and closely related context. Estimated Motion-Compensated Compounding (EMC²) is an ultrasound image formation strategy in which partially formed images acquired across multiple transmit events are first used to estimate inter-acquisition motion, then spatially realigned, and finally combined by coherent summation. In the literature summarized here, the term appears in two closely related but architecturally distinct settings: as an extension to Recursive Aperture Decoded Ultrasound Imaging (READI) for Fast Orthogonal Row-Column Electronic Scanning (FORCES), and as a real-time framework for ultrafast echocardiography that combines motion-corrected coherent compounding, harmonic imaging, and angular-coherence weighting (Henry et al., 10 Sep 2025, Mougharbel et al., 2023). In both settings, EMC² addresses the same central problem: motion between constituent acquisitions degrades compounding gain, tissue speckle, contrast, and sharpness unless the sub-images are aligned before summation.

1. Terminological scope and problem setting

EMC² is motivated by the motion sensitivity of coherent compounding. In FORCES, motion sensitivity arises from ensemble size and aperture encoding; in ultrafast echocardiography, it arises because successive steered diverging-wave transmissions are acquired at different angles while tissue continues to move. The common remedy is to estimate displacement between sub-images or between successive pulse-echo data, warp the data into a common reference geometry, and preserve complex phase so that coherent summation remains valid.

A concise comparison of the two documented formulations is useful.

Context	Motion-estimation basis	Final compounded image
READI with FORCES	Block-matching with normalized cross-correlation on low-resolution READI sub-images	$F_{\mathrm{EMC}^2}(r)=\sum_{s=1}^S \hat{R}_s(r)$
Ultrafast echocardiography	Cross-correlation of complex baseband signals or Doppler-based ensemble autocorrelation	$$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$

The literature therefore does not present EMC² as a single immutable algorithm. Rather, it designates a class of motion-aware coherent-compounding procedures whose concrete implementation depends on the acquisition scheme. This suggests that the acronym names a pipeline role—estimated motion followed by motion-compensated compounding—more than a unique estimator or beamformer.

2. READI-based EMC² in FORCES imaging

In the READI formulation, EMC² is built on a decomposition of an $N$-line FORCES acquisition $G(t)\in\mathbb{R}^N$ with $N=S\cdot Q$ into $S$ groups of $Q$ Hadamard-encoded receive signals. For the $s$-th block $g'_s(t)\in\mathbb{R}^Q$, partial decoding is performed as

$d'_s(t)=H_Q^{-1} g'_s(t),$

where $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$0 is the Sylvester Hadamard matrix of order $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$1. Each partially decoded group is then beamformed into $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$2 sub-images by a modified delay-and-sum operator

$$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$3

with $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$4. These sub-images are combined using the $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$5-th row of $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$6,

$$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$7

so that, by linearity, the full FORCES image is recovered as

$$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$8

EMC² modifies this workflow by retaining each $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$9 in complex analytic (RF) form. That choice is structurally important: later coherent summation is intended to preserve phase, so motion correction is applied before envelope detection. The pipeline proceeds by selecting one low-resolution image, typically $N$0, as a reference; estimating motion from each remaining $N$1 relative to that reference; interpolating the resulting sparse displacement estimates to a dense field; warping each low-resolution image; and summing the aligned complex images coherently (Henry et al., 10 Sep 2025).

This architecture exploits a specific property of READI: the low-resolution images are less susceptible to motion than the full FORCES reconstruction, yet they sum to form the complete FORCES image. The reported purpose of EMC² is therefore not merely post hoc registration, but restoration of a motion-corrupted encoded-aperture reconstruction by operating on intermediate sub-images that remain physically tied to the original acquisition.

3. Motion estimation, outlier rejection, and warping in READI EMC²

For READI-based EMC², motion estimation is performed by block matching with normalized cross-correlation (NCC). At each motion-grid location $N$2 in the reference image, a template patch

$N$3

is compared against candidate patches within a target-image search window

$N$4

The NCC score at offset $N$5 is

$N$6

and the integer-pixel displacement is chosen as

$N$7

Sub-pixel refinement is obtained by fitting a $N$8-D paraboloid to the $N$9 neighborhood around the integer-pixel peak and taking the maximum of that fitted surface. Spurious or unreliable vectors are rejected if any of three criteria hold: peak $G(t)\in\mathbb{R}^N$0, $G(t)\in\mathbb{R}^N$1, or the smallest eigenvalue of the paraboloid Hessian at $G(t)\in\mathbb{R}^N$2 is below $G(t)\in\mathbb{R}^N$3. The paper reports typical parameter ranges tuned per experiment: motion grid spacing $G(t)\in\mathbb{R}^N$4–$G(t)\in\mathbb{R}^N$5 px, template size $G(t)\in\mathbb{R}^N$6–$G(t)\in\mathbb{R}^N$7 px, search extension $G(t)\in\mathbb{R}^N$8–$G(t)\in\mathbb{R}^N$9 px, $N=S\cdot Q$0–$N=S\cdot Q$1, $N=S\cdot Q$2–$N=S\cdot Q$3, and $N=S\cdot Q$4–$N=S\cdot Q$5 (Henry et al., 10 Sep 2025).

After sparse displacements $N=S\cdot Q$6 are estimated, they are interpolated to full image resolution $N=S\cdot Q$7. Each low-resolution image is then warped according to

$N=S\cdot Q$8

with complex-valued bilinear interpolation. If $N=S\cdot Q$9 lies between integer samples, one writes $S$0 and evaluates

$S$1

using the usual bilinear weights. The final motion-compensated reconstruction is the phase-coherent sum

$S$2

followed optionally by envelope detection $S$3 and post-processing such as coherence-factor weighting or log compression.

4. EMC² in ultrafast echocardiography

A separate EMC² formulation was introduced for ultrafast echocardiography as a unified framework combining coherent compounding, second-harmonic imaging, and angular coherence for simultaneous high-quality B-mode and tissue Doppler. Here the acquisition consists of $S$4 steered diverging-wave transmissions emitted at successive angles. Motion between the $S$5-th and $S$6-th acquisitions misaligns the echo fields and degrades compounding gain, so EMC² estimates inter-pulse motion either by cross-correlation of complex baseband signals or, more efficiently, by Doppler-based ensemble autocorrelation (Mougharbel et al., 2023).

The Doppler-based estimator uses slow-time autocorrelations over ascending and descending halves of a triangular angular sweep,

$S$7

From the resulting phase $S$8, the radial Doppler velocity is implemented in pseudocode as

$S$9

and the spatial delay per transmit is

$Q$0

After beamforming each diverging wave into a complex IQ field $Q$1, motion-compensated coherent compounding is

$Q$2

In practice, the realignment is implemented by $Q$3D linear interpolation along the radial coordinate followed by phase correction $Q$4.

The same framework incorporates second-harmonic imaging via pulse inversion. The receive signal is modeled as

$Q$5

so that transmitting alternating positive and negative pulses and summing the echoes yields

$Q$6

after which a band-pass filter around $Q$7 isolates the second harmonic. Both fundamental and second-harmonic channels undergo the same motion estimation and compounding steps.

Residual misalignment and clutter are further handled through a lag-1 angular coherence factor,

$Q$8

which weights the motion-compensed IQ image to produce

$Q$9

For B-mode display, $s$0 is envelope-detected and logarithmically compressed. In this formulation, EMC² therefore denotes not only motion-compensated compounding but a compound pipeline in which motion correction, harmonic clutter suppression, and angular-coherence weighting are explicitly coupled.

5. Quantitative performance across reported implementations

The two EMC² instantiations report quantitative gains under different motion regimes and acquisition models. In the READI/FORCES setting, the reported benchmarks emphasize recovery from probe motion, performance of low-resolution READI sub-images relative to sparse STA baselines, and speckle preservation in flow. In the ultrafast echocardiography setting, the reported benchmarks emphasize gCNR under controlled in vitro motion, Doppler accuracy, and in vivo performance during low- and high-velocity phases of the cardiac cycle.

Setting	Condition	Reported result
READI/FORCES	Lateral probe motion at $s$1 cm/s	Standard FORCES gCNR fell by $s$2–$s$3; EMC² restored gCNR to within $s$4–$s$5 of the static reference
READI/FORCES	Same $s$6 transmit count as uFORCES	A single READI sub-image achieved $s$7–$s$8 higher generalized CNR than uFORCES sparse-Hadamard
READI/FORCES	$s$9, $g'_s(t)\in\mathbb{R}^Q$0, $g'_s(t)\in\mathbb{R}^Q$1 MHz, PRF $g'_s(t)\in\mathbb{R}^Q$2 kHz	EMC² recovered coherent blood-speckle flow at $g'_s(t)\in\mathbb{R}^Q$3 cm/s
Ultrafast echocardiography	In vitro spinning disk	Standard compounding gCNR fell from $g'_s(t)\in\mathbb{R}^Q$4 at $g'_s(t)\in\mathbb{R}^Q$5 cm/s to $g'_s(t)\in\mathbb{R}^Q$6 at $g'_s(t)\in\mathbb{R}^Q$7 cm/s
Ultrafast echocardiography	In vitro EMC²	gCNR $g'_s(t)\in\mathbb{R}^Q$8 up to $g'_s(t)\in\mathbb{R}^Q$9 cm/s in fundamental and $d'_s(t)=H_Q^{-1} g'_s(t),$0 cm/s in second harmonic
Ultrafast echocardiography	Doppler estimation	NRMSE $d'_s(t)=H_Q^{-1} g'_s(t),$1 for $d'_s(t)=H_Q^{-1} g'_s(t),$2 cm/s in fundamental and $d'_s(t)=H_Q^{-1} g'_s(t),$3 cm/s in second harmonic
Ultrafast echocardiography	In vivo ventricular filling	Fundamental gCNR $d'_s(t)=H_Q^{-1} g'_s(t),$4; second-harmonic gCNR $d'_s(t)=H_Q^{-1} g'_s(t),$5
Ultrafast echocardiography	In vivo diastasis	Fundamental gCNR $d'_s(t)=H_Q^{-1} g'_s(t),$6; second-harmonic gCNR $d'_s(t)=H_Q^{-1} g'_s(t),$7

In the READI paper, EMC² is also described qualitatively as fully recovering images corrupted by probe motion and restoring tissue speckle and sharpness to an image of a beating heart. In the echocardiography paper, the unified framework is reported to increase gCNR from $d'_s(t)=H_Q^{-1} g'_s(t),$8 to $d'_s(t)=H_Q^{-1} g'_s(t),$9 during ventricular filling when compared against coherent compounding, and the conclusion states that real-time implementation at $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$00-angle compounding and $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$01 Hz yields high-contrast B-mode images and reliable tissue Doppler quantification even during fast myocardial motion up to $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$02 cm/s (Henry et al., 10 Sep 2025, Mougharbel et al., 2023).

6. Computational profile, methodological implications, and recurring misconceptions

The computational cost of READI-based EMC² can be decomposed into partial decoding, Hilbert transforms, beamforming, NCC block matching, and warping with coherent compounding. The paper specifies: $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$03 matrix multiplies of size $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$04 using cublasSgemm, with complexity $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$05; Hilbert transforms costing $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$06 per channel; beamforming over $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$07 subsets with $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$08 delay-and-sum operations; NCC block matching over $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$09 targets times $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$10; and warping plus compounding with $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$11 interpolations and additions. On a modern NVIDIA GPU using CUDA, cuBLAS, cuFFT, and NPP, real-time performance greater than $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$12 Hz was reported for $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$13 TOBE arrays and $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$14 transmit events (Henry et al., 10 Sep 2025).

In ultrafast echocardiography, the real-time pipeline is summarized at the algorithmic level rather than through asymptotic complexity terms. The implemented parameters are a $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$15 sector, $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$16 diverging waves, angular step $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$17, transmit PRF $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$18 Hz, frame rate $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$19 Hz, demodulation to IQ at $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$20 MHz for the fundamental or $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$21 MHz for the second harmonic, and grid $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$22, $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$23. The processing sequence consists of beamforming each diverging wave into IQ lines, computing the autocorrelations $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$24 and $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$25, deriving $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$26, estimating $$y_{\mathrm{EMC}^2}(r)=\mathrm{ACF}(r)\,y_{\mathrm{comp}}(r)$$27, warping and phase compensating each angle, summing the motion-corrected fields, computing the angular coherence factor, and applying envelope detection with log compression (Mougharbel et al., 2023).

Several recurring misconceptions can be addressed directly from these formulations. First, EMC² is not synonymous with a particular motion estimator: one implementation uses NCC block matching on low-resolution sub-images, whereas another uses ensemble autocorrelation and Doppler phase. Second, EMC² is not always limited to geometric realignment alone: in ultrafast echocardiography, the final reported EMC² image includes angular-coherence weighting, and the framework supports both fundamental and second-harmonic imaging. Third, motion compensation is not applied after irreversible magnitude formation in the READI setting; the low-resolution images are retained in complex analytic form specifically so that coherent compounding can preserve phase. A plausible implication is that EMC² is best understood as a phase-aware alignment-and-summation paradigm whose performance depends on the reliability of motion estimation in the relevant signal domain.