Recursive Aperture Decoded Ultrasound Imaging

• READI is a recursive decoding strategy for Hadamard-encoded synthetic aperture data that partitions full ensembles into shorter groups to reduce motion-induced artifacts.
• It reorganizes the FORCES imaging pipeline by factorizing the Hadamard matrix to produce low-resolution subframes which are coherently summed to achieve full-aperture performance.
• EMC² builds on READI by using internal motion estimation with NCC-based block matching to align subframes, thereby enhancing SNR, penetration, and contrast in dynamic scenes.

Searching arXiv for READI, FORCES, and related synthetic-aperture ultrasound papers to ground the article. Recursive Aperture Decoded Ultrasound Imaging (READI) is a decoding and beamforming strategy for Hadamard-encoded synthetic transmit aperture acquisitions, introduced for Fast Orthogonal Row-Column Electronic Scanning (FORCES) on bias-sensitive Top-Orthogonal to Bottom Electrode (TOBE) arrays. Its defining operation is to factor a full coded ensemble into shorter sequential groups, partially decode each group into a low-resolution image, and then coherently sum those images so that, in the absence of motion, the result is exactly equivalent to the full FORCES reconstruction. In the same framework, Estimated Motion-Compensated Compounding (EMC²) uses the low-resolution READI images as internal subframes for motion estimation, warping, and coherent recombination, with the stated objective of preserving FORCES’ high SNR and penetration while mitigating ensemble-length motion sensitivity (Henry et al., 10 Sep 2025).

1. Definition and conceptual scope

READI belongs to the class of recursive, extended, and synthetic aperture methods that construct a “virtual” aperture from a sequence of smaller encoded or partially decoded acquisitions. In the specific formulation reported for FORCES, the method does not alter the underlying transmit count required by the full Hadamard-encoded sequence; instead, it reorganizes decoding and beamforming so that the same ensemble yields multiple shorter-footprint low-resolution images whose coherent sum reproduces the full FORCES image (Henry et al., 10 Sep 2025).

The term “recursive aperture decoded” refers to the aperture mapping induced by the Sylvester structure of the Hadamard matrix. The full aperture is partitioned into groups, partial decoding is performed within each group, and sub-images associated with different element groups are combined with Hadamard-derived weights. This suggests that “recursion” in READI is primarily algebraic rather than iterative optimization: the aperture is decomposed hierarchically through the factorization of the encoding matrix, and the image is recovered by summing recursively defined partial reconstructions.

A common misconception is to treat READI as synonymous with generic synthetic transmit aperture imaging. The underlying papers distinguish these notions. Classical synthetic transmit aperture imaging provides a multistatic dataset that is beamformed after acquisition, whereas READI is specific to a coded acquisition and decoding strategy for FORCES on TOBE arrays, with motion compensation built around low-resolution internal subframes rather than around direct external tracking or standard post hoc compounding (Henry et al., 10 Sep 2025).

2. FORCES and the problem READI addresses

FORCES is a Hadamard-encoded Synthetic Transmit Aperture imaging sequence using bias-sensitive TOBE arrays. TOBE arrays are row-column arrays in which a DC bias along a row or column makes the corresponding elements active, and reversing the bias flips the phase of their acoustic output or received signal. This enables elementwise $+1/-1$ modulation of transmit and receive phase via the DC bias pattern, which FORCES exploits through an $N \times N$ Hadamard matrix $\mathbf{H}_N$ (Henry et al., 10 Sep 2025).

If the encoded RF data are denoted $G(t)$ and the underlying multistatic signals are denoted $S(t)$, FORCES is expressed as

$$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$

Since

$$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$

decoding recovers the multistatic dataset, up to scaling, and the decoded data are then beamformed as in synthetic transmit aperture imaging (Henry et al., 10 Sep 2025).

The motivation for FORCES is that each coded shot uses the entire aperture rather than a single element or small sub-aperture. The reported consequence is that energy per transmit is multiplied by $N$ versus a single-element STA transmit, and SNR improves roughly by $\sqrt{N}$. TOBE hardware also permits elevational transmit focusing and steering in the orthogonal direction, with the stated effect of improving in-plane SNR and penetration (Henry et al., 10 Sep 2025).

The weakness is motion sensitivity. Standard FORCES decoding assumes that the object remains static over the $N$-event ensemble. Because FORCES measurements are linear combinations of all elements at each event, motion corrupts not only beamforming coherence but also the coding assumption itself: the decoded $N \times N$0 becomes a time-averaged multistatic dataset with motion-induced inconsistencies. READI is designed precisely to preserve FORCES’ full-aperture transmit benefits while reducing the effective temporal footprint of the intermediate images used for compounding (Henry et al., 10 Sep 2025).

3. Mathematical formulation of recursive aperture decoding

READI uses the Sylvester factorization of the Hadamard matrix. For powers of two,

$N \times N$1

Accordingly, for $N \times N$2 with $N \times N$3 and $N \times N$4 both powers of two,

$N \times N$5

This permits a re-indexing of transmit elements and events into element-group and event-group coordinates, so that the full coded ensemble can be reshaped into $N \times N$6 groups of $N \times N$7 events (Henry et al., 10 Sep 2025).

The encoded data $N \times N$8 are reshaped into grouped data

$N \times N$9

Standard FORCES reconstruction is then algebraically rearranged into a sum over groups: $\mathbf{H}_N$0 Defining the READI low-resolution image from group $\mathbf{H}_N$1 as

$\mathbf{H}_N$2

one obtains

$\mathbf{H}_N$3

The full FORCES image is therefore the coherent sum of $\mathbf{H}_N$4 READI low-resolution images, each derived from only $\mathbf{H}_N$5 transmits (Henry et al., 10 Sep 2025).

The grouped beamformer $\mathbf{H}_N$6 differs from standard delay-and-sum in that it applies delays corresponding to the $\mathbf{H}_N$7-th element group after partial decoding: $\mathbf{H}_N$8 This definition is required because the partially decoded group signal contains contributions from all transmit elements, intermixed through Hadamard structure, rather than from a single isolated sub-aperture (Henry et al., 10 Sep 2025).

The low-resolution images are called “low-resolution” because each is formed from only $\mathbf{H}_N$9 events and therefore exhibits READI artifacts due to residual cross-terms. The appendix analysis in the reported work shows that these cross-terms are distributed with opposite signs across groups and cancel when all $G(t)$0 images are coherently summed. In static conditions, this yields exact recovery of the standard FORCES image; in moving scenes, the cancellation becomes imperfect unless motion is corrected before compounding (Henry et al., 10 Sep 2025).

4. Estimated Motion-Compensated Compounding (EMC²)

EMC² is the motion-compensation procedure built on top of READI. Its purpose is to use the set of low-resolution READI images from a single FORCES ensemble as internal motion-estimation frames: motion is estimated between those images, the images are warped into alignment with a reference, and the aligned images are then coherently summed (Henry et al., 10 Sep 2025).

Let $G(t)$1 denote the complex RF or envelope low-resolution images. EMC² chooses a reference image $G(t)$2 and estimates a displacement field $G(t)$3 for each target image such that

$G(t)$4

The reported implementation uses block matching with normalized cross-correlation (NCC). Motion is estimated on a coarse grid with spacing 4–16 pixels. At each grid point, a template patch of size $G(t)$5 up to $G(t)$6 pixels is extracted from the reference image, a search window is defined in the target image, the NCC map is computed over candidate displacements, and the displacement is taken from the NCC peak. Sub-pixel refinement is then performed by fitting a 2D paraboloid to a $G(t)$7 neighborhood around the maximum (Henry et al., 10 Sep 2025).

The method applies three reliability tests to the NCC estimate: an absolute peak threshold, a relative peak threshold with respect to the no-motion correlation value, and a curvature requirement based on the Hessian eigenvalues of the paraboloid fit. Where these tests fail, the motion estimate is rejected or treated as no motion. The accepted vectors are interpolated to a dense displacement field, and each low-resolution image is warped as

$G(t)$8

Coherent compounding then gives

$G(t)$9

Because compounding remains coherent, EMC² is intended to preserve the cross-term cancellation and full-aperture focusing of static FORCES while reducing motion-induced inconsistencies (Henry et al., 10 Sep 2025).

A second misconception is that EMC² makes READI motion-invariant in an unrestricted sense. The reported limitations are explicit. Large displacements can move structures outside the overlapping field of view of the grouped images, low-SNR regions can yield ambiguous NCC surfaces, and out-of-plane motion or complex three-dimensional deformation cannot be corrected by the 2D local translational model used in the experiments (Henry et al., 10 Sep 2025).

5. Image formation pipeline, performance, and trade-offs

The reported CUDA implementation proceeds in a fixed sequence. FORCES data $S(t)$0 are acquired over $S(t)$1 encoded transmits. A choice of $S(t)$2 satisfying $S(t)$3 is made, the data are reshaped into $S(t)$4 groups of $S(t)$5 events, each group is partially decoded by $S(t)$6 using cuBLAS, analytic signals are produced by a Hilbert transform using cuFFT, grouped beamforming $S(t)$7 is performed for each element group, and the weighted sum over $S(t)$8 gives one low-resolution READI image per group. Summing those images yields the FORCES-equivalent image. EMC² adds NCC-based motion estimation using NPP, interpolation of motion fields, warping, and coherent recombination (Henry et al., 10 Sep 2025).

The central trade-off is between $S(t)$9 and $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$0. Smaller $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$1 means that each low-resolution READI image spans fewer transmits, so the temporal footprint is shorter by a factor $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$2, and motion blur in each low-resolution image is correspondingly reduced. However, smaller $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$3 also means lower per-image SNR and stronger READI artifacts, since cross-term cancellation occurs only after summation over all groups. Larger $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$4 improves low-resolution image quality but reintroduces more motion blur into each constituent image (Henry et al., 10 Sep 2025).

The static-case equivalence of READI and FORCES was validated in Field II point-spread-function simulation. A FORCES PSF and READI reconstructions with $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$5, $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$6, and $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$7 were reported to match to numerical precision after compounding, confirming the algebraic equivalence in static conditions (Henry et al., 10 Sep 2025).

The same work reports several motion-sensitive application regimes. In a static anechoic cyst phantom, a single READI low-resolution image was compared with uFORCES at equal transmit counts of 32, 16, and 8 transmits. The observation reported is that READI low-resolution images showed visibly better SNR and contrast than uFORCES at all three counts, and generalized contrast-to-noise ratio outperformed uFORCES at all depths and transmit counts, with the advantage increasing as transmit count decreased. In a lateral probe sweep at approximately $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$8, uncompensated FORCES showed severe blur, while READI with $$G(t) = \mathbf{H}_N S(t), \qquad S(t) = \mathbf{H}_N^{-1} G(t).$$9 plus EMC² recovered cyst visibility and background speckle nearly to the static FORCES baseline, except beyond approximately 70 mm depth where SNR and low-resolution image visibility were too poor to recover the cysts (Henry et al., 10 Sep 2025).

In a beating-heart phantom at 120 bpm and PRF $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$0, READI with $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$1 and EMC² restored speckle patterns and sharpened myocardial boundaries relative to uncompensated FORCES, although residual blur persisted in low-SNR, heavily moving regions where the motion estimator failed its reliability thresholds. In a 6 mm vessel flow phantom using an 8 MHz TOBE array, PRF $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$2, and blood-mimicking fluid at $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$3, standard FORCES yielded little coherent blood speckle, whereas READI with $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$4, followed by acquisition of 16 FORCES frames and basic SVD clutter filtering across the resulting 256 low-resolution frames, recovered clear blood speckle and a visible parabolic flow profile (Henry et al., 10 Sep 2025).

6. Relation to broader extended- and coded-aperture ultrasound methods

READI sits at the intersection of two broader lines of work: aperture extension by coherent multi-aperture compounding, and coded acquisition with explicit decoding of a latent multistatic dataset.

The first line is exemplified by coherent multi-transducer imaging, where the effective aperture is extended by coherently compounding RF data from multiple synchronized probes with partially shared field of view. In that framework, only one transducer transmits at a time while all $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$5 transducers receive, and the current transducer locations and the speed of sound are deduced by optimizing cross-correlation of echoes from targeted scatterers, without external tracking. The reported result was improvement in lateral resolution, contrast, and contrast-to-noise ratio from $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$6, $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$7, and $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$8, respectively, for a single probe, to $$\mathbf{H}_N \mathbf{H}_N^T = N \mathbf{I}, \quad \mathbf{H}_N^{-1} = \frac{1}{N}\mathbf{H}_N^T,$$9, $N$0, and $N$1 in coherent multi-transducer imaging. This suggests a conceptual parallel to READI: both treat the aperture as a latent structure to be reconstructed or decoded from distributed measurements, and both rely on coherent summation over geometrically or algebraically factorized components (Peralta et al., 2019).

The second line is exemplified by optimization of array encoding for ultrasound imaging, which formulates synthetic transmit aperture acquisition as a coded linear measurement model

$N$2

with decoding via a Tikhonov-regularized pseudoinverse

$N$3

That work uses a differentiable beamformer and machine learning to optimize transmit delays and apodization weights under image-domain losses, while retaining REFoCUS as the decoder. It frames the multistatic dataset as a basis representation of the scene and practical transmit sequences as coded measurements of that basis. READI can be situated naturally in that coded/decoded-aperture perspective, but with a specific Hadamard factorization and a recursive groupwise reconstruction that exposes internal motion-estimation frames rather than only a final decoded dataset (Spainhour et al., 2024).

A plausible implication is that READI, coherent multi-transducer compounding, and optimized coded acquisition can be viewed as three complementary strategies for virtual-aperture formation. Coherent multi-transducer imaging emphasizes geometric aperture extension and self-calibration from RF data; optimized array encoding emphasizes the design of $N$4 and efficient end-to-end differentiation; READI emphasizes algebraic factorization of a fixed encoded ensemble into shorter-footprint subframes that can be motion-compensated before coherent summation. The current READI formulation is specific to Sylvester Hadamard structure, TOBE-compatible FORCES encoding, and 2D NCC-based motion compensation, and its reported limitations remain tied to field-of-view overlap, SNR in low-resolution subframes, and unmodeled out-of-plane motion (Henry et al., 10 Sep 2025).