SlingBAG Pro: 3D PAI Reconstruction Accelerator

• SlingBAG Pro is a point cloud-based iterative reconstruction algorithm that adapts localized spherical or Gaussian-like elements to model the initial pressure in 3D photoacoustic imaging.
• It employs a hierarchical optimization strategy with zero-gradient filtering and progressive temporal sampling to reduce computational cost and accelerate convergence.
• By replacing dense voxel grids with a sparse point cloud, the method achieves high image quality with fewer transducers and reduced memory requirements.

Searching arXiv for the specified paper and its predecessor to ground the article in the current literature. SlingBAG Pro is a point cloud-based iterative reconstruction algorithm for three-dimensional photoacoustic imaging (3D PAI) designed to accelerate reconstruction under arbitrary transducer array geometries while maintaining high image quality with fewer transducers. Introduced in "SingBAG Pro: Accelerating point cloud-based iterative reconstruction for 3D photoacoustic imaging under arbitrary array" (Li et al., 2 Jan 2026), it extends the earlier SlingBAG framework (Li et al., 2024) from mainly regular cubic imaging regions to irregular, conformal, sparse, and geometry-agnostic acquisition settings. Its central premise is to reconstruct the initial pressure distribution $p_0(\mathbf{r})$ not on a dense voxel grid but as an adaptive set of localized spherical or Gaussian-like basis elements stored as a point cloud, and to combine this representation with zero-gradient filtering and hierarchical temporal sampling to reduce memory usage, accelerate convergence, and shorten wall-clock reconstruction time.

1. Position within 3D photoacoustic imaging

In 3D PAI, pulsed optical illumination generates an initial pressure distribution $p_0(\mathbf{r})$ through thermoelastic expansion, and ultrasound transducers record the time-varying acoustic pressure $p(\mathbf{r}_d,t)$ on the boundary. The inverse problem is to recover $p_0(\mathbf{r})$ from measurements $p(\mathbf{r}_d,t)$. Analytic methods such as universal back-projection (UBP) work well for regular, dense arrays and ideal acoustics, but degrade when the detection view is limited, the array geometry is irregular or conformal, or the measurements are noisy and the model is imperfect (Li et al., 2 Jan 2026).

These conditions motivate iterative reconstruction (IR), typically posed as minimization of a least-squares mismatch between a forward model $F(p_0)$ and measured data. The difficulty is that conventional voxel-grid IR scales poorly with imaging volume, detector count, and temporal sampling density. For arbitrary layouts, the Green’s function or time-of-flight from each voxel to each detector differs, symmetries cannot be exploited, explicit system matrices become impractical, and even matrix-free forward or adjoint operators remain expensive. SlingBAG Pro addresses this computational bottleneck by replacing dense volumetric parameterization with an adaptive point-cloud representation and by restructuring optimization so that many irrelevant degrees of freedom are discarded early (Li et al., 2 Jan 2026).

Relative to the original SlingBAG algorithm, the Pro variant generalizes the method to arbitrary array geometries and introduces a hierarchical optimization strategy that combines zero-gradient filtering with progressively increased temporal sampling rates during iteration. This suggests that the method is not merely an implementation refinement but a reformulation of how geometry, initialization, and multi-resolution temporal information are coupled in point cloud-based 3D PAI reconstruction.

2. Point-cloud representation and iterative formulation

SlingBAG Pro inherits the core representation of SlingBAG, in which the initial pressure field is modeled as a collection of localized “balls” rather than a dense voxel lattice (Li et al., 2024). In the formulation described for the Pro method, each ball $i$ has a position $$\mathbf{r}_s^{(i)} = (x_s^{(i)}, y_s^{(i)}, z_s^{(i)})$$, a peak initial pressure $p_0^{(i)}$, and a standard deviation or size $a_0^{(i)}$, yielding

$p_0(\mathbf{r})$0

A differentiable fast forward operator, termed the rapid radiator in the SlingBAG line of work, computes simulated signals from the ball parameters and detector coordinates on the fly, without storing a system matrix. In vector form, the simulated data are written as

$p_0(\mathbf{r})$1

and the optimization objective at temporal level $p_0(\mathbf{r})$2 is

$p_0(\mathbf{r})$3

No explicit regularization term such as total variation is included in the described version. Instead, the sparse adaptive point cloud acts as a structural prior (Li et al., 2 Jan 2026).

The iterative process has two stages. In coarse reconstruction, only $p_0(\mathbf{r})$4 and $p_0(\mathbf{r})$5 are optimized while positions remain fixed. In fine reconstruction, all parameters $p_0(\mathbf{r})$6 are updated. Throughout optimization, the point cloud evolves by destroying, splitting, and, in the fine stage, duplicating balls. Destroying removes negligible contributions; splitting increases local spatial resolution where a large ball is still significant; duplicating increases point density in detailed regions. The “sliding” in SlingBAG refers to the motion of ball centers during fine reconstruction (Li et al., 2024).

A technical distinction between the predecessor and the Pro variant is that the earlier paper describes the basis in terms of spherical sources implemented via a 10th-order spherical decomposition and differentiable analytic radiation formulas, whereas the Pro exposition emphasizes Gaussian-ball parameterization while still relying on the differentiable rapid radiator (Li et al., 2024). A plausible implication is that the essential abstraction for both papers is the same: a sparse, differentiable, localized source representation whose parameters can be optimized directly against measured acoustic time series.

3. Extension to arbitrary array geometries

The defining architectural contribution of SlingBAG Pro is its extension from mainly regular cubic imaging regions to arbitrary array geometries (Li et al., 2 Jan 2026). Rather than assuming a fixed cubic reconstruction volume, the method constructs a polygonal envelope mesh $p_0(\mathbf{r})$7 from the detector coordinate list $p_0(\mathbf{r})$8. The reconstruction region is then defined inside this mesh, slightly inward from the detector surface normals, and ray casting is used to initialize random points only within the enclosed imaging volume.

This geometry handling is matrix-free and coordinate-driven. The forward model accepts the detector list directly, and for each ball–detector pair computes travel time from Euclidean distance under a constant-speed-of-sound assumption,

$p_0(\mathbf{r})$9

Because no geometry-specific system matrix is precomputed, arbitrary, sparse, curved, conformal, and miniaturized layouts can be handled through coordinates alone. The paper illustrates this with two classes of arrays: irregular sparse envelope arrays around simulated hand vasculature with 505, 1009, and 2006 detectors, and a hemispherical 1024-element array used for in vivo rat imaging (Li et al., 2 Jan 2026).

This geometry-agnostic formulation is significant because irregular arrays arise from clinical hardware constraints, flexible PMUT or CMUT technologies, wearable or miniaturized probes, and cost pressures that favor fewer but better-placed elements. In such settings, traditional analytic inversions are poorly matched to the acquisition physics, while dense matrix-based iterative solvers become prohibitive. SlingBAG Pro recasts the difficulty as a sparse parametric optimization problem over an irregularly initialized point cloud, thereby aligning the reconstruction representation with the nonuniform support of practical measurements.

4. Hierarchical optimization strategy

SlingBAG Pro introduces two algorithmic accelerations beyond the original SlingBAG method: zero-gradient filtering for initialization and progressive temporal sampling during optimization (Li et al., 2 Jan 2026).

Zero-gradient filtering begins with a random uniform point cloud of $p(\mathbf{r}_d,t)$0 balls inside the reconstruction envelope. The method temporarily sets all ball pressures to zero, computes the loss against downsampled measured data, and evaluates the zero-pressure gradient for each ball,

$p(\mathbf{r}_d,t)$1

Balls with $p(\mathbf{r}_d,t)$2 are retained and their original pressures are restored; balls with $p(\mathbf{r}_d,t)$3 are discarded. The retained set is therefore

$p(\mathbf{r}_d,t)$4

In the hand vessel simulation, this reduced 200,000 initial balls to 106,924 at the coarsest temporal sampling level. The filtered cloud already outlined vascular structures qualitatively, and correspondence improved as the number of detectors increased from 505 to 1009 to 2006. This suggests that zero-gradient filtering acts as a data-driven pruning mechanism that concentrates optimization capacity near likely sources before heavy reconstruction begins (Li et al., 2 Jan 2026).

Progressive temporal sampling implements a coarse-to-fine schedule over the measured time series. If $p(\mathbf{r}_d,t)$5 denotes the full-rate signals and $p(\mathbf{r}_d,t)$6 is the downsampling factor at level $p(\mathbf{r}_d,t)$7, then

$p(\mathbf{r}_d,t)$8

with corresponding simulation performed on a temporally matched grid using $p(\mathbf{r}_d,t)$9 and $p_0(\mathbf{r})$0. In the hand simulation the schedule is $p_0(\mathbf{r})$1; in vivo it is $p_0(\mathbf{r})$2. Early iterations use heavily downsampled data to obtain a cheap rough solution, and later stages progressively restore temporal resolution as the point cloud becomes spatially denser, especially through duplication in the fine stage (Li et al., 2 Jan 2026).

At the final fine stage, positivity of ball size is enforced by softplus reparameterization,

$p_0(\mathbf{r})$3

so that optimization proceeds over unconstrained $p_0(\mathbf{r})$4 while guaranteeing $p_0(\mathbf{r})$5. This is a technical but important refinement because it avoids ad hoc projection steps and preserves differentiability in the final stage (Li et al., 2 Jan 2026).

5. Computational characteristics

The computational profile of SlingBAG Pro is governed by the number of balls, the number of detectors, and the number of time samples at the current hierarchical level. Per-iteration complexity is described as roughly proportional to

$p_0(\mathbf{r})$6

Its acceleration mechanisms target exactly these terms: zero-gradient filtering reduces $p_0(\mathbf{r})$7 from the beginning, while hierarchical temporal sampling reduces $p_0(\mathbf{r})$8 in early iterations (Li et al., 2 Jan 2026).

Memory efficiency derives from abandoning the voxel-grid parameterization of conventional IR. In the hand simulation, the evaluation grid corresponds to $p_0(\mathbf{r})$9 million voxels, whereas SlingBAG Pro manages only tens to hundreds of thousands of balls, such as 200,000 initially and about 100k after filtering. The system matrix is never stored, and forward or adjoint operations are computed on the fly from ball and detector coordinates. The predecessor paper quantified the broader advantage of this representation: for a large-scale setup, k-Wave required 46.7 GB GPU memory while the differentiable rapid radiator required 0.32 GB; for full reconstruction with 238 sensors, SlingBAG needed less than 3 GB GPU memory (Li et al., 2024).

In the irregular-array hand-vessel simulations reported for SlingBAG Pro, runtime reductions are substantial. For 2006 sensors, SlingBAG Pro required 3.79 h versus 8.32 h for SlingBAG, corresponding to a 2.2-fold speed improvement, while also slightly improving PSNR and SSIM. For 1009 sensors, times were 2.47 h versus 4.23 h, and for 505 sensors, 1.71 h versus 2.25 h (Li et al., 2 Jan 2026). The paper attributes this to three factors: pruning by zero-gradient filtering, cheap early iterations enabled by aggressive temporal downsampling, and geometry-aware initialization that avoids placing points in irrelevant regions outside the practical imaging area.

6. Validation, limitations, and prospective directions

Validation was performed in both simulation and in vivo settings (Li et al., 2 Jan 2026). In the 3D hand-vessel simulation, the imaging region comprised $p(\mathbf{r}_d,t)$0 grid points at 0.2 mm spacing, corresponding to $p(\mathbf{r}_d,t)$1, with density $p(\mathbf{r}_d,t)$2, sound speed 1500 m/s, temporal sampling at 40 MHz, and 4900 time points. Three detector counts were evaluated on irregular envelope arrays: 505, 1009, and 2006. Across all three, SlingBAG Pro outperformed UBP by a wide margin and modestly improved on SlingBAG while reducing runtime.

Setting	UBP	SlingBAG	SlingBAG Pro
505 sensors	PSNR 18.75 dB, SSIM 0.144	PSNR 26.38 dB, SSIM 0.53, 2.25 h	PSNR 26.44 dB, SSIM 0.565, 1.71 h
1009 sensors	PSNR 20.54 dB, SSIM 0.181	PSNR 27.92 dB, SSIM 0.589, 4.23 h	PSNR 28.20 dB, SSIM 0.638, 2.47 h
2006 sensors	PSNR 21.95 dB, SSIM 0.216	PSNR 28.86 dB, SSIM 0.652, 8.32 h	PSNR 29.22 dB, SSIM 0.722, 3.79 h

Qualitatively, UBP under sparse irregular arrays exhibited strong streaking and failed to recover vasculature, whereas SlingBAG Pro reconstructed detailed vessels even with 505 sensors. Line profiles through MAP images showed good amplitude agreement with ground truth and improved further with more sensors (Li et al., 2 Jan 2026).

In vivo experiments used a hemispherical 1024-element array with 60 mm radius, 2.02 MHz center frequency, 54% bandwidth, field of view approximately $p(\mathbf{r}_d,t)$3, and sampling at 8.33 MHz with 896 samples. Rat liver and rat kidney reconstructions were compared using CNR and runtime. For rat liver, SlingBAG Pro achieved CNR 44.69 in 3.37 h, compared with SlingBAG’s CNR 42.01 in 4.75 h and UBP’s CNR 30.62. For rat kidney, SlingBAG Pro achieved CNR 28.51 in 1.81 h, compared with SlingBAG’s CNR 30.26 in 2.90 h and UBP’s CNR 20.18. The kidney result is an explicit trade-off case in which SlingBAG had slightly higher CNR, but SlingBAG Pro remained much faster and still far exceeded UBP (Li et al., 2 Jan 2026).

The method assumes acoustic homogeneity, using a constant speed of sound and no modeling of refraction, scattering, or strong attenuation variations. It relies on $p(\mathbf{r}_d,t)$4 data fidelity without an explicit robust noise model or added regularizers such as $p(\mathbf{r}_d,t)$5 or TV. Detector position accuracy is important, and no explicit study is presented for severe model mismatch such as heterogeneous media. Hyperparameters include the initial ball count, thresholds for splitting, destroying, and duplicating, and the temporal downsampling schedule. Sensitivity is not exhaustively analyzed, although the authors report consistent behavior across different datasets (Li et al., 2 Jan 2026).

Several future directions are identified or implied. These include further acceleration through better GPU implementations, integration with deep learning or model-based learning, extension to heterogeneous media and full-wave solvers, hardware–algorithm co-design for arbitrary arrays, and incorporation of non-smooth regularization. The source code is publicly available at https://github.com/JaegerCQ/SlingBAG_Pro, which situates the method within an open and reproducible line of development extending from the original SlingBAG repository (Li et al., 2 Jan 2026).