UltraGauss: Ultrafast Gaussian Reconstruction of 3D Ultrasound Volumes
Authors: Mark C. Eid
Ana Namburete
Joao F. Henriques
Venue: ICLR 2026 Poster (2026)
Keywords: Ultrasound
3D Reconstruction
Gaussian
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Ultrasound imaging is widely used due to its safety, affordability, and real-time capabilities, but its 2D interpretation is highly operator-dependent, leading to variability and increased cognitive demand. 
We present $\textbf{UltraGauss}$: an ultrasound-specific Gaussian Splatting framework that serves as an efficient approximation to acoustic image formation. Unlike projection-based splatting, UltraGauss renders by $\textit{probe-plane intersection}$ with in-plane aggregation, aligning with plane-based echo sampling while remaining fast and memory-efficient. A stable parameterisation and compute-aware GPU rasterisation make this method practical at scale. On clinical datasets, UltraGauss delivers state-of-the-art 2D-to-3D reconstructions in minutes on a single GPU (reaching 0.99 SSIM within $\sim$20 minutes), and a clinical expert survey rates its reconstructions the most realistic among competing methods. To our knowledge, this is the first Gaussian Splatting approach tailored to ultrasound 2D-to-3D reconstruction. Our code is available at: https://www.robots.ox.ac.uk/~vgg/research/UltraGauss/

Published as a conference paper at ICLR 2026
ULTRAGAUSS: ULTRAFAST GAUSSIAN RECONSTRUCTION OF 3D ULTRASOUND VOLUMES
Mark C. Eid 1.2.
Ana I.L. Namburete 2
João F. Henriques
Visual Geometry Group (VGG), University of Oxford, U.K.
2 Oxford Machine Learning in Neurolmaging Lab (OMNI, University of Oxford, U.K.
ABSTRACT
Ultrasound imaging is widely used due to its safety, affordability, and real-time capabilities, but 2D interpretation highly operator-dependent, leading to vari- ability and increased cognitive demand. We present UltraGauss: specific Gaussian Splatting framework that serves as an efficient approximation to acoustic image formation. Unlike projection-basee based splatting, UltraGauss ren- ders by probe-plane intersection with in-plane aggregation, aligning with plane- based echo sampling while remaining fast and memory-efficient. A stable pa- rameterisation and compute-aware GPU rasterisation make this method practical scale. On clinical datasets, UltraGauss delivers state-of-the-art 2D-to-3D re- constructions in minutes on single GPU (reaching 0.99 SSIM within ~22 min- utes), and a clinical expert survey rates its reconstructions the most realistic among competing methods. To our knowledge, this is the first Gaussian Splatting ap- proach tailored to ultrasound 2DDto-33 reconstruction. Our code is available at: http:://wwww..ooooooooooooooaa//
ultrasound-
INTRODUCTION
Ultrasound (US) is mainstay in medical imaging: it is real-time, low-cost, portable, and non- ionising. Yet routine use still asks clinicians to infer 3D anatomy from 2D slices: cognitively de- Nelson manding step that introduces operator-dependent variability (Benacerraf | & Pretorius) 1998), undermining reproducibility and hindering standardised assessment. While volumetric (3D) probes exist, their workflows are largely offline and the hardware is costly, making them uncom- mon outside well-resourced centres (Merz & Welter, 2005). Enabling 3D reasoning from routine 2D acquisitions offers software-oni path to scale volumetric assessment across sites and resource levels. As 2D is the universal denominator in global healthcare, 2D-to-3D reconstruction will yield standardised volume and surface metrics that are difficult to obtain consistently from single slices.
Recent learning-baan based approaches span implicit NeRF-style representations (Eid al.. 2025 Yeung et al. 2024; Gaits et al. 2024] Wysocki et al.) 2023) and explicit voxel grids (Solberg et al. 2007). However, implicit fields are computationally heavy; voxel grids incur memory and resolution limits; and many methods adopt light-transport assumptions (ray accumulation with transmittance), which fundamentally mismatch US physics, where waves propagate into tissue and return to the probe (Powles et al.) 2018 Aldrich 2007). Classical Gaussian Splatting (GS) for cameras achieves fast, high-quality rendering in optical settings by projecting 3D Gaussians onto the image plane and blending splats in depth order using alpha compositing (Kerbl et al., 2023). US images, the other hand, are not perspective renderings: they sample echo intensities within the probe plane (with attenuation), so the camera-style projection and occlusion paradigm does not apply.
We introduce UltraGauss: an saassound-specffi GS reconstruction framework that serves as an efficient approximation to the US image formation model, replacing projection-basee based rendering with probe-plane intersection rendering, consistent with wave-based acquisition. Instead of marching rays projecting Gaussians into 2D, UltraGauss evaluates anisotropic 3D Gaussians where they intersect the probe plane and aggregates intensities in-plane. This removes the need for depth- based occlusion, matches the acquisition geometry of linear and curvilinear probes, and enables resolution- free slicing at arbitrary orientations. This approximation preserves plane- plane-based sampling and dominant attenuation behaviour while avoiding expensive wave simulation.

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Published as a conference paper at ICLR 2026
US Probs
Ray
Image plane
TE
OR
OR
Image
Image
plane
plane
a) NeRF on RGBXX-Ray/CT
by Grusssins Splatting on RGRX-RRy/CT c) Ultrasound (US)
ci) US Vonel Grid
cii) US Physier-
ciiii US
Based NoRF
UltraGause
Figure 1: Image formation mechanisms, (a, b) Projection-based (e.g. camera) rendering with depth- ordered compositing and (c) ultrasound. (c.i) Slice sampling from a voxel grid or continuous field. (c.ii) Physies-based ultrasound: path integration with attenuation/backscaaa (c.iit) UltraGauss: plane-intersection of 3D Gaussians with in in-plane aggregation.
Our technical contributions are:
Efficient forward-model approximation: UltraGauss renders by probe-plane intersection rather than camera projection, capturing plane-based sampling and attenuation while side-stepping costly wave simulation.
Stable parameterisation: A triangular mnvers-ccoariinn (precision) factorisation guarantees and well-conditioned gradients for reconstruction and densification. Compute-aware rasterisation: Closed-form x² ellipsoidal bounds in probe coordinates yield tight per-plane 2D) boxes, coupled with two-phase, load-halanced CUDA pipeline that rejects non- intersecting Gaussians and confines work to those boxes.
Beer-Lambert term models acoustic shadowing when probe geom- etry is available, adding realism at negligible cost.
Lightweight attenuation:
Experiments on clinical fetal datasets and freehand cinesweeps show high- fidelity 3D reconstructions (strong SSIM) within minutes single GPU, with clinician-perceivell realism comparable to native 3D scans.
Clinical validation:
RELATED WORK
2D-to-3D reconstruction in medical imaging. Classical pipelines segment 2D slices, register them (e.g. using B-splines), and interpolate a 3D volume (Sarmah et al. 2023). These approaches require substantial manual intervention. struggle with irregular sampling. and scale poorly to high-resolution data. Beuuttre-ffrm--oent
(SfM) methods (c.g. COLMAP (Schönberger & Frahm! 2016)) trian- gulate depth from photometrically consistent views, but ultrasound violates these assumptions and precise multi-view registration is often impractical.
Learned volumetric representations. In US imaging. three requirements shape learned volumetric methods: coping with irregular plane sampling, avoiding external probe tracking, and adopting a rendering model aligned with plane-based echo acquisition (Fig. [T] rather than camera projection (Fig. Db-b). Yeung et al. (2024) (ImplicitVol) reconstructs volumes from routine sweeps without IMU/vision tracking, and Eid et al. (2025) (RapidVol) introduces tensor-factorised hybrid that accelerates training. These advances improve data efficiency and robustness, yet the underlying rendering remains tied point/voxel sampling using neural networks.
Uttrasound-specffee forward models. Physics-basee renderers estimate position-dependent acous- tic parameters (e.g. attenuation, scattering) and synthesise images along wave paths (Wysocki al,, 2023 Guo et al. 2024b) (Fig. [Te.i)). These models can be accurate with known probe poses and source geometry, but they necessitate precise acquisition; curvilinear probes (common in obstetrics) further require cone-to-Cartesian transforms, increasing complexity and sensitivity to error.
GS in medical imaging. GS replaces implicit fields with explicit 3D Gaussians and differen- tiable rasterisation, enabling fast, high-quality rendering. Medical adaptations have largely targeted projection- modalities:
endoscopy (RGB), MRI, and CT (Guo et al.) 2024a| Liu et al. 2024]
Xie et al. 2024 Bonilla et al. 2024 Peng et al. 2025 Zha et al.) 2024 Cai al. 2025 Nikolakakis
et al. 2024). These, therefore, retain camera- projection and depth compositing (Fig, [p). Zha
al. (2024) introduce CUDA-based Gaussian voxel former, but only regulariser at small
spatial resolutions, and plane sampling from voxels reintroduces interpolation artefacts.
Ultrasound-adaptee GS. As far know, ditrasound-adaplee GS method exists for 2D- to-3D reconstruction. UltraGauss extends GS to volumetric US by replacing projection based ren- dering with prohe-plane intersection rendering (Fig. [e.iii): efficient approximation to the US
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image formation model that aligns with plane-based echo sampling and attenuation. The continuous moo-vooeelised) Gaussian representation supports arbitrary resolution and orientation slicing without external probe tracking, yielding fast, memorry-eficiee and clinically relevant reconstructions.
3 BACKGROUND
For cameras in the visible spectrum, one can render the color CRGB for given pixel by evaluating volumetric model along the corresponding ray, sampling the model's opacities ((x) and colors e(x)) at m ordered sample points z on the ray:
i-1 CRGB = (T(())))())) T, II - à (xx)), k=1
(1)
where T denotes the accumulated transmittance of the material. The main purpose of the accumu- lated transmittance is to model occlusions (see Fig. T points at the back are blocked by those in front, so given a lower T, /.iimportancc). In the case of GS (Kerbl et al. (202)), the volumetric model a weighted sum of n Gaussian functions with means Hi, covariances Y, colors G. and coefficients (maximum opacities) ag. In contrast NeRFs (Mildenhall et al.] 2020), the sampled points Tj can now be reduced to the projections of close Gaussians onto the image plane, which can be made more efficient. The opacity of the :th Gaussian at a 2D (image-space) point æ is then calculated
di (1) ai exp (-1 (5 2D= ( (2 (x~14²) (E
2D squared Mahalanobis distance
with the mean 20 and covariance >? projected to the 2D image. To achieve this, the 3D parame- ters (ng 20,) are translated and rotated to the camera's reference frame by view transformation (ex- trinsic camera matrix) W, and projected to 2D with the affine approximation of projective transfor- mation (using intrinsic camera matrix K) (Zwicker al., (001). We can express this formally with operators to convert to and from homogenous coordinates, fi(u)= = [ug....,up,1 1]" for RP, and h
(wt) h='((')
[u' /u//u//4 //1.......... /'b/ub- ]" for u' E R+1 respectively:
14 = proj (14) = h- (KWh(H4)
(3)
MND J.WE.WT JT, J, @proj(us)
Ops
One can then combine the opacities and colors all close Gaussians to obtain the rendered opacity à (x:) and color è (zx) for in Eq.
2 [ (Ex (1)
à (a) Σ (x), ((x) à (x)
(5)
In this summary we leave out spherical harmonics, which support directionally-dependen colors (Ramamoorthi 2006 Yu al. 2021).
3.1
OPTIMIZATION
While Eqs. 5 can render a Gaussian model, practical use requires optimizations avoid costly nested iterations. Namely, Gaussians are tiled and depth-sorted, then distributed across GPU threads for rasterization onto pixels (Kerbl al. 2023). Depth sorting and cut-off distances for Gaussians in the image plane make rendering only approximate (Huang et al. (2024). Heuristics are necessary to remove/resample Gaussians in overly sparse/dense regions (Rota Bulô et al. 2025 Yu al. 2023).
METHOD
4.1 IMAGE FORMATION MODEL ULTRASOUND VS. VISUAL SPECTRUM
While fast splatting is well-developed for RGB cameras, for ultrasound it requires rethinking many design choices due to differences image formation. US probes use reflected ultrasound waves to measure the response of materials at dense range of depths from the probe (Fig. [[]). Image
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cameras, in contrast, measure light that typically reflects off solid objects, and thus are more affected by occlusions and most often measure surfaces (Fig. b)... Accumulated transmittance Tÿ in Eg is therefore unsuitable, as it would treat opaque volumes occlusions (see T(x) subplot in Fig.
Instead, the key mechanism for ultrasound is detecting intersections with the probe plane (Fig.) projections, while accumulated transmittance plays only secondary role (discussed in Sec. 4.5)
We do this by first opting to include an additional uniform component (background) with color CBG and coefficient ong, which improves numerical stability by avoiding division by zero. By reusing the definition of (x) from Eq. 5 combine the gaussian and background components at 2D point (pixel) z, rendering equation simply becomes:
à (x) = Σ de (x) + ABG
(6)
n
CUltrasound (x) = - ā (x)
Σ ai (x) &BGCBG
(7)
A large conceptual difference from rendering the visual spectrum is that, instead of projecting the Gaussian parameters to 2D, we need them to intersect, or touch, the probe plane in 3D. The opacity à; (z) of the 5th Gaussian 2D point (necessary for Eqs. 6 7 is then evaluated Mahalanobis distance in 3D space, by lifting the 2D image point to 3D in the coordinate- frame of the probe:
â (x) = a; exp (-2 (Il 3D) T (2
(m - -4,
squared Mahulanobis
where æio = [x1, x2, 0j" and the Gaussian's parameters are moved to the probe's coordinate-frame using its inverse transform matrix W:
3D = h- (Wh())), ΣN WE.T".
Contrast Eq. 2 to Eq. 8 despite the similarities, the former (for RGB images) evaluates 2D Gaus- sian after projecting it to image-space, while the later evaluates 3D Gaussian by doing the reverse operation (for ultrasound images). Using this model, we can now design fast splatting strategy avoid the computational expense of simply evaluating Eqs. 6 and Eq. (which would result in nested iterations over all pixels and all Gaussians).
4.2 TRIANGULAR COVARIANCE PARAMETERISATION FOR EFFICIENT INVERSION AND GAUSSIAN SAMPLING
One challenge in optimizing representations with covariance matrices Σ (omitting the subscript for conciseness) is that they must remain positive-definite (PD, all eigenvalues strictly positive), while gradient-based optimization methods typically only support unconstrained optimization. Kerbl et al. (2023) achieved this by reparameterising the covariances as a product of scaling vector s and a -derivee rotation matrix R, as Σ = R diag (s²) R² When a Gaussian is projected onto 2D plane, its covariance becomes 2D, which easily invertible for use in Eq. 2 However, we found that for our setting the inversion of 3DD (as opposed to Σ²D) when it is formed by quaternion (which to represent 3D rotation must first be normalized), resulted numerical instabilities. Being
3 x 3 matrix, inversion also takes longer. Hence, we propose to learn (MDD)= directly. We ensure is PD by parameterising it as a product of a matrix M with itself (which is saative-ssmidefinite)) and adding small multiple of the identity I to ensure positive eigenvalues (with 3 > O):
M1
M12
M13
M'' = MM- + BI, =
Mi2
M22
Mza
(10)
[M13
M23
Msa
Note that M itself is symmetric, to achieve the minimal number of degrees of freedom 3D covariance (6). No normalization is needed. A remaining challenge is that, in addition requiring the inverse covariance E-1 frequently to render pixels (Eq.8) we also occasionally (e.g. every 100 iterations) must perform heuristic resampling of some Gaussians. This operation requires inverting Σ-1 explicitly to obtain the original covariance Σ, as well as factorizing it draw a sample from the multivariate Gaussian (Sec. 3.1). Both operations can be numerically unstable for ill-conditioned
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Non con-inteeseeeinn
ZA
Rendered US absoorpion
Warped image of US absorption
1. Warp
Intersecting
Warp grid
bg
MyD x
Probe plane (XY)
Probe
Unwarp
b
mm
2. Cumulative sum absorptions
1:22
1:22
Probe
Pixels XXXXXXX
MIN
Figure 3: Approximating acoustic shadow- ing via warp integrate-urwert Left: ren- dered ultrasound absorption in image space. Right: the same field warped to beam- aligned grid. (1) Warp to beam space; (2) take cumulative sum along the beam di- rection (path integral of absorption); (3) un- warp back to image space obtain the tenuation map.
Figure Side view of the Gaussian bounding- box/plane intersection used to derive 2D rasterisa- tion bounds (Sec. 4.3). Boxes that do not intersect the probe plane rejected early (Sec. 4.4 inter- secting boxes are rasterized only over their 2D inter- section window (Sec. 4.3). Pixels not iterated over are marked with x. The probe plane (XY) points into the page.
E-1. Therefore, we propose instead a more efficient parameterisation, as a product of a lower- triangular matrix L:
Li +
0
Σ- LL", L=
L12
L + 3
(11)
L13
L23
L
+
Eq. [1 guarantees that Σ-1 PD, since triangular matrix's 's eigenvalues are its diagonal elements, and it be that these strictly positive (with 3 > 0). Moreover, lower-triangular matrix extremely efficient to invert via forward substitution, which implemented in most numerical packages. This allows easily computing Σ (I.-1)² L-1 Finally, another advantage is that this factorization allows sampling from a Gaussian (for the resampling heuristic, Sec. 3.T), by simply projecting a standard normal sample z with the (efficiently) inverted L:
μ = 2, ~00
(12)
Eq. 11 supports efficient calculation of Σー1, and thus opacity å; (x:) (Eq. 8), 1.40x faster than had used Σ = R diag (s2) RT. Similarly Σ and the resampling heuristics computed 1.25x faster (see Appendix A for more details). Yet for most Pixel- Gaussian pairs, opacity is near 0 and so can be ignored. Sec. 4.3 discusses this.
4.3 RASTERIZATION BOUNDARIES
To avoid evaluating Eq. when it is known to yield opacities close to 0, we compute it only inside bounding box for each Gaussian. The bounding box is defined around the ellipsoid that encompasses p% of the Gaussian's probability density. This ellipsoid can be expressed squared Mahalanobis distance from Eq. [8] and the chi-squared distribution value for 3 degrees of freedom (omitting the subscript for clarity):
function of the 3D
(11o-1127) ) (ΣaD) (z 2D) S X3.1-p'
-
(13)
which evaluates to X X3.1-p 7.815 for 95% (note that it is a bound on squared distance; see Appendix G.3 for ablation on p value). To calculate this ellipsoid's bounding box, its bounds are 3D vectors Jumi and jmax,
mmin / 3D + λe,
X3.1-p
(14)
det
(E
Σ. 3DD53D
3DD
2
12
Σ
-EED
2
Σ1 22
1212
using Dijk
YBD to index the (j. th element of matrix. Eq.
14
can be obtained by writing Eq.
13 as
function of each element of zij 3D Fig.
iillustrates these bounds.
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4.4 LOAD BALANCING ACROSS RENDERING THREADS
Note that the bounding box (Eq. 14) can be partitioned into 2 components:
1. The 2D bounding box in the probe plane (first two elements of bmin max i.e. b max
2. The ID segment orthogonal the probe plane (the third element, min max
This suggests two-phase process for efficient rendering:
Reject any Gaussians whose cut-off boundaries do not intersect with the probe plane: bysin
or b < 0.
2. For each accepted Gaussian, only iterate over 2D pixels æ inside the bounding box of the plane: 8 S 11 < 1/pax and Ugin 322 S Umaxx
This process shown in Fig. 2 and naturally balances the load across parallel GPU threads. Phase I requires iterating through all the Gaussians (but not the pixels), marking them as accepted jected based on the perpendicular distance, which can be equally partitioned between all the threads. Standard buffer compaction (Corp. (2200) can then reduce this list of Gaussians to only the accepted ones. Phase 2 then requires iterating through the compacted list of only accepted Gaussians, and rasterizing each one only onto the corresponding bounding box of the image buffer, by atomically adding to pixel accumulators for color è (I) and opacity á (I:) (implementing Eqs. 6 7 The com- pacted list in Phase 2 is again equally partitioned among the threads, ensure optimal throughput.
4.5 SHADOW MODELLING
While other NeRF-based methods model more complex ultrasound physics (Guo et al.) 2024b Wysocki et al. 2023), we found that simple and fast approximation can account for the attenu- ation of ultrasound waves they traverse from the probe to a point æ; producing shadows along the way. Based on the Beer-Lambert Law, we calculate an intensity reduction factor I for a pixel located at position x:
T (x: = expli-o a(s)ds Rs exp Σj=0 6gbb
(15)
where di is the opacity of pixel as computed in Eq. 6 and 8, is the distance between successive dyy's, 's, which in our case is simply the pixel spacing (= (/lmageHeighh) This can be efficiently com- puted for all pixels by using a cumulative sum (cumsum) along the row dimension, after warping (bilinearly interpolating) the ultrasound cone into a square image (illustrated in Fig. 33. We then multiply CUltrasound (z) from Eq. by T (z) to give the updated image pixel color.
4.6 OPTIMIZATION
We train UltraGauss end-to-end by backpropagation with the Adam optimizer, optimizing per- Gaussian parameters {us, Lis G. oi). From our triangular parameterisation, precision and covari- ance are linked by
A = = = +el (cf. Eq.
so we optimize L; (lower triangular) directly and recover E, A as needed. Initialisation. Unlike camera GS, we do not use COLMAP initialisation (which yields surface-only points). Instead, we sample μ4 uniformly within the acquisition volume, and intialise & 0.5 and a; = 0.731 (constrained to [0, 1) via a sigmoid). For the precision factor, we draw entries Lij ~ U(4, 5); under our normalisation this corresponds to small initial marginal variances for E; (approximately (1.7-4.3) 10-3), A small jitter ensures A
Training details. Sparsification and densification heuristics follow (Kerbl et al. 2023 Ye et al. 2024) with small adaptations for plane-intersection rasterisation. We evaluate N E [100k, 200k, 2M} Gaussians; our CUDA kernels sustain N with minute-level reconstructions (fewer Gaussians trade a little accuracy for speed). Learning rates 0.05 for all parameters ex- cept H4. which uses an exponentially decaying schedule starting at 1.6x10 -4 (Yu et al. 2021 Kerbl et al. 2023). All experiments
single NVIDIA RTX-A4000. Various ablations can be found in Appendix G Our code (Py (PyTorch and custom CUDA kemels for forward process and gradients) is available at:

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5 EXPERIMENTS
5.1 DATASETS
Two clinical datasets were curated to validate UltraGauss in different ultrasound acquisition settings. Dataset A consists of volumetric scans, allowing assessment of reconstruction fidelity across multi- ple orthogonal views. Dataset B comprises freehand US video sequences, evaluating reconstruction quality in the absence of full 3D coverage, mimicking real-world fetal monitoring scenarios.
Dataset A 3D Ultrasound Volumes: This dataset includes twelve 3D fetal brain ultrasound vol- umes (160 x 160 x 160 voxels. 0.6 0.6 x 0.6 mm3 resolution), obtained from the INTERGROWTH.
21 study (Papageorghiou et al. 2018), 8). Acquisitions were performed between 14 and 26 gestational weeks: spanning critical period of brain maturation (Namburete al. 2015 2023), and the stan- dard time for fetal anomaly screening (Salomon et al., (022). The scans were collected using a Philips HD9 curvilinear probe (2.5 MHz wave frequency) by multiple sonographers, which intro- duces variability in probe positioning and image appearance.
Dataset B 2D Freehand Video Sequences: Three freehand 2D ultrasound videos of fetal brain acquisitions were collected at 19 and 20 weeks' gestational age at Leiden University Medical Center using a GE Voluson E10 ultrasound scanner. Each video consists of ~ 1000 frames, with each frame cropped and resized 160 160 pixels, and resampled a resolution of 0.6 x 0.6 mm? Fig. D1 (Appendix) illustrates how UltraGauss uses Datasets A and B.
5.2 EVALUATION OF RECONSTRUCTION QUALITY AND SPEED
To assess maximum achievable reconstruction quality, we provide UltraGauss with oracle cover- age: 160 evenly spaced axial slices (sampled from a 3D Scan in Dataset A). This exposes the full 160x160xx 160 volume. We evaluate reconstructions by rendering 160 slices in each of the axial, coronal, and sagittal cross-sections, comparing against the native slices using SSIM (t) (Wang et al.) 2004), PSNR (f) and LPIPS (L) (Zhang et al. 2018). As reconstruction time is key consideration for clinical adoption, we report metrics at f = 5 and = 20 minutes, and at convergence (t = o0).
Limited and perturbed inputs. We repeat the evaluation with only 50% coverage (80 evenly spaced axial slices), and add small, random rotations about the x and y axes (9 (0~Wu((5)))) +59)) to mimic the sonographer's 's hand motion and fetal movement encountered in practice.
Baselines and protocol. We benchmark against three SOTA ultrasound reconstruction models: UltraNerf (Wysocki et al. (2023) (physics-informed ray tracing); Rapid (Eid et al., 2025) (hy- brid implicit-explicit) and Implicit (Yeung et al., (024) (fully implicit, NekF-like). To satisfy UltraNerf's parallel-ray assumption, curvilinear inputs are converted from polar to Cartesian.
5.3 CLINICIANS' SURVEY OF RECONSTRUCTION QUALITY
We assess perceived clinical realism and fidelity of UltraGauss' reconstructions a two-part reader study with expert sonographes. All survey material was de-identified.
Study A: Pairwise realism preference. Participants compared natively acquired 3D ultrasound images with reconstructions from RapidVol, ImplicitVol, and UltraGauss variants. We evaluated four routinely assessed fetal brain planes at the 20-week anomaly scan: mid-axial, transthalamic, transventricular, and mid-coroni (Salomon et al. 2022). Assessments spanned five models (Implic- iol, RapidVol, UltraGauss- Mocuuss-100Ke 100K, UltraGauss- 300K, UltraGauss-2MM
four fetal scans, and three training budgets t {5, 20, oo} minutes. Each participant viewed 10 randomized image pairs (or- der of questions and left/right placement shuffled) and selected the most realistic scan, or indicated "no preference" if indistinguishable. Respondents reported speciality training, years in practice, and confidence in fetal brain assessment.
Study B: Temporal fidelity ("Turing"") test. To examine convergence over time, a second survey presented nine UltraGauss reconstructions of a mid-coronal plane generated at = (0.5, 1, 2, 3, 4, 5, 10, 15, 20} minutes, alongside two identical ground-truth images. Inputs com- prised 80 axial slices, covering only 50% of the 3D volume and orthogonal to the target plane. Participants labelled images either real ultrasound scan an "Al recosttuucttin" We randomly included duplicated ground-truth images as a control, to quantify variability in expert judgements. The survey given clinicians can be seen in Appendix C.1
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1.0
0.9
Mode's
0.6
Implettall
0.7
Ulraaeed
65M
UltraGauss 100K
Ultra UltraaGuss- 300K
Ultra
US Scans and used the ming set
5-5
1-5
20
1-5
t-
C-
Figure 4: Reconstruction results for different models (ours in bold) on different training datasets. Shown 3 time points the duration of reconstruction. Higher SSIM is better. Errors bars show #1 standard deviation amongst fetuses.
5.4 ULTRAGAUSS AN END-TO-END CLINICAL PIPELINE
We evaluate UltraGauss in practical point-of-care workflow: a clinician acquires freehand video (cinesweep) with a standard sensorless 2D probe, and then reconstructs full 3D volume for retro- spective multiplanar review. This enables imputation of planes that missed during scanning. Pipeline. Each video frame first receives 6-DoF pose via ultrasound pose-estiee eese-estimation model (e.g.. Ramesh et al. (2024); Di Vece et al. (2024); Yeung et al. (2022). Frames and predicted poses are then fed into suitable 3D reconstruction model (i.e., UltraGauss), or to baseline (RapidVol, ImplicitVol, UltraNerf).
adopt frame hold-out cross-validation scheme: for each video, we randomly partition the video frames into 80% training and 20% testing. After reconstruction, poses and compare the syntheses against the corresponding acquired frames. The residual captures aggregate error from both pose estimation and reconstruction inaccuracies. We run this end-to-end pipeline on all three fetal cinesweeps in Dataset B.
Evaluation protocol. Ground-truth 3D for these fetal cinesweeps is unavailable,
render the held-out slices their predicted
5.5 SHADOW MODELLING
We compare UltraGauss with the shadow modelling feature enabled against UltraNerf (Wysocki al. 2023), using their publicly available Synthetic Liver linear-probe dataset.
Generated or Real Scan?
100
Percentage who thought UltraGauss' reconstruction was a real US scan (f)
80
60
0.0
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Time spent reconstructing (mins)
Figure Clinicians' survey results, asking whether an image is real "AI Generated" (i.e. UltraGauss UltraGauss-eeneratee])
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6 RESULTS
6.1 QUALITY AND SPEED EVALUATION
Summary. We evaluate on three test sets of increasing difficulty (Fig. 4 more in Fig. B Across all datasets and training budgets E {5, 20, 0o), UltraGauss outperforms RapidVol, Implicit Vol and
UltraNerf. It also has high anatomical accuracy and is not organ-specific (Appendices E and F
Accuracy VS. time. For near-real-time use (t=5 min), UltraGauss exceeds the best baseline by 2 0.20 SSIM. At convergence, UltraGauss-2M attains a mean SSIM of 0.995. Variability across gestational ages is markedly lower: at least 10x smaller variance than all baselines.
Capacity-time trade-offs. Model capacity interacts with the time budget: (i) at t=5 min, 1000 Gaussians perform best; (ii) for long runs (Ohours)), 2M Gaussians yield the highest final accuracy; (iii) 3000 Gaussians offer strong balance across budgets.
6.2 CLINICIANS' SURVEY
Participants. We invited 12 expert sonographers specializing in fetal, pediatrie, and general ultra- sound imaging, from hospitals in four countries (UK, Ghana, Denmark, and the Netherlands). Ten experts responded (consultant fetal surgeons and senior sonographers), with mean of 18 years' experience (range: 7-30 years).
Under matched short training budgets (t S 20 minutes), all clinicians preferred UltraGauss over RapidVol and Implicit At convergence (!=x)), no clin- ician rated UltraGauss worse than the alternatives (Appendix C.2 Within UltraGauss, preference increased with capacity (2M > 300K > 100K), indicating that larger Gaussian sets yield more real- istic reconstructions.
Comparison of reconstruction methods:
Temporal realism. In the progressive ("Turing") test, where the rendered novel-view was orthogo- nal to the input data, 70% of clinicians selected UltraGauss more realistic than the ground-truth after just 4 minutes of training, rising to 80% after 15 minutes (Fig. 5 examples in Fig. B2), As a control for judgement variability, two identical ground-truth images were included; 40% of partici- pants labelled one of these as AI, underscoring the inherent variability in expert assessments.
Table 1: Quantitative results of reconstruction performance on cinesweep videos. The SSIM scores shown are the average across all the frames held-out for testing/used in training, at t = cc. Best scores are highlighted in bold. indicates higher better.
Model
Video
Video 2 Video 3
Avg.
Std.
ImplicitVol
0.674
0.797
0.772
0.747
0.065
RapidVol
0.745
0.799
0.760
0.768
0.028
Test SSIM f
UltraNerf
0.446
0.626
0.521
0.531
0.091
UltraGauss
0.928
0.905
0.910
0,914
0.012
ImplicitVol
0.914
0.899
0.893
0,902
0.010
RapidVol
0.871
0.884
0.880
0.878
0.007
Train SSIM
UltraNerf
0.489
0.591
0.598
0.559
0,061
UltraGauss
0.959
0.957
0.939
0.952
0.011
G.T.
UltraGauss
UltraNerf
Diff. UGauss
Diff. UNerf
1.0
Absolute Difference
0.75
0.5
0.25
Figure 6: UltraGauss and UltraNerf being used to predict one of the withheld US video frames at its estimated pose. The pixel-wise absolute difference between each prediction and the ground truth image is also shown in the perceptually uniform i ferno colored heatmap (alongside its color bar). UltraGauss produces reconstruction that is nearly identical to the Ground Truth (G.T.) Image, as reflected by the near-black absolute difference image ('Diff. UGauss").

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Published as a conference paper at ICLR 2026
UltraGauss Shadow w/o Densification
UltraGauss w/o Shadow
UltraGauss w/ Shadow
Ground Truth
UltraNerf
w/ Densification
w/ Densification
b)
Figure 7: Test-set images rendered with UltraGauss (toggling Gaussian densification or shadow modelling), and UltraNerf. Without densification, speckle is lacking (a). Without shadow modelling, acoustic shadows can abruptly stop (b), bulge out and become curved instead of straight (c).
6.3 END-TO-END PIPELINE RESULTS
We evaluate end-to-end by rendering held-out frames their predicted poses and comparing against the acquired images (Tab. D examples in Fig. 6 temporal curves and full visualizations in Ap- pendix D.2). Across videos and training budgets (t E (5, 20, 20}), UltraGauss achieves the highest SSIM/PSNR and lowest LPIPS, with especially large gains at short budgets which is relevant for time-critical scans and brief consultations. Averaged over all held-out frames, UltraGauss reaches SSIM 0.91. Note that these scores reflect the combined error from both pose estimation and recon- struction.
6.4 SIIDDOW MODELLING
Enabling shadow modelling on UltraGauss improves SSIM by 0.005, PSNR by 0.2 dB and LPIPS by 0.004, with practically time penalty. Whilst the metric gains minimal, it can be seen in Fig. 7 that it leads to straighter, and more realistic acoustic shadows. More are shown in Fig. G2 still lack some of the speckle arising from multiple-scattering which UltraNerf is able to capture, thanks to their inclusion of scattering density and intensity parameters. However, when the ground truth poses are not known due to acquisition from sensorless 2D probes (Sec. 5.4). or if the ultrasound rays are not perfectly straight (e.g. due to using a curvilinear probe and then transforming the image from polar to Cartesian space, in Fig. 3}. then UltraGauss can yield more robust results, since it is less dependent on calibrated physics model. This is evident by UltraGauss also much quicker 6.94x faster than UltraNerf.
Despite these improvements,
results Secs. 6.1 6.3
CONCLUSION
We introduced UltraGauss, an ultrasound-seecffe Gaussian Splatting framework that serves as an efficient approximation to acoustic image formation via probe-plane intersection rendering. We derived closed-form x2 bounds for inter-intersectoon rasterisation and two-phase, load- balanced CUDA pipeline, and proposed a triangular inverse-covariance (precision) parameterisation that sta- bilises optimization scale. On clinical datasets, UltraGauss delivers minute-level 2D - 3D recon- structions with high fidelity, and expert sonographers consistently prefer its realism to prior methods. In end-to-end cinesweep workflow, it enables retrospective multiplanar review from routine 2D acquisitions without additional hardware. By aligning the rendering model with ultrasound physics while retaining the efficiency of Gaussian splatting, UltraGauss provides practical route to stan- dardised volumetry and more accessible 3D ultrasound, particularly valuable across diverse clinical settings. Future work includes joint resetrrccntttuutionn optimization, richer acoustic effects beyond Beer-Lambert attenuation, and broader evaluation across anatomies and scanners.
ACKNOWLEDGMENTS
M.E. is kindly supported by the EPSRC CDT in AIMS (EP/S024050/1) and AWS; A.N. by the Bill & Melinda Gates Foundation; J.H. by the Royal Academy Engineering (RF/201819/188/11
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A DETAILED ANALYSIS OF OUR COVARIANCE PARAMETERIZATION
A.1 SPEED IMPROVEMENT
The number of FLOPs (Floating Point Operations) to compute (ΣSD) which is required every iteration, and E3D which is only required every N Nemmiiicattinn iterations, can be calculated theoreti-
cally (Hunger 2007) as follows:
EXISTING PARAMETERIZATIONS APPLIED TO 3D
1. To form the Rotation matrix R from the 4 quaternions {9r: 9t. 9z- RN (where N is the Number of Gaussians), we apply the standard quaternion to rotation matrix conversion and see that it requires 30N FLOPs. Forming the diagonal Scaling matrix S from the 3 scaling values (8x, Sy: 82) E RN requires O FLOPs.
2. Next, to form yED from 30 RTTSSSSS SR, we are first required to form A SR. Since S is diagonal, this only requires 3 x 3 = 9 FLOPs. We now have Σ = A and can make use of the symmetric nature of the product's 's result, one of the off- diagonall triangles. This means for 3 3 matrix, 30 FLOPs are required to compute this. Thus in total 9 30 39N FLOPs for this step.
only have to compute the leading diagonal terms and
Finally, to compute (>>D)= we can exploit the properties of the Covariance matrix, namely that it is symmetric and positive definite, and use Cholesky decomposition to invert it in 39 FLOPs. This means that overall, Σ3D requires 30N + ON 9N 30N 69.V FLOPs, and form 3D)
69N + 39N = 108N FLOPs.
OUR PARAMETERIZATION
To form
L1 +
L
L12
T B
0
L13
L23
T23 8
from our 6 covariance parameters (L11, L12; Lis, Lrz, L2s, Las} E RR- (where is the Number of Gaussians), requires 3N multiplications and 3N additions, 6N FLOPs in total.
Next, form (5 from (E -1 LIIT" as ii a x Lower Triangular Matrix with M = 3, we only require M + M + M = 133 + 1 + 3 = 14 FLOPs. So 14N FLOPs in
total. Therefore every iteration, we require 6NV 14N 20N FLOPs to compute (Σ3D)-1
-1 from
our 6 parameters and use it in in Eq.
Finally, to form 3 from E (L-1)" L-1 we first need invert L. However since L is Lower Diagonal, this can be done very efficiently using forward substitution and only takes 3 + 3 . 133 + 23 = FLOPs. The product of two lower triangular matrices (this time I, rather than L.) is, as before, 14 FLOPs. As such, we need 11N + 14N 25N additional FLOPs to form DBD
Importantly, however, is that during densification we only need dwz, dyy & oaz. and not the full
covariance matrix. This means we can simply take the squared L2-norm of (L-1)² along the column dimension, only requiring 9N FLOPs. When then need sample from the gaussian
distributions, we can also make use of Eq. 12 which again does not require 23D but (L-1)² and that has already been calculated. Theretore, every Nacnsiñeationn iterations we only require 20 additional FLOPs.
The above can be nicely summarised in Al
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Published as a conference paper at ICLR 2026
FLOPs for E-1 (computed every iter)
FLOPs for (computed every
Method
denniffccation iters)
Original 3DGS Formulation (e.g., Kerbl et al. Zha et al.)
108
69.V
UltraGauss (ours)
20/V
+20N in practice*
(+N if Σ is actually needed*) )
Table Al: Comparison of theoretical FLOPs required to compute > and - for different parame- terizations. "For densification, only the variances needed not the full covariance matrix can take the squared L.2 norm of L saving FLOPs.
As can be seen, our formulation is much more efficient than had used the 3D version of 3DGS' formulation. Not only do require 5.4X less FLOPs to compute 2 each iteration, even if we need the full covariance matrix E, our formulation still computes this using 1.73x less FLOPs. In practice, measured speed-ups of 1.4x the forward pass and 1.2x the backward pass when calculating Σ-1 As Σ is required every iteration, this is meaningful speed improvement. When Σ is required for densification every DDensification iterations, 1.25x faster the forward pass and 6.65x faster in the backward pass than had we used the 3DGS formulation (although in practice we only need L-" not the full covariance matrix, speed-up is even quicker).
formulation computes this
16

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Published as a conference paper at ICLR 2026
A.2 NUMERICAL STABILITY
In terms numerical stability, we also performed a numerical sensitivity analysis on the two formulations. Each input parameter was subjected to small perturbation, to simulate numer- ical errors/rounding, and relative Frobenius Norm difference between the error-containing inversc/covariance matrix and the error-free matrix is shown.
Original 3DG5 Formulation Inverse Covariance
UltraGauss Formulation Inverse Covariance
0.35
0.35
Relatve change IF iFobenus Normi
Relatvv change IF (Fobeenss Norm!
0.3
0.3
0.2
0.2
0.15
0.15
0.1
0.02
0.04
0.06
0.08
0.04
0.06
0.08
011
Emor
Erore
Original Formulation
UltraGauss Formulation Covariance
0.4
0.4
Reatve change (Frobemus Norm)
Relaivv change (Fobbensis Norm)
0.3
0.3
0.25
0.25
133
0.15
0.15
0.1
0.1
0.06
Figure A2: Numerical analysis of the effect a small perturbation error in each parameter has the final Covariance or Inverse Covariance Matrix, for the original parameterisation and our proposed one.
evident that our proposed formulation is robust numerical error, when computing both the inverse covariance matrix and covariance matrix from the input parameters. We also found empirically that sometimes, when trying to invert the 3 3 covariance matrix formed by the four quaternions and three scaling parameters (which is never done in classical GS, as only the projected 2 2 covariance matrix is inverted), if any of the eigenvalues became close zero would get numerical instabilities. Our formulation (by adding a small (), ensures all eigenvalues always 2 3 never into this issue.
From Fig.
17

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Published as a conference paper at ICLR 2026
A.3 PHYSICAL MEANING OF Lij TERMS IN EQ.
In order to gain an intuitive understanding of the physical meaning of the components of Eq. 11 the 3 x 3 covariance matrix Σ needs to be algebraically calculated, as only then can the standard deviations and correlation coefficients the z; y, z dimensions extracted and provide us with geometric insight. Ignoring the small value 3 from Eq. [1 for simplicity:
L
0
L =
L12
L22
0
(A1)
L13
L23
L33.
1.11
T.11 T-12
Ti 1.33
Σ- -I- =
L L12
Li2 + L22
Lzs 122 + L12L13
(A2)
1 Lı3
122 + L12L13
[ + L23 + Li33
PayTnn nm yuu
Paz zn 0zz
Σ [--]]-
PzyOxz oz yy
o
Py:T Oxx
(A3)
Pr 2z
Py=0vy 2z
2
By inverting Σ of Eq. A2 and comparing to the standard format of Σ as in Eq A3 we are then able relate how Gaussian size the æ, y, dimensions (governed by the three standard deviations), and shape/rotation (governed by the three correlation coefficients) are affected by the six Lij terms of
Unfortunately, due to the nature our efficient decomposition, and which is optimized to primarily learn E rather than Σ, as can be seen below the six Lij terms are all heavily coupled and it is hard to get physical intuition of what each term is responsible for. Nevertheless, the sensitivity graphs centered around 4 (the initial values of Liy) do provide some insight (see Fig. A3).
is also evident from the equations below that six of a Gaussian's defining parameters are solely function of L, and so L can be thought of the 6 Degree of Freedom matrix directly responsible for defining a 3D Gaussian's 's DoF shape and size. Whilst the individual terms of not interpretable as the original gaussian splatting's ' parameterization, which used parameters 9r, qs qj 9k. S2, Sy: Sz to rotate and scale the gaussian, L as a whole is still responsible for the shape and size of Gaussian, and nothing else. We believe this slight loss in interpretability is overwhelmingly outweighed by the efficiency gains demonstrated (see Appendix A.T).
For choosing initialisation values, it also helpful to note that generally, increasing all six Lij values causes L and subsequently E-1 to be larger. The covariance E, equal to the inverse of Σ Σ-1, will thus be smaller, and the 3D Gaussians will start off smaller.
L Lg 22
Li L2n Lig
L2 L3
T2g + 1
Oyg
L22
1
0ss 12
L [2g + I2a)) Ibg L13
Pzy
Pyx
VD Lig L3g + L72L20
22 2 Li Lig Li L20
L1 + 1.22
Paz
Pux
V Li + L3
2L L Lo3 + Lig + Lig L3
L23
Pys Pay / D2g + L
18

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Published as a conference paper at ICLR 2026
beaetteity Exprite: Aqp Aip. Agp Agp far Pay
1850
0/835
0.00
0.00
00654
18600
00833
6.60
3.30
3.34
010855
402
4.5
22
Viried by Volne (ceabee
Figure A3: Sensitivity table showing the effect each Lij term in Eq. 11 has on a Gaussian's 's defining parameters it is perturbed and all other Lij values are held constant. Perturbations are shown here centered about as that is the initialisation value of all Lays, however during the optimization process each Tig term is free take any real value.
19

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Published as a conference paper at ICLR 2026
B EXPERIMENT 5.2 (3D SCANS)- ADDITIONAL RESULTS
+0,000
0000
+0.024
0.0
005
0
005
0,09
4 043
000
001920
0012
+0.00
00033
#0.094
0:30:0000)
00716-0000
000
+ 0..00
025
0.53-6000 00.0
0,0
40 005
0.00
-0.025
0.255 0.119
0000
005
0,310
033
0.00
+ 009
00.08
0.025
10 ONS
025
00.77
=0 ON5
0,001 = 0.000
40.016
922
0.080 mos
- 0.07
+ 8.005
LPWP (4)
0.12
0,310
0,0020
0.033
0.42
Avergg Accoss all 3wews
0,202
0.52
0.43
0,25
0.00
0228
0,238
0432
0,088
0.278
0021
0,111
0,005
0,53
0.499
0,227
0,04
0,281
0021=
0.048
0222
0,029
0,12
0:210
0,618
0,471
0
0.554
0,240
0,422
0,008
0,089
0.035
0,207
0
0
0.08
PEERR [e)
2%
23.10-:3000
15.00 +1.47
+:.45
16:08
1.52
40.00
60,06
280-2 2238
25.05
4,28
17:44
44.01-2.00
45:19
40.06
-0.86-1.744
1.74
29,05
5.05
42.02
23.18
22.08
+2.99
50.05
1.00
15.05
11448
15,25
22.7
39
38,00
2247
06.00
39.00
-2.3
2
17.00-1.37
40.73
=2.33
36.00
2.05
25,55 +2.38
18:37=1.00
06
40.27 8.18
45.01
77.00
1.51
16.08 1
15.98
0.332
31.97
1.11
37:76
37.00
-2.12
30,25
2.55
4.00
25,34
- 1.07
15.00
30.88:0.88
60.02
30.95
+2.12
57,00
-1.41-2.56
31.41
-2.5
33.40
17:01
#1.35
00.85
30.54
1.05
36.71
0.97
S31M (+)
0.008
0.702
0.20-
0.052
0.389
- 0.088
0.094
0.800 0.004
5.00
07758 0.090
6.077 0.786 0.038
0.595 G.013 D H 0.047
0.000 0.00
0.000
0.001 = 5.000
0.585
- 0.00
0.773
0.577
0.008
0.988-
0.00
0.500
6.001
0.585
0.03
0,500
0.653+ 0.085 02252
0.00)
0.5AA 0.0RL
C 007 0.572 0.077
0.970
0.000
(177=
0.0177
0
0.730 0.58
D425= 0.088
= 0.000 0:755 0.00
8.004
D.570
0.04
0.01
0.909
0.470=
= 0.00)
0.081 0.010 0.961
0.570
0.007
0.905: - 0.008
0.200
0.00
0 0.081
0.04
0.00S D.500 0.008
6.007
0.80
0.00
0.543
0.0%
0.065 0.503- 0.098
0,027 0.31 0.0
0.821 0:0
0.00
0.583
0.964
6.007
0.038
08L1=00
0440=
0.033
0.91
0.580- 0.007
8.004
0.440
LPPPS [1]
0.000
0.055
0.685 c.015
0.007
0005=00.000
0.000
0.00H
0.000
D 0.0
0..03
0.055
+ 0.005
0000=0000
0.000
0.004
O 0.002
0225
6.033
0.000
0 AN
0.037
0.127
0.004 0.928
0.005
0.3033 C 0.00
0,036:00000
0,000
0.74:00
033
0.042 0.007
0 + 0.089
0.050
0.004
0 ove
030
0.571
009
0.001 e 0.005
0.005
0.551 C.032
0.113
O
0.000
0.006
8.005
0:200
0.677
0.685
0.273
0,227
0.025
6.021
0.0
0.245-
0.027
0
0.023
0
0.308
0.564
0.037
0.00
0.531
0.289
0003
0.052
0.058
0.134
0.277
0.615
0,600
0.054
0.057
0,225-
0.675
0.022
0.087
0.085-
0.056
Sagital
PENR (1)
29.78 2,000
16.30
1:74
34.47
0.54
44.20
2275
1.88
42.30
40:11+2.00
2230
20.22
29:6
54.47
0.78
45.113.88
45.51
22.0
40.15
26,70
3.28
3.45
45.82
4.00
45.20
48.63
26,37=2.00
10.25
20.25+1.70
1.79
$4.06
1.51
41.37
40, 2.05
40.28
98.48-2.1
98
-2.1
27.17
24.20
0.88
40.99
42:12
40:28+24::
25.30 2.77
30.81-3.00
14.04:0777
40.30
8.74
3.81
46:89
20030
+2.30
1.81
16.36
24.8
0
31.50
1.06
1.06
30.20
38.35
22.00
+
$2.35
29.70
3.80
1.09
23.74
20:50
00.54
4B 00.78
38.56
2.50
88.72
3.19
3..72-3.11
3,48
24/00
200000
20.55
0.26
0.99
20
00.000
00.62
0.252 0.088
0 000 0.57 00.004
0.24S U62 0.708 +0.051
0.788 0..86
0.13 0.6 0,250 +0.055
0 20,004
102 6.0N0 -0-003
0.98S 00,005
0.577 00.00
0.579 000044
0,550 00.002
00.044
0URIS 10.600
G.417 C 0,0 0 00.71
0.030 & 000 D.BRS =0 0.000
0.8/4 19400.00 10,006
0 1000010.000 1O. 6.075 20
0 00,500
0.721 00177
C. 0.046 0.200 00,049
0.5 0.RS -0.008
0.975 10,005
DUSIE +0,002
OURSS 10,220
1051=0.00 DO 0.5D 00.000
00.004
(. 000
00.622
0.259 0.029
00000
00 QUETT 00000
6,555 005 0.808 00,00
0 C. 000 0.756 -0.058
0,897 O. 002 OUESE -0,041
0.093 AOD 0.800 0:0000
0.057 O. 000 0.882 00.008
0.008 0.997 00,007
0.84E +0,053
0.51 10.0
0.050- C 007 0.000 00008
0.00000.00 0,006 0.945 0.004
(1)
0.574
0228
0,302 002 0277 =
0,034 O 003 0.960 +0
0.504 110400.00
0:24
0.007
0.258
000 0,089
0.555
0.000
0.008 0.577
0.6ED C.OST 0
0.800
0.244
0.000
0.008
2 0.08
0.000
0.01 000
0.000
O 001
0,358 0OS
0.681 0.DND
0.035 0.0
0.343 0..0
0.111
0.363 & 00D
0,03 0,033
1017-0011N
0.440 0,005
C.0ESS A 007
A 0.005
0.919
0
0.101
Test Store 160 Linearly Spocoed
LPPS
0,299
0.651
0-112
0.00
0.251
0.012
0,34S
0.0244
0.015
0.005
0,007
0.935
0.023
0.050
0.137
0.051
0.0E2
0.717
0.043
0.076-0
0.074
C,LAS
0.127
0.
0.054
Caronal
(1)
23:07 2.04
122
19.71
1.48
40.86
30-06:
1.20
30.1+125
23
1.55
3.54
23.6-
43.44
1.78
42.4411
1.30
434411.9
30.31
1.29
23.60
1,22
40.53-241
17:0+1.0
4050
1.79
46.20
1.60
22:80
1,70
15.75
1:27
56.25
1.8
37
2.08
30 2.20
37.30 1.89
2RSS 1.78
23.06
3.50
19
1.12
58
2.10
37.56+
1,82
22.57
1.77
30:00
2.53
17.071138
38-22
1.81
37.71
2,36
40.244
2.17
1.24
15.87+
1.50
17.00
1.22
20
1.01
38.00-
1.50
23.02
1.85
24.57
3.58
19:88
1.39
29:5
0.50
36.00
B0:07-
1.87
31.92+250
17:30
1.04
20.55
20.35- 0.96
29.3 29.1240.78
17
00,02
0.062
0.422 1102200..0 0.049
0.378 O 204
0.909
00047
0,789 0.087
0.42 0.044L
0,807 00,00
0.901 =0.002
0.385 0.003
0.977 0
0020
00000
00002
0.896 0.001
0897
0.648 0.00 0.24D 00.60
0.572 0.087
0.055 - 0.003 0.5T0 0.007
0,977 699750.00
0,650 0.252
0-189
0,44 0,44=00.0 -0.699
0:301 =000
0.975=00000
00,00
9.005
0,03 00019 0,899
2000
0.804
002
0,061 00.000
0.584 0.584-0.00
204
0.061
0.534 0.04
000
0.561-0.00
=0.00
00,02
0.756 00.98
-0.06
0.401=0.00
0.91=00000
0,0500 00000
0.985 0.000
0.842 0.088
0,600
00:000
0.017 n220
0.827 0.008
0.004
0.083
0.974
0,600
0.235
0.382
0.704
0.557
0.088
0.969
0.723
0.875
0,532
0.571
0L780
0,246
0,501
0.964
0.065
0,635
0,401
0,505
0.400
0,807
0.000
0.518 3.027
0.881 0.000
0.000
0.086
0.005
0.053
20001
=00000
0.024 0.003
0.33 3.000
-000
0,011 -0.00
0.000
0009
0,537 0.032
DARP 2000
0.000
0.119
0307= 0.049
0.0 g
0.000
3.004
0217= 0.0
00003
0,005 0.001
0.64D 0.042
0.57 0.034
C.PDD 3.009
0.057 0.0
0.00
+0007
-0000
0004
0285 0.095
0.661 0.008
8.007
(4)
0013
0,227=
0,000 1000000000
0.M 1
0,106
6:003
06:6-
0.13
0.000
0.540
0,25
0.89
0,005
0.02
06411
0.001-
0.600
0:227
0,172
0.21
0,000
0,69
0.37
0.012
0.25
0,23
0.70
0
0.067
0,02
0.0%
0.770
0.004
Axial
(-)
28-44 0.07
1:30
15.08
13:39
2.35
222
40.73
1.75
01.77=105
4.19
23.57
24.67
4,8
1.91
20.6-
45:27=
3222
46.11
2.08
1.85
41.77
33.35
42001
3772
22,78=32788
45.25 3222
50.38 3,50
2.06
64.25
2472+
1.30
15:28
18:49
39.59
2.89
40:00
2:34
1181
20.00
248
28.00
23.88
5.07
2:35
21 11700222
41.45 5,00
20.68
2200
20.33
32-35-3222
32:35
23-42
303
59:50
3:18
59:50
40.98
3.26
44.40
2281
20.00
1-42
15.00
13.02
38.34 1.28
98:50
252
23:47
2.55
4.70
24.54
0.25
18.97
31.82
0.84
52.50
1.25
1.21
56.29
2.52
80-42-
2.9
22.28
2.17
20.00
31,00
52.54- 1.85
38.11
1.22
(P)
0.788 1000--000 00000
0.087
0.8822 0.000
0.000
0.655 0.05
9.067
0.047
0.000
0.986 0.000
17--0000 0.000
0.004
an0
0.008 0.000
0.000
0000
9.085
0.004
0005:00.00 0.000
(UNN 0.00
0.033
0,578 0778-0.0
0.097
U.90 0.00
0.085= 0.003
0.745 0.044
0.014
0:000 0.302
0.974 0.970.00000
0.005
0.99 - 0.002
0J00 O.RR
0.088
0.65- 0.067
0,509 0.007
0.065-008
0.000
0.555-9.2
0.026
0.817 0.006
0.040
0.800 0.000
1.988-0.00
0.00
1564-0..22
0.03
0008
0.684- 0.005
0.004
0.240
0.444
0.880
0.807
0.208
0.SEL
0,891
0.387
0.560
0.882-
0.69:
0,882-
1.600
0,752
0.248
0.428
0,800
0.805
0.760
0.745
0.661
0.584
0.822
0,253-
0.000
0.065-
0.855
0.477
0,588
0.985
0.664
0.825
0,500=
0.509
0.842
Training SSW
0.880 040)
0.240 0.047
C_533 00,38
0.985 00:03
1.500 10 00.002
0.998 0 0,060
0.880 0 00002
0.000 a 0.0011
0.366-- 002002
0.960 0.215
0.982 1 0:004
0.000 00000
0.008 0.000
0.999 10 0.000
0.248 0.055
1.000 0.344
0.950 10 0,002
0.992 08022000000 0.001
0::00: 00:00
0. 0.02205 0.322-00:333
0:315
0.011 00:81
0.004 10,00
0.0088 0.000
0.006 0.000
0.365 0,000
0,906 0,002
0.088 0 107
0..00
0.008 10.08
C. 00:36
0,508 0.082
0.994 00001
0.001
0.950 0,001
0,855 0.361
0:987 0.001
0.950 0.000
0.990 00.01
0.977 00307
0.97 0,005
0,710 0:300
0.000
0.000
1. 1.000 0
0.000 10000-00.00
0.988
0.000
0.807-
0:000
0,752
0.999
0.094
0.6225
0.007
0.998
1,000
UttraGauuss 100K
untovtauss 300K
ultrwGouss 190K
ultroBauuss S00K
Uttroaauss uttoaauus 100k
SO0K
Uttranaauss 100K
300K
UntawGuurs 100K
untraaauss SB0K
uttwwauss- 100K
Uttraaauss 380K
Untestauuss 100K
UttanGurs 300K
untreGeuss- 100K
uttrabaus 360K
UltaGauus 180K
ultrodauus 380K
UttraCaurs 20M
untradauss 2H
Uttradauss 20
UltrwGauss ultraaauss 2MM
uttrotaurs 204
untrassurss 2H
ultraduuss 201
utraaauss -M
Ultrafaurs 2M
Model
Implictilll
Radidoa Uttrakert
in implictee
Rocidval
Uttrakiert
implicttel
Raskevot
Uttraheni
Implicanno
Rastovar
Uttrakerr
Raplosst
Uttrahert
In piiieeeee
Uttrakent
implicnia
Raskoval
Uttraker
Rastoval
Uttralerr
Implicivol
Rapiossl
UttraNee
Convergence
Convengince
Convergince
Timex
Smins
mins
20 mins
Emins
20 mins
Inpur Set
150 Lincrrty
spaced fecal
Sioes
ao sticed
Sices
Bolinarryy spaced Slices with 25
kittur
Figure BI: Reconstruction results for 6 different models on 3 different training sets, shown at 3
time points. 1 indicates higher is better. "UltraGauss N indicates UltraGauss with N starting
Gaussians, Best scores are highlighted in bold.
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Reconstruction Time (mins)
Rendered Image
Absolute Difference Image
Ground Truth
N/A
0.5
t=1
t=2
1=3
t=
t=
t= 10
[ - 15
t= 20
Figure B2: Progression of UltraGauss as 3D reconstruction is formed from 80 Axial Slices (sam- pled from a GW 3D fetal scan). The image requested mid-coronal cross-section. Clinicians were then asked to chose whether each one of these was a real or "AI Generated" sean. The results to this are in Fig. 5
rendered image an orthogonal,
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C EXPERIMENT 5.3 (CLINICIANS' SURVEY)
C.I THE SURVEY
Below is the questionnaire which Clinicians answered:
PRELIMINARY
1. What is your current position?
2. How many years of medical experience do you have?
On a scale of 1 to 5, how would you rate your confidence in assessing Fetal Brains Ultra- sound? (5 is very confident, is not at all)
PART
Participants answered a random selection of 10 pair-wise comparisons. An example of one is shown
below:
Please choose the most realistic ultrasound image out of the two.
o Option
o Option 2
o No Preference
PART II
Participants then answered 10 of the following questions. The order of these 10 questions and the order the two options within each question was randomly shuffled for each participant.
The ultrasound scans below show mid-sagittal views of the same fetal brain. Some are real scans acquired using standard ultrasound equipment, while others are Al-generated reconstructions. For each image, please identify whether the sean is real or Al-generated.
Real ultrasound scan, or Al reconstruction?
Real Al-generated
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C.2
CLINICIANS' SURVEY ADDITIONAL RESULTS
Compas feora Fin Modela each Temepsin!
100
750
400
Purcerlage St
60
4G3
416
Comnergenoe
-25
-75
1680
lepilttes
uttradause 1ook
Rapiifaal
Ultradaus 300
Uttaaaasss -2
Impilaionn
Repidiil
Ultra dauns 200
Ultracauss 100k lltrraGauns B00k
Repittid
Uttrataasss 2M
Ulrraaass
Uftra
Comparison of 5 models at each time point. Actively choosing a model gave it 1 point, not choosing -1, and choosing "no preference" gave it 0 points. Each model's score was then divided by the total number of points it could have gained, to give a percentage. Thus +100% means that all participants actively selected and preferred that model. -100% means that all participants actively disliked this model (and selected the other one). 0% means that the model is neither liked or disliked, as the participant chose the "no preference" option.
Figure C2: Survey results:
Comparisons the Models at each Timepoint
At Time:
20 Convergence
Percentage Score (t)
25
0,0
-28 5
-75
100
100.0
100.0
100.0
.000
Imptiattal
RapetVal
UltraGause
Impliaa
Rapid Vol
UltraGauss
ImplicaVol
RapidVall UltraGauss
Figure C3: Survey results: Comparison of Implicit RapidVol, and the better of the three Ultra- Gauss Models at each time point. The same scoring system applies as Fig.
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D EXPERIMENT 5.4 (US VIDEOS)
D.1 FURTHER EXPERIMENTAL SETUP DETAILS
In this experiment we use ultrasound cinesweep videos acquired using standard sensorless probes. As such, the poses their frames have first be estimated using suitable pose estimation that works with sensorless probes (c.g. (Ramesh et al. 2024 Di Vece et al. 22224] Yeung et al. (0222). ) 20% of the frames are then randomly removed and witheld for testing. The remaining frames (blue frames in Input Video on Fig. D1) and their corresponding poses (blue probes/planes) are then inputted into UltraGauss to form 3D reconstruction of the individual fetal brain within minutes. Images are then sampled from the reconstructed volume at the poses of the withheld video frames (green probes/planes), and are shown green the Output Video tape.
similar method also applies to forming 3D reconstructions from Dataset A. Rather than the blue Input Views being video frames and their predicted poses, they are instead slices sampled from the volumetric scans of Dataset A and their corresponding ground truth poses.
New, reconstructed view
New. reconstructed nonnstructd view
Input Views
New, reconstructed view
Input Video:
Output Video:
Figure D1: Input images of the fetal brain acquired (blue Input Video frames) and their poses estimated (blue probes/planes). UltraGauss then uses these to form a total 3D reconstruction within minutes. few of the 3D Gaussians unseen poses can then be sampled from the reconstructed 3D volume to form a complete cinesweep or be viewed individually. These can be seen in the green Output Video cinefilm frames.
shown as gray ellipses. Cross-sectional views at previously
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D.2 ADDITIONAL RESULTS FROM VIDEOS
V1
V1 12
V2 I1
V2 12
V3 11
V3 12
Ground Truth
ImplicitVol
RapidVol
UltraNerf
UltraGauss
ImplicitVol Diff.
RapidVol
Diff.
UltraNerf Diff.
UltraGauss Diff.
Continued below...
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ImplicitVol
Diff.
Grayscale
Vop
Diff. Grayscale
UltraNerf
Diff.
Grayscale
UltraGauss
Diff.
Grayscale
Figure D2: For each video (Vn), two ground truth images (1j) from the test set shown, alongside the corresponding image generated by ImplicitVol, Rapid UltraNerf and UltraGauss 300K (ours). The absolute difference (diff.) between the predicted and ground truth image is also shown, in both grayscale and the perceptually uniform inferno colored heatmap. The inferno color bar ranging from 0.0 to 1.0 is shown the right-hand side. A pure black diff. image means perfect similarity, and so the darker the diff image the better.
Accuracy of US Video Frames Withheld for Testing
1.0
Vidco UltraGauss
SSIN (1)
Video 2- UltraGauss
Video 3- UltraGauss
Video RapidVol
Video 2 RapidVol
Vidco RapidVol
Video ImpliciVVol
Video 2 ImpliciiVol
Video Implicitool
Video UltraNerf
Video 2- UltraNerf
Video UltraNerf
1000
1500
2500
3000
3500
4000
Epochs
Figure D3: Test accuracy over the course reconstrucion//raiiing for UltraGauss 300K (ours, shown in blue), UltraNerf, RapidVol and ImplicitVol, reported on ultrasound cinesweeps. Higher SSIM is better.
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E ANATOMICAL ACCURACY OF ULTRAGAUSS' 3D RECONSTRUCTIONS
In Figs. E1 and E2 we overlay segmentation masks (produced automatically using a convolutional neural network (Hesse al., 2022, Namburete et al. 2023)) over the UltraGauss reconstructions in hardest of our scenarios using 80 Axial slices with =550 Jitter (to mimic the sonographer's natural hand motion). We also segment the original 3D volume benchmark. We show mid-axial, mid-coronal and mid-sagittal views, at both 20 and 26 Gestational Weeks (no segmentation masks were available for 14 GW). All the UltraGauss reconstructions show high anatomical accuracy.
Segmentation of Reconstructed Volume from 80 Axial Images with 5 Jitter, 20 GW
Mid-Axial
Mid Coronal
Mid-Sagittal
Figure E1: Segmentation mask applied the Original and
volumes of
20 GW fetus. The input to the reconstructions was 80 Axial slices with +5 of rotational jitter in the plane.
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Segmentation of a 3D Reconstructed Volume from 80 Axial Images with #5 Jitter, at 26 GW
Mid-Axtial
Mid-Coronal
Mid-Sagittal
Figure E2: Segmentation mask applied to the Original and UlraaGaus-recoonstrtteed volumes of 26 GW fetus. The input the reconstructions was 80 Axial slices with 5 of rotational jitter in the plane.
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F PRELIMINARY RESULTS ON ULTRASOUND CINESWEEPS OF FOREARMS
To further demonstrate that UltraGauss is not organ-specific and can be applied to general 2D ultrasound cinesweeps, we also validate on the TUS-REC Dataset (Li et al. 2025). This dataset of adult forearm ultrasound cinesweeps was acquired using an Ultrasonix machine equipped with curvlinear probe (4DC7-3/40). A NDI Polaris Vicra optical tracking system was used to provide ground truth poses, and standard ultrasound post-procsssing speckle reduction was not applied. shape trajectory scans and poses were used imput into UltraGauss, and then novel views were sampled from the 3D) reconstruction at the poses of the straight line trajectory scans. Preliminary results can be found in Fig. F1 It is evident that UltraGauss still forms good 3D reconstructions, and that the main structural content and anatomy is modelled accurately. Most of the pixel-wise difference
in Fig. F1 is in extremely small Gaussians. In practice there would not be this much speckle in the ultrasound cinesweeps, as post-procsssing speckle reduction would be applied, and so the reconstructions would be even better. It is only in this dataset the authors intentionally chose not to do this, in order help the participants in their pose-prediction challenge.
with lot of fine speckle and which would require many,
UltsaGau
UltraGauss-300K UltraGauss-1008
Ground Truth
Predicted
Difference
Ground Truth
Predicted
Difference
Figure F1: UltraGauss (2M, 300K, 100K) applied form 3D reconstruction from input cinesweep of a human participant's 's forearm. Two novel views were then sampled and are shown here alongside their ground truth. The pixel-wise absolute difference between the predicted and ground truth images are shown in the perceptually uniform inferno colored heatmap. with the color bar ranging from 0.0 1.0 shown on the right-hand side. A pure black difference image means perfect similarity, and so the darker the difference image the better.
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G ABLATIONS
G.1 CAND A OR ONE COMBINED PARAMETER?
Since ultrasound is grayscale modality, requiring only one color channel, it is possible to combine each Gaussian's 's single-channe color C (which ranges from to and opacity a, (also ranges from O to 1) into a single learnable parameter, say Pt. as for example is done by Zha et al. (2024). We could then multiply this by Gaussian i's probability at point x. This would then be repeated for all the other Gaussians and they can then either be summed together (but stopping if the sum reaches 1), or have the average taken. Since we do not splat Gaussians onto an image plane and sort them by depth from front to back, it is not obvious how to decide which 3D Gaussians to exclude from the sum if it reaches I, so we would have to take the average. Expressed formally this potential approach is: P (x) P exp (-2 [1 14 2 (2
(I m
squared Mahalanobis distance
8(x) > (x(x)
(GI)
7
Instead, propose a formulation in Eq. Ignoring ang and onG for now, we are essentially proposing:
a (z) Σ (z) Σ ā; (2)
(G2)
which mathematically is a weighted average. Compare this with the arithmetic average when one parameter is used. The former is naturally powerful, and Table GT confirms this. In application, intuitively, a dictates the importance each Gaussian, and how strongly its color should feature in the final pixel color ė
Table G1: Using C; and a; vs. only one parameter. Scores are the avg. across all three standard test views, for four 20 week fetuses from Dataset A.
Input Sct 80 Axial Slices
Input Sct 160 Axial Slices
SSIM (?) PSNR (?) LPIPS (U)
SSIM (?) PSNR (?) LPIPS (L)
One Param, 0.931 0.0012 3.42 100 0.026 +0.01 3.66 0.085 0.026
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G.2 THE IMPACT SHADOW MODELLING AND DENSIHICATION
Fig. visibly shows the beneficial impact adaptive Gaussian densification and sparsification, once modified and implemented appropriately, can have. Greater sharpness and speckle can be seen in the top quarter of the images (a) when Densification turned on (c.f. 2nd column with 3rd column). When Shadow Modelling is turned on, view-dependent acoustic shadows become more realistic. Comparing columns 3 and 4, one can see that shadows or dark patches no longer appear where they shouldn't t (b), shadows longer bulge out and curve, but rather become straight & e), and they also no longer abruptly stop (d).
UltraGauss w/o Shadow
UltraGauss w/o Shadow w/ Densification
UltraGauss Shadow w/ Densification
Ground
UltraNerf
Truth
w/o Densification
a)
Figure G2: Test-set images rendered using UltraGauss-30OK with (w/) and without (w/o) shadow modelling or adaptive Gaussian densification & sparsification. A fully ultrasound physics based model, UltraNerf (Wysocki et al. 2023) serves as a baseline.
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G.3 THE EFFECT OF GAUSSIAN'S PROBABILITY DENSITY P VALUE
We expected that as the gaussian probability density threshold value was increased, the size of Gaussian's 8 ellipsoid of influence would increase, and so more pixels would consider it to be "close" Applied all N Gaussians, this would mean more atomic additions and so increase the time taken, but lead to potentially higher accuracy each pixel's final grayscale value is now controlled by more Gaussians. To confirm this we conducted an ablation on UltraGauss-300K applied to of the 3D scans from Dataset A, and varied the p% value (see Eq. (3). Metries are averaged across all test views. Our ablation confirmed our expectations and as such we chose p 95% good compromise between the number of Gaussians accumulated each iteration and reconstruction accuracy (considering all three metrics).
and accumulate
Effect Gaussian Probability Density Threshold
1,000.000
0.974
39.0
0.993
0.970
000,000
0.972
0.969
Number of Gausslars Accumulated (1)
38.
0.906
SSNN
0.907
35.2
0.900
600 200
-0.985
0.065
500,000
0.9964
37.8
SSW
0.063
00
02
98
100
5% value
Figure G3: Ablation study on UltraGauss in which the value of the Gaussians' bounding proba- bility density ellipsoids are increased. Metrics averaged across all 3 test views.
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G.4 THE EFFECT OF POSE ESTIMATION ERROR
To attempt disentangle the enrrr as consequence of having inaccurate poses from the error due to the 2D)to-3D reconstruction process, we conducted the following ablation. For 80 linearly spaced axial slices sampled from the 3D volumes of Dataset A, rather than providing these images along did in Sec. 5.2
with their ground truth poses noise to the poses, to simulate UltraGauss being given inaccurate poses as would be the case when using the cinesweeps in our end-to-end pipeline (Sec. 5.4).
intentionally add some uniform random
Effect of Pose Estimation Error on SSIM (t)
1.0
UltraGauss 100K
0.9
UltraGauss 300K
Avg. Test SSIM (t)
UltraGauss 2M
0.8
0.7
0.6
0.5
0.4
0.5
1.5
2.5
3.5
Error in Pose (=%)
Effect of Pose Estimation Error PSNR (t)
35.0
UltraGauss 100K
UltraGauss 300K
Avg. Test PSNR (t)
UltraGauss 2M
30.0
27.5
25.0
22.5
20.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Error in Pose (#%)
Effect of Pose Estimation Error on Inverted LPIPS (1)
0.00
Avg. Test LPIPS (invorted, 1)
UltraGauss 100K
-0.05
UltraGauss 300K
-0.10
UltraGauss 2M
-0.15
-0.20
-0.25
-0.30
-0.35
-0.40
0.5
1.5
2.5
3.5
Error Pose (#%)
Figure G4: Ablation study on how the different variants of UltraGauss cope as error in the pose of the reconstruction input images increases. LPIPS is shown inverted here, so that like SSIM and PSNR, higher value signifies better accuracy.
From Fig. G4 it can be seen as is typical with reconstruction methods which are overfitted to particular scene and are not pre-trained, they do require fairly accurate registration of images to fixed co-ordinate system. For UltraGauss the error limit is about 1.5% before reconstruction accuracy degrades.
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G.5 THE EFFECT OF SCAN COVERAGE
Here we investigate how well UltraGauss copes with incomplete scan coverage. We train on less and less axial slices, and report the average SSIM, PSNR and Inverted LPIPS averaged across the novel axial, coronal and sagittal views generated. Generally, quality is good until around 25% coverage of the subject, beyond which it drops significantly. Since UltraGauss is not pre-trained, and like most other 3D reconstruction methods over-optimizes to the specific scene, this behaviour is somewhat expected. Crucially, it also ensures that UltraGauss does not hallucinate create existent anatomy, something that crucial for use medical settings and we actively considered when tuning heuristies.
Effect of Scan Coverage SSIM (1)
1.0
0.8
Avg. Test SSIM (t)
0.6
0.4
UltraGauss 100K
0.2
UltraGauss 300K
UltraGauss 2M
0.0
100
80
60
40
20
Percentage of Brain Scanned and Used for Reconstruction
Effect of Scan Coverage PSNR (t)
45
UltraGauss 100K
UltraGauss 300K
Avg. Test PSNR (1)
35
UltraGauss 2M
30
25
20
100
80
60
40
20
Percentage of Brain Scanned and Used for Reconstruction
Effect of Scan Coverage Inverted LPIPS (t)
Avg. Test LPIPS (inverted, ()
0.0
-0.1
-022
-0.3
-0.4
UltraGauss 100K
-0.5
UltraGauss 300K
UltraGauss 2M
-0.6
100
80
60
40
20
Percentage of Brain Scanned and Used for Reconstruction
Figure G5: Ablation study the quality of UltraGauss reconstructions
coverage decreases.
LPIPS is shown inverted here, so that like SSIM and PSNR, a higher value signifies better accuracy.
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G.6 THE EFFECT OF SCAN SIZE ON RECONSTRUCTION TIME
To confirm our expectation that reconstruction time scales linearly with the number of pixels (i.e. the area of the 2D scans), we conducted an ablation using UlrraGaus-3000. Fig. Gojevidently confirms this.
Effect of Scan Size on Reconstruction Time
160
140
Time for 2000 Epochs (mins)
120
100
80
60
40
20
0
3
4
Number of Pixels (Millions)
Figure G6: Ablation study how scans size (and thus the number of pixels), affects reconstruction time of UltraGauss (shown here for UltraGauss with 300,000 initial Gaussians).
35


Reviews:
[{"review_id": "6sRNUV2aSH", "reviewer": "Reviewer_A5Tv", "summary": "The paper proposes UltraGaussian, the first Gaussian Splatting method tailored for ultrasound reconstruction. It introduces a rendering strategy and rasterization design guided by ultrasound imaging principles. Experiments on two datasets demonstrate improved performance over previous methods.", "questions": "Refer to the weakness section.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The rendering strategy is thoughtfully designed based on the principles of ultrasound imaging.\n- The paper provides a thorough discussion of related work.\n- It clearly introduces the background of 3D Gaussian Splatting and ultrasound imaging to aid understanding.\n- The datasets and experimental settings are described in detail, including a clinician survey for evaluation.", "weaknesses": "1. **Further clarification on efficiency**: UltraGauss is described as \"efficient\" in terms of both memory and computation time (lines 013, 044, 107, 122, 211， 477， etc.). However, this claim is not supported by any quantitative results in the current version of the paper.\n- Although the time consumption of various methods is presented in Fig. 4, Fig. B1 (Appendix), and Fig. D3 (Appendix), these results do not clearly support the claimed efficiency of UltraGauss.\n\n2. **Clarification for the proposed method**:\n- The rendering process described in Eq. 6 and Eq. 7 appears to be weakly motivated, as it mainly adds a background term to fill in empty regions.\n- The matrix ${M}$ used in Eq. 10 is semi-definite and cannot represent complex transformations involving non-uniform scaling combined with rotation. Additionally, an ablation study for different $\\beta$ should be added to show the effectiveness of the proposed method.\n- The lower triangular matrix ${L}$ lacks clear interpretability. It would be helpful to provide some geometric insight or intuition behind its role in the model.\n- The rasterization boundaries defined in Eq. 12 only operate in 3D space, which may not accurately correspond to the 2D image space. When the probe pose varies significantly or the viewing angle becomes complex, the projected 3D bounding box may become imprecise or inefficient. The effectiveness of the method should be further validated on datasets with more diverse and challenging probe poses.\n- The effect of the different threshold $p\\%$ for Gaussian;s probability density utilized in Eq. 12 on time efficiency should be discussed.\n\n3. **Engineering implementation and open-sourcing**: Engineering implementation is an important contribution of this work. It would be valuable to know whether the code will be open-sourced in the future.\n\n4. **Use of heuristics**: Heuristic methods are discussed in Sec. 3.1. However, it is unclear whether any heuristics are actually employed in the proposed method. Clarification on this point would be helpful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "W3Xcnqq6pi", "reviewer": "Reviewer_oeeh", "summary": "This paper introduces UltraGauss, a Gaussian-splatting framework tailored to ultrasonic image formation. Instead of camera-style projection and depth-ordered alpha compositing, the method renders by probe–plane intersection with in-plane aggregation, which better matches plane-based echo sampling. The authors propose a triangular precision (inverse-covariance) parameterization for stability/efficiency, and χ²-based ellipsoidal bounds to confine per-plane rasterization, and a two-phase, load-balanced CUDA pipeline to reject non-intersecting Gaussians and rasterize only relevant 2D windows, as well as a lightweight Beer–Lambert attenuation to approximate acoustic shadowing. On fetal ultrasound, UltraGauss is reported to reach SSIM≈0.99 in minutes on a single GPU and to be preferred by expert sonographers in a reader study.", "questions": "Please refer to weaknesses part, especially points 1&3.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The χ² ellipsoidal bounding and two-phase GPU pipeline are clearly motivated and provide a practical path to minute-level reconstructions with up to ~2M Gaussians. The exposition around Fig. 2–3 and Sec. 4.3–4.4 is crisp.  \n2. The lower-triangular precision factorization (Σ⁻¹=LLᵀ with positive diagonals) is a sensible alternative to quaternion-scaled covariances in this setting; it simplifies inversion and Gaussian resampling and appears faster in ablations.\n3. Beyond 3D volume metrics, the reader study with experienced sonographers (10 respondents, multi-country) is valuable; the progressive Turing test design is thoughtful and reveals interesting perceptual dynamics.", "weaknesses": "1. The Beer–Lambert shadow term is extremely lightweight; while speed is compelling, the paper concedes deficits in speckle and multiple-scattering effects, where physics-informed baselines sometimes produce more realistic artifacts. The method’s robustness when probe geometry is imperfect is discussed qualitatively but not deeply quantified.\n2.  Core quantitative results hinge on 12 fetal brain volumes (Dataset A) and 3 freehand videos (Dataset B). That is narrow in anatomy, vendor, and probe diversity; generalization to abdomen/heart/musculoskeletal, different curvilinear geometries, and pathology is not demonstrated. Claims of broad clinical benefit feel premature without wider coverage.\n3. UltraNeRF assumes straight rays/known poses; here, curvilinear inputs are converted and poses are predicted for cinesweeps. It is unclear whether each baseline is given the same pose treatment, regularization, and time/capacity budgets across settings (e.g., RapidVol capacity vs. UltraGauss-300K/2M).\n4. The cinesweep evaluation conflates pose and reconstruction error. While that is realistic, the paper does not separate pose noise vs. reconstruction error (e.g., by repeating with GT poses on a subset, or sweeping pose perturbations). Consequently, it is hard to attribute gains to UltraGauss vs. better tolerance to pose jitter.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PNDaQXP6fQ", "reviewer": "Reviewer_Qd6J", "summary": "This paper presents UltraGauss, an ultrasound-specific Gaussian Splatting framework that efficiently approximates acoustic image formation via probe–plane intersection rendering instead of traditional camera projection. It introduces a triangular precision parameterization for stable optimization, a compute-aware χ²-based rasterization pipeline for fast and balanced GPU rendering, and a lightweight Beer–Lambert attenuation model for realistic acoustic shadowing. Validated on clinical fetal datasets and freehand sweeps, UltraGauss achieves minute-level, high-fidelity 3D reconstructions (SSIM≈0.99) on a single GPU, with clinician-rated realism comparable to native 3D ultrasound scans.", "questions": "Could you clarify how UltraGauss differs conceptually and technically from R2Gaussian and X-Gaussian, given that both works also adopt Beer–Lambert–style attenuation modeling and Gaussian-based volumetric rendering?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper shows several notable strengths.\n\nFirst, its novelty is good — it is the first to adapt 3D Gaussian Splatting to ultrasound reconstruction, proposing a probe–plane intersection rendering mechanism that better captures ultrasound’s plane-based sampling and attenuation behavior. This represents a well-motivated and technically elegant reformulation of Gaussian splatting aligned with acoustic image formation.\n\nSecond, the performance is solid and the experiments are very comprehensive. The authors conduct extensive evaluations on both clinical fetal ultrasound volumes and freehand cine sweeps, achieving high-fidelity 3D reconstructions (SSIM≈0.99) within minutes on a single GPU. The ablation studies are detailed and convincing, isolating the effects of the precision parameterization, rasterization bounds, and attenuation modeling, all of which contribute measurable gains.\n\nFinally, the writing is clear and the presentation is well-organized and polished. The figures effectively visualize the differences between camera-based and ultrasound-based rendering, and the overall exposition makes the paper easy to follow even for readers outside the immediate subfield.", "weaknesses": "(1) The paper lacks sufficient discussion and comparison with two highly related works, X-Gaussian and R2Gaussian. In particular, its Beer–Lambert–based attenuation formulation is conceptually similar to the radiative attenuation modeling in R2Gaussian, yet this connection is not explicitly acknowledged. The writing style and overall pipeline design also bear strong resemblance to these two prior works. Without a detailed discussion or quantitative comparison against them, the novelty claim of UltraGauss becomes weaker, as it appears to build upon established ideas without clearly articulating its distinct contributions.\n\n\n(2) The code and pre-trained models are not submitted, making it impossible to verify the reported performance or reproduce the results. Given that the paper claims substantial efficiency gains and clinical realism, open-sourcing the implementation would be crucial for reproducibility and community adoption.\n\n\n(3) Ablation interpretation and runtime trade-offs. While the ablation studies are extensive, the analysis remains mostly quantitative. A deeper discussion of why specific design choices (e.g., triangular precision vs. quaternion covariance) improve stability or speed would enhance the reader’s understanding.\n\n(4) Dependence on probe pose estimation. In the end-to-end pipeline, UltraGauss still relies on an external pose estimation module. Errors in pose prediction could propagate into reconstruction quality, yet this dependency and its potential failure modes are not deeply analyzed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RydZH6UM3W", "reviewer": "Reviewer_e5ae", "summary": "This work introduces a specialized Gaussian Splatting framework for ultrasound imaging, designed to reconstruct 3D volumes from traditional 2D scans. Instead of using conventional camera-style projection, it models the acoustic image-formation process via probe-plane intersection rendering. This approach ensures physical consistency with ultrasound acquisition while achieving high computational efficiency.\n\nEmpirical results from fetal ultrasound datasets show that UltraGauss consistently outperforms recent models, such as ImplicitVol, RapidVol, and UltraNeRF, in both reconstruction fidelity and speed. When integrated into an end-to-end clinical pipeline using freehand 2D video sweeps, UltraGauss achieved the highest structural similarity and perceptual quality among all models tested.", "questions": "In addition to the weaknesses that need to be addressed, the authors may want to consider addressing the following questions:\n\n1. How sensitive is UltraGauss to pose estimation noise and incomplete scan coverage?\n2. Can the model handle more complex acoustic effects or generalize to cardiac and abdominal ultrasound?\n3. How does it compare with ultrasound-NeRF baselines such as AcousticNeRF?\n4. Is there a way to estimate reconstruction uncertainty or confidence for clinical safety?\n5. What are the typical compute requirements, memory usage, and runtime scaling with scan size?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed UltraGauss is an interesting technical advance in 3D ultrasound reconstruction. It replaces traditional ray-based rendering with a probe-plane intersection model that matches real ultrasound physics, giving more accurate and consistent reconstructions. \n2. The method is extremely fast, achieving near–real-time 3D volume generation on a single GPU while maintaining high image fidelity. \n3. Its design is simple and effective, combining Gaussian splatting with a lightweight attenuation model for stable optimization.\n4. Experiments show strong quantitative and visual gains over existing methods, and expert evaluations confirm clinical realism.", "weaknesses": "1. The Beer–Lambert shadow model ignores scattering, refraction, and probe beam patterns, which may reduce physical accuracy; the method also assumes precise probe poses;\n2. Experiments are limited to fetal ultrasound, leaving its generalization to other organs or imaging conditions uncertain;\n3. Evaluation focuses mostly on visual and SSIM metrics, with limited anatomical accuracy analysis;\n4. The paper omits details on GPU memory, runtime scaling, and failure cases, which are important for clinical reliability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)