R$^2$-Gaussian: Rectifying Radiative Gaussian Splatting for Tomographic Reconstruction
Authors: Ruyi Zha
Tao Jun Lin
Yuanhao Cai
Jiwen Cao
Yanhao Zhang
Hongdong Li
Venue: NeurIPS 2024 poster (2024)
Keywords: 3D Gaussian Splatting
3D Reconstruction
CT Reconstruction
Tomographic Reconstruction
Primary area: machine_vision
TL;DR: We discover an inherent problem in 3DGS and develop a novel 3DGS-based framework for tomographic reconstruction.

Abstract: 3D Gaussian splatting (3DGS) has shown promising results in image rendering and surface reconstruction. However, its potential in volumetric reconstruction tasks, such as X-ray computed tomography, remains under-explored. This paper introduces R$^2$-Gaussian, the first 3DGS-based framework for sparse-view tomographic reconstruction. By carefully deriving X-ray rasterization functions, we discover a previously unknown \emph{integration bias} in the standard 3DGS formulation, which hampers accurate volume retrieval. To address this issue, we propose a novel rectification technique via refactoring the projection from 3D to 2D Gaussians. Our new method presents three key innovations: (1) introducing tailored Gaussian kernels, (2) extending rasterization to X-ray imaging, and (3) developing a CUDA-based differentiable voxelizer. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art approaches in accuracy and efficiency. Crucially, it delivers high-quality results in 4 minutes, which is 12$\times$ faster than NeRF-based methods and on par with traditional algorithms.

Gauuusssi: Rectifying Radiative Gaussian Splatting for Tomographic Reconstruction
Ruyi Zha'
Tao Jun Lin't
Yuanhao Cai?.* Jiwen Cao Hongdong Li
Yanhao Zhang
The Australian National University Johns Hopkins University Robotics Institute, University of Technology Sydney
(ruyi zha, taojun. lin, jiwen. cao, hongdong, 1inuanu. edu. caiyuanhao19980gmai com yanhao zhangduts edu, au
Abstract
3D Gaussian splatting (3DGS) has shown promising results in image rendering and surface reconstruction. However, its potential in volumetric reconstruction tasks, such as X-ray computed tomography, remains under-explored. This paper intro- duces R²-Gaussian, the first 3DGS- framework for sparse-view tomographic reconstruction. By carefully deriving X-ray rasterization functions, we discover a previously unknown integration bias in the standard 3DGS formulation, which hampers accurate volume retrieval. To address this issue, we propose novel recti- fication technique via refactoring the projection from 3D to 2D Gaussians. Our new method presents three key innovations: (1) introducing tailored Gaussian kernels, (2) extending rasterization to X-ray imaging, and (3) developing a CUDA- dif- ferentiable voxelizer. Experiments synthetic and real- world datasets demonstrate that our method outperforms state-of-the-ar approaches in accuracy and efficiency. Crucially, it delivers high-quality results in 4 minutes, which is 12x faster than NeRF-based methods and on par with traditional algorithms. Code and models are available on the project page https: h/th:::://ggttbb com/ /aam//uu---- Wwwaaaaaaaaseeeeeeee
Introduction
Computed tomography (CT) is an essential imaging technique for noninvasively examining the intemal structure of objects. Most CT systems use X-rays the imaging source thanks to their ability to penetrate solid substances (20). During CT scan, an X-ray machine captures multi-angle 2D projections that measure ray attenuation through the material. As the core of CT, tomographic reconstruction aims to recover the 3D density field of the object from its projections. This task is challenging in two aspects. Firstly, the harmful X-ray radiation limits the acquisition of sufficient and freee-free projections. making reconstruction a complex and ill- ill-posed problem. Secondly, time-sensitive applications like medical diagnosis require algorithms to deliver results promptly.
Existing tomography methods suffer from either suboptimal reconstruction quality or slow processing speed. Traditional CT algorithms [13] 2
55 deliver results in minutes but induce serious artifacts. Su-
pervised learning-based approaches 32
achieve promising outcomes by learning semantic priors but struggle with out-of-distribution objects. Recently, neural radiance fields (NeRF) 43 have been applied to tomography and perform well in per-case reconstruction [67 66 48 6 54). However, they are very time-consuming (> 30 minutes) because huge amount of points have be sampled for volume rendering.
33
10
35)
Yuanhao Cai is the corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Intralomo 33.19 dB, 138
SAX-NeRF
Ours (Iter=10k) 34.56
Oure (Iter=30k)
Chest (50-view) Ground Truth
min
min
Figure I: We compare our method to state-of-the-an NeRF-based methods (IntraTomo [66] NAF 67 SAX-NeRF (6) in terms of visual quality, PSNR (dB), and training time (minute). Our method achieves the highest reconstruction quality and is significantly faster than other methods.
Recently, 3D Gaussian splatting (3DGS) 2 has outperformed NeRF both quality and efficiency for view synthesis 64 38 31 and surface reconstruction 16
However, attempts to apply the 3DGS technique to volumetric reconstruction tasks, such as X-ray tomography, are limited and ineffective. Some concurrent works [4] empirically modify 3DGS for X-ray view synthesis, but they treat it solely as data augmentation tool for traditional tomography algorithms. date, there no 3DGS-based method for direct CT reconstruction.
18 65
In this paper, reveal inherent integration bias in 3DGS. This bias, despite having negligible impact on image rendering, critically hampers volumetric reconstruction. To be more specific, we will show in Sec. 44.2.1 that the standard 3DGS overlooks covariance-related scaling factor when splatting 3D Gaussian kernel onto the 2D image plane. This formulation leads to inconsistent volumetric properties queried from different views. Besides the integration bias, there are other challenges in applying 3DGS to tomography, such as the difference between natural light and X-ray imaging and the lack of an effective technique to query volumes from kernels.
We propose R R²-Gaussian (Rectified Radiative Gaussians) to extend 3DGS to sparse-view tomographic reconstruction. Ra-Gaussian achieves bias-free training pipeline with three significant improve- ments. Firstly, we introduce a novel radiative Gaussian kernel, which acts as local density field parameterized by central density, position, and covariance. We initialize Gaussian parameters using the analytical method FDK [13] and optimize them with photometric losses. Secondly, we rectify the 3DGS rasterizer to support X-ray imaging. This is achieved by deriving new X-ray rendering functions and correcting the integration bias for accurate density retrieval. Thirdly, we develop CUDA-based differentiable voxelizer, which not only extracts 3D volumes from Gaussians but also enables voxel- based regularization during training. We evaluate R e2-Gaussian on both synthetic and real-world datasets. Extensive experiments demonstrate that our method surpasses state-of of-the-ar (SOTA) methods within minutes, which is 12x faster than the most efficient NeRF-based solution, NAF 667 and comparable to traditional algorithms. converges to optimal results 15 minutes, improving PSNR by 0.6 dB compared to SOTA methods. A visual comparison is shown in Fig. [
Our contributions can be summarized as follows: (1) We discover a previously unknown integration bias in 3DGS that impedes volumetric reconstruction. (2) We propose the first 3DGS-based tomogra- phy framework by introducing new kernels, extending rasterization to X-ray imaging, and developing differentiable voxelizer. (3) Our method significantly outperforms state-of-the the-art methods in both reconstruction quality and training speed, highlighting its practical value.
2 Related work
Tomographic reconstruction Computed tomography (CT) is widely used for non-intrusive inspec- tion in medicine [2] biology [12 39 24), and industry [1]. Conventional fan-beam CT produces 3D volume by reconstructing each slice from 1D projection arrays. Recently, the cone-beam scanner has become popular for fast scanning and high resolution (52) leading to the demand for 3D tomography, i.e., recovering the volume directly from 2D projection images. Our work focuses on 3D sparse-view reconstruction where less than a hundred projections are captured to reduce radiation exposure. Traditional algorithms mainly grouped into analytical and iterative methods. Analytical

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methods like filtered back projection (FBP) and its 3D variant FDK [3] produce results instantly (< second) by solving the Radon transform and its inverse 46). However, they introduce serious formulate tomography as maximum-a pootttrioro problem and iteratively minimize the energy function with regularizations. They successfully suppress artifacts but take longer time (< 10 minutes) and lose structure details. Deep learning methods can be categorized as supervised and self-supervised families. Supervised methods learn semantic priors from CT datasets. They then use the trained networks to inpaint projections [ [51, denoise volumes T0, 28 35 37 or directly output results [9
streak artifacts in sparse-view scenarios. Iterative methods [2 55 40
51
[ I 32, 33
Supervised learning methods perform well in cases similar to training sets but suffer from poor genera- tion ability when applied to unseen data. To overcome this limitation, some studies [67
66
48
6 54]
handle tomography in a self-supervised learning fashion. Inspired by NeRF 4 they model the density field with coordinate-base. networks and optimize them with photometric losses. Although NeRF-based methods excel in per-case reconstruction, they time-consuming (>30 minutes) due to the extensive point sampling in volume rendering. Our work can be put into the self-supervised learning family, but it greatly accelerates the training process and improves reconstruction quality.
3DGS 3D Gaussian splatting 23 outperforms NeRF in speed by leveraging highly parallelized rasterization for image rendering. 3DGS represents objects with a set of trainable Gaussian-shaper primitives. It has achieved great success RGB tasks, including surface reconstruction [T6] T8| (65], dynamic scene modeling [60
34 61]. human avatar [36 30 271. 3D generation [57]
62 91. etc.
Some concurrent works have extended 3DGS to X-ray imaging. X-Gaussian 7 modify 3DGS to
synthesize novel-view X-ray projections. Gao et al. [4] improve X-Gaussian by considering complex noise-inducing physical effects. While they produce plausible 2D X-ray projections, they cannot directly extract 3D density volumes from trained Gaussians. Instead, they first augment projections with 3DGS, and then use traditional algorithms such as FDK for CT reconstruction, which is neither efficient nor effective. Li et al. [29] represent the density field with customized Gaussian kernels, but they replace the efficient rasterization with existing CT simulators. In comparison, our work can both rasterize X-ray projections and voxelize density volumes from Gaussians.
3 Preliminary
3.1 X-ray imaging
A projection I E #x measures ray attenuation through the material as shown in Fig. 2] For an X- ray r(t) o + td E R with initial intensity To and path bounds in and ly, the corresponding raw pixel value (r(r) is given with the Beer-Lambert Law (20) by: ('(r) = To exp(- f.d a<((()) df). Here, axx) is the isotropic density (or attenuation coefficient in physics) at position x E IR3. Tomography typically transforms raw data to the logarithmic space for computational simplicity, i.e.,
Projection 1
t
Source
ff" ((r(t))dt, (1)
(4
[(r) = log Iu - log I'(r) ((r)==(
Circular Scanning
Object a(x)
where each pixel value ((r) represents the density integral along the ray path. Except otherwise spec- ified, we use the logarithmic projections as inputs. The goal of tomographic reconstruction is to esti- mate the 3D distribution of (x). output as a discrete volume, with X-ray projections i=,..,,.... 1....... captured from N different angles. Note that real-world projections contain minor anisotropic physical
Figure 2: A detection plane captures the atten- uation X-rays emitted from different angles.
effects such as Compton scattering. Following previous works [3 2 55 67], we do not explicitly
model them but treat them as noise during the reconstruction.
3.2 3D Gaussian splatting
3D Gaussian splatting 23 models the scene with a set of 3D Gaussian kernels G3 (G J-=1.... -......M .Ms each parameterized by position, covariance, color, and opacity, A rasterizer R renders an RGB image
3

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Radiative Caussian
(K) A (x-pp) (e-pu)
Density
P.. Position
-. Coveriance
Dimnnnsion
Density Field Modelling
Training
Vowelization
xxx)
Σ Gl
Gaussians
Optimited Geussians
Density
Figure 3: represent the scanned object as a set radiative Gaussians. We optimize them using
real X-ray projections and finally retrieve the density volume with voxelization.
Lngh E &RHxWx x3 from these Gaussians, formulated as
Lgb = = Co
where T, P, and C are the transformation, projection, and composition modules, respectively. First, T transforms the 3D Gaussians into the ray space, aligning viewing rays with the coordinate axis to enhance computational efficiency. The transformed 3D Gaussians are then projected onto the image plane: @ P(GS). The projected 2D Gaussian retains the same opacity and color as its 3D counterpart but omits the third row and column of position and covariance. An RGB image is then rendered by compositing these 2D Gaussians using sipha-blending 45] L C(G2). ) The rasterizer is differentiable, allowing for the optimization of kernel parameters using photometric losses. 3DGS initializes sparse Gaussians with structure- fromtmro-roommmotion (SfM) points (53. During training, an adaptive control strategy dynamically densifies Gaussians to improve scene representation.
4 Method
In this section, we first introduce radiative Gaussian as novel object representation in Sec. 4.1 Next, we adapt 3DGS to tomography in Sec. 4.2 Specifically, we derive new rasterization functions and analyze the integration bias of standard 3DGS in Sec. 4.2.1 We further develop a differentiable voxelizer for volume retrieval in Sec. 4.2.2 The optimization strategy is elaborated in Sec. 4.2.3
4.1 Representing objects with radiative Gaussians
As shown in Fig. we represent the target object with a group of learnable 3D kernels G {G .M that we term as radiative Gaussians. Each kernel G. defines a local Gaussian-shaped density field, i.e.
1 G: E,) Pi cxp (- (x p:) Ex- p.)).
(3)
where Ps Pt E R and E; E 3x3 are learnable parameters representing central density, position and covariance, respectively. For optimization purposes, we follow 23 to further decompose the covariance matrix Σ, into the rotation matrix R, and scale matrix S, Σ, = R.S.S, R.T. The overall density at position 3 is then computed by summing the density contribution of kernels:
Af
(x) = Σ G2 Pa, E.).
==1
Compared with standard 3DGS, our kernel formulation removes view-dependent color because X-ray attenuation depends only on isotropic density, shown in Eq. (I. More importantly, we define the density query function (Eq. (4) for radiative Gaussians, making them useful for both 2D image rendering and 3D volume reconstruction. In contrast, the opacity in 3DGS is empirically designed RGB rendering, leading to challenges when extracting 3D models such as meshes from Concurrent work (29] also explores kernel-based representation but uses simplified isotropic Gaussians. Our work employs general Gaussian distribution, offering more flexibility and precision in modeling complex structures.
Gaussians [6
g 65

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(8) Training Pipeline
X-ray Rasterizer
L1
Randered
Adaptive Control
FDK Volume
Dimension
Density Voxelizer
Initiliiation
Redistive Caursions
Volume
Rogularization
(b) X-ray Rasterizer (84.2.1)
dity
Transformation
Projection
Composition
Rendered
Space
(c) Density Voxelizer (84.2.2)
(d) Adaptive Control (34.2.3)
Split
Prune
Clone
Radiative Gaussians
Density Volume
Figure 4: Training pipeline of -2-Gaussian. (a) Overall training pipeline. (b) X-ray rasterization for projection rendering. (c) Density voxelization for volume retrieval. (d) Modified adaptive control.
Initialization 3DGS initializes Gaussians with SfM points, which is not applicable volumetric tomography. Instead, we initialize our radiative Gaussians using preliminary results obtained from the analytical method. Specifically,
FDK [13] reconstruct low-quality volume in less than I second. We then exclude empty spaces with a density threshold t and randomly sample M points kernel positions. Following 23 we set the scales of Gaussians the nearest neighbor distances and assume no rotation. The central densities are queried from the FDK volume. We empirically scale down the queried densities with ls compensate for the overlay between kernels.
4.2 Training radiative Gaussians
Our training pipeline is shown in Fig. Radiative Gaussians first initialized from FDK volume. We then rasterize projections for photometric losses and voxelize tiny density volumes for 3D regularization. Adaptive control is used densify Gaussians for better representation. After training, we voxelize density volumes of the target size for evaluation.
4.2.1 X-ray rasterization
This section focuses on the theoretical derivation of X-ray rasterization R. As discussed Sec. 3.1 the pixel value of projection the integral of density along the corresponding ray path. We substitute Eq. (4) into Eq. ( yielding
(,(r) / Σ G)((())ep... Ed Σ (i((())ps P. E,)dt,
(5)
where (x) is the rendered pixel value. This implies that we can individually integrate each 3D Gaussian rasterize X-ray projection. Note that f,, and ty in Eq. (T) are neglected because assume all Gaussians are bounded inside the target space.
Transformation Since a cone-beam X-ray scanner can be modeled similarly to pinhole camera, follow [69] to transfer Gaussians from the world space to the ray space. In ray space, the
5

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viewing rays are parallel to the third coordinate axis, facilitating analytical integration. Due to the non-Cartesian nature of ray space, we employ the local affine transformation to Eq. ( yielding
M
/ (x ( J.WE,W dr
(
(6)
P.
Σ
where x [2o, I1, 22] is point in ray space, P: R is the new Gaussian position obtained through projective mapping o, and Σ; E R is the new Gaussian covariance controlled by local approximation matrix J, and viewing transformation matrix W. Refer to Appendix for determining 6, J, and W from scanner parameters.
Projection and composition A good property of normalized 3D Gaussian distribution is that its integral along one coordinate axis yields normalized 2D Gaussian distribution. Substitute Eq. into Eq. 6 and we have
M
(r)) ~ Σ
(2T)
exp
(x-))))
|
(x- p.)
dx²
(2m)
Normalized 3D Gaussian distribution
M
exp (- 1/2 & p.) ( p.)
Σ (2
2T Σ Σ
2
Normalized 2D Gaussian distribution
M Σ (
2T D
Pi, Di, E.), Σ1
1=1
_ E 2 are obtained by dropping the third rows and columns of their counterparts x. p. and E, respectively. Eq. ( shows that an X-ray projection can be rendered by simply summing 2D Gaussians instead of alpha-compositing them in natural light imaging.
where & E R22 p RR2
Integration bias During the projection, a key difference between our 2D Gaussian and the original one in 3DGS is the central density (opacity) d.. As shown in Eq. (7). we scale the density with a covariance-related factor 14 (2 E 1,) P. p.P;, while 3DGS does not. This implies that 3DGS, in fact, learns an integrated density in the 2D image plane rather than the actual one in 3D space. This integration bias, though having a negligible impact on imaging rendering, leads to significant inconsistency density retrieval. We demonstrate the inconsistency with simplified 2D-to-1D ID projection in Fig.
5 When attempting to recover the central density p in 3D space with pi = px/P.. we find different views (1z) lead to different results. This violates the isotropic nature of pt. preventing us from determining the correct value. In contrast, our method assigns the actual 3D density to the kernel and forwardly computes the 2D projection, thus fundamentally solving the issue. While conceptually simple, implementing our idea requires substantial engineering efforts, including reprogramming all backpropagation routines in CUDA.
p= M/Ps or p 0/P2
Figure 5: Density incon- sistency in 3DGS.
4.2.2 Density voxelization
We develop a voxelizer V to efficiently query a density volume V E XXxxx from radiative Gaussians: V V(G Inspired by voxelizers used RGB tasks (57). our voxelizer first partitions the target space into multiple 8 x 8 x 8 3D tiles. It then culls Gaussians, retaining those with a 99% confidence intersecting the tile. In each 3D tile, voxel values are parallelly computed by summing the contributions of nearby kernels with Eq. 4 We implement the voxelizer and its backpropagation in CUDA, making it differentiable for optimization. This design not only accelerates the query process (> 100 FPS) but also allows to regularize radiative Gaussians with 3D priors.

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Table Quantitative results on sparse-vies tomography. We colorize the best
second-ber -bsst
and
third-best numbers.
75-view
50-view
25-wiew
Methods
PSNRT
SSIM
Timel
PSNRT
SSIMI
Time|
PSNR
SSIM
Time.
Synthetic dataset
FDK 13
28.63
0,497
26,50
0,422
22.99
0.317
SART 2
36.06
0.897
4m41s
34.37
0.875
3m366
31.14
0.825
Im47s
ASD-POCS [55
36.64
0,940
2m25s
34.34
0.914
lm52s
30.48
0.847
56s
Tomo 666
35.42
0.924
2h7m
35.25
0.923
2h9m
34.68
0.914
2h 2h19m
NAF 67
37.84
0,945
30m43s
36.65
0,932
32m4s
33.91
0.893
31mls
SAX-NeRF 6
38.07
0,950
13h5m
36.86
0.938
13h5m
34.33
0.905
13h3m
Ours (iter=10 (0k)
38.29
0.954
2m38s
37.63
0.949
2m35s
35.08
0.922
2m35s
Ours (Iter=3kk)
38.88
0.959
8m21s
37.98
0.952
8m14s
35.19
0.923
8m228
Real-world dataset
FDK 13
30,03
0.535
27.38
0.449
23.30
0.335
SART 2
34.42
0,845
Smlls
33.61
0.827
3m28s
31.52
0.790
Im47s
ASD-POCS 55
36.33
0.868
2m43s
34.58
0.861
Im49s
31.32
0.810
56s
IntraTomo
66
36.79
0.858
2h25m
36.99
0.854
2h19m
35.85
0.835
2h18m
NAF 67
38.58
0.848
51m28s
36.44
0.818
51m31s
32.92
0.772
51m24s
SAX-NeRF 6
34.93
0.854
13h21m
34.89
0.840
13h23m
33.49
0.793
13h25m
Ours (0ter=10k)
38.10
0.872
3m399
37.52
0.866
3m37s
35.10
0.840
3m23s
Ours (iter=30k)
39.40
0.875
14m16s
38.24
0.864
13m52s
34.83
0.833
12m56s
4.2.3 Optimization
We optimize radiative Gaussians using stochastic gradient descent. Besides photometric LI loss Li and D-SSIM loss Cassm [59], we further incorporate a 3D total variation (TV) regularization [49] Liv homogeneity prior for tomography. At each training iteration, randomly query tiny density volume Vt E R> Dx D (same spacing as the target output) and minimize its total variation.
The overall training loss is defined
Liotal (Ir, + Csim (I,, Im) + dev Cev (Vio),
where I,., Im. Assim and Aev are rendered projection, measured projection, D-SSIM weight, and TV weight, respectively. Adaptive control is employed during training to enhance object representation. We remove empty Gaussians and densify (clone or split) those with large loss gradients. Considering objects such as human organs have extensive homogeneous areas, we do not prune large Gaussians. As for densification, we halve the densities of both original and replicated Gaussians. This strategy mitigates the sudden performance drop caused by new Gaussians and hence stabilizes training.
Experiments
5.1 Experimental settings
Dataset We conduct experiments on both synthetic and real-world datasets. For the synthetic dataset, we collect 15 real CT volumes, ranging from organisms to artificial objects. We then the tomography toolbox TIGRE S to synthesize X-ray projections and add Compton scatter and electric noise. For real-world experiments, we use three cases from the FIPS dataset (56), each with 721 real projections. Since ground truth volumes are unavailable, we use FDK 13 to create pseudo pseudo-ground truth using all views and then subsample views for sparse-view experiments. We set 75, 50, and 25 views for both synthetic and real- world data as three sparse-view scenarios. Refer to Appendix for more details of datasets.
Implementation details Our R-- e2-Gaussian is implemented in Py Torch 44 and CUDA 50), and trained with the Adam optimizer (25) for 30k iterations. Learning rates for position, density, scale, and rotation are initially set as 0.0002, 0.01, 0.005, and 0.001, respectively, and exponentially to
0.1 of their initial values. Loss weights are Ausim 0.25 and A 0.05. We initialize M 50k Gaussians with density threshold 0.05 and scaling term A 0.15. The TV volume size is
7

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Pepper
25 68
34.0
36.5 92
33 95
37 18
36 87
39 45
75 view
Chest
24
31 86
18
33. 99
34 45
35 01
view
Bockpat
39.19
29.51
32.05
32. 72.70
32 24
35.25
view
Fine
30.48
32.41
33.50
38
38.35
36.96
40. 40.38
50-view
SAX-NeRF
Figure 6: Colorized slice examples different methods with PSNR (dB) shown at the bottom right of each image. The first three rows are from the synthetic dataset and the last row is from the real-world dataset. Our method recovers more details and suppresses artifacts.
D 32. Adaptive control runs from 500 to 15k iterations with gradient threshold of 0.00005. All methods run on a single RTX3090 GPU. We evaluate reconstruction quality using PSNR and SSIM [59], with PSNR calculated in 3D volume and SSIM averaged over 2D slices in axial, coronal, and sagittal directions. We also report the running time as a reflection of efficiency.
5.2 Results and evaluation
For faimess,, do not compare methods that require external training data but focus those that solely use 2D projections of arbitrary objects. We compare R² 22-Gaussian with three traditional methods (FDK [33. SART (2). ASD-POCS (55)) and three SOTA NeRF-based methods (IntraTommo (66), NAF 67 SAX-NeRF (6). Tab. reports the quantitative results on sparse-view tomography. Note that do not report the running time for FDK it is instant. R²-Gaussian achieves the best performance across all synthetic and most real-world experiments. Specifically, our method delivers 0.93 dB higher PSNR than SAX-NeRE, on the synthetic dataset, and 0.95 dB improvement over IntraTomo on the real-worke dataset. It is also worth noting that our 50-view results are already on par with the 75-view results of other methods. Regarding efficiency, our method converges to optimal results in 15 minutes, which is 3.7x faster than the most efficient NeRF-based method, NAF, Surprisingly. it takes less than minutes surpass other methods, which even faster than the traditional algorithm SART. Fig. shows the visual comparisons of different methods. FDK and SART introduce streak artifacts, while ASD-POCS and IntraToom blur structural details. NAF and SAX-NeRF are better than other baseline methods but have salt-and-pepper noise. In comparison, our method successfully recovers sharp details, e.g., ovules pepper, and maintains good smoothness for homogeneous arcas, e.g., muscles in the chest.

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Table 2: Quantitative results X-3DGS and our method on the synthetic dataset.
75-view
50-view
25-view
X-3DGS
Ours
X-3DGS
Ours
X-3DGS
Ours
2D PSNR
49.97
50.54
47.26
49.70
39.84
46.28
2D SSIM
0.987
0.986
0,984
0.986
0.967
0.982
3D PSNR
23.40
38.86
21.24
37.98
14.07
35.17
3D SSIM
0.660
0.959
0.562
0.952
0.408
0.923
5.3 Ablation study
Integration bias To demonstrate the impact of integration bias discussed in Sec. 4.2.1 1, we develop an X-ray version of 3DGS (X-3DGS) that uses X-ray rendering while retaining the biased 3D-to-2D Gaussian projection. We use the same voxelizer in Sec. 4.2.2 extract volumes. Before voxelization, we divide the learned density of each Gaussian by the mean scaling factor μ of all training views. Tab. shoos that rectifying integration bias benefits both 2D rendering (+3.15 dB PSNR) and 3D reconstruction (+17.77 PSNR). Fig. 7 visualize rendering and reconstruction results. While X-3DGS renders reasonable 2D projections, its reconstruction quality is significantly worse than ours. Besides, there are notable discrepancies slices queried from different views. The conflicting 2D and 3D performances indicate that X-3DGS, despite fitting images well, does not accurately model the density field. In contrast, our method learns the actual new-inddeenndett density, climinating inconsistencies and ensuring unbiased object representation.
Component analysis We conduct ablation experiments assess the effect of FDK initialization (Init.), modified adaptive control (AC), and total variation regularization (Reg.) on performance. The baseline model excludes these components and uses randomly generated Gaussians for initialization. Experiments are performed under the 50-view condition, evaluating PSNR, SSIM, training time, and Gaussian count (Gau.). Results are listed in Tab. FDK initialization boosts PSNR by 0.9 dB. Adaptive control improves quality but prolongs training due more Gaussians. TV regularization increases SSIM by reducing artifacts and promoting smoothness. Overall, our full model outperforms the baseline, improving PSNR by 1.51 dB and SSIM by 0.018, with training time under 9 minutes.
Parameter analysis We perform parameter analysis on the number of initialized Gaussians M, TV loss weight Atv. and TV volume size D. The results are shown in the last three blocks of Tab. 3 R²_ 2P-Gaussian achieves good quality-efficiens of Ato 0.05 improves reconstruction, but larger values can lead to degradation. The training time increases with TV volume size while the performance peaks at = 32.
balance 50k initialized Gaussians, A TV loss weight
Convergence analysis Fig. 8 compares the results of NeRF-based methods and our R²-Gaussian method different iterations. Our method, both with and without FDK initialization, converges significantly faster, displaying sharp details by the 500th iteration when other methods still exhibit artifacts and blurriness. Notably, the FDK initialization offers rough structure before training, which
Rendered Projections (30*)
Reconstructed Volumes
150
90
46,38
48.20
34.97
34.72
34.78
42.39
30
Beelle (50-vieri
X-3DGS
Ours
X-3DGS (30%)
(-3DGS.909)
X-3DGS (150")
Ours
Ground Truth
Figure 7: Results of X-3DGS and our method with PSNR (dB) indicated on each image. We show slices of X-DDGS queried from three viewing angles. Although X-3DGS can produce plausible X-ray projections, its reconstructed volume lacks view consistency and exhibits poor quality.

---

Engine (50-vlew)
Table Ablation results with our choices in bold.
1etter
Synthetic
PSNR1
SSIM
Timel
Gau.
Baseline (B)
36.47
0.934
4m57s
50k
280 Iter
B+Init.
37.37
0.944
5m29s
50k
B+AC
37.33
0.942
7m33s
70k
B+Reg.
36.79
0.943
6m30s
50k
Full model
37.98
0.952
8m37s
68k
500
M=5k
37.44
0,946
9m18s
32k
M=10k
37.56
0.948
8m59s
35k
M=50k
37.98
0.952
8m14s
68k
750 her
M=100k
38.03
0.953
9m4s
112k
M=200k
37.82
0.949
9m54s
206k
A
37.66
0.948
7m9s
68k
1000P ter
=0.01
37.88
0.950
8m21s
68k
37.98
0.952
8m14s
10=0.11
37.73
0.951
8m11s
68k
Nr. 0.15.15
37.40
0.949
8m27s
69k
Emnl
D=8
37.74
0.949
7m56s
68k
D=16
37.94
0.950
8m 5ms8s
68k
D=32
37.98
0.952
8ml4s
68k
IntraTomo
NAF
SAX-NeRF Ours wio Init.
Ours
na 176mmn
37.15 56 mi1min 38.80
D=48
37.90
0,951
9m34s
Results of NeRF-based methods and our Ra-aausiaan different iterations.
Figure
D=64
37.82
0.949
11m35s
67k
further accelerates convergence and enhances reconstruction quality. Finally, our method outperforms others in both performance and efficiency, achieving the highest PSNR of 38.90 dB in 9 minutes.
Discussion and conclusion
Discussion R RR-Gaussian inherits some limitations from 3DGS, such as varying training time across modalities, needle-like artifacts under extremely sparse-view conditions, and suboptimal extrapolation for other tomography tasks. Besides, we have not considered calibration errors regarding the scanned geometry and anisotropic physical effects such as Compton scattering. More details are discussed in Appendix B Despite these limitations, our method's superior performance and fast speed make valuable for real-world applications for medical diagnosis and industrial inspection.
framework for sparse-view tomographic reconstruction. We identify and rectify a previously overlooked integration bias of standard 3DGS, which hinders accurate density retrieval. Furthermore, enhance 3DGS for tomography by introducing new kernels, devising X-ray rasterization functions, and developing a differentiable voxelizer. Our R R-Gaassian surpasses state-of- the methods in both reconstruction quality and training speed, demonstrating its potential for real-world applications. Crucially, we speculate that the newly found integration bias may be pervasive across all 3DGS-related research. Consequently, our rectification technique could benefit more tasks beyond computed tomography.
Conclusion This paper presents R² RR-Gaussian, novel
Acknowledgments
The research is funded in part by ARC Discovery Grant (grant ID: DP220100800) of the Australia Research Council.
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A Transformation module in X-ray rasterization
The configuration of a cone beam CT scanner is shown in Fig. 9 The X-ray source and detector plane rotate around the z-axis, resembling pinhole camera model. Therefore, can formulate the field-of-view (FOV) scanner as
Dz Dz-,, FOY, = 2 arctan Duy 2LSD
FOV, = 2 arctan
(9)
2Lsn
Here, (Dz, Dy) is the physical size of the detector plane, and LsD the distance between the source and the detector. Following [23], then use FOVs to determine the projection mapping ø.
To get Gaussians the ray space, we first transfer them from the world space the scanner space. The scanner space is defined such that its origin is the X-ray source, and its z-axis points to the projection center. The transformation matrix T from the world space to the scanner space
sin A cos 0
0
W
W =
-1 0
(10)
cos sin 0
Lso
Here, is the rotation angle, and Lso is the distance between the source and the object. Next, we apply local approximation each Gaussian. The Jacobian of the affine approximation J, is the same as Eq. (29) in [69]. Finally, we have the Gaussian in the ray space with new position p and covariance Σ, as
p: = d -.W. J.
(11)
Toleetory
Saires
Lsp
Figure 9: Configuration of a cone-beam CT scanner.
B Details of dataset
Synthetic data We evaluate methods with various modalities, covering major CT applications such as medical diagnosis, biological research, and industrial inspection. The synthetic dataset consists of 15 cases across three categories: human organs (chest, foot, head, jaw, and pancreas), animals and plants (beetle, bonsai, broccoli, kingsnake, and pepper), and artificial objects (backpack, engine, present, teapot, and mount). The chest and pancreas scans are from LIDC-IDRI 4 and Pancreas-CT CT
respectively. Broccoli and pepper are obtained from X-Plant [58] and the rest are Following 67 6), we preprocess raw data by normalizing densities to [0, and resizing volumes to 256 x 256 256. We then use the tomography toolbox TIGRE [5] to capture 512 512 projections
47
from Vis 26
the range 0 360 We add two types noise: Gaussian (mean 0, standard deviation 10) as electronic noise of the detector and Poisson (lambda le5) as photon scattering noise. volumes and their projection examples are shown in Fig.
Real-world data We use FIPS [66 a public dataset providing real 2D X-ray projections. FIPS includes three objects (pine 41, seashell [21] and walnut [2 ). Each case has 721 projections in range of 0 -306 We preprocess 2D projections by resizing them to 560z560 and normalizing them to [0. 1]. Since ground truth volumes are unavailable, we use FDK to create pseudo-ground truth with all views and then subsample 75/50/25 views for sparse-view experiments. The size the
target volume is 256 x 256 256.
15

---

Marmurhaaw
Serri Piieto
Miker Divan
Exgo
Figure 10: Datasets used for experiments. We show half volume and projection examples for each case.
C Implementation details of baseline methods
For fairness, we do not compare methods that require external training data but focus on those that solely use 2D projections of arbitrary objects. We run traditional algorithms FDK, SART, and ASD- POCS with GPU-accelerated tomographic toolbox TIGRE [5], and select three SOTA NeRF-based tomography methods. IntraToomo models the density field with large MLP. NAF accelerates the training process by hash encoding. SAX-NeRF achieves plausible results with a line segment-based transformer. We use the official code of NAF and SAX-NeRF and conduct experiments with default hyperparameters. The IntraTommo implementation is sourced from the NAF repository. The training iterations of NeRF-based methods set to 150k (default of NAF and SAX-NeRF). All methods run on single RTX 3090 GPU.
D More qualitative results
Main results We visualize more reconstruction results in Fig. and Fig. 12 FDK and SART in- troduce notable streak artifacts, while ASD-POCS and Intra Tomo blur structural details. NeRF-based solutions perform better than traditional methods but exhibit salt-and and-pepper noise. comparison, our method successfully recovers sharp details and maintains smoothness in homogeneous areas.
Integration bias We show more qualitative comparisons of X-3DGS and ours in Fig. 13 method outperforms X-3DGS in both 2D rendering and 3D reconstruction.
Our
Components and parameters Our newly introduced components improve the reconstruction quality. Our parameter setting also yields the best performance.
We visually compare different components and parameters Fig.
We show the PSNR and SSIM plots in Fig. 15 Our method converges significantly faster than NeRF-based methods and outperforms them in only 3000 iterations.
Convergence analysis
E
More discussion of limitation
Varying time We present the training times for all cases Fig. 16 The training time varies across cases, primarily due to the different numbers of kernels used. Our method takes more time on objects
16

---

75-view
view
view
14
44
31.0
22.45
34.0
4.
IntraToro
NAF
SAX-NeRF
Ours
IntraTormo
NAF
SAX-NeRF
Ours
IntraToroo
NAF
SAX-NeRF
Ours
GT
Figure 11: Reconstruction results of NeRF- methods and our method on the synthetic dataset.
with large homogeneous areas, such the chest, pancreas, and mount, and less time those with sparse structures, such as the beetle, backpack, and present.
Needle-like artifacts While our method achieves the highest reconstruction quality, it introduces needle-like artifacts, especially under the 25-view condition (Fig. (7). This suggests that some Gaussians may overfit specific X-rays. Similar artifacts also observed in 3DGS (6).
Extrapolation ability While this paper focuses sparse-view CT (SVCT), we also test R Gaussian on limited-angle CT (LACT), where the scanning range constrained less than 180". Unlike SVCT, which highlights the interpolation ability of methods, LACT challenges their extrapo- lation ability, i.e., estimating unseen areas outside the scanning range. We generate 100 projections within ranges of (0 150", 0" 120" and 0 90" The quantitative results in Tab. show that
17

---

Waimut
25. 13
31.32
48
32.11 17
88
30.91
33.34
25-view
Pine
30.4 4.
41
33 50
38 73
38 39.35
36. 96
40.
50-viow
Sagshell
24:54
34.07
32 84
39 00
35.04
55.55
36. 74
25 V--vV
FDK
SART
ASD-POCS
Introloma
NAF
SAX-NNRF
Ours
Ground Truth
Figure 12: Reconstruction results the real-world dataset.
our method has slightly lower PSNR but higher SSIM than the NeRF-based method NAF. Visualiza- tion results in Fig. indicate that method recovers more details in scanned areas but exhibits blurred artifacts in unseen areas. We attribute this performance drop to the nature of the networks and kernels. Given the gradient of ray, NeRF updates the entire network while 3DGS individually optimizes intersected kernels. Thus, NeRF has better global awareness and consistency, while 3DGS more local-oriented and has suboptimal extrapolation ability.
Calibration error In real-world applications, calibration errors can affect reconstruction quality. For example, tomography requires reference image lo in Eq. ( represent the illumination pattern without the object. This reference image may have artifacts, such as intensity dropoff towards the image boundaries and other non- uniformiiies illumination. Additionally, scanner extrinsics and intrinsics may vary during scanning due to heat expansion and mechanical vibrations. Addressing these practical challenges will be the focus of future work.
Anisotropic effects Following existing CT reconstruction methods, we work under the isotropic assumption of X-ray imaging. However, the real world, some X-ray transport effects, such as Compton scattering, are anisotropic. We do not explicitly model them but treat them as noises on X-ray projections. This necessary simplification for CT reconstruction but may induce inaccuracy for novel-view X-ray synthesis. Readers may refer to
14 for using 3DGS for X-ray view synthesis.
Table Quantitative evaluation on limited-angle tomography. We colorize the best and third-best numbers.
second-best
0" 150"
0"
120
90"
Methods
PSNR1
SSIMI
Time!
PSNR
SSIMT
Time.
PSNR1
SSIM
Time!
FDK
13
26.83
0.570
24.00
0.566
21.22
0.547
SART 2
33.34
0.883
7m9s
30.21
0.847
7m8s
26.71
0.795
7m59s
ASD-POCS [55
33.16
0.913
3m41s
29.76
0.875
3m39s
26.34
0.812
4m8s
NAF 67
36.29
0.940
27m18
33.35
0.922
27m6s
29.89
0.884
27m25s
Ours
36.12
0.948
9m3s
32.68
0.923
8m36s
29.21
0.886
8m28s
18

---

Rendered Projections (0*)
Reconstructed Slices
52.09
52.72
28.68
48.31
49.60
26.33
34.41
50 view
38.89
53.11
23.92
36.47
25 55-view
X-3DGS
Ours
Ground Truth
X-3DGS
Ours
Ground Truth
Figure 13: Qualitative comparison of X-3DGS and our method.
Broader impacts
Impacts real-world applications Computed tomography is essential imaging technique that widely used in fields including medicine, biology, industry, etc. Our R2 RR-Gaussian enjoys superior reconstruction performance and fast convergence speed, making it promising to be implemented in real-world applications such as medical diagnosis and industrial inspection.
Impacts research community We discover previously unknown integration bias problem in currently popular 3DGS. we speculate that this problem could be universal across all 3DGS-related works. Therefore, rectification technique may apply wider practical domains, not limited to tomography but also other tasks such as magnetic resonance imaging (MRI) reconstruction and volumetric-based surface reconstruction.
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37.83
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34.7 74
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==
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Figure 14: Quantitative comparison of different components and parameters.
10.0
PONH (801
Curs (Fina))
Durs
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10k
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Reration
Figure 15:
and ASIM-iteration plots of case engine, 50-view.
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15
75-view 50-viw
26-view
Tima
Chess
Foot
Hoad
Panereaa
Beaate
Bornn
Brocol
Kingenaki
Pepper
Backpack
Engine
Present
Tespel
Mount
Figure 16: Training time on the synthetic dataset.
75-view
50-view
25-view
75-view
50-view
25-view
39.45
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34.32
40.08
38 38.90
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Figure 17: 3DGS-based methods tend to introduce needle-like like artifacts when there are insufficient amounts images. PNSR (dB) is shown at the bottom right of each image.
Mount
22.53
33.09
36.47
36.82
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Pancreas
23.24
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NAF
Ours
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Visualization of reconstruction results under limited-angle scenarios. PNSR (dB) is shown Figure 18:
at the bottom right each image.
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discovery of integration bias problem in 3DGS and novel 3DGS- based framework
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Reviews:
[{"review_id": "BU7VJIW9nZ", "reviewer": "Reviewer_sKx6", "summary": "This paper presents a 3D reconstruction method for sparse-view computed tomography using 3D Gaussian Splatting. The core contribution is the reformulation of the volumetric rendering equation to include view-independent central density estimation. Additionally, the paper introduces a differentiable voxelizer that converts a set of 3D Gaussians into a voxel grid of densities, proving effective in computed tomography tasks.", "questions": "I have a couple of questions.\n\nThe paper focuses on sparse-view computed tomography. I am curious about how the method would perform in dense-view scenarios. In industrial CT scanning, capturing a few thousand projections for high-quality microscale geometric details is not uncommon. With a sufficient number of projections, the FDK algorithm typically performs well. How does the proposed method compare to FDK when a large number of projections are used? At what point might it start to underperform, if at all? Would it still outperform in dense-view scenarios?\n\nAdditionally, what are the implications of correcting the integration bias in image-based 3D reconstruction tasks? Would this correction lead to improved 3D reconstructions as well? If not, why?", "limitations": "The only limitation I can think of is the scope of this work; computed tomography represents a relatively niche area in the fields of machine learning and computer vision.", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper reveals the view-dependent integration bias in 3D Gaussian Splatting (3DGS), which, to my knowledge, has not previously been reported in the community. This discovery may have a high impact to computer vision. \n- While this is not the first paper to apply 3DGS to computed tomography, it is the first to accurately reconstruct 3D volumes with an image formation tailored specifically for this task. \n- The proposed method is well evaluated and compared against other baseline methods. \n- Exposition is clear. The paper really reads well.", "weaknesses": "I don’t see any particular weakness of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your recognition of our work and your valuable feedback.\n\n> Q1.1: How does the proposed method compare to FDK when a large number of projections are used? \n\nWe further evaluate FDK and our method with 500 to 2000 views. Results in Tab. A show that our method outperforms FDK by a large margin in all settings. Additionally, our method achieves a peak PSNR of around 39.85 dB while FDK is at approximately 37 dB.\n\n*Table A. Quantitative results of FDK and our method under dense-view scenarios.*\n\n| No. views      | PSNR (FDK)$\\uparrow$ | SSIM (FDK)$\\uparrow$ | PSNR (Ours)$\\uparrow$ | SSIM (Ours)$\\uparrow$ | Time (Ours) |\n| -------------- | -------------------- | -------------------- | --------------------- | --------------------- | ----------- |\n| 50 (reference) | 26.5                 | 0.422                | **37.98**             | **0.952**             | 8m14s       |\n| 500            | 34.04                | 0.755                | **39.73**             | **0.964**             | 8m33s       |\n| 1000           | 36.67                | 0.899                | **39.84**             | **0.963**             | 9m5s        |\n| 1500           | 36.89                | 0.913                | **39.84**             | **0.963**             | 8m49s       |\n| 2000           | 37.00                | 0.919                | **39.85**             | **0.963**             | 9m22s       |\n\n\n> Q1.2: What are the implications of correcting the integration bias in image-based 3D reconstruction tasks?\n\nWhile RGB-based 3D reconstruction is out of our scope, we share some preliminary findings. We compare vanilla 3DGS and rectified one (R-3DGS) on NeRF synthetic dataset. We define the geometry field as the sum of Gaussian opacities, the same as SUGAR (CVPR'24). For vanilla 3DGS, we compute the mean of recovered 3D opacities of all training views. We then use our voxelizer (Sec. 4.2.2) to query opacity volumes and extract meshes using marching cubes (MC). Note that because the actual iso-value of the surface is unknown, we report chamfer distances (CD) with three MC thresholds. \n\nResults in Tab. B and Fig. 3 (PDF) show that correcting the integration bias does not harm 2D rendering. Furthermore, it improves 3D reconstruction, though less significantly than in volumetric CT reconstruction. We suspect three reasons. First, for opaque objects, Gaussians are trained to be flat, so integration values ($\\mu$ in L182) do not change significantly in front views. Second, Gaussians are close to the surface, allowing for reasonable surface extraction using only positions. Third, the splatting technique involves many simplifications in rendering equations, which may have more impact than integration bias on 3D reconstruction.\n\nSince these findings are preliminary, we do not include them in this paper. We will make efforts to develop a bias-free 3DGS for RGB-based reconstruction in future research.\n\n\n*Table B. Quantitative results of vanilla 3DGS and rectified one (R-3DGS) on NeRF-synthetic dataset.*\n|                          | Vanilla 3DGS | R-3DGS     |\n| ------------------------ | ------------ | ---------- |\n| 2D PSNR$\\uparrow$        | **31.46**    | 31.28      |\n| 2D SSIM$\\uparrow$        | 0.966        | **0.967**  |\n| No. Gaussians            | 285k         | 345k       |\n| CD (MC=5.0)$\\downarrow$  | **0.0182**   | 0.0202     |\n| CD (MC=10.0)$\\downarrow$ | 0.0179       | **0.0147** |\n| CD (MC=20.0)$\\downarrow$ | 0.0172       | **0.0141** |", "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": "FRFuQkMsOu", "reviewer": "Reviewer_w7Ci", "summary": "### Motivation\n- The authors propose to adapt 3D Gaussian Splatting (3DGS) to sparse-view tomographic reconstruction, i.e., to recover a radiodensity 3D volume from a small set of X-ray images and corresponding sensor information. This is relevant for various clinical and industrial applications.\n\n### Contributions\n- Their _R2-Gaussian_ model iterates over existing 3DGS solutions tuned for XR/CT imaging, proposing a 3DGS initialization scheme better suited for tomographic reconstruction.\n- The authors also correct an integration bias in 3DGS (meant for faster image inference but causing ambiguities in volumetric reconstruction) and provide the corresponding CUDA patch.\n- The proposed _R2-Gaussian_ includes other adaptations (custom densification parameters, voxel-based regularization, simplified isotropic kernels), resulting in an end-to-end CT reconstruction system.\n\n### Results\n- The authors provide extensive qualitative evaluation and quantitatively compare to other NeRF-based and traditional CT reconstruction methods, showing that their method provides a better trade-off between volume accuracy and reconstruction time.\n\n### Relevance\n- Effort in applying 3DGS to XR/CT imaging has grown the past year [6, 27, 39], as 3DGS appears to be a well-suited representation for such applications (due to its compactness, fast convergence, etc.). This work is a meaningful iteration and could benefit the community.", "questions": "_see **Weaknesses** for key questions/suggestions._\n\n### Q1. Typo?\n- [L223] Do the authors mean \"Compton scatter\" rather than \"ponton scatter\"?", "limitations": "Some limitations and societal impacts are discussed in detail (see **S4** above).", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "_(somewhat ordered from most to least important)_\n\n### S1. Convincing Comparative Evaluation and Qualitative Results\n\n- The quantitative comparison to state-of-the-art CT reconstruction methods appear convincing, with the proposed solution demonstrating a better trade-off between volume accuracy and reconstruction time.\n- The authors provide a lot of meaningful qualitative results, to illustrate theoretical contributions, to highlight the benefits of their method, but also to showcase its limitations. This makes reading this paper the more interesting.\n- An ablation study w.r.t. some of the key contributions and w.r.t. some hyperparameters is also provided.\n- Though limited in number (15 volumes), the authors evaluate on different categories (animal, vegatal, and synthetic targets).\n\n\n### S2. Iterative yet Meaningful Contributions Towards 3DGS for XR/CT\n\n- The discussion w.r.t. the integration bias in vanilla 3DGS is interesting, and the technical solution brought by the authors appear valuable to the community. As mentioned in the paper, their corrected CUDA implementation could benefit other 3DGS works targeting volumetric reconstruction.\n- The authors propose an initialization scheme dedicated to volumetric tomography, as usual 3DGS initialization techniques (e.g., SfM) are not applicable here. This is a relevant contribution, properly described and evaluated (qualitative + quantitative evaluation).\n- The proposed system, tackling xrays-to-CT reconstruction in an end-to-end differentiable manner, is indeed novel. Existing 3DGS models [6, 27, 39] for XR/CT imaging rather focus on digitally-reconstructed-radiograph (DRR) novel-view synthesis (NVS) rather than CT reconstruction.\n\n\n### S3. Sound Theory and Reproducibility\n\n- Background theory is well described by the authors, and the scientific/technical insight of the authors w.r.t. the identified integration bias could benefit the community.\n- The authors provide their model implementation, which appears sound and well-structured (note that I did not try to run the code, but had a look at key files).\n\n\n### S4. Detailed Discussion of Limitations and Contributions\n\n- The authors put significant effort in discussing and illustrating some of their method's limitations (needle-like artifacts inherent to 3DGS, varying convergence time, limited extrapolation ability, etc.), as well as summarizing its impact (possible clinical/industrial applications, benefit of their CUDA code to the CV community, etc.) in Appendices G and H.\n\n\n### S5. Well-Written and Illustrated Paper\n\n- Overall, the paper is nicely structured, written, and illustrated. E.g., Figures 2-4 are helpful to understand the methodology at a glance.", "weaknesses": "_(somewhat ordered from most to least important)_\n\n\n### W1. Lack of Consideration for Real-World Noise and Anisotropic Effects\n\n- The authors claim that \"X-ray attenuation depends only on isotropic density\" [L140] to justify their radiodensity-based model, but this is not entirely correct. While most models generating digitally reconstructed radiographs (DRRs) from CT volumes indeed consider x-ray attenuation as an isotropic phenomenon, this is a approximation. Some x-ray transport effects, such as Compton scattering, are actually anisotropic (but because CT volumes do not inherently contain the material information necessary to accurately simulate Compton scattering, DRR models ignore the anisotropy part) [a]. \nHowever, the authors do claim that they trained their model on DRRs generated using TIGRE [5] configured to simulate Compton scattering [L217-224]. Does it mean that the authors preprocessed the CT volumes to replace the attenuation values by material information (e.g., mapping HU values to a set of predefined materials)? More information on the data generation process would be helpful here.\nIf indeed the model was trained on DRRs containing anisotropic residual noise (c.f. Compton effect's residual impact on XR imaging), but the proposed algorithm itself only consider isotropic attenuation, how does it impact the results? E.g., it could be interesting to generate 2 sets of input DRRs (one generated with the approximated/simplified attenuation model and one more realistic) and compare the final accuracy of the reconstructed CT volumes.\n- The fact that the method is only applied to synthetic inputs (DRRs generated by TIGRE) rather than real, usually noisier, X-ray images is also problematic. The paper would benefit from a real-world evaluation, or at least a discussion on why it was not performed.\n\n\n### W2. Lack of References/Comparisons to SOTA on XR-3DGS\n\n- The authors mention some of the existing 3DGS solutions applied to CT/XR imaging (e.g., X-Gaussian [6], GaSpCT [39], Li et al.'s model [27]) [L89-94] but do not perform any form of comparison with those. The lack of qualitative/quantitative comparison is fair (the authors argue that these models \"cannot generate CT models\" [L91-92], which is somewhat true ; though a comparison on the XR-NVS task could have made this paper stronger). But I would argue that the authors should have better referenced some of these works in the Methodology. E.g., even if less formalized, radiative Gaussians are already presented in X-Gaussian [6] ; and even though it is performed in the pixel domain rather than voxel one, GaSpCT [39] already proposes a total variation loss to regularize their XR-3DGS.\nWhile I still believe that the proposed work is a valuable iteration over these works (by better formalizing and adapting the Gaussian properties and rasterization to CT data), I think the authors should be more transparent w.r.t. the SOTA.\n- References w.r.t. total variation (TV) theory are also missing, making it hard to contextualize the scope of the authors' contribution w.r.t. the objective function.\n\n\n### W3. Somewhat Overstated Contributions\n\n- The CUDA implementation of the radiodensity voxelization seems to be more a technical feat rather than a scientific contribution. While possibly valuable to the community, I do not see any novelty in this module (maybe not as GPU-optimized, but differentiable point-cloud-to-voxel-grid tools already exist, e.g., in PyToch3D).\n- Changes to the adaptive control are unclear/minor, according to the authors' descriptions [L210-214] (i.e., changing the size threshold w.r.t. pruning large Gaussians, editing the density of cloned/split Gaussians).\n- The positive impact of the TV regularization does not appear that statistically significant (+0.32dB for PSNR, +0.009 for SSIM, +1m33s for convergence). Maybe some qualitative results could help grasp its contribution?\n\n\n### W4. Limited Dataset Size\n- The quantitative evaluation is performed on only 15 samples, even though varied. Larger CT datasets are available, e.g. CTPelvic1K [b]. \n\n### W5. Limited Impact of Integration Bias Correction (?)\n- The authors claim that \"this integration bias, though having a negligible impact on imaging rendering, leads to significant inconsistency in density retrieval\" [L184-186], but they only provide qualitative imaging results (Fig. 6) to justify their un-biasing contribution. Additional results could better contextualize the corresponding claims.\n\n\n----\n\n### Additional Reference:\n\n[a] Gao, Zhongpai, et al. \"DDGS-CT: Direction-Disentangled Gaussian Splatting for Realistic Volume Rendering.\" arXiv preprint arXiv:2406.02518 (2024).\n\n[b] Liu, Pengbo, et al. \"Deep learning to segment pelvic bones: large-scale CT datasets and baseline models.\" International Journal of Computer Assisted Radiology and Surgery 16 (2021): 749-756.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your positive review and valuable feedback.\n\n> Q2.1: Did the authors preprocess the CT volumes? \n\nYes we convert raw volumes from HU to attenuation coefficients. Following [62, 7, 27], we then normalize voxel values to [0,1] for balanced evaluation across modalities. We will add more details in the revised manuscript.\n\n\n> Q2.2: How does anisotropic residual noise impact the results?\n>\n> Q2.3: The paper would benefit from a real-world evaluation.\n\nWe address two questions together since they all relate to scattering effects. We agree that anisotropic effects, such as Compton scattering, occur in real-world X-ray imaging. We follow most CT reconstruction work [13,2,50,61,62,7], assuming that the target radiodensity field to be isotropic, and treating scattering as a noise source on the detector. \n\nWe do consider scattering in the experiments. \n* When preparing X-ray projections, we follow conventions [a] to model scattering noise with Poisson. \n* As requested, we further evaluate our method on real-world data containing scattering effects. We use FIPS [b], a public dataset providing real **2D** X-ray projections. FIPS includes three objects (pine, seashell, and walnut). Each case has 721 projections in the range of $0^{\\circ}\\sim 360^{\\circ}$. Since ground truth volumes are not available, we use FDK to create pseudo-ground truth with all views and then subsample 75/50/25 views for sparse-view experiments. We report the quantitative and qualitative results in Tab. 1 (PDF) and Fig. 1 (PDF). Our method outperforms baseline methods in 75- and 50-view scenarios. In 25-view, our method slightly underperforms IntraTomo but is 11$\\times$ faster. Overall, our method shows superior performance and efficiency in the presence of real-world noise and scattering effects.\n\n> Q2.4: The authors should better reference existing X-ray 3DGS works.\n\nWe will enhance the comparison with existing X-ray 3DGS methods by adding more details in the related work section. Please note that all X-ray 3DGS works were preprinted/under review before the NeurIPS submission deadline.\n\nWe would also like to clarify the following points regarding the comments:\n\n* Radiative Gaussian: Although X-Gaussian and our method coincidentally use the same term, the motivations and formulations are quite different.\n  \n  * X-Gaussian replaces view-dependent spherical harmonics with a view-independent feature vector based on the isotropic assumption. It retains color and opacity, which do not physically represent radiodensity field. Besides, it uses alpha-blending, which contradicts the unordered nature of X-ray imaging.\n  * We define the Gaussian kernel as a local radiodensity field and derive new rendering equations (Eq. 7), demonstrating that summation should be used instead of alpha-blending (L176-178). See Q3.1 (Reviewer P8A3) for more details.\n  \n* Total variation (TV): Our work does not claim contributions to TV regularization. Instead, we use TV to demonstrate the possibility of applying voxel-based supervision to Gaussians, thanks to the differentiable voxelizer. To our knowledge, we are the first to do so.\n\n> Q2.5: References w.r.t. TV are missing.\n\nWe will add reference [c] w.r.t. TV.\n\n> Q2.6: I do not see novelty in the radiodensity voxelization.\n\nWe conclude the technical novelty of our voxelizer as the first differentiable CUDA-accelerated one for 3D Gaussians. The voxelizer offers opportunities to apply other voxel-based losses (such as SDF supervision) to 3D Gaussians, which can benefit the community.\n\n> Q2.7: Changes to the adaptive control are unclear/minor.\n\nWe made minor modifications to adaptive control to suit X-ray imaging. We do not intend to claim a contribution regarding adaptive control. We will clarify it in the revised manuscript.\n\n\n> Q2.8: Some qualitative results could help grasp the contribution of TV regularization.\n\n\nWe show qualitative results with and without TV in Fig. 12. It is clear that adding TV promotes smoothness and homogeneity. However, needle-like artifacts still persist, consistent with the general acknowledgment that high-level priors such as TV do not significantly improve performance. As mentioned in Q2.6, we do not claim contributions to TV but rather to a novel strategy for applying 3D losses.\n\n> Q2.9: Larger CT datasets are available.\n\nWhile there are larger CT datasets, such as CTPelvic1K, they only cover human organs with similar structures and materials. We focus on method's representation capability of arbitrary objects. Therefore, we chose data across various modalities, aiming at diversity rather than quantity. Besides, NeRF-based baseline methods typically require hours for training (SAX-NeRF needs 13 hours). We unfortunately do not have sufficient resources to support thousand-level experiments.\n\nCompared with previous work, our dataset has the same size as SAX-NeRF (15), and is larger than NAF (5) and X-Gaussian (5).\n\n> Q2.10 Additional results could better contextualize the claims of integration bias correction.\n\nWe show more quantitative and qualitative results regarding integration bias in Tab. 2 (PDF) and Fig. 2 (PDF). Our method achieves better results than X-3DGS in both 2D and 3D. This suggests that correcting integration bias improves both image rendering and volume reconstruction in CT. Note that this conclusion slightly differs from L253-254, and we will update it in the revised manuscript.\n\n\n\n> Q2.11: Typo: \"Compton scatter\" or \"ponton scatter\"?\n\nWe use Poisson to model photon statistics on the detector, which also includes Compton scattering. We will use \"photon statistics\" or \"Compton scattering\" for clarity.\n\n\n---\n\n**Reference**\n\n*[a] Zhu, Lei, et al. \"Noise suppression in scatter correction for cone‐beam CT.\" Medical physics (2009).*\n\n*[b] Siltanen, Samuli, et al., \"FIPS: Open X-ray Tomographic Datasets.\", Zenodo (2022)*\n\n*[c] Rudin, Leonid I., et al. \"Nonlinear total variation based noise removal algorithms.\" Physica D: nonlinear phenomena (1992).*", "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": "6VZB5wBzX9", "reviewer": "Reviewer_P8A3", "summary": "The paper aims to achieve high tomographic reconstruction performance with a limited number of views in a time-efficient manner.\nTo this end, the paper modifies 3DGS for X-ray projection by adjusting the rendering equation, correcting 2D projection errors, and using voxelizers for regularization.\nThe experiments demonstrate the efficacy of the proposed method.", "questions": "In the SAX-NeRF paper, the performance gap between SAX-NeRF and NAF is significant.\nHowever, in the R^2 Gaussian paper, the performance gap between them is not significant.\nCould you elaborate on why this happens?", "limitations": "The paper clearly states the limitations and the potential societal impact of the work.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "First of all, the paper is well written.\nThe problem in the current tomographic representation is clearly stated, and the authors' objectives are sufficiently addressed.\nThe overall structure is easy to understand, and the ablation study covers most of the arising questions.", "weaknesses": "One critical weakness of this paper is the existence of prior work using 3DGS for X-ray projection.\nAlthough the paper seems structurally well-written, the paper seems less novel due to the presence of X-Gaussian.\nX-Gaussian, which has been accepted to ECCV 2024, employs a similar method.\nMore importantly, X-Gaussian achieved a PSNR of 43 in Human Organ reconstructions, whereas this paper (R2 Gaussian) achieved a PSNR of 36, despite slightly different experimental settings.\nIf the authors provide persuasive explanation on the novelty of this paper, I am willing to raise my score.\n\nAdditionally, it would be helpful to augment the related works and baseline models.\nFor example, C^2RV (CVPR 2024) is another recent tomographic representation model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed review. The comments and suggestions are helpful in improving our paper.\n\n> Q3.1: Novelty comparison w.r.t. X-Gaussian.\n\nOur method demonstrates considerable novelty compared to the concurrent work X-Gaussian for the following reasons:\n\n* **Broader task scope**: Our R$^2$-Gaussian is designed for both X-ray view synthesis and direct 3D CT reconstruction, whereas X-Gaussian only supports 2D X-ray novel view synthesis.\n\n* **Theory-supported model design**: Our R$^2$-Gaussian has successfully extended 3DGS to 3D CT reconstruction with a theoretically sound approach,   including new Gaussian kernels, new splatting equations, and a voxelization strategy.  All of these are grounded with careful theoretical derivations. In contrast, X-Gaussian empirically modifies and extends 3DGS for X-ray view synthesis applications, with limited novel theoretical contribution.\n \n* **Theory contribution**: Our method provides novel and original theoretical results,  including the new derivation of X-ray rasterization and the identification (and the remedy) of the previously overlooked integration bias in the standard 3DGS technique. X-Gaussian, on the other hand, does not offer theoretical contributions in this regard,  despite that it indeed represents  the first successful yet empirical adaptation and application of 3DGS to X-ray view synthesis. \n\n* **Efficient CT reconstruction**:  Our method can directly output 3D CT volumes.  In contrast, X-Gaussian augments novel-view projections first,  and then relies on other existing CT algorithms for CT reconstruction.\n\nIn summary, our method offers notable technical and theoretical contributions compared to X-Gaussian. The methodological comparison between the two methods has been discussed in L89-94. We will further highlight our novelty and contributions w.r.t. X-Gaussian in the revised manuscript.\n\n\n> Q3.2: X-Gaussian achieved a PSNR of 43, whereas this paper achieved a PSNR of 36.\n\nActually, the reported 43 (in Tab. 1 of the X-Gaussian paper) is the **2D PSNR** of novel-view image rendering quality, rather than the **3D PSNR** of the CT volume reconstruction quality.  As a mater of fact,  X-Gaussian only reported their 3D PSNR as 30.56 dB (from 5+95 view, as shown in Tab. 2 in the X-Gaussian paper),  which is significantly lower than our 3D PNSR at 36 (or 36.89 dB, for human organ, in Tab. 3 of our paper).  \n\n\n> Q3.3: It would be helpful to augment the related works and baseline models, such as C$^2$-RV (CVPR'24).\n\nThank you for this suggestion. We will include these SOTA related works in our revised manuscript.  These will add value and authority to our current paper. \n\nRegarding baseline selection, we did not include supervised learning methods like C$^2$-RV because they require external datasets for pre-training. Our primary focus is on evaluating the method's representation capability for arbitrary objects without pre-training.  For a fair comparison, we choose self-supervised learning methods that require only X-ray projections of objects (L483-484).  To our knowledge, SAX-NeRF [7] (CVPR'24) is the latest SOTA work, and we have included it in our baseline methods.  Experiments show that our method also outperforms SAX-NeRF, with a 0.93 PSNR increase and 78$\\times$ faster training speed.\n\nPlease also note that C$^2$-RV has not released their code and models (empty GitHub repository). We tried to contact the authors but received no response. Due to the limited time available for rebuttal, it is unfortunate that we could not reproduce their method and compare it with our method experimentally.\n\n\n> Q3.4: Could you elaborate on the performance gap between SAX-NeRF papers and your paper?\n\nWe use the official code of SAX-NeRF [7] and NAF [62] to perform experiments without changing the network or hyperparameters. We show human organ results from the SAX-NeRF paper [7] and our paper, which use the same source (Tab. C). SAX-NeRF performs consistently in both papers, while NAF in our paper has a higher PSNR. Nevertheless, both the SAX-NeRF paper and our paper conclude that SAX-NeRF achieves better results than NAF. Therefore, the slight performance inconsistency does not harm the arguments made in our paper.\n\n*Table C. PSNR values in SAX-NeRF paper [7] and our paper.*\n\n|         | NAF in [7] | SAX-NeRF in [7] | NAF in ours | SAX-NeRF in ours |\n| ------- | ---------- | --------------- | ----------- | ---------------- |\n| Jaw     | 34.14      | 35.47           | 35.01       | 35.37            |\n| Foot    | 31.63      | 32.25           | 31.65       | 31.90            |\n| Head    | 36.46      | 39.70           | 38.90       | 39.51            |\n| Chest   | 33.05      | 34.38           | 33.99       | 34.45            |\n| Average | 33.71      | 35.44           | 34.85       | 35.29            |", "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": "1RGE5Trdma", "reviewer": "Reviewer_DGKr", "summary": "The paper introduces R2-Gaussian, a framework for tomographic reconstruction using 3D Gaussian splatting (3DGS). This framework aims to address the limitations of traditional 3DGS in volumetric reconstruction, specifically for tasks like X-ray computed tomography (CT).", "questions": "- After the voxelization from Gaussians, can the final density volume be compatible with traditional CT scans and be viewed in CT software?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Identifying and addressing the integration bias in standard 3DGS formulation for volumetric reconstruction\n- The proposed R2-Gaussian framework is developed with tailored Gaussian kernels, rectified projection techniques, and a CUDA-based differentiable voxelizer.\n- The paper provides simulated X-ray validation, comparing the proposed method against state-of-the-art techniques", "weaknesses": "- The proposed method shows results in experimental settings. However, its performance in real-world clinical or industrial scenarios is not thoroughly examined. For instance, real X-ray images are given to reconstruct the CT scan. \n\n- I am not sure if 75, 50, and 25 views of X-rays are considered sparse-view reconstruction and if it is practical to have 75, 50, or 25 views of X-rays for CT reconstruction. \n\n- In the proposed method, the kernel formulation removes view-dependent color. However, it cannot model the scattering effect in X-ray.\n\n- In Fig. 7, it is unclear how to get X-3DGS slices and what \"queried from three views\" means in detail. Why not implement the voxelization on X-3DGS (or vanilla 3DGS) for a fair comparison?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed review.\n\n> Q4.1: The performance in real-world clinical or industrial scenarios is not thoroughly examined. For instance, real X-ray images are given to reconstruct the CT scan.\n\nWe further evaluate our method on real-world data. We use FIPS [b], a public dataset providing real **2D** X-ray projections. FIPS includes three objects (pine, seashell, and walnut). Each case has 721 projections in the range of $0^{\\circ}\\sim 360^{\\circ}$ captured by Hamamatsu Photonics C7942CA-22. Since ground truth volumes are unavailable, we use FDK to create pseudo-ground truth CT volumes with all views and then subsample 75/50/25 views for sparse-view experiments. We report the quantitative and qualitative results in Tab. 1 and Fig. 1 in the attached PDF file. Our method outperforms baseline methods by a large margin in 75- and 50-view scenarios. In 25-view, our method slightly underperforms IntraTomo but is 11$\\times$ faster. Overall, our method shows superior performance and efficiency in the presence of real-world noise and scattering effects. We will include these results in the revised manuscript.\n\n> Q4.2: I am not sure if 75, 50, and 25 views of X-rays are considered sparse-view reconstruction and if it is practical to have 75, 50, or 25 views of X-rays for CT reconstruction.\n\n* Practicality: As mentioned by Reviewer sKx6, in industrial and medical applications, CT machines typically take hundreds to thousands of X-ray projections for high-quality details [a]. Therefore, modifying existing CT machines to 75/50/25 views is practical and convenient by setting different scanning intervals.\n* Rationale: The community considers fewer than 100 projections as sparse-view CT (SVCT). We list the numbers of projections used in some published papers, as shown in Tab. D. These papers use 20-180 views. Accordingly, we set our projections to 75, 50, and 25 views. Additionally, there are works investigating extremely sparse-view CT (ESVCT), which uses 2-10 views [58,30,10]. ESVCT is a severely ill-posed problem that cannot be solved without fine-grained prior knowledge, such as pretraining with external datasets. Therefore, ESVCT is out of the scope of our study, as we only use projections (completely self-supervised). Overall, our experimental setting aligns with previous work and should be considered sparse-view.\n\n*Table D. Number of projections used in published papers.*\n\n| Paper          | Publisher | Number of projections |\n| -------------- | --------- | --------------------- |\n| DD-NET [b]     | TMI'20    | 60-180                |\n| IntraTomo [61] | ICCV'21   | 20                    |\n| NEAT [c]       | TOG'22    | 25-50                 |\n| NAF [62]       | MICCAI'22 | 50                    |\n| SAX-NeRF [7]   | CVPR'24   | 50                    |\n\n\n> Q4.3: It cannot model the scattering effect in X-ray.\n\nWe follow most CT reconstruction work [13,2,50,61,62,7], assuming that the target radiodensity field to be isotropic, and treating scattering as a noise source on the 2D detector. \n\nAlthough we do not explicitly model scattering effects, we take it into consideration in the experiments. When generating X-ray projections for synthetic datasets, we model scattering noises with Poisson. We also evaluate our method in the real-world data which contains scattering effects (Q4.1). All results demonstrate our method's superior performance and robustness to scattering noise.\n\n> Q4.4: It is unclear how to get X-3DGS slices and what \"queried from three views\" means in detail. \n\nWe will improve Fig. 7 captions and relevant descriptions for better understanding.\n\n* X-3DGS slice: After recovering the 3D density of each Gaussian (L251-252), we use the same voxelizer (Sec. 4.2.2) as R$^2$-Gaussian to extract CT volumes. We then show slices of these volumes in Fig. 7 to demonstrate the reconstruction quality.\n* \"Queried from three views\" means that we show reconstruction results from three different views to demonstrate the view inconsistency in X-3DGS.\n\n> Q4.5: Why not implement the voxelization on X-3DGS (or vanilla 3DGS) for a fair comparison?\n\nFor X-3DGS, we implement the same voxelizer (Sec. 4.2.2) in R$^2$-Gaussian to extract a CT volume. We will clarify it in the revised manuscript.\n\n> Q4.6: After the voxelization from Gaussians, can the final density volume be compatible with traditional CT scans and be viewed in CT software?\n\nYes the reconstructed volume is compatible with traditional CT scan viewers. We show a screenshot of inspecting reconstructed volumes with the Weasis DICOM medical viewer in Fig. 4 (PDF).\n\n> Q4.7: Ethics review.\n\nAll data used in our experiments are from open-source datasets. We have properly cited the relevant references (Appx. B) and adhered to the respective data licenses.\n\n---\n\n**Reference**\n\n*[a] Villarraga-Gómez, Herminso, and Stuart T. Smith. \"Effect of the number of projections on dimensional measurements with X-ray computed tomography.\" Precision Engineering 66 (2020): 445-456.*\n\n*[b] Zhang, Zhicheng, et al. \"A sparse-view CT reconstruction method based on combination of DenseNet and deconvolution.\" _IEEE transactions on medical imaging_ 37.6 (2018): 1407-1417.*\n\n*[c] Rückert, Darius, et al. \"Neat: Neural adaptive tomography.\" _ACM Transactions on Graphics (TOG)_ 41.4 (2022): 1-13.*", "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:
Dear Reviewers,

Thank you for your insightful comments and constructive suggestions. We appreciate Reviewers sKx6 and w7Ci for recognizing our paper's solid technical contribution, high impact on related areas, excellent evaluation, and good writing. We are grateful for the positive feedback from all reviewers on our novel CT reconstruction framework and the discovery of integration bias.

Based on these valuable suggestions, we have added more experimental analysis and clarified important concepts. Please note that we add some figures and tables in the **PDF** file. Here is a summary of the changes:

* **Experiments on real-world data (Reviewer w7Ci and P8A3)**. We further evaluate our method on the real-world dataset FIPS [a]. The results in Tab. 1 (PDF) and Fig. 1 (PDF) show that our method outperforms baseline methods in the presence of real-world noise and scattering effects.
* **Clarification of our contribution w.r.t. existing X-ray 3DGS works (Reviewers w7Ci and DGKr)**. We summarize our primary contribution as the first theory-supported 3DGS framework for direct CT reconstruction, and the discovery and remedy of previously overlooked integration bias. We provide a detailed comparison between our method and X-Gaussian in Q3.1 (Reviewer P8A3).
* **Demonstration of integration bias (Reviewers w7Ci and DGKr)**. We have added more quantitative and qualitative results (Tab. 2 and Fig. 2 in PDF) to demonstrate the necessity of correcting integration bias.

We hope our response has addressed the initial concerns. Please let us know if you have any other questions.

Kind Regards,

Authors

---
**Reference**

*[a] Siltanen, Samuli, et al., "FIPS: Open X-ray Tomographic Datasets.", Zenodo (2022)*

Meta-review: This paper initially received mixed scores with two postive (strong accept, accept) and two negative (borderline reject, borderline reject) reviews. Some reviewers praised the clarity of the writing and felt the proposed application of 3DGS to medical imaging could have significant impact. Reviewers also raised concerns about novelty compared to X-Gaussian and evaluation on real-world data. The authors provided a rebuttal, and after the discussion period the reviewers felt that their concerns were addressed. The paper is accepted, and the authors should incorporate the results and clarifications from the rebuttal into the paper.
Decision: Accept (poster)