X-Field: A Physically Informed Representation for 3D X-ray Reconstruction
Authors: Feiran Wang
Jiachen Tao
Junyi Wu
Haoxuan Wang
Bin Duan
Kai Wang
Zongxin Yang
Yan Yan
Venue: NeurIPS 2025 spotlight ()
Keywords: 3D Reconstruction; X-ray
Primary area: applications

Abstract: X-ray imaging is indispensable in medical diagnostics, yet its use is tightly regulated due to radiation exposure. Recent research borrows representations from the 3D reconstruction area to complete two tasks with reduced radiation dose: X-ray Novel View Synthesis (NVS) and Computed Tomography (CT) reconstruction. 
However, these representations fail to fully capture the penetration and attenuation properties of X-ray imaging as they originate from visible light imaging.
In this paper, we introduce X-Field, a 3D representation informed in the physics of X-ray imaging. 
First, we employ homogeneous 3D ellipsoids with distinct attenuation coefficients to accurately model diverse materials within internal structures. Second, we introduce an efficient path-partitioning algorithm that resolves the intricate intersection of ellipsoids to compute cumulative attenuation along an X-ray path.
We further propose a hybrid progressive initialization to refine the geometric accuracy of X-Field and incorporate material-based optimization to enhance model fitting along material boundaries.
Experiments show that X-Field achieves superior visual fidelity on both real-world human organ and synthetic object datasets, outperforming state-of-the-art methods in X-ray NVS and CT Reconstruction. Our code is
available on the project page: https://github.com/Brack-Wang/X-Field.

X-Field: A Physically Informed Representation for 3D X-ray Reconstruction
Feiran Wang"
Jiachen Tao
Junyi Wu
Haoxuan Wang'
Bin Duan ?
Kai Wang?
Zongxin Yang'
Yan Yan
University of Illinois Chicago National University of Singapore
Universityy of Michigan Harvard Medical School
Imaging
Image
Representation
(a) Visbble Light
O
Reffection
Soattering
Pixei
3D6S
NeRF
(b) X-ray
Moterial
Source
Penstrotion
Material
Material
Altenuction
X- Field
Figure 1: Comparison Imaging Processes and Corresponding 3D Representations. (a) Visible light interacts with surfaces mainly through scattering and reflection. 3D representations of NeRF [ and 3DGS [2) model this process by accumulating directional light rays. (b) Unlike previous approaches, our model represents internal structures using homogeneous ellipsoids capture X-ray attenuation and penetration properties.
Abstract
X-ray imaging is indispensable in medical diagnostics, yet its use is tightly regu- lated due to radiation exposure. Recent research borrows representations from the 3D reconstruction area to complete two tasks with reduced radiation dose: X-ray Novel View Synthesis (NVS) and Computed Tomography (CT) reconstruction. However, these representations fail fully capture the penetration and attenuation properties of X-ray imaging as they originate from visible light imaging. In this paper, we introduce X-Field, 3D representation informed in the physics of X-ray imaging. First, we employ homogeneous 3D ellipsoids with distinct attenuation coefficients to accurately model diverse materials within internal structures. Sec- ond, we introduce an efficient auth-parittioinn algorithm that resolves the intricate intersection of ellipsoids to compute cumulative attenuation along an X-ray path. We further propose hybrid progressive initialization to refine the geometric curacy of X-Field and incorporate material-based optimization to enhance model fitting along material boundaries. Experiments show that X-Field achieves superior visual fidelity on both real-world human organ and synthetic object datasets, out- performing state-of-the-arr methods in X-ray NVS and CT Reconstruction. Our wwtpeauu..ooooooaaa.oee
code is available on the project page:
Introduction
X-ray imaging is cornerstone of Computed Tomography (CT) reconstruction 3 providing critical insights into internal structures for clinical diagnostics. X-rays undergo progressive attenuation
"qqual Contribution |Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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while penetrating materials until the residual energy reaches the detector, forming an X-ray projection. Traditional CT reconstruction relies on hundreds of X-ray projections acquired from various angles to recover a density field 7 g 9 However, acquiring such vast number of projections exposes patients Consequently, X-ray reconstruction 17 aiming to synthesize novel X-ray views from a sparse set of input projections and facilitate the reconstruction of high-quality CT volumes.
to high doses of ionizing radiation, posing health risks
[0] [1]
has gained increasing attention [12 13
[4
15, 16
Recent advances in 3D reconstruction 16]. Yet, current 3D reconstruction techniques were originally designed for visible light imaging. as illustrated in Figure aa). a). When visible light rays interact with a surface, despite minor absorption, most wavelengths are reflected or scattered into multiple directions [99]. As a result, light rays from different directions can accumulate to form a single pixel, producing varying appearances of the same surface depending on the viewpoint. Neural Radiance Fields (NeRF) [ employ deep networks to implicitly encode directional light information at each sampled point, conditioned by spatial positions and view directions to model appearance variation. Similarly, 3D Gaussian Splatting (3DGS) 2 projects anisotropic ellipsoids onto the image plane and accumulates the projected Gaussian splats to determine the pixel colors. Both methods reconstruct scenes considering the accumulation of multi-directional rays, thus well-suited for visible light imaging.
[8] have laid the groundwork for X-ray reconstruction 14
However, X-ray imaging operates on fundamentally different principles from visible light imaging, as depicted in Figure [b.. First, X-rays are a form of high-energ clectromagnetic radiation that is more capable of penetrating objects [20], whereas visible light, emitted from natural sources, has lower energy and is primarily reflected by surfaces. Second, X-rays progressively attenuate as they pass through an object a nearly straight line before reaching the detector on the opposite side. Third, the pixel intensity is primarily determined by energy absorption from the corresponding X-ray, rather than by the combined contributions of visible light from multiple directions.
Therefore, X-ray imaging reveals materials composition with distinct densities distributed throughout the object's interior structure. As X-rays penetrate materials, their energy attenuates proportionally to both material density and path length through each substance. Recent X-ray reconstruction methods build upon 3DGS and achieve efficient imaging. For instance, X-Gaussian [15] replaces the spherical harmonics function with an intensity function to capture X-ray grayscale values, while R²- R²-Gaussian [6] extends this by correcting integration bias to enhance CT reconstruction quality. Yet, these methods represent each material as a Gaussian ellipsoid, depicted as 3DGS representation in Figure [ with maximum density at the center that gradually diminishes toward the periphery, and render through ellipsoid superposition, without accounting for the critical factor of penetration length through each material. To faithfully adhere to the principles of X-ray imaging, X-ray reconstruction demands representation that accurately models the radiological density across various materials.
In this paper, we present X-Field, a physically informed ellipsoid representation specifically designed to model X-ray attenuation as it penetrates materials during imaging. To differentiate various materials, X-Field represents an object's 's structure using ellipsoids with distinct attenuation coefficients. shown as X-Field representation in Figure [] Then, to track the distance an X-ray travels through each material, we derive an explicit form for segment length.
For precise pixel intensity calculation, we express the attenuation integral along the ray path as a finite Riemann sum of the attenuation coefficients and segment lengths. Yet, during optimization, ellipsoids with different attenuation coefficients may overlap. To ensure that each position in 3D space corresponds to a single material, we further design first-pass precedence strategy, resolving ellipsoid ambiguities in overlapping regions. Furthermore, we propose Material-Bann -Based Optimization, which adaptively refines ellipsoid placement by splitting them along material boundaries guided by the local attenuation gradient.
We evaluate X-Field on X-ray reconstruction across various modalities, including real-world human organ datasets and simulated object datasets. Furthermore, we compare X-Field with existing meth- ods in vierse-view CT reconstruction based on Novel View Synthesis (NVS) results. Experiments demonstrate that X-Field outperforms state-of-the-art (SOTA) methods in both NVS and CT recon- struction. Notably, our method achieves optimal results within 10 minutes using only 10 input X-ray projections, delivering substantial PSNR improvements of 2.44 dB for X-ray NVS and 3.98 dB for CT reconstruction over SOTA methods.
2

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Hybrid Progressive Initiolization
World
Physically Informed Ellipsoid Representation
Inpet X-ray Images
Material
Materiil
Intorsectton
Reg
Comaeed
Ne
Maserial-Adaptive Ellipooid
Coloalo Lengh
(c) Material- Bassed Optimization
Filber
gind
Rondar
Initialized
Distribution
Segnent Length
Segment lengtha
Fitting
Figure Overview of X-Field. (a) Hybrid Progressive Initialization. We begin with X- ray images construct a coarse initialization using combined iterative methods. (b) Physically Grounded Ellipsoid Representation. transform initialized ellipsoids into NDC space and associate them with pixels. then compute the attenuation integral along the ray, considering segment length and ellipsoid intersections. (c) Material- -ased Optimization. Our optimization captures detailed material boundaries for high-quality rendering.
Related Work
X-ray Novel View Synthesis.
3D X-ray reconstruction consists of two tasks: X-ray novel view synthesis and CT reconstruction [1]. X-ray NVS [14 13 [3 16 is essential for medical diagnostics, as it reconstructs images from sparse-view inputs, significantly reducing patient exposure X-ray radiation. Motivated by the fact that visible light and X-rays are both electromagnetic waves, two representation methods originally designed for visible light have been adopted for X-ray reconstruction: Neural Radiance Fields (NeRF) M. which employ multi-layer perceptrons for implicit encoding, and 3D Gaussian Splatting (3DGS) which uses Gaussian ellipsoids for explicit representation. NeRF-base methods, such as NAF 13, improve efficiency by incorporating hash tables, while SAX-NeRF [14 introduces transformer architecture better model 3D structural dependencies.
3DGS-based approaches, such as X-Gaussian [5], replace the spherical harmonics color in 3DGS with response function, addressing differences between visible light color and X-ray intensity. R2_ Gaussian [6] further corrects integration bias for more accurate density retrieval. However, neither representation has a comprehensive consideration the attenuation and penetration properties the X-ray. Instead, we introduce connuattio-adaptii cllipsoids with distinct attenuation coefficients and consider the X-ray penetration length within the materials, ensuring more accurate reconstructions.
Computed Tomography Reconstruction, such medical diagnosis 66
reconstruction is of vital importance for domains 24 biology 22] 26 27), industrial inspection [28] 29)]
21
22
23
and security screening 30 Early CT reconstruction methods 20 can be categorized into analytical approaches (3 and iterative techniques
32 33) Analytical methods such FDK perform a filtered back-projection of acquired projections. Iterative methods such as SART 4 and CGLS (34
reconstruct CT images by iteratively refining the solution through algebraic projection corrections and solving least-squares optimization problem. respectively. However, these methods typically require hundreds thousands of X-ray images, leading to increased radiation exposure.
Fortunately, X-ray NVS offers an effective alternative to CT reconstruction. By generating diverse it enables reconstruction of high-quality CT volumes from limited data. We further reduce the number of required X-ray images to as few as five while achieving superior performance both qualitatively and quantitatively compared to other baselines, highlighting its practicality for real-world applications.
novel views from sparse X-ray inputs [5] 16] 14
I7 35
3 Method
In this section, we introduce our Physically Informed Ellipsoid Representation Sec. 3.1 including (1) the formulation of X-ray Physical Field, (2) the Attenuation-Adan Adaptive Ellipsoids, (3) the algorithms for calculating segment lengths with intersections, and (4) the overlap filtering method. Then, we propose Hybrid Progressive Initialization in Sec. 3.2 and Material-BB -Based Optimization in Sec. 3.3

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3.1 Physically Informed Ellipsoid Representation
X-ray Physical Field. X-ray imaging quantifies the cumulative attenuation X-rays as they traverse an object, governed by the material-dependent absorption properties of the medium. To formally describe this process, we define the X-ray Physical Field, where each spatial position X R is
characterized by the local energy absorption rate o (x) E R+ of X-ray (20 S 36]. When X-rays with an initial intensity To propagate through the X-ray Physical Field, their energy progressively attenuated by the materials they traverse. Ultimately, the remaining intensity forms projection image I E H Mathematically, we represent an X-ray path as r(t) o o+ id E R where denotes the X-ray source position, d is the unit view direction vector, and f varies between the entry lo and exit in points the object. According to the Beer-Lambert law [37] the X-ray intensity l'(r) after attenuation is given by:
(((r) - exp (((")((()))
In practice, the data is typically processed in logarithmic space for computational efficiency (00
(r) log Lo log (or) 1/mm (r(()) dt.
Thus, each pixel intensity (x) in the projection image aggregates the material absorption along the X-ray path, providing the foundation for reconstructing internal structures.
Attenuation-Adane daptive Ellipsoids. Inspired by the infinitesimal method spatial distribution of materials using set of N ellipsoids {E₃}. Each ellipsoid E; encodes material properties and contains center position Pe and covariance matrix ΣsD. To maintain physical consistency with X-ray imaging, we further assign distinet, non-negative attenuation coefficient to characterize the local absorption behavior. Given an X-ray path that traverses multiple ellipsoids, the corresponding pixel intensity is determined by the accumulated attenuation along the path:
we represent the
((r) f/m a(((t)) dt
1'h a(r(t)) dt +....
o (r(()) dt An-
frmm dt
tu
00 dl on-
to
kn-
oolo + o1h + On-1hn-1,
4++1 4 2 0 represents the segment length of the X-ray path within ellipsoid E, Therefore, accurately measuring the accumulated attenuation requires precise determination each segment length 4, In the following, we derive explicit formulation for 4.
where 4;
Explicit Form of Segment Lengths. Our objective is to compute the segment length 4 for ellipsoid E, along the view direction d, process illustrated in Figure (b). To achieve this, we first transform E, from world camera coordinates and then into Normalized Device Coordinates (NDC) space, aligning X-rays with the coordinate axes for computational simplicity. To establish reference for general segment length computation, we define the maximum segment length /max as a pivot, which corresponds to the ray path passing through the center of E,:
mmxx
V d ddTE, I
Given Inax- we then compute the segment length 4 for an arbitrary point u inside the projected ellipse of E, in NDC space. The relationship is formulated as:
14 = Imax x V1 10-(c-1)) B²
where
(5)
A
A A Eddd B B C= a Tyja.
(6)
Here, a u Pe is the displacement vector from the center of the ellipsoid Pe to the point u. The full derivation of mmxx and I, is provided in the supplementary materials.

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Segment Lengths with Intersections. As ellipsoids adjust their positions and sizes during opti- mization, intersections may occur, leading ambiguities the attenuation rates within overlapping regions. To ensure that each location corresponds to a single dominant material, resolving these complex overlaps is crucial to maintaining physically consistent representation.
For optimization stability, we adopt simple yet effective first-pass prece- dence strategy. Given two intersecting ellipsoids along the X-ray path, we first sort them in ascending depth or- der, denoted by E, and E where E, appears first along the ray. The overlapping region 6, then exclu- sively assigned to E, by adjusting the effective region of E to E Cz. This modification directly impacts the segment length computation, requir- ing additional updates to ensure con-
Algorithm Segment Length Correction with Intersections
Input: (zo, R1, An-1 ): Sorted depths
(lo /1 11....................... n Initial segment lengths
Output: Updated segment lengths i, 4....,, -
I:
for = 0 to n do
2:
if 0 then
3:
l - In
4:
20, lo
5:
else
if 2, < 2 + / then
7:
1. + max (0, (z+ + 44) - (= + 4)
sistency in attenuation accumulation, as shown in Figure 2b.). To system- atically handle different intersection scenarios, introduce an efficient correction function f. as outlined in Algorithm 1. The final accumulated attenuation along the X-ray path is then expressed as:
8:
else
9:
4 5 4-min(4, (54 + 21) - (=+ + 20)
10:
end if
11:
if 4 # 0 then
12:
[z, + 44 - 1,, 21 + [44 i + 2z, 1 +
13:
end if
14: end if
15:
end for
n-1 ((r)
Σ
0. f
2 VA
C B²
Physically Faithful Overlap Filtering. While resolving segment length inconsistencies ensures a physically valid attenuation model along the X-ray path, accurately associating ellipsoids with their corresponding pixels equally critical for maintaining consistency the projected space. Current methods often rely on coarse bounding-box approximations, leading to unintended pixel assignments that compromise reconstruction fidelity. To address this, we introduce a refined Oriented Bounding Box (OBB) strategy that establishes precise pixel-ellipsoid associations. As illustrated in Figure 3. existing approaches 2 [5] [6] typically rely on Axis-Alignn Aligned Bounding Boxes (AABB) [39], which approximate the projected ellipse using circumscribed circle. However, this method includes extrameous pixels (red region) that do not truly overlap with the projected ellipse, leading to physically inconsistent assignments. In contrast, we first determine the OBB via eigenvalue decomposition, aligning it with the principal axes of the projected ellipse. Then, we filter out irrelevant pixels while preserving only the valid tiles (green region) that closely related the actual projected ellipse. By eliminating spurious pixel association, our method enhances the physical consistency and numerical stability of the reconstruction process.
3.2 Hybrid Progressive Initialization
It is widely acknowledged that initializing point clouds with detailed geometric information accel- crates model convergence [5] 40]. However, conventional initialization techniques fail in the context of X-ray imaging. Specifically, SIM-based methods [41 2 estimate point clouds by feature matching across multiple images. While in X-ray imaging, variations
ray paths significantly alter the remaining energy intensity, leading feature mismatches and unreliable correspondences.
Similarly, point cloud regression methods such as Dust3R 42
43 44 directly infer 3D structures from images but fail in X-ray reconstruction due domain shifts, as they are trained natural images and do not account for X-ray-speet -rra-sic attenuation properties.
To address this challenge, we design a Hybrid Initialization strategy that progressively refines the geometric structure through a sequence of iterative methods. We compare the results in Figure
Our process begins with Conjugate Gradient Least Squares (CGLS) [5] 34 20), which efficiently
5

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Baseline Methods
(b) With OBB
Method
Random
FDK
Ours
Figure 3: Pixel-Ellipse Association. (a) AABB 33 results in incorrect associations. (b) OBB pixels. (c) Ours keeps only aligned pixels.
Figure 4: Initialization Comparisons. Ours produces informative geometry prior, compared to Random X-Gaussian [5 and FDK of R ---uussiann [T]]
provides coarse global estimate. Next, we refine this estimate using Simultaneous Algebraic Reconstruction Technique (SART) 4 leveraging its capability to enhance local details and correct early-stage inconsistencies. Finally, we incorporate Total Variation (TV) regularization [46] into the initialization process to suppress noise and artifacts, preserving critical structural edges. For each 3D point obtained from our initialization, we define an ellipsoid centered that position, assigning it a random covariance matrix ΣιD and an attenuation coefficient o₁. We subsequently devise an optimization method for these parameters for better modeling the detailed material distribution.
3.3 Material-Based Optimization
Mainstream optimization strategies 2
15
16
Geometry Based Opt.
Material- Based Opt.
40] involve splitting and cloning in regions with poor structural fidelity while pruning with neg- ligible opacity. However, this geometry- based optimization fails capture the material com- position along materials' edge. As depicted in Figure the ground-truth object consists of three distinct materials, represented by different optimization
Split and Clone
split
Fitting
Initializationn Fitting Geometry
Distribution
Figure 5: Illustration of Adaptive Optimization
colors. While the geometry- successfully reconstructs the overall shape, it captures only two intermediate material types, as shown in Figure 5)b), highlighting the limita- tions of purely geometry centric approaches in faithfully modeling material compositions.
Strategy. Ground-truth Material Distribution with different colors indicating distinct materials. (b) Geometry-Ban -Bsed Optimization (2). which fits the ellip- soids closely to the object geometry. (c) Our Material- Based Optimization, which further refines the ellip- soids capture the material distribution.
The challenge arises from the continuous fluctuation of an ellipsoid's attenuation coefficient, which hinders stable convergence and prevents accurate material differentiation. To address this, we propose a material-basee distributions inherent in the X-ray Physical Field. Our approach is designed to identify material boundaries through local density estimation. Concretely, we randomly sample a subset of ellipsoids and compute the k-nearest neighbors [48), averaging their distances and analyzing the attenuation gradient. Regions with high density and steep gradients indicate material transitions, necessitating finer adjustment. Therefore, to enhance fidelity, selectively split ellipsoids in these complex regions, scaling each by a factor of 1.6, which follows empirical heuristics in prior studies [5] 16]. strategy significantly improves material separation, faithfully
optimization strategy that explicitly accounts for the heterogeneous material
As presented in Figure 5)c). capturing the internal composition of objects.
4 Experiment
4.1 Experiment Settings
we conduct experiments on large-scale X3D dataset containing 15 scenes with two collections: Human Organs, derived from real-world medical datasets, to evaluate model performance the medical domain; and Daily Objects, generated from synthetic datasets, to assess generalization ability. Specifically, chest scans are sourced from LIDC-IDRI [49,, pancreas CT scans from Pancreas-CT [50]. The remaining objects are obtained from VOLVIS [51 and the open scientific visualization dataset [52] We adopt the tomography toolbox TIGRE 53 to capture projections from CT volumes the range of 0' 180 with minor
Dataset. Following the literature convention 14
scatter and electronic noise. For highly sparse-view novel view synthesis, 5 and 10 views used for

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GT
Sax-NeRF
X- G-Gaussian
R-Gaussian
Ours
Head
Figure 6: Qualitative Comparison of NVS. We present visual examples of reconstructed images across four cases trained with views. Our results demonstrate superior visual quality, richer details, and fewer artifacts.
G G coc
O
GooDo
GOODO
Head
Teapot
X-Gaussian
R--Gousian
Figure 7: Qualitative Comparison of CT Reconstruction. Our method produces clearer textures, more refined anatomical structures, and fewer artifacts, particularly in high-contrast regions such as the cranial cavity.
training and 50 views are used for testing. To further assess model performance and scalability, we generate 50, 25, and 15 views for evaluating the performance under sparse-view synthesis.
Baselines. We compare X-Field with state-of-the-art 3D X-ray reconstruction methods, including TensoRF [12. NeAT 177. NAF 13), SAX-NeRF [14). X- X-Gaussian (15), and R R-GGaussian [16].
TensoRF, NeAT, NAF, and SAX-NeRF are SeRF-based methods designed for efficient reconstruc- tion, with SAX-NeRE achieving SOTA performance among them by incorporating transformer architecture as the model backbone. X-Gaussian and R²-Gaussian are 3DGS- methods, where X-Gaussian focuses NVS, and R²-Gaussian extends its applicability to CT reconstruction by introducing voxelization. We also compare against traditional CT reconstruction methods including FDK and SART The novel view images are obtained by leveraging TIGRE for rendering.
ratio (PSNR) [54] to assess the quality rendered images, structural similarity index measure (SSIM) [55 to measure consistency between predicted images and ground-truth and Learned Perceptual Image Patch Similarity (LPIPS) [56] to analyze the perceptual quality in high-level feature space. For clarity, report LPIPS LPIPS* LPIPS x x10º instead.
Metrics. We adopt peak signal-to-nn
Implementation Details. X-ray novel view synthesis, we evaluate learning- based methods using their official implementations. For CT reconstruction, we follow [17
14 15] to synthesize
novel-view X-ray images from sparse inputs and reconstruct CT volumes using total of 100 images. All experiments were conducted single RTX 6000 Ada GPU.

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Human Organ t0 views Lts
Daily Object tO views LG)
Hummn
Orean 5 views [6]
Daily Object 5 views [16]
Method
PSNRT
SSIMT
LPIPS* L
PSNRT
SSIMT
LPIPS"L
PSNRT
SSIMT
EPIPS' 3
PSNRT
SSIMT
1PPPS
Truditional Machodd
HOK
3
12.35
0.675
291.2
16.52
0.716
259.1
8.15
0.618
310.6
14.42
0.688
283.7
SART
1
13:25
0
264.8
17.69
0,724
247.3
9.31
0.634
203.4
15.65
0.663
290.5
Derp based Methods
Temani
B.98
1E2.
18.27
210.3
NeAT
0.924
155.2
0.857
17.29
211.3
17.89
8.925
193.2
25.44
151.9
0.894
17:02
208.5
SAX-NeRF
14]
19.32
0.945
186.
25.38
0.979
143.6
14.18
0.901
191.
19:09
0.948
204.9
X Gusssicn
13
22.88
0.947
130.3
22.91
0.982
79.12
17.23
0.947
176.4
20.31
0.961
108.1
R
Guussinn
Lo
38.72
B1967
85.97
41.93
0,986
54.31
31.12
0.956
109.7
34.52
0.965
82.46
Ours
35.71
0.580
21.03
42,50
0,953
45.64
32.34
0.963
103.2
37,61
0,970
51.02
Table I: Quantitative Comparison of NVS. We compare our X-Field with: (a) Traditional analytical method:
FDK, SART. (b) Deep Learning- Searning-based methods: TensoRF, NeAT, NAF SAX-NeRF, X Gausssian and
R²-Gaussian. We report LPIPS* LPIPS x10?. We mark out best and second best method for all metrics.
4.2 Comparison with State-of-the-Art Methods
We present the quantitative results of X-ray NVS and CT reconstruction, discussing the qualitative results across all scenes, highlighting the superior performance of X-Field.
Discussion on NVS Quantitative Results. We compare X-Field with two traditional methods (FDK, SART), three NeRF-based methods (TensoRF, NeAT, NAF), and three SOTA methods (SAX- NeRF, X-Gaussian, and R²-Gaussian). Table reports the quantitative results of highly sparse views (10 views and 5 views) X-ray NVS. Note that we report quantitative results as the mean results across all scenes under the same setting, and scene-wise results are presented in the supplementary material. X-Field demonstrates superior performance in reconstructing X-ray novel views across most scenarios, surpassing the state-of-the-a R² RP-Gaussian in all metrics. Specifically, in the human organ reconstruction setting, X-Field consistently outperforms the state-of-the-ard R² R~-Gaussian, achieving higher SSIM scores and competitive PSNR and LPIPS values, highlighting its effectiveness for complex reconstruction scenarios.
Discussion on NVS Qualitative Results. Figure 6 presents visual comparisons between X-Field and multiple methods, including SAX-NeRF, X- X-Gaussian, and R R=-Gaussian. These highly sparse view settings provide limited information, resulting in artifacts of varying severity across all methods. SAX-NeRF reconstructs the overall structure but introduces noticeable blurry artifacts, particularly in the bonsai scene. X-Gaussian produces line and wave-pattern artifacts, which are prominent in the head scene. While R² R--Gaussian performs better than the other basclines, it exhibits flaws in the fine details of the bone structure. For example, R² Gaussiin introduces black linear artifacts in the bone region in the foot scene, where ours produces smoother textures. In summary, our method is able to effectively mitigate blurry, line, and wave-pattern artifacts while maintaining smoothness in distinct areas and on object surfaces. More visual comparisons are provided in supplementary video.
Discussion on CT Reconstruction Results. Following the tradition [4 15 16), we also compare X-Field with traditional learning-free algorithms. FDK (3 and ASD_POCS [57]. as well as learning- based methods including X- X-Gaussian [15] and R Ra-Gaussia [66] for sparse-view CT reconstruction. Specifically, we assess the performance of JDGS-basce methods and X-Field by generating novel view images from sparse input projections (5 and 10 views) and reconstructing CT scans using ASD PDC_ with a total of 100 views. The quantitative results, presented in Table 2 demonstrate that X-Field consistently achieves the best performance across all scenarios. In particular, when generating novel X-ray projections from 10 input views, X-Field combined with ASD POCS achieves PSNR of 33.26 and SSIM of 0.911, a comparable result with the SOTA method, RaGasssin
Figure 7 shows qualitative results of vier CT reconstruction of foot, head, and teapot scans using input views. Without novel view projections, FDK introduces streak artifacts, while ASD POCS results in blurred structural details. The NVS of X-Gaussian produces clearer recon- structions but introduces shadow artifacts. In contrast, our method and R2 RGGaassia achieve superior visual quality, with X-Field delivering better results by recovering smoother textures and reducing niedle-like artifacts, especially in delicate structures such as the cranial region.
4.3 Ablation Study
To comprehensively assess the performance of X-Field, we evaluate the impact of the proposed components, compare different initialization strategies, and evaluate X-Field under various input view settings from 5 to 50 views.
8

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iVViews
Method
PSNR
SSIM 1
LPIPS"
Method
PSNR 1
SSIM
PSNR
SSIM T
Opt.
34.78
FDK
15.20
0.144
19.95
0.257
w/o Overlap Filter
34.59
0.937
74.32
ASD_POCS POCS
24.89
0.731
27.50
0,787
X-Gausiia
17.57
0.688
18.44
0.537
w/o Segment Length
w/o Intersection
27.48
33.84
0.929
0.875
76.60
87.85
R² Gaussian
26.83
0.804
29.28
0.946
Ours
35.03
0.953
72.12
Ours
28.04
0.815
33.26
0.911
Table 3: Ablation on Components. Best performance
Table 2: CT Reconstruction Comparison. Ours
when all components included.
achieve comparable results under 5 and 10 views.
#Views
PSNR f
SSIM
LPIPS"
Initialization
PSNR
SSIM |
LPIPS*
5
31.61
0.933
92.23
Random
37.85
0.966
60.02
10
35.11
0.959
83.72
FDK 3
37.96
0.967
59.81
25
41.71
0.979
75.26
Ours
38.67
0.969
59.95
50
42.61
0.993
61.19
Table 4:
Initialization Methods. Our initialization
improves reconstruction accuracy.
Table 5: Number of Input Views.
Performance im-
proves input views increase.
Component Analysis. Table 3 evaluates the effects of individual components on reconstruction performance. We observe that Material-Basee Optimization and Overlap Filter have limited impact on reconstruction quality. Removing the Intersection Module (Sec. 3.1) leads to substantial performance drops, with PSNR drop of 1.19 dB, and an SSIM decrease of 0.024, underscoring its importance in preserving structural integrity.
Segment Length is fundamental component of our ellipsoid representation, defined in Sec. 3.1 for capturing the distance each ray traverses within the ellipsoid. Removing also severely impacts the model's s ability to render novel views, leading to significant performance degradation, with 7.6 dB drop in PSNR and an 11.71 increase in LPIPS. These results underscore the importance of Segment Length in enabling X-Field to achieve high-quality reconstructions.
Initialization Analysis We compare our hybrid initialization with random initialization used in X- X-Gaussian 155 and FDK 3 introduced in q2-Guassian [16]. As shown in Figure ours provides a point cloud with clearer boundaries and reduced noise, facilitating initialization. Table shows that both FDK and our hybrid initialization outperform random initialization. While FDK achieves slightly lower LPIPS, also results in lower PSNR and SSIM. contrast, ours improves both PSNR and SSIM, demonstrating its effectiveness in enhancing reconstruction quality.
Input View Number Analysis. To further demon- strate the scalability of X-Field, we conduct experi- ments to assess the effect of the input view number on reconstruction performance. As shown in Table as the number of input views increases, the performance is consistently enhanced. When using 50 views as input, X-Field achieves performance comparable work designed for sparse view X-Ray reconstruction 14]. We further show Figure that, with the increase in the input number, the synthesis X-ray images exhibit smoother and clearer bone textures.
10
Figure Input View Numbers. Five views able good NVS, additional input views en- hance texture smoothness.
5 Conclusion
We present X-Field, a physically informed representation derived from the penetration and attenuation properties of X-rays for X-ray imaging. To model diverse materials within the internal structures of objects, we introduce attenuation-aaaatiie ellipsoids with distinct attenuation coefficients. To compute pixel intensity, we design a segment length algorithm that incorporates ellipsoid intersections, enabling accurate estimation of each material's 's X-ray energy absorption. To improve reconstruction performance, we optimize the ellipsoids along material boundaries and refine the geometric properties during initialization. Our proposed X-Field significantly outperforms state-of-the-an methods in both X-ray NVS and CT reconstruction, demonstrating strong potential for medical applications. Furthermore, we provide insights into the fundamental design of X-ray imaging, which can be extended to other tasks such as the reconstruction of translucent objects.

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Acknowledgments: This research is supported by NSF IIS-2525840, CNS CNS-2432534, NIH IRFIMH133764-01 and Cisco Research unrestricted gift. This article solely reflects opinions and conclusions of authors and not funding agencies.
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20

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Technical Appendices and Supplementary Material
Overview
Detailed Derivations for Segment Lengths $
3DGS Preliminaries $
Implementation Details $ C
Dataset C.I
Slyper-parameters
Additional Quantitative Results:
Ablation on 50 Views $ D.I
Ablation on Computational Efficiency $ D.2
Statistical Significance Tests for Hybrid Initialization
D.3
Component Efficiency Analysis $ D.4
World Data Validation $ D.5
Quantitative Comparison across Scenes $ D.6
Additional Qualitative Results
More Discussions
F
Limitations
Ethics Considerations $
F2
A Detailed Derivations for Segment Lengths
providing a convenient framework for deriving explicit segment length formulas. In this space, the segment length 1; for each ellipsoid E; is calculated along the view direction d of a given ray path. Leveraging the orthographic property of NDC space, the 3D ellipsoid E; can be projected onto 2D plane, forming an ellipse. The segment lengths are then constrained within this 2D ellipse, with values outside the ellipse
In this work, we define the ray space using normalized device coordinates (NDC) 16
Our goal is to find the relationship between the segment length and the point u = (zz y, 0) on the ellipse traversed by the ray. Below, first derive the special value 4max. which is the maximum distance a ray travels through the ellipsoid when originating from the center the ellipse [58]. Then, use Imux bridge express & for rays originating from other points the ellipse.
Consider a ray originating from the center of the ellipse u, and traveling along direction d. The modeling of the ray is: R(s) = ue + sd E IR3 where s is scalar parameter. Meanwhile, the ellipsoid centered at Pe can be described as
(p pp-Pel Pe) E (p Pc) = 1.
(8)
Substituting p = =R(s) into above equation, we have s as in
(ue sd Pe) "EE (ue (ue + sd pc) =
In the NDC space, translating an object does not affect its geometric length. Therefore, without loss of generality, we can assume that Pe lies on ues and then have:
(ue + sd P.) En
(ue (ue + sd Pa) sdd E =
(10)
Solving this simplified equation, we can obtain the roots:
where
A=d d=d- I.
(11)
VA
The segment length corresponding u. is the difference between the two roots; Imax
|s2
81
21

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Next, we consider a general point u on the ellipse plane. The ray originating from u in the direction can be represented by R(s) = u + sd. Similarly, we define a u Pe and substitute R(s) into
the ellipsoid equation:
(a + sd) T EDD'le (a sd) =
(12)
Expanding the equation results in:
s²d Σ 3D d + 2sa DnD'd + a- 1.
(13)
For simplicity, we denote B=a T En and C=a T En a. Thus, the equation becomes As 2Bs C - = 0, whose roots are:
B + B A(C-1)
81,2
(14)
The segment length then be derived as:
C B² A
4 =
82
81
Imxx
X
V
(15)
B 3DGS Preliminaries
3D Gaussian Splatting (3DGS) 2 represents a scene collection of anisotropic Gaussian kernels in 3D space, denoted as Q [ Each Gaussian is parameterized by its center position, covariance matrix, color, and opacity. By splatting these kernels onto the image plane using a differentiable rasterizer, 3DGS enables efficient view synthesis and supports end-to-end optimization.
Ji=1
Formally, an RGB image Lgb E #x rendered from the 3D Gaussians as
Lrgh C(P(T)((((T
(16)
where T, P, and correspond to the transformation, projection, and composition modules.
Transformation. The set of 3D Gaussians is first transformed into the viewing ray coordinate system:
G. =T(+)). This step aligns the kernels with the camera rays to reduce computation overhead.
Projection. The transformed Gaussians are then mapped onto the image plane: G = P(G)). The resulting 2D Gaussians retain the opacity and color of their 3D counterparts, while the depth dimension in position and covariance is discarded.
Composition. Finally, image is synthesized by alpha-compositing the 2D Gaussians in a back- to-front order: Lb C(G²). The differentiable rasterizer enables the optimization the kernel parameters using gradient-based losses.
C Implementation Details
C.1
Dataset
We conduct experiments the large-scale X3D [14] dataset with multiple modalities across two categories: Human Organ (Chest, Head, Foot, Jaw, Pancreas, Abdomen, aneurism, Leg) and Syn- thetic Object (Bonsai, Teapot, Engine, Backpack, box, carp, Pelvis). The chest scans sourced from LIDC-IDRI [49), the pancreas scans from Pancreas-CT [59]. and the remaining cases from SciVis 600. Following prior works [15 16 we employ the TIGRE tomography toolbox (53) to capture 512x512 projections across a range of 02 to 180%, including both in in-distribution and out-of-distribution views.
View Distribution. For training, we uniformly capture 5, 10, 15, 25, and 50 views within 09 to 180 For testing, randomly capture 50 views in the same range.
Views. We also evaluate robustness of our method on out-of-distribution views. To construct these views, we apply consistent algorithm across all cases. Specifically, for each case, we calculate the Oriented Bounding Box (OBB) [47] of the trained ellipsoid distribution to determine its center and principal axis. Next, select direction d orthogonal to this principal axis.
22

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Starting from the center, we move along d by radius r to establish the initial camera position. The camera then rotates 360 around the principal axis form the view distribution. Due to differences in direction and radius, the resulting new view distribution is entirely distinct from both the training and testing sets. This setup comprehensively evaluates the model's robustness and consistency under unseen views.
C.2 Hyper-parameters
In our experiments, the position learning rate density and scaling learning rates were both initialized and maintained at 0.01 and 0,005, respectively. The structural dissimilarity loss weight (ADSSIM) was 0.25. To refine the density of kernels, minimum density threshold of 1 x 10 was applied. The total number of kernels was capped at 500,000 to balance computational efficiency with accuracy.
an initial and final value of 0.0002, while the
D Additional Quantitative Results
D.1 Ablation on 50 views
We evaluate our method under the 50-view setting and compare with state-of-the-art baselines. As shown in Table our approach achieves the best performance in PSNR and LPIPS while maintaining SSIM comparable to X-Gaussian. These results, together with the 5- and 10-view experiments, demonstrate that our method consistently outperforms existing approaches across different input sparsity levels, from highly sparse moderately dense scenarios.
Method
SSIM
PSNR
IPIPS
SAX-NeRF
0.975
28.53
124.10
X- X-Gaussian
0.997
32.78
70.52
R² Gaussian
0.982
42.61
64.38
Ours
0.993
42.65
61.19
Table 6: Novl View Synthesis with 50 input views.
D.2 Ablation on Computational Efficiency
We compare the computational efficiency of different approaches as shown in Table 7 Our method achieves training and memory consumption on the same order as 3DGS- based methods, while being significantly more efficient than NeRF-based baselines. In terms of inference, X-Field supports real-time rendering, offering practical balance between quality and speed for clinical deployment.
NAF
SAX-NeRF
X-Gaussian
Rd Gausssin
Ours
Training Time
38m 57s
307m 50s
1m 55s
3m 52s
4m
Inference Speed
2.07 fps
0.36 fps
132 fps
72 fps
45 fps
Memory Usage
24 GB
27 GB
824 MB
968 MB
1456 MB
Table Computational profile comparison. Training and inference statisties on RTX 6000 Ada GPU,
D.3 Statistical Significance Tests for Hybrid Initialization
As shown in Table all tests indicate statistically significant improvements of our hybrid initialization over both Random and consistency across all evaluated scenes demonstrates the reliability of the initialization strategy. This suggests that even small improvements in early-stage initialization can provide stable advantage for subsequent optimization.
baselines. While the average gains are relatively modest, their
Metric
Ours vs Random
Ours vs FDK
p-val
p-val
PSNR 1 SSIM LPIPS
+0.84
0.0002
+0.42
0.0016
+0.0071
0.0005
+0.0035
0.0012
-0.53
0.0007
-0.29
0.0013
Table 8:
Statistical significance of hybrid initialization.
23

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D.4 Component Efficiency Analysis
Table reports runtime and memory for different ablation settings. Removing segment length yields the largest savings but leads to severe quality degradation, accurate wath-dependent attenuation cannot be computed. Excluding material optimization provides moderate efficiency gains but com- promises material-aware reconstruction. Overlap filter removal has little effect training time yet increases memory usage, mainly affecting efficiency. Removing intersection computation offers mod- est speedup with reduced memory, indicating moderate overhead. Overall, the full model introduces limited additional cost while preserving reconstruction fidelity.
Method
R GGusssin
w/o Material Opt.
w/o Overlap Filter
w/o Intersection
w/o Segment Length
Full
Training Time
3m52s
3m158s
4m12s
4m04s
3m146s 945 MB
4m15s
Memory Usuge
968
1174 MB
1359 MB
1368 MB
1456 MB
Table 9: Training time and memory usage under different ablations.
D.5 Real-World Data Validation
We evaluate our method on the FIPS dataset [61], which provides real 2D X-ray projections. As shown in Table TO X-Field consistently outperforms baseline approaches under both 5-view and 10-view settings. Moreover, the performance trends observed on synthetic benchmarks generalize well to real projection data containing scattering, beam hardening, and detector noise, demonstrating that our approach remains robust and effective under practical real-world conditions.
Method
5-view SSIM
10-view SSIM
PSNR
IPIPS
PSNR
LPIPS
0.963
162.23
0.978
141.50
X- Gaussian
25.68
0,967
121.62
29.24
0.980
76.94
R² Gaussian
36.42
0.970
71.25
38.87
0.984
34.77
Ours
37.56
0.981
70.41
40.26
0.989
34.22
Table 10: Performance on the FIPS real projection dataset.
D.6 Quantitative Comparison across Scenes
Tables aa and present the scene-wise guantitative results for 10-view and 5-view training data, respectively. In the 10-view setting (Table IT), our method achieves the best performance in most scenes across all metrics, including PSNR, SSIM, and LPIPS. Specifically. for challenging scenes such as the pancreas and bonsai, our method consistently outperforms all others, demonstrating its ability to handle complex structures effectively. The R Gaussian method performs well in certain scenes, such as the foot and engine, achieving competitive SSIM and LPIPS scores. However, our method maintains an edge overall quality, solidifying its superiority in reconstructing finer details.
Foot SSIMT
Head
Chest
Jaw SSIM7
Methal
PSNRT
LPIPS*
PSNRT
SSIMI
LPIPS"
PSNRT
SSIMT
LPIPS'J
PSNRT
1 1PPPS
SAX-NNRF
0.943
17.25
18,07
367.9
X-Ginssiin
17.66
8.042
120.6
17.30
136.4
0.930
17.93
237.2
k" Coussion
[b
34.46
0.911
80.75
34.00
0.972
$6.25
35.46
0.949
100.4
30.15
0.909
175.3
Ours
35.54
0.999
75.56
35.71
0.975
76.84
34.03
0.944
91.15
29.32
0.10
164.3
Bonsai
Toaget
Fagine
Method
PSNR1
SSIM
LPIPS*
PSNR1
SSIM
LPIS*1
PSNR1
SSIM
LPPSS*
PSNRF
SSIM1
1PPSS
NcRE
18.78
0.999
21.28
174.5
23.67
0,983
21.89
127.2
X-Cuassin
22.14
Guassinn
8.98
7933
42.12
Ours
0.069
8125
43.06
45.50
0.088
41.79
20.18
Table 11: Results of Quantitative Comparison on 10 views (8
D.6).
We compare our X-Field on 10 views with
Deep Learning- based methods: SAX-NeRF [14), X-Gaussian 15
and RD Ra--ssan [6] LPIPS = LPIPS
x x10 best and second best methods are marked.
In the 5-view scenario (Table 12]. the performance gap between methods decreases due to the limited training data. Nonetheless, our method achieves the best results in most scenes, particularly excelling in the chest, bonsai, and teapot, where it surpasses other methods across all metrics. While the Ra-Gssssian method demonstrates strong performance in specific scenes, our approach consistently proves to be more robust. These results underscore our method's adaptability to reduced training data while maintaining ffate-of-the-n performance.
24

---

Foot
Head
Chess
Jaw SSIMT
Method
PSNRT
SSIMT
L
PSNRT
SSIMT
LPISS"
PSNRT
SSIMT
EPIPS'J
PSNRT
1PEPS
16.16
5224
X Guassian
13
18.88
0.955
147.8
13.19
0.929
1772
13.01
0.824
201.1
16.13
0.918
306.7
Caussion
[b
30.32
8.9
117.2
31.25
0,958
108.6
27.83
0.862
169.1
26.43
0.850
232.9
Ours
3038
0.901
1162
31.85
0.900
97.53
29.73
0.899
146.4
25.16
0833
227.5
Tageet
Frpgine
Method
PSNR1
SSIM
LPIPS*
PSNR1
SSIM
LPIES1
PSNR1
SSIMS
LPIPS
PSNRF
SSIM
LPIPS*
ScRE
18.57
0,924
19.25
20966
0,975
21.87
116.2
X-Caussion
8:927
145.5
148.2
23.25
8937
Ours
30.99
0.943
1002
34.73
0.965
82.64
26.30
0,400
76.45
32.13
0.970
52.13
Table 12: Results of Quantitative Comparison on 5 views ($
D.6)
We compare X-Field on 5 views with
Deep Learning-basse methods: SAX-NeRF [4]. X-Gaussian x x10 best and second best methods are marked.
[5] and R R=-Gaussian [66] LPIPS* LPIPS
E
Additional Qualitative Results
Out-of-Distribution Results To further evaluate X-ray reconstruction capabilities, we rendered 360-degree images around the scene, We compared X- X-Gaussian, R² Ra-Gaussian and our method with viewpoints being out-of-distribtioon in 5 different scenes, including foot (Figure D), head (Figure TO jaw (Figure I1. and teapot (Figure ??).
Each figure presents five out- oistriiiittttiionn views, with representing methods and columns corresponding to the same view. In the foot scene (Figure 9 X X-Gaussian exhibits black linear artifacts in views and 4. While R²-Gaussian reconstructs the general foot structure, method delivers smoother and more visually consistent results. In the head scene (Figure 10), X-Gaussian displays wave-like artifacts the top of the head, whereas both R²-Gaussian and method achieve improved overall reconstructions. Notably, our method captures finer details, showcasing superior reconstruction quality and generalization. Similarly, in the jaw (Figure ITI scenes, method produces clearer and more complete reconstructions compared to the other approaches.
View
View
View 3
View 4
View
R FFGGaussian X- G-Gaussian
Ours
Figure 9: Qualitative Comparison of Foot on Out-of-Distribation Views.
2
View
X-Gaussian
R Gausssis
Ours
Figure 10: Qualitative Comparison of Head Out-of-Distribution Views.
25

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View
View
4
View
RiGGausiian X- X-Gaussian
Ours
Figure 11: Qualitative Comparison of Jaw on Views.
F
More Discussions
F1
Limitations
Despite the significant advancement achieved by X-Field X-ray sparse-view reconstruction, is not without limitations:
The ellipsoid representation may not be the most efficient. We use ellipsoids to capture the internal material distribution of objects; however, there could exist more efficient form that can reduce the number of parameters while also requiring fewer elements to effectively learn the material distribution.
Lack of leveraging large model prior knowledge. Many recent methods for sparse view reconstruction have utilized prior knowledge from large models to guide their learning processes. In our task, we didn't find suitable large model for X-ray data introduce into our framework. is possible that some large-scale medical models could provide prior knowledge enhance convergence speed and improve overall performance.
F.2 Ethics Considerations
Ethical considerations are critical in developing X-ray reconstruction technologies. Ensuring in- formed consent for transparent communication about and limitations of X-ray reconstruction methods essential to respect privacy and prevent misrepresentation. Secure handling and storage of medical imaging data are paramount safeguard against unauthorized access and misuse. Furthermore, recognizing the potential risks associated with the misuse of advanced imaging technologies, we emphasize the importance of establishing robust ethical guidelines to ensure their responsible applica- tion. Our commitment is to uphold the highest ethical standards in all aspects of X-ray reconstruction. protecting the integrity and confidentiality of patient data.
26


Reviews:
[{"review_id": "rOGOpI3Bd9", "reviewer": "Reviewer_eKjo", "summary": "The paper introduces X-Field, a 3D representation tailored for X-ray imaging tasks. X-Field leverages physically grounded ellipsoids with distinct attenuation coefficients to accurately model material interactions and X-ray attenuation properties. The method employs an efficient path-partitioning algorithm and a material-based optimization strategy to refine the accuracy of reconstructions, significantly outperforming existing methods in X-ray Novel View Synthesis and CT reconstruction.", "questions": "1. What is the computational overhead of X-Field compared to existing methods, particularly for real-time or near-real-time scenarios?\n\n2. How robust is X-Field against varying noise levels and inaccuracies typically encountered in practical medical and industrial imaging applications?\n\n3. Can the authors clarify potential limitations and scenarios where X-Field might significantly underperform or require extensive tuning?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed representation uses ellipsoids with distinct attenuation coefficients to model diverse materials within internal structures.\n\n2. The proposed path partitioning algorithm and hybrid progressive initialization effectively improve the performance.\n\n3. The experiments show a large improvement of the proposed method compared with the other state-of-the-art methods.\n\nWeaknesses:\n\n1. The claim of being “physically grounded” may be somewhat overstated. Representing internal structures with homogeneous ellipsoids does not fully capture the complex physical properties and heterogeneity of real-world materials.\n\n2. A table of comparison under 50 views is desired to further show the ability of the proposed method.\n\n3. Some comparison methods appear to exhibit over-exposure in Figures 6 and 7. It might be beneficial to double-check the normalization and visualization settings to exclude artifacts introduced by inconsistent processing or suboptimal parameter tuning.", "was_revised": true, "final_justification": "The paper introduces X-Field, a 3D representation tailored for X-ray imaging tasks, which shows great results in X-ray Novel View Synthesis and CT reconstruction. During the rebuttal, the authors effectively addressed the initial concerns; thus, the reviewer would like to raise the rating to accept.", "rebuttal": "We appreciate your encouraging assessment and the detailed suggestions.\n# Weakness 1: Clarification on \"Physically Grounded\"\n\"Physically grounded\" reflects our motivation to design a 3D reconstruction system that more closely simulates the actual X-ray imaging process, rather than claiming perfect replication of all physical complexities. Our approach assigns homogeneous attenuation coefficients to each 3D ellipsoid to model diverse materials within internal structures, and our rendering pipeline explicitly calculates X-ray path lengths through each ellipsoid while modeling overlap situations according to Beer-Lambert law. In contrast, previous representations (NeRF and 3DGS) were designed for visible light imaging—NeRF implicitly encodes directional light information through deep networks, while 3DGS uses Gaussian ellipsoids with center-peaked density distributions. Our method specifically incorporates X-ray imaging principles, providing a more physically informed approach.\n\nIf the reviewer's concern relates to the scope implied by \"Physically Grounded,\" we would be open to clarifying this by replacing \"Physically Grounded Representation\" with \"Physically Informed Representation\" in the title and throughout the paper. We welcome the reviewer's feedback on whether this adjustment would better reflect the intended scope of our contribution.\n# Weakness 2: A table of comparison under 50 views\nWe conducted supplementary experiments for novel view synthesis with 50 input views and compared our method against state-of-the-art approaches. The results are presented in the following table:\n\n| Method       | SSIM ↑   | PSNR ↑    | LPIPS ↓   |\n|--------------|----------|-----------|-----------|\n| SAX-NeRF     | 0.975    | 28.53     | 124.1    |\n| X-Gaussian   | **0.997** | 32.78     | 70.52     |\n| R²-Gaussian  | 0.982    | 42.61     | 64.38     |\n| **Ours**     | 0.993    | **42.65** | **61.19** |\n\nOur method achieves the best performance in PSNR and LPIPS metrics, with SSIM performance comparable to X-Gaussian. Combined with the results in Table 1, these findings demonstrate that our method consistently outperforms existing methods across different input sparsity levels, from highly sparse scenarios to moderate input settings.\n\n# Weakness 3: Over-exposure in Comparison Methods\nWe appreciate the reviewer's observation regarding the apparent brightness differences in Figures 6 and 7. We have carefully verified our experimental setup and can confirm that all methods use identical data preprocessing, visualization parameters, and evaluation metrics.\n\nSpecifically:\n\n1. For novel view synthesis in Figure 6: We run SAX-NeRF, X-Gaussian, and R²-Gaussian following their recommended settings and official implementations.\n\n2. For CT reconstruction in Figure 7: All methods follow identical processing pipelines as described in our paper. We synthesize novel view images from sparse input projections and reconstruct CT scans using ASD-POCS with 100 total views.\n\nWe emphasize that all methods employ: (1) identical data preprocessing, (2) consistent visualization parameters, and (3) the same evaluation toolbox. The observed brightness differences reflect the inherent reconstruction characteristics of each method under challenging sparse-view conditions, rather than artifacts from inconsistent processing or parameter tuning.\n\n\n\n\n# Question 1: Computational Overhead of X-Field\nRegarding computational overhead for real-time scenarios, we provide comprehensive timing analysis for reconstructing 10 input views with 5,000 iterations on an RTX 6000 Ada GPU, including training time, inference speed and memory usage:\n\n|                | NAF        | SAX-NeRF   | X-Gaussian | R²-Gaussian | Ours      |\n|----------------|------------|------------|------------|-------------|-----------|\n| Training Time  | 38 min 57s| 307 min 50s | 1min 55s | 3 min 52s  | 4min 15s |\n| Inference Speed |  2.07 fps | 0.36 fps | 132 fps | 72 fps | 45 fps |\n| Memory Usage   | 24GB       | 27GB       | 824MB      | 968 MB      | 1456 MB   |\n\n\n**Real-time Performance:** Our inference speed satisfies real-time requirements for clinical applications while providing substantial quality improvements (Table 1, Figures 6-7).\n\n\n**Training and Memory Efficiency:** Our X-Field achieves similar level in training time and memory consumption with 3DGS-based methods, suitable for clinical workflows demanding rapid deployment and interactive responsiveness.\n\n*We thank the reviewer for this important question and will include this computational analysis in the final version.*\n\n\n# Question 2:  Robustness of X-Field against noise \nOur experiments incorporate realistic noise conditions commonly encountered in practical applications.\n\n**Clinical Data Source:** Following established X-ray reconstruction literature  (SAX-NeRF, X-Gaussian, R²-Gaussian) , we conduct experiments on the large-scale X3D dataset [1], which contains clinical data sourced from real medical CT scans, including chest scans from LIDC-IDRI [2] , pancreas scans from Pancreas-CT [3], and foot/head scans from VOLVIS [4]. These datasets inherently contain practical imaging artifacts and noise from clinical acquisition systems, representing real-world conditions with varying noise levels typically encountered in medical practice.\n\n**Additional Noise Simulation:** The X3D dataset includes added Poisson noise to simulate quantum noise typical in practical medical scenarios, providing comprehensive noise evaluation beyond the inherent clinical noise.\n\nOur evaluation on X3D data, which contains both real clinical noise and simulated quantum noise, demonstrates X-Field's robust performance across multiple noise scenarios, confirming its noise resilience and practical applicability in medical imaging.\n\n# Question 3: Potential Limitations and Scenarios for Underperformance\nWe acknowledge that X-Field, like other 3D reconstruction methods, has certain limitations that may affect performance in specific scenarios.\n\n**Camera Pose Dependency:** X-Field requires accurate camera positions and calibration for precise reconstruction. Errors in camera positioning or pose estimation can degrade reconstruction quality, as inaccurate camera poses lead to incorrect calculations of X-ray path lengths through ellipsoids, preventing accurate optimization of ellipsoids.\n\nThis limitation is common across most 3D reconstruction methods that require precise spatial relationships. Developing robust pose estimation and joint optimization approaches remains an active research direction across the entire field.\n\nWhile our current implementation assumes reliable calibration, integrating pose refinement capabilities represents a valuable extension for handling challenging real-world scenarios with calibration uncertainties.\n\n\n[1] Structure-Aware Sparse-View X-ray 3D Reconstruction. CVPR 24.\n\n[2] The lung image database consortium (lidc) and image database resource initiative (idri): a completed reference database of lung nodules on ct scans. Medical physics 2011.\n\n[3] Data from pancreas-CT. The Cancer Imaging Archive 2016.\n\n[4] Teem nrrd volume visualization datasets. 2022."}, {"review_id": "D8rXZNYsRO", "reviewer": "Reviewer_VmXW", "summary": "This work proposes X-Field, a 3D Gaussian Splatting (3D GS)-based method for reconstructing 3D CT volumes from sparse-view X-ray projections. Technically, the main contributions of this work are as follows: 1) proposing a material-adaptive ellipsoids representation for improved accuracy; 2) building a hybrid progressive initialization pipeline for enhancing initial point clouds of 3D GS; 3) using a material-based optimization to improve compact and efficient representations. On X3D, a large-scale simulated CT dataset, the empirical results show that the proposed X-Field produces state-of-the-art (SOTA) performance in both X-ray novel view synthesis and CT reconstruction.", "questions": "See Strengths and Weaknesses above, please.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n+ In the proposed material-adaptive ellipsoids, the algorithm to partition rays into segments intersecting overlapping ellipsoids is elegant and practical. It can ensure correct X-ray integration and eliminate complex overlaps.\n+ The proposed hybrid initialization is well-engineered and reasonable.\n+ On a simulated dataset, the proposed X-Field performs better compared to the latest 3D GS-based methods, such as X-Gaussian and R^2-Gaussian.\n+ This work conducts comprehensive ablation studies, confirming the effectiveness of several components of the proposed X-Field.\n\n---\n\n### Weaknesses\n\n+ This work claims that the proposed X-Field is a physically grounded representation. However, it merely converts the continuous integral transform (Eq. 2) into a discrete summation (Eq. 3), which is a common practice in previous NeRF-based CT reconstruction methods (e.g., NAF and SAX-NeRF). Moreover, the term “material-adaptive” is very vague. Specifically, by using the segment length, any location in the imaging space is dominated by a single ellipsoid. But why does this make it “material-adaptive”? The term “material” is commonly used in quantitative CT, such as in material decomposition. Here, does “material-adaptive” refer to material decomposition?\n+ The hybrid initialization is fully empirical and involves multiple pre-processing methods. From the results in Table 4, its effectiveness seems to be very minor.\n+ All the experiments are conducted on a single simulated dataset. As a new reconstruction technique for medical imaging, I think it is necessary to test its effectiveness on clinical data.\n+ The results do not report the computational costs (running time and memory footprint) of the proposed X-Field. As a 3D CT reconstruction technique, its computational efficiency is critical for clinical use.\n+ The writing could be improved for clarity. In particular, readers who are not familiar with 3D GS may find the paper hard to follow.", "was_revised": true, "final_justification": "During the rebuttal, the authors have addressed my concerns about \"material adaptive\" and evaluation on real-world data well. Therefore, I have raised my rating to accept.", "rebuttal": "We thank the reviewer for the detailed review. The comments and suggestions are helpful in improving our paper.\n # Weakness 1.1: Physically Grounded Representation\n\nWe acknowledge that discrete summation is common across methods, but our contribution lies in the representation design and rendering approach specifically tailored for X-ray imaging physics.\n\n**Different Origins and Motivations:**\n\n- NeRF-based methods (NAF, SAX-NeRF): \n\nThese originate from NeRF, which is designed for volumetric rendering. They implicitly encode information with networks, predicting radiodensity along rays and accumulating for rendering, rather than explicitly representing material distribution.\n\n- 3DGS-based methods (X-Gaussian, R²-Gaussian): \n\nThey explicitly model with Gaussian ellipsoids, where density peaks at ellipsoid centers and decreases toward edges. They render images by projecting 3D Gaussian ellipsoids into 2D planes in a splatting manner, instead of considering physically accurate ray traversal through 3D space.\n\n- Our approach: \n\nOur work originates from X-ray imaging principles and extends 3DGS for physically grounded X-ray rendering. We use homogeneous 3D ellipsoids with distinct attenuation coefficients representing different materials. Following Beer-Lambert law, we explicitly calculate the path length X-rays traverse through each ellipsoid and compute cumulative attenuation along the entire X-ray path for physically accurate rendering. During optimization, ellipsoids are updated based on X-ray physics constraints to ensure the representation matches the actual X-ray attenuation process.\n\n**Key Physical Modeling Differences:** Our method explicitly implements physically meaningful parameters with homogeneous attenuation coefficients and precise path lengths, rather than generic volumetric rendering or splatting operations. While the mathematical form appears similar (discrete summation), the underlying physical modeling, representation design, and parameter optimization are fundamentally different and specifically grounded in X-ray imaging principles.\n\n\n# Weakness 1.2:  \"Material-Adaptive\" Definition\nFrom a high-level perspective, \"material-adaptive\" refers to our method's ability to automatically adjust ellipsoid representation to capture different material distributions within the X-ray imaging space. Unlike traditional material decomposition in quantitative CT (which separates known materials like bone and tissue), our approach adaptively refines the ellipsoid placement and attenuation coefficients to model diverse internal materials without prior knowledge of their specific types.\n\nThe \"adaptive\" aspect is achieved through our use of homogeneous 3D ellipsoids with distinct attenuation coefficients, positions, and covariances. During optimization, the model updates ellipsoids’ properties, and adaptively clones and splits ellipsoids to model diverse materials within internal structures, ensuring the representation matches the actual material distribution.\n\nThis differs from material decomposition in that we do not decompose into predefined material types, but rather adaptively discover and model the material distribution patterns present in the data through our ellipsoid-based representation and optimization process.\n\n# Weakness 2: Hybrid Initialization \nOur hybrid approach follows a principled coarse-to-fine strategy: CGLS provides global structural estimation, SART refines local details, and TV regularization preserves edges while suppressing noise. Each component serves a specific purpose based on established CT reconstruction principles rather than empirical selection.\n\nOur method requires comparable computational time to FDK but achieves better performance and provides improved geometric priors for the subsequent optimization process.\n\n# Weakness 3: Effectiveness on Clinical Data.\n\nFollowing previous X-ray reconstruction works (SAX-NeRF, X-Gaussian, R²-Gaussian), we use the X3D dataset [1], which includes substantial real clinical data evaluation rather than being limited to simulated data. The X3D dataset contains two collections: Human Organs derived from real clinical medical datasets, and Daily Objects from synthetic datasets, with projections captured using the TIGRE tomography toolbox with realistic scatter and noise modeling.\n\n**Real Clinical Data Sources in X3D:**\n\n- Chest scans from LIDC-IDRI [2]: A widely-used clinical lung CT database containing real patient data\n\n- Pancreas scans from Pancreas-CT [3]: Clinical pancreatic CT data from real medical cases\n\n- Additional human organ data from VOLVIS [4]: Clinical foot, head, and jaw CT scans\n\n**Clinical Data Performance:** Our results on real clinical data are reported in Tables 1, 6, and 7, with detailed scene-wise breakdowns in supplementary materials. The human organ results demonstrate our method's superior performance across most metrics for both 5-view and 10-view inputs, confirming effectiveness on clinical data.\n\n\n\n\n# Weakness 4: Computational Costs \n\nWe provide comprehensive computational cost analysis comparing our method with state-of-the-art approaches for reconstructing 10 input views with 5,000 iterations on an RTX 6000 Ada GPU, including two NeRF-based methods (NAF, SAX-NeRF) and two 3DGS-based methods (X-Gaussian, R²-Gaussian):\n\n|                | NAF        | SAX-NeRF   | X-Gaussian | R²-Gaussian | Ours      |\n|----------------|------------|------------|------------|-------------|-----------|\n| Training Time  | 38 min 57s| 307 min 50s | 1min 55s | 3 min 52s  | 4min 15s |\n| Inference Speed |  2.07 fps | 0.36 fps | 132 fps | 72 fps | 45 fps |\n| Memory Usage   | 24GB       | 27GB       | 824MB      | 968 MB      | 1456 MB   |\n\n\n**Clinical Efficiency:** Our method achieves real-time inference, delivering substantial speedup over NeRF-based methods while requiring significantly less memory. Compared to 3DGS methods, we maintain comparable training efficiency and memory usage while providing superior reconstruction quality (Table 1).\n\n**Clinical Deployment:** The combination of fast training, real-time inference, and moderate memory requirements makes our approach well-suited for clinical workflows where both reconstruction accuracy and computational efficiency are essential.\n\n*We thank the reviewer for this important suggestion and will include this computational analysis in the final version.*\n\n\n\n# Weakness 5:  3DGS Preliminary\n\nThank you for this suggestion. We will add detailed 3DGS background material in the supplementary materials to improve clarity.\n\n\n\n[1] Structure-Aware Sparse-View X-ray 3D Reconstruction. CVPR 24.\n\n[2] The lung image database consortium (lidc) and image database resource initiative (idri): a completed reference database of lung nodules on ct scans. Medical physics 2011.\n\n[3] Data from pancreas-CT. The Cancer Imaging Archive 2016.\n\n[4] Teem nrrd volume visualization datasets. 2022."}, {"review_id": "RUSXnfRS8Y", "reviewer": "Reviewer_u2wa", "summary": "This paper presents X-Field, a novel physically grounded 3D representation for X-ray imaging. The key contributions include the use of homogeneous 3D ellipsoids with distinct attenuation coefficients to model diverse materials within internal structures, an efficient path partitioning algorithm to resolve the intersection of ellipsoids and compute cumulative attenuation along an X-ray path, a hybrid progressive initialization to refine the geometric accuracy, and a material-based optimization to enhance model fitting along material boundaries.", "questions": "Please see the Weaknesses 1 2 3\n1. Clarification of OBB and X-Field's Physically Faithful Overlap Filtering Differences.\n2. Qualitative Results for Ablation Studies.\n3. Consideration of Polychromatic X-ray Sources: Would the  X-Field results in artifact when face the polychromatic X-ray sources?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "### **Strengths**:\n\n1. The uses of material-adaptive ellipsoids with distinct attenuation coefficients, is a novel and physically meaningful approach tailored for X-ray and CT. \n2. The proposed segment length calculation with intersections and physically faithful overlap filtering help calculate pixel values more accurately.\n3. The hybrid progressive initialization helps for efficient optimization and better convergence.\n3. The method demonstrates superior performance in both X-ray NVS and CT reconstruction tasks across various datasets compared to state-of-the-art methods, especially under sparse input view.\n### **Weakness**\n1. Clarification of OBB and X-Field Differences during the splatting process:\nHow to \"filter out the irrelevant pixels while preserving only the valid tiles\" after OBB process.\n2. Lacking qualitative results for ablation studies, which would offer more comprehensive insights into the impact of each component on the overall performance and visual quality of the reconstructions.\n3. The method does not account for the polychromatic nature of X-ray sources, which can lead to beam hardening artifacts.", "was_revised": true, "final_justification": "This paper present a novel design for 3DGS in CT novel view sythnsis.\nConsidering the views of other reviewers that this work seems to combine NeRF's ray tracing integration with 3DGS, I think this is an innovative combination. Because unlike some works that combine NeRF and GS that simply and directly introduce the dual-stream rasterization and ray tracing process, X-Field ingeniously makes up for the shortcomings of 3DGS by integrating only within the ellipsoid. At the same time, it does not have the drawback of slow training introduced by NeRF.\nMy concern of Weakness 1,2,3 has been solved by authors response.  \nTherefore my score is accept(5)", "rebuttal": "Thank you very much for your acknowledgement of our work and your constructive feedbacks and suggestions.\n\n# Weakness 1 & Question 1: OBB vs. Physically Faithful Overlap Filtering \n**Our process after OBB:**\nBuilding upon the eigenvalue decomposition performed during OBB computation, our X-Field method utilizes the extracted principal axis lengths (eigenvalues $\\lambda_1$, $\\lambda_2$) and rotation angle $\\theta$ from the 2D covariance matrix. We compute directional projection extents using the formula:\n\n$$\\text{proj}_{x} = \\max(|\\lambda_1 \\cos(\\theta)|, |\\lambda_2 \\sin(\\theta)|)$$\n\n$$\\text{proj}_{y} = \\max(|\\lambda_1 \\sin(\\theta)|, |\\lambda_2 \\cos(\\theta)|)$$\n\nThis determines the maximum extents in horizontal and vertical directions, enabling us to establish tight-fitting rectangular boundaries that conform to the ellipse's actual shape. We then precisely identify relevant pixel regions while filtering out irrelevant areas by iterating only through tiles within the computed rectangular bounds, thereby directly excluding tiles that fall outside the ellipse's effective coverage without requiring explicit per-pixel testing.\n\n**Difference from OBB:** Unlike OBB which uses axis-aligned rectangular bounds ignoring ellipse orientation, our method incorporates the computed ellipse orientation and principal axes information. This determines orientation-aware bounds that more accurately capture the ellipse's spatial footprint, achieving tighter boundaries and reducing unnecessary tile processing.\n\n\n\n\n\n\n\n\n\n# Weakness 2 & Question 2: Qualitative Results for Ablation Studies\n\nThanks for your suggestion.\n\nOur current manuscript includes qualitative comparisons across different input views (5, 10, and 15 views), showing consistent performance enhancement with increased input data. We plan to conduct the following qualitative comparison experiments:\n\n**Component Ablation Visualization** (corresponding to Table 3):\n\n(1) w/o Material-Based Optimization\n(2) w/o Physically Faithful Overlap Filter (using AABB instead of our approach)\n(3) w/o Intersection Module\n(4) w/o Segment Length\n(5) Full method\n\n**Initialization Method Visualization** (corresponding to Table 4):\n\n(1) Random initialization\n(2) FDK initialization\n(3) Our hybrid initialization\n\nWe will include these additional qualitative ablation results in the camera-ready version.\n\n\n\n\n\n# Weakness 3 & Question 3: Polychromatic X-ray Sources\n\nThanks for raising the important point about polychromatic X-ray sources. Our method, following current works on ray-level modeling (SAX-NeRF, X-Gaussian, R2-Gaussian), focuses on monochromatic X-ray reconstruction rather than polychromatic X-ray sources. The X3D dataset we used also adopts monochromatic assumptions, generated using the TIGRE tomography toolbox with single-energy X-ray simulations.\n\nTo better model polychromatic X-ray sources, photon-level modeling of X-ray imaging physics would be a promising direction to capture energy-dependent attenuation effects. This represents an excellent future research direction that we plan to explore in our subsequent work. Thanks for your insightful suggestion."}, {"review_id": "cne3LWauQg", "reviewer": "Reviewer_Xpaz", "summary": "This paper proposes a 3DGS variation adapted for 3D X-rays, by representing an object's structure using ellipsoids with distinct attenuation coefficients. This strategy models the radiological density across various materials and accounts for penetration properties, unlike previous 3DGS approaches like $\\text{R}^2$-Gaussian, which achieves maximum density at the ellipsoid center and gradually decays outwards. Additionally, the authors introduce an intersection algorithm to handle overlapping ellipsoids and employ hybrid initialization and material-based optimization strategies. The experiments on real and synthetic datasets demonstrate that X-Field outperforms current methods in 3D CT reconstruction and X-ray novel view synthesis.", "questions": "- Could the authors clarify the computational runtime and memory for each component, and also with respect to approaches such as $\\text{R}^2$-Gaussian.\n- How robust is the method to initialization, e.g. if the ellipsoid positions are substantially off? And how robust is the method to different hyperparameters?\n- What cases does the method struggle with, e.g. how does it perform in vascular networks or heterogeneous tissues? Clarifying potential failures would be beneficial.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The main strengths are:\n\n- The paper is well-written, easy to follow, with clear handling of ellipsoid intersections.\n- The work is clearly motivated by the physics of X-ray imaging, addressing limitations of previous visible-light-based representations.\n- The paper has appropriate experiments, showing consistent improvements across multiple challenging datasets, both quantitatively and qualitatively.\n\nThe main weaknesses are:\n\n- The results only offer a small visual improvement over $\\text{R}^2$-Gaussian, yet the paper doesn't sufficiently detail performance/memory requirements, making it difficult to assess how beneficial this work will be to the community. However, the supplementary material video helps convince of the quality improvement.\n- The new parameterization and pipeline stages seem slightly more involved (and potentially slower) than for example $\\text{R}^2$-Gaussian. While an ablation study is included, it's not clear if there are elements here that are dependent on the full new pipeline or can be used in isolation with concurrent 3DGS strategies, and how they contribute to the performance timings/memory. Similarly, as 3DGS benefits from more iterations/elements, it's not clear how hyperparameter tuning influences the results, given the small visual improvements.", "was_revised": true, "final_justification": "The authors have addressed my concerns during the rebuttal, conducting new experiments that demonstrate reasonable computational requirements and robustness in different scenarios.", "rebuttal": "We thank the reviewer for detailed review and constructive suggestions, which are helpful in improving our paper.\n# Weakness 1.1: Visual improvement\nWe acknowledge that the visual improvements in Figure 6 appear modest compared to R²-Gaussian. However, as you noted, the supplementary video demonstrates more significant quality differences, particularly in terms of smoother reconstructions and reduced artifacts. The varying visual improvement magnitudes may be attributed to different evaluation conditions: Figure 6 results are obtained from test distribution views, while the supplementary video showcases out-of-distribution (OOD) views, which present greater reconstruction challenges and thus highlight our method's advantages more clearly.\n\nThe supplementary material video showcases our method's performance on out-of-distribution (OOD) views. As detailed in Section C.1 and visualized in Supplementary Material E, we evaluate 360-degree novel view synthesis where cameras are positioned orthogonally to the principal axis at varying radii, creating viewpoints entirely distinct from the training distribution.\n\nUnder these challenging OOD conditions, our method demonstrates superior generalization capabilities and enhanced robustness to unseen viewpoints. This stems from our physically grounded representation that better captures X-ray physics, enabling consistent performance across diverse viewing scenarios.\n\n\nThe supplementary material video showcases our method's performance on out-of-distribution (OOD) views. As detailed in Section C.1 and visualized in Supplementary Material E, we evaluate 360-degree novel view synthesis where cameras are positioned orthogonally to the principal axis at varying radii, creating viewpoints entirely distinct from the training distribution.\n\nUnder these challenging OOD conditions, our method demonstrates superior generalization capabilities and enhanced robustness to unseen viewpoints. This stems from our physically grounded representation that better captures X-ray physics, enabling consistent performance across diverse viewing scenarios.\n\n\n\n# Weakness 1.2: Computational requirements\nWe provide detailed computational analysis for reconstructing 10 input views with 5,000 iterations on an RTX 6000 Ada GPU:\n \n|                | NAF        | SAX-NeRF   | X-Gaussian | R²-Gaussian | Ours      |\n|----------------|------------|------------|------------|-------------|-----------|\n| Training Time  | 38 min 57s| 307 min 50s | 1min 55s | 3 min 52s  | 4min 15s |\n| Inference Speed |  2.07 fps | 0.36 fps | 132 fps | 72 fps | 45 fps |\n| Memory Usage   | 24GB       | 27GB       | 824MB      | 968 MB      | 1456 MB   |\n \nOur method significantly outperforms NeRF-based approaches across all metrics: substantially faster training, considerably faster inference, and significantly lower memory usage.  Compared to 3DGS methods, we achieve comparable computational efficiency while delivering superior reconstruction quality. We appreciate the reviewer's valuable feedback and will incorporate the computational analysis in the final version.\n\n\n# Weakness 2: Component Dependencies\n\nOur method introduces a physically-grounded representation using homogeneous 3D ellipsoids with distinct attenuation coefficients to model diverse materials. This foundational representation requires\n\n **Tightly-coupled specialized components:**\n\n- Segment length calculation: Computes X-ray path lengths through ellipsoidal volumes\n\n- Intersection correction: Handles overlapping ellipsoid scenarios \n\n- Material-based optimization: Refine ellipsoid distribution along material boundaries.\n\nThese components are intrinsically interdependent as they collectively implement our physics-based ellipsoidal representation. Ablation results in the Table 3 confirm this coupling - removing segment length calculation drops SSIM from 35.03 to 27.48.\n\n\n**Modular Components:**\nWhile the core representation requires full pipeline integration, our Faithfully Overlap Filter is modular and can be independently applied to other 3DGS-based X-ray reconstruction methods.\n\n**Performance Timings/Memory:** We provide detailed analysis of computational runtime and memory usage for each component in **Question 1** below.\n\n**Hyperparameter:** For all training processes, we trained all methods using the same number of iterations (20,000) to ensure fair comparison. We provide more comprehensive analysis on hyperparameter sensitivity in **Question 2.2** below.\n\n\n\n\n\n# Question 1: Computational runtime and memory for each component\n\n\nWe appreciate this important question and provide detailed computational analysis for each component in the table below.\n\n| Method | R²-Gaussian | w/o Material Opt. | w/o Overlap Filter | w/o Intersection | w/o Segment Length | Full |\n|---------|-------------|-------------------|-------------------|------------------|-------------------|------|\n| Training Time | 3 min 52s | 3min 58s | 4min 12s | 4min 04s | 3min 46s | 4 min 15s |\n| Memory Usage | 968 MB | 1174 MB | 1359 MB | 1368 MB | 945 MB | 1456 MB |\n\nComponent Analysis: Removing segment length calculation results in the most significant computational savings in both training time and memory usage. However, this severely degrades rendering quality as the model cannot accurately compute X-ray attenuation along ray paths, which is fundamental to our method. Material optimization removal provides moderate computational benefits but compromises material-aware reconstruction capabilities. Overlap filter elimination shows minimal training time impact while slightly increasing memory usage, indicating this component primarily affects memory efficiency. Intersection calculation removal offers modest speedup with reduced memory consumption, demonstrating its moderate computational overhead. We will include this computational analysis in the final version of the paper.\n\n\n\n\n# Question 2.1: Initialization Robustness\n\nOur method demonstrates robust performance across different initialization strategies. As illustrated in Figure 4, we evaluate three initialization approaches: (1) random Gaussian initialization of ellipsoids (producing scattered, uninformative starting positions), (2) FDK-based initialization providing coarse geometric structure, and (3) our hybrid initialization with detailed geometry information.\n\nAs shown in Table 4, even with random initialization representing substantially poor starting positions (Figure 4, left), our method achieves competitive performance, demonstrating strong convergence capability from uninformative initial states. Progressive improvement is observed with better initialization: FDK initialization achieves 37.96, while our hybrid approach reaches optimal performance.\n\nThis robustness stems from our physics-based optimization framework, which guides ellipsoids toward correct positions through X-ray consistency constraints regardless of starting positions. However, more informative geometry initialization further accelerates convergence and helps the model achieve superior final performance by providing better starting points closer to optimal solutions.\n\n\n\n# Question 2.2: Robustness to hyperparameters\n\n\nTo evaluate hyperparameter robustness, we conducted two experiments on human organ data using 10 input views, analyzing the sensitivity to training iterations and learning rates across different hyperparameter configurations.\n\n**Training Iteration Analysis:**\n\nOur first experiment examines performance across different training iterations:\n\n| Iteration | 5000  | 10000 | 20000 | \n|---------|-------|-------|-------|\n| PSNR    | 34.78 | 35.18 | 35.71 | \n| SSIM    | 0.960 | 0.943 | 0.980 | \n| LPIPS   | 73.26  | 72.14  | 71.03 |\n\nThe results demonstrate stable performance across different iteration counts, with all configurations achieving competitive quality metrics. \n\n**Learning Rate Sensitivity:**\n\nOur second experiment evaluates robustness to learning rate variations:\n\n| Learning Rate | PSNR | SSIM | LPIPS |\n|---------------|---------|---------|-------|\n| × 0.5         | 33.540  | 0.962   | 71.59 |\n| × 1 (default) | 34.801  | 0.971   | 71.26 |\n| × 2           | 34.767  | 0.966   | 71.48 |\n\nThe results show consistent performance across different learning rates, with only modest variations in quality metrics. Both halved and doubled learning rates maintain competitive reconstruction quality compared to the default setting.\n\n**Conclusion:** These experiments demonstrate that our method exhibits strong robustness to key hyperparameters, maintaining stable performance across different training iterations and learning rate configurations.\n\n\n# Question 3: Cases method struggle with.\n\nFor vascular networks, the low inherent contrast in conventional X-ray images poses challenges for our reconstruction method, as fine vascular structures may not be adequately captured. However, given contrast-enhanced imaging, the underlying formulation of our method is readily compatible, offering a promising solution.\n\nFor heterogeneous tissues, as demonstrated in our foot (Figure 6) and head (Figure 7) examples, our method successfully differentiates between distinct tissue types such as bone and muscle. However, achieving cellular-level precision in highly heterogeneous regions remains an area for future exploration."}]

Author rebuttal:

Meta-review: The paper introduces X-Field, a 3D representation tailored for X-ray imaging tasks, which shows great results in X-ray novel view synthesis and CT reconstruction. The proposed method is novel and experiments show superiority over state-of-the-art, especially with sparse inputs. The reviews are consistent and positive. Since the authors have addressed most concerns, many reviewers raised their scores.
Decision: Accept (spotlight)