3DGEER: 3D Gaussian Rendering Made Exact and Efficient for Generic Cameras
Authors: Zixun Huang
Cho-Ying Wu
Yuliang Guo
Xinyu Huang
Liu Ren
Venue: ICLR 2026 Poster (2026)
Keywords: Volumetric Rendering
Differentiable Rendering
Novel View Synthesis
Radiance Fields
Neural Reconstruction
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: Can Gaussian rendering be both exact and fast without relying on lossy splatting? Checkout our 3D-GEER!

Abstract: 3D Gaussian Splatting (3DGS) achieves an appealing balance between rendering quality and efficiency, but relies on approximating 3D Gaussians as 2D projections—an assumption that degrades accuracy, especially under generic large field-of-view (FoV) cameras. 
Despite recent extensions, no prior work has simultaneously achieved both projective exactness and real-time efficiency for general cameras. We introduce 3DGEER, a geometrically exact and efficient Gaussian rendering framework. From first principles, we derive a closed-form expression for integrating Gaussian density along a ray, enabling precise forward rendering and differentiable optimization under arbitrary camera models. To retain efficiency, we propose the Particle Bounding Frustum (PBF), which provides tight ray–Gaussian association without BVH traversal, and the Bipolar Equiangular Projection (BEAP), which unifies FoV representations, accelerates association, and improves reconstruction quality. Experiments on both pinhole and fisheye datasets show that 3DGEER outperforms prior methods across all metrics, runs 5x faster than existing projective exact ray-based baselines, and generalizes to wider FoVs unseen during training—establishing a new state of the art in real-time radiance field rendering.

Published as a conference paper at ICLR 2026
3DGEER: 3D GAUSSIAN RENDERING MADE EXACT AND EFFICIENT FOR GENERIC CAMERAS
Zixun Huang, Cho-Ying Wu', Yuliang Guo', Xinyu Huang, Liu Ren Bosch Research North America & Bosch Center for Al https: / 2 xunh. gf thub. 3o/3d-ger
ABSTRACT
3D Gaussian Splatting (3DGS) (Kerbl et al. 2023) achieves an appealing balance between rendering quality and efficiency, but relies on approximating 3D Gaus- sians as 2D projections an assumption that degrades accuracy. especially under generic large field-of-view (FoV) cameras. Despite recent extensions, no prior work has simultaneously achieved both projective exaciness and real-time effi- ciency for general cameras. We introduce 3DGEER, geometrically exact and efficient Gaussian rendering framework. From first principles, we derive a closed- form expression for integrating Gaussian density along a ray, enabling precise forward rendering and differentiable optimization under arbitrary camera mod- els. To retain efficiency, we propose the Particle Bounding Frustum (PBF), which provides tight vay-Gaussian association without BVH traversal, and the Bipolar Equiangular Projection (BEAP), which unifies FoV representations, accelerates association, and improves reconstruction quality. Experiments both pinhole and fisheye datasets show that 3DGEER outperforms prior methods across all metrics, runs 5x faster than existing projective exact ray-based baselines, and generalizes to wider FoVs unseen during art real-time radiance field rendering.
a new state of the
INTRODUCTION
volumetric particle rendering meth- ods formulated under splatting Yu et al. 2024a Yan et al. 2024 Lu et al.) 2024 Kheradmand let al. 20244) Charatan et al.] 2024 Mallick et al. 2024) have gained significant popularity due to their impressive visual fidelity and high rendering speed. The core efficiency stems from EWA splatting (Zwicker al. 2001), which uses a local affine approximation to project a 3D Gaussian as a 2D Gaussian on the image plane. While effective in narrow field-of-view (FoV) scenarios, this approximation introduces substantial errors in wide-FoV settings, where nonlinear distortions cause the true projection to deviate sigmificantly from a symmetric 2D Gaussian especially in fisheye images commonly encountered in roboties and autonomous driving (Rashed et al., 2021| Courbon et al., 2007 Kumar et al.) 2023 ZZhnn et al. 2016 Yogamani et al., 2019 [024]] Seekat et al., 2022). Although recent works have attempted to enable differentiable rendering under specific fish- eye camera models (Liao et al., 2024) or more general camera models (Huang et al.) (2024)), these approaches remain fundamentally constrained by their reliance on 2D projective-space approxima- tions and have yet to yield exact closed-form solution.
Initiated by 3D Gaussian Splatting (3DGS) (Kerbl et al. 2023
Meanwhile, Celarek et al. (2025) questions the necessity of exact rendering by scaling up the num- ber of Gaussians (up to IM) to mitigate projective approximation in splatting. However, the results are limited to small object-centric scenes. In large-scale scene reconstructions, the required number of Gaussians varies significantly across objects and frustums, making scaling up to larger scenes prohibitively costly. Further, we show projective errors under generic distorted cameras are substan- tially larger, where simply scaling up the Gaussian counts cannot close the quality gap (see Fig. (5).
A parallel line of work tackles volumetric particle rendering directly in a ray-traciig//marchin for- 2024b Condor al. 2025 Mai et al.) 2025) Moënne-Loccoz et al., 2024 Tale- gaonkar et al. 2025 Gu et al. 2025). ). These approaches naturally support large-FoV camera models
mulation (Yu et al.
*Corresponding Author: (yuliang-guee guo2, cho-ying.wu hos.hhoch.com

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Published as a conference paper at ICLR 2026
3DGUT
Ours
Approx. Cov. E'EwA
Missnathhdd Culling
True Cov. Σ
PBF
ScanNet++ zapner
EWA
UT
Ours (asso.)
Mismatched Culling
Approx. Cov. E'y
Exact Contour
Figure Linear Approximation Error in Ray-Particle Association. (Left) Grid-line artifacts caused by inaccurate UT association. (Righ() Diagram illustrating our association and comparison with others. Our method avoids the intermediate conic approximation by directly computing the exact bounding structure from the true 3D covariance.
and are free from projective approximation enror. However, achieving high efficiency remains in associating each rendering ray with most relevant 3D particles. To address this, significant effort has been devoted to accelerating it, such adopting advanced Bounding Volume Hierarchies (BVH) employed in EVER (Mai et al.] (025) and 3DGRT (Moênne- Loccoz al. 2024). Nevertheless, due to the inherent algorithmic complexity and difficulty in parallelization, pure ray-based methods have yet to match the efficiency 3DGS. Alternatively, a hybrid approach has been proposed in 3DGUT (Wu al., 2025) which adopts ray-tracing render- ing with rasterization to solve ray eay-particle association efficiently. However, the inherited projective approximation error from the association stage still compromises rendering exactness in projective Left). To date, no existing method achieves
key
geometry and even risks grid-line artifacts (see Fig. [ both the geometric exactness of ray-based rendering and the high efficiency of splatting.
In this work, we propose 3DGEER-a novel 3D Gaussian Exact and Efficient Rendering method that achieves exactness in projective geometry (see definition in Sec. A while maintaining high efficiency comparable to splatting for generic cameras (both perspective and a variety of fisheye projection models). 3DGEER is built upon three key contributions: (i) We revisit the heuristic maximum response function (Keselman & Hebert [002] of Gaussians from first principles and reveal its connection with the projective exactness. Based on it, we further derive the differen- tiable rendering framework with numerical stability. (ii) To perform efficient ray ray-particle associa- tion without compromising geometric exactness, we reformulate the problem as associating Camera Sub-frustums (CSF) with Partiele Bounding Frustums (PBF)-analogous to the tile-AABB map- ping used in 3DGS. For each 3D Gaussian, we derive a closed-form solution to compute the PBF tightly bounding it, enabling highly efficient association. (iii) In addition, propose uniformly sample rays in novel Bipolar Equiangular Projection (BEAP) space to apply color supervision. This ray sampling strategy not only aligns image-space partitioning and the underlying CSF but also improves conventional pinhole or equidistant projections in reconstruction quality.
Extensive experiments on both fisheye and pinhole datasets datasets-iincuuding ScanNet++ (Yeshwanth et al. 2023), Zip-NeRF (Barron et al.] 2023), Aria (Lv et al.] 2025), and MipNeRF (Barron et al. 2021 show that 3DGEER consistently outperforms prior and concurrent methods, achieving 0.9 4.8 PSNR gains under challenging wide-FoV conditions, and 0.3-2.0 on pinhole views. Re- markably, our approach even surpasses splat-based methods on wide-FoV cameras when trained only with narrow-FoV data, highlighting its strong generalization to extremely wide-FoV rendering.
RELATED WORKS
2.1 SPLATTING-BASER GAUSSIAN RENDERING
Most 3DGS variants Yu al. 2024a
Zhang et al.
2024
Rota Buld al.) 2024 Gao et al. 2025
Zhao et al. 2025 Kerbl et al., 2024
Liu et al.) (025) adopt the original EWA splatting strategy (wwickrr et al. 2001 Ren et al. 2002), ) projecting 3D Gaussians onto the image plane and approximating their appearance as 2D conics. To support differentiable rendering under large-FoV camera models with distortion, methods such as FisheyeGS (Liao et al. (024) and GS++ (Huang et al. (024b) estimate the Jacobian of equidistant spherical projections to adjust the 2D mean and covariance the projected Gaussians. However, these splatting palsed based approaches still rely first-order Taylor expansion of the highly non-linear projection function. In wide-FoV
Hou et al. 2025
2

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Published as a conference paper at ICLR 2026
Table 1: Summary of Particle Rendering Methods w.r.t. Projective Exactness. The top section lists splatting-bai based methods, while the bottom section presents ray-based approaches some of which still rely on projective approximated association (e.g.. EWA or Unscented Transform).
Method
Render
Particle Asso
Particle
Large FaV
High
Exact
Method
Scope
Type
FPS
Meth.
3DGS
Kerbl
2023
Splat.
Img.
3D Gauss.
x
FishoyeGS
[L:o 0 0
2024
Splat.
EWA
Img.
3D Gauss.
4
/
GS++
Hunng et m..
20240
Splat.
EWA
Img.
3D Gauss.
x
2DGS
Tuang ml.
2024g
Ray
Splat Bound.
Img.
2D Gauss.
K
/
m. A124h
Ray
Img.
Gauss.
X
EVER
Mar
2026
Ray
BVH
Scene
3D Ellip.
5
K
3DGKT Moenne Laceuz et al.
2024
Ray
BVH
Sceno
3D Gauss.
3DGUT
ai 2025
Ray
UT
3D Gauss.
HTGS
Haniihhm ar
2025
Ray
Splat Bound.
Img.
3D Gauss.
X
JDGEER((Un)
Ray
PBF
Gauss.
settings, higher-order terms are non-negligible, resulting in significant approximation errors and degraded reconstruction quality.
Fundamentally, the true projection of a 3D Gaussian under non-linear camera models-whethher ideal pinhole distorted projections is not symmetric 2D Gaussian. In other words, any frame- work that relies on linear projective geometry to approximate inherently nonlinear transformations will inevitably introduce approximation errors that cannot be fully eliminated. In contrast, our approach avoids such projective approximations entirely by aggregating Gaussian density directly along rays in volumetric rendering formulation.
2.2
GAUSSIAN RENDERING
Keselman & Hebert (0022) introduces an assumption that a ray intersects the Gaussian density function at the maximum intensity response. Building on this idea, 3DGRT (Moënne- Loccoz et al. 2024) adopts this heuristic intersection to establish 3D Gaussian ray-tracing framework, which enables ray-space sampling without explicit projection and alleviates wide-FoV distortions. How- ever, 3DGRT introduces substantial computational overhead, as the ray ray-particle association relies on costly scene-level spatial partitioning structures for BVHs, ultimately resulting in low frame rates.
Alternatively, EVER (Mai et al., 2025) replaces the Gaussian representation with simplified ellip- ray integra-
soids of constant density, enabling closed-form solution for transmittance tion. However, this substitution reduces the expressive capacity of the representation compared to full 3D Gaussians and still suffers from limited runtime efficiency due to its reliance on BVH-based structures. 2DGS (Huang et al. 2024a) substitutes 3D Gaussians with 2D surfels, allowing exact ray- surfaee intersections. HTGS (Hahlbohm et al. 2025) reintroduces the full Gaussian expres- sions while adopting 2DGS's ray-splat association. Yet, because its association remains restricted to screen space, it falls short of ensuring projective exactness under wider FoVs. Our contribution lies in deriving closed-form bounds and full rendering pipeline directly in a camera-agnostic angular domain (see details in Sec. D) and we especially discuss the relation and differences computation to prior 2DGS and HTGS parametrization and rendering in Sec. D33
3DGUT eW al. 2025), a follow-up to 3DGRT, improves runtime efficiency by adopting splatting- based particle association, achieving frame rates comparable to 3DGS. However, its acceleration relies on the Unscented Transform (UT), sampling-buis based approximation used estimate 2D con- ics in projective image space. A similar limitation exists in GOF (Yu et al. 2024b), which directly applies the EWA trick to fit 2D bounding conics and projective centers for association. Notably, un- der extreme-FoV conditions, imprecise AABB estimation and the resulting inaccurate tile-to-AABB associations can lead to visible grid-line artifacts (see Fig, Left).
In contrast to these methods, our approach solves the ray-particle association problem directly at the frustum level in 3D space (see Fig. I Right), preserving the geometric exactness ray-based rendering across arbitrary camera models while maintaining rendering speeds comparable to 3DGS. Tab. I summarizes representative methods that aim to address projective approximation errors, high- lighting key differences across several important dimensions.
3

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Published as a conference paper at ICLR 2026
Grad. forward
Grad. backward
Learnable
Sampled
Axx
Direc.
Color
Scaling
d
Diree.
(x)
d
Mah. Dist.
Rotation
'
Mome
D p
World Frame
q
Origin
Origin
my
Transin
O
T
Mean
Opac
|Aul
f
Canonical Ray
Oro(u)
Gaus. Params
Ray-Gaus. Integral
Canonical Frame
Figure 2: Gradient Flow and Canonical Transtormation. (Left) Illustration of forward and back- ward gradient propagation from the Gaussian parameters s, q, μs, and the opacity coefficient the transmittance. (Right) Integral of 3D Gaussian density along a ray, where the green box highlights product of density and ray segment length (see Eq. B.5).
METHOD
We propose 3DGEER, a Gaussian rendering framework that achieves both geometric exactness and real-time efficiency under arbitrary FoV cameras. Our method addresses the two key bottlenecks of prior coork-rrojection approximation in splatting-hased rendering and BVH overhead in ray-based wendering-throwee three complementary components:
Ray-based rendering formulation (Sec. 3.1 closed-form differentiable solution for integrating Gaussian density along 3D rays, eliminating linearization error from projection-based splatting.
Particle Bounding Frustum (PBF) (Sec. 3.2
novel ray-particle association scheme that replaces BVH traversal with exact frustum-particle tests, yielding both accuracy and efficiency.
Bipolar Equiangular Projection (BEAP) (Sec. 3.3 an image representation unifying arbitrary FoV camera models, which accelerates PBF computation and improves reconstruction quality.
The Appendix further provides: the necessary preliminaries and notations (Sec. (ii) the complete proof to Sec. 3.1 establishing the connection between maximum response and projective exactness (Sec. B). (iii) the full derivations backward gradients (Sec. and its numerical stability,
and (iv) the full derivations of geometrically exact association (Sec. D
3.1 RENDERING WITH EXACT PROJECTIVE GEOMETRY
One key challenge geometrically exact Gaussian rendering is to obtain a closed-form expression for integrating density and color along ray. Our strategy is to map each anisotropic Gaussian in world space into a canonical coordinate system where all Gaussians become isotropie. This reduces the transmittance integral to prreservrr change of variables, admitting simple closed- form solution.
Canonical transformation. For each Gaussian with mean μ4, rotation R, and scaling S, we define the mapping
x
RS
which maps canonical coordinate to world coordinates This implies Ax RSAu and absorbs the Jacobian determinant into the density measure (see Fig, Right). Consequently, each Gaussian reduces to the shared isotropic form
1 (r.
exp( -
(2)
0
4

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Closed-form transmittance. with
For ray r(t) + td. its canonical form is ru(() Ou + td
O₂ = (o- -e), =
(3)
The accumulated transmittance then
d) = J JIER gT,0 + (d,) dl o -
(4)
where
( d) 1o,, x d.ll
(5)
||d
the squared perpendicular (Mahalanobis) distance from the ray to the Gaussian center.
Discussion. This result reveals that the transmittance depends only on the ray's distance to the Gaussian in the canonical space, yielding an efficient and geometrically exact closed-form solu- tion. Interestingly, this expression equivalent to the "maximum response" heuristic used in prior works (Keselman & Hebert 2022, Moénne-Loccor et al. (024). while our mathematical deriva- tion reveals its underlying projective exactness (see details Sec. B). Furthermore, by following our closed-form derivation process and the corresponding backward gradient chain (see Fig. 2 Left and details Sec. C) we explicitly avoid the numerical instabilities that prior ray-based work (Yu et al., (024b) encounters and eliminates the need for redundant proseprocessing such filtering degenerate Gaussians.
3.2
ASSOCIATION WITH EXACT PROJECTIVE GEOMETRY
Given ray-based Gaussian rendering method, under
both its final accuracy-partice large-FoV camera models and its runtime ef- ficiency depend heavily on the ray-particle as- sociation strategy, which has been largely over- looked in prior works. To determine which Gaussians contribute to each ray, we sim- plify the association problem by mapping Cam- era Sub-Frustums (CSFs) to Particle Bounding Frustums (PBFs) the tight frustums enclosing each Gaussian at a certain distance from its cen- ter (Fig. 3). Gaussian is assigned CSF if its PBF intersects the CSF All Gaussians asso- ciated with CSF are then considered contrib- utors to every ray within that CSF.
fir sphere
PRF
PBF
Intersection
0
imagonng splire
FaVx
Figure 3: PBF-CSF Association. (Left) PBF de- fined by four planes tangent to 3D Gaussian el- lipsoid. (Right) The intersection between the PBF and CSF, unfolded onto the BEAP imaging plane.
To efficiently compute PBF, we first define two spherical angles (see Fig. E.
the camera space:
da
d
0 = arctan
Φ arctan
d
where (de.z, de.g de.z) denotes the normalized direction vector pointing to the camera (view) space point de. The angles 0 and Φ correspond the ray's incidence angles in the horizontal (æz) and vertical (yz) planes, respectively, each ranging from - to We define PBF by four planes two aligned with the angle and with the & angle (see Fig. 3):
g+ = (-1,0, tan 01.2, T)T,
g01 = (0, 1, tan $1,2, 0)T,
0)
where 01 < 0 0g and Φ1 < o d2 specify the angular limits.
Given a camera with extrinsic matrix V = [R te] E SE(3) and a mation matrix H (See Eq. B.4) apply the inverse transpose of the camera-to snnera-to-canonicll matrix (V to parameterize the corresponding canonical planes using the angular limits defined in view space. This transformation enables bounds of 0 as an example, the canonical plane is given by:
transfor-
effectively derive the constraints for the PBFs. Take the
g (VH) ge. - (V H)o tan (V H)2.
(8)
5

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Published as a conference paper at ICLR 2026
where (VH), denotes the i-th row of the matrix, expressed column vector.
contour, we compute the angular bounds of the local frustum by solving the following quadratic equations, which enforce that the transformed planes are tangent to the ellipsoidal contour:
Assuming a Gaussian ellipsoid defined by
((>2, x2: x² -1) o g..) gu = O,
where denotes the Hadamard (element-wise product, and À is parameter that controls the spread of the contour, typically set to 3. An identical constraint applied to g. in order to compute the corresponding angular bounds 01.2- The roots of these quadratic equations determine the tangent values of the angular bounds [01, 02] and [01, Ø2]. which define the PBF angular limits. Notably, solving these quadratic equations under canonical space constraints respects the true 3D covariance Eg and 3D mean He of the Gaussian in the camera space. Specifically, for the bounds 6, we have tan 01,2 satisfying:
T22c² 2To2c Too = 0, where = (m.M)
(10)
This formulation is efficient to solve (See details in Sec. (.I) and eliminates the need for interme- diate conic approximations, as in EWA or UT, as well as the screen-space constraints in HTGS and 2DGS that limit their FoV.
3.3 BIPOLAR EQUIANGULAR PROJECTION (BEAP)
We introduce novel BEAP image representa- tion for several reasons. First, it effectively rep- resents images from large-FoV cameras with- out introducing FoV loss. Second, it allows im- age tiles and their corresponding CSFs share an identical parameterization in terms of two angles, enabling highly efficient
spanser
Perspective
Equidistant
Equiangular (ours)
0 and PBF association and optimal GPU paralleliza- tion. Third, the resulting ray sampling achieves more uniform distribution in the 3D space compared to conventional projective rendering. which empirically improves training efficiency and reconstruction quality.
Figure 4: Comparison of Ray Distributions un- der Varying Projections. We project pixels from three imaging spaces onto the unit sphere to com- pare ray distributions. Among them, our BEAP projection achieves the most uniform coverage.
Specifically, we evenly sample the rays using the spherical angular coordinates (0, Φ) (See Eq. and then map the rays back to a discrete w x image space using a linear transformation consisting of scaling factor and center shift as:
W
W +1
h+1 2 FoV, 2
he
(a, v)
(1I)
FoV.
where FoV., Vo indicate the camera's horizontal and vertical field of view. As illustrated in Fig. 4 sampling rays uniformly in (0, Φ) leads to more balanced spatial coverage within the view frustum. Our sampling strategy contrasts with projective sampling, which disproportionately allo- cates samples toward peripheral regions, and equidistant projection, which oversamples near the image center.
EXPERIMENTS
In this section, we evaluate our proposed method, 3DGEER, across multiple datasets featuring both pinhole and fisheye camera models. All experiments are conducted using an RTX 4090 GPU and 64GB of RAM as the default configuration, unless otherwise stated. Full implementation details are provided in the Sec. G
Datasets. We evaluate performance on generic distorted cameras, where projective errors are crit- ical, on all scenes from Fisheye ZipNeRF (Barron et al. 2023), five from ScanNet++ (Yeshwanth et al., 2023), and two from Aria (Lv et al. 2025). ZipNeRF/ScanNet++ cameras have 180 di- agonal Fo V, and Aria has 110 circular Fov. For pinhole settings, we use Pinhole ZipNeRF and MipNeRF360 (Barron et al. 2021)

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Published as a conference paper at ICLR 2026
Bedroom (1
Office Room (2ala3afad9)
Kitchen
Laboratory (1d003607bd)
(4ef75031e3)
30
32.5
30.0
PSNR
PSNR
PSNR
26
90
10"
10P
10a
10a
HsheycGS
RsheycGS w/ lower thres.
3DGEER
Figure 5: PSNR Trends When Scaling-Up Gaussian Counts. Brute-force scaling fails close
the gap in projective exactness (see full metrics in Fig. K.3 and Tab.
K.3
Treining Time Fov
FisheyeGS
3DGEER (Ours)
Training Tine Fov
FisheyeGS
3DGEER (Ours)
Bedroom
Kitehen
(Original FOV)
Gaus Num: 985k
Gaus. Nun: 664k
(Originsl FOV)
Gaus. Nun: 768k
Gaus. Num: 552k
Laboratory
OfficeRom
(Original FOV)
Gaus. Num: 848k
Gaus.Numt 582k
(Driginel FoV) Gaus.Num: 1872k
Gaus. Num: 732k
Figure 6: Extreme-Fov Generalization. 3DGEER reconstructs complete scenes with fewer Gaus- sians, while FisheyeGS suffers from severe quality degradation.
Evaluation Metries and Protocol. We report PSNR1. SSIM1, LPIPS runtime performance (FPST) and Gaussian counts|. To avoid representation bias, all results are projected back to the image space for full-FoV evaluation after 30k iterations.
4.1 SCALING GAUSSIANS VS. PROJECTIVE EXACTNESS IN WIDE-FOV CAMERAS
(Celarek al. 2025) argues that simply scaling the number of Gaussians (up to 1M) suffices to mit- igate the linear approximation gap between splat-based and ray-based methods. However, this strat- egy has only been demonstrated on small object-centric scenes and does not address the challenges posed by Wide-FoV distortions. To further investigate, scale up the splat-based fisheye method (FisheyeGS) on ScanNet++ scenes by lowering its densification threshold, producing 3.6-5M Gaus- sians (H200).
As shown in Fig. 5) the red curve (FisheyeGS) and blue curve (3DGEER) show a clear PSNR gap under the same Gaussian count, implying approximation gap in projective exactness (from 4k to 30k iterations, after which densification typically saturates). The light-blue curve further shows the scaled-up FisheyeGS, despite its substantial increase in memory and computation, performance saturates at 29.3 PSNR, whereas ours achieves 32.1 with only 550-700K Gaussians. Full statistics are reported in Tab.
4.2 COMPREHENSIVE EVALUATION ON ALL-FOV NOVEL VIEW SYNTHESIS
We first evaluate 3DGEER against representative baselines on both fisheye and pinhole benchmarks.
To ensure comprehensive evaluations, we incorporate additional analysis tailored to each dataset.
ScanNet++. As shown in Tab 3DGEER consistently outperforms all baselines across metrics. To
examine distortion effects, we separately evaluate central and peripheral regions (details in Sec. G
Competing methods perform reasonably at the center but degrade at the periphery, whereas 3DGEER maintains high fidelity throughout, producing sharper boundaries and fewer artifacts (Fig. K.4 Notably, even when trained only on central crops, 3DGEER still surpasses FisheyeGS across all regions.

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Published as a conference paper at ICLR 2026
Table Comparison on the ScanNel++ Dataset. Each method is trained either on the full FoV or only the central region, and then evaluated on the full, central, and peripheral regions to assess ro- bustness under varying distortions. Region-wise metrics are weighted by pixel masks, while LPIPS is excluded since it is not directly separable by region (Tab. 2.2 shows full stats).
Method
Training
Training
Full FoV
Central
Peripheral
FoV
Space
PSNR f SSIM LPIPS 4
PSNR t SSIM 1 PSNR f SSIM
3DGS
Central
Persp.
31.26
0.948
FisheyeGS
Full
Eq.
27.81
0.946
0.139
32.44
0.956
23.28
0.914
EVER
Full
KB
29.47
0,924
0.167
29.93
0.924
28.72
0,925
3DGUT
Full
Eq.
30.64
0.944
0.150
31.87
0.945
28.84
0.937
3DGEER (Ours)
Central
BEAP
29.93
0,949
0.130
32.86
0.954
26.64
0,930
3DGEER (Ours)
Full
BEAP
31.50
0.953
0.126
32.64
0.955
28.94
0.945
Table 3: Comparison the ZipNeRF Dataset. The experiments follow cross-camera general- ization setup, where each method is trained on either Fisheye (FE, 1/8 resolution) or Pinhole (PH, 1/4 resolution) data and evaluated separately on both Fisheye and Pinhole test sets (Tab. K.9 shows full stats).
Train Data
1/8 FE
1/8 FE
1/8 FE
1/4 PH
1/4 PH FE SSIM f
1/4 PH FE
Test Data
FE
PH PSNR f
FE SSIM f
PH
FE LPIPS
PH
FE
PH
PH
PH
Method
PSNR f
LPIPS L
FisheyeGS
23.18
26.44
0.858
0.868
0.211 0.239
19.43
26.61
0.791
0.889
0.247
0.212
EVER
25.17
26.14
0.880
0.851
0.153
0.237
22.90
26.45
0.835
0.862
0.207
0.222
3DGUT
24.77
25.59
0.879
0.804
0.183
0.324
18.61
25.96
0.662
0.841
0.300
0.270
3DGEER (Ours)
26.24
27.62
0.897
0.888
0.140 0.214
23.39
27.61
0.846
0.863
0.209 0.254
Qualitatively (Fig. 6). we evaluate extreme-Fo FoV generalization by training on the original dataset FoV (180% diagonal V) but testing at much wider FoVs (180 FoV, circular shape). 3DGEER constructs complete scenes with fewer Gaussians, while splatting-basee methods such as FisheyeGS degrade significantly at the periphery (see more visuals in Fig. K. K.2
with extreme FoV).
ZipNeRF. The dataset provides frame-wise aligned pinhole and fisheye views. Be- yond standard fisheye evaluation, we con- duct cross-camera genevalization-trranne on fisheye and testing on pinhole, and vice versa to assess robustness. As shown in Tab. 3. 3DGEER consistently outperforms all methods across metrics, fully exploit- ing large-FoV training data via geometri- cally exact rendering. Trained on pinhole, 3DGEER still yields notable PSNR gains on both views. In contrast, FisheyeGS per- forms well only on pinhole-to-pinhole, and 3DGUT drops sharply when tested on fish- eye after pinhole training. Overall, EVER and 3DGEER generalize better due to geo- metric exactness, while 3DGEER superior in all cases. Qualitatively (Fig. [7]. 3DGEER recov- ers finer details and shows stronger resistance to artifacts in the periphery in the most challenging pinhole-to-fie fisheye setting.
Table 4: Quantitative Results on the MipNeRF360
Dataset. In addition to quality metrics, FPS is re- ported using an RTX 4090 as the default benchmark. Numbers sourced from original works are marked with (measured on RTX 6000 Ada) and I (mea- sured on RTX 5090).
Method
PSNR 1 SSIM t LPIPS 1
FPSt
3DGS
27.21
0.815
0.214
343
EVER
27.51
0.825
0.233
36
3DGRT
27.20
0.818
0.248
52 / 682
3DGUT
27.26
0.810
0.218
265 / 317
3DGEER (Ours)
27.76
0.821
0.210
327
We further evaluate 3DGEER on the MipNeRF360 dataset under standard narrow-FoV conditions. As shown in Tab. 4 3DGEER sets the new state-of-the-art across all the metrics. It is over 5x faster than exact ray- wer--mmaaarchcciiiininn methods such as EVER and 3DGRT, and the only projective- exact method with runtime comparable to 3DGS. Moreover, it consistently surpasses projective- approximation baselines (3DGS, 3DGUT) in PSNR, SSIM, and LPIPS. Additional quantitative sults on Aria, an egocentric fisheye dataset (moderate 110 FoV, circular shape) with low-light imaging, and more visuals are provided in the Sec.
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Training Fov
FS
EVER
3DGUT
3DGEER (Ours)
Figure 7: Qualitative Results the ZipNeRF Dataset with Performance differences are pronounced in the most challenging Pinhole (PH) train - Fisheye (FE) test setup, where our method demonstrates a sigmificant advantage.
Generalization.
Table 5: Training- Time Transmittance Replacement ScanNet++ Using splatting based trans- mittance increases the Gaussian count due to larger approximation error, yet the perceptual gap re- mains compared with our projective-exact formulation (Tab. K.7 shows the full stats).
Transmittance
Full FoV
Central
Peripheral
Avg. Gaussian Number (k) |
Method
PSNR + SSIM LPIPS
PSNR 1 SSIM
PSNR 1 SSIM
3DGS Splats
22.86
0.878
0.177
27.16
0.884
18.56
0.855
1396.4
FisheyeGS Splats
27.90
0.948
0.141
32.49
0.957
23.36
0.917
920.6
Projective-Exxal (Ours)
31.50
0.953
0.126
32.64
0.955
28.94
0.945
591.8
4.3 COMPREHENSIVE RUNTIME ANALYSIS
We further analyze runtime RTX 4090 by comparing 3DGEER with splatting- methods (3DGS on MipNeRF360 and FisheyeGS on ScanNet++). Although our ray-based rendering incurs slightly higher cost in the rendering stage, PBF's efficient ray- ray-particle association greatly reduces Gaussian duplication and sorting overhead (See detailed runtime in Tab, K.4 resulting in overall efficiency comparable to optimized splatting pipelines.
Specifically, PBF yields very tight bounds, with each tile associated with only 4455 Gaussians on average- 3-5x fewer than EWA or UT (Tab, K.6). This directly explains the 2.5-5x runtime speedup in the association stage (Tab. K.5p. In contrast, projection- -ased schemes (EWA, UT) con- servatively overestimate screen-space extent, leading to excessive overdraw and inefficient memory access. Even when accelerated by SnugBox (Hanson et al., 2025), they remain constrained by mis- centered density approximations, often exacerbating artifacts (Fig. [ Beyond speed, PBF preserves geometric accuracy and numerical stability, eliminating projective approximation errors. As shown in Tab. K.6 even when ray-based rendering Gaussian fields trained under other association schemes, PBF remains artifact-free.
4.4 LOCAL AFFINE APPROXIMATION Vs. PROJECTIVE-EXIE INTEGRAL
To isolate the effect of our projective-exact transmittance formulation (Eq. 4 we replace it with the commonly used local affine splats (i.e., conic as approximated 3DGS and FisheyeGS).
Rendering- Time Approx. To better highlight the difference between the two rendering formula- tions, we replace the transmittance with the splatting-based approximation only at test time, while keeping all projective-exact components during training, As shown in Fig. g when the transmit- tance relies on the splat approximation but the associations are computed using our exact projec- tive geometry, their inconsistency leads to mismatched culling. This experiment magnifies and explains the gridline artifact reported in 3DGUT (Fig. [ where the mismatch arises in the opposite direction-rra-bbssee transmittance combined with conic-approximate association.

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3DGS Splats PBF
FisheyeGS Splats PBF
3DGEER (Ours)
GT
Figure 8: Rendering- Time Transmittance Fune. Replacement. Replacing projective-exate trans- mittance with the splatting approximation only at test time leads to mismatched culling.
Table 6: BEAP vs. Alternative Supervision Schemes. BEAP consistently outperforms other
projections across all metrics on ScanNet++. (Tab. K.8]shows full stats).
Method
Training
Training Space
Full FoV
Central SSIM 1 LPIPS PSNR SSIM
Peripheral PSNR SSIM t
FoV
PSNR
3DGEER w/o BEAP Central
Persp.
29.84
0.943
0.131
32.21
0.951
26.23
0.915
3DGEER w/o BEAP
Full
Persp.
21.11
0.853
0.300
21.32
0.855
20.46
0.846
3DGEER w/o BEAP
Full
Eq.
31.05
0.948
0.135
32.21
0.952
28.56
0,935
3DGEER (Ours)
Full
BEAP
31.50
0.953
0.126
32.64
0.955
28.94
0.945
Training- Training-Time Approx. As shown in Tab. experimented on ScanNet++, when splatting-based transmittance is also used during training while association remains projective-exact, the optimizer and densification partially compensate for the approximation сπтоr and the cross-view inconsistency introduced by splatting. The larger loss induces stronger gradients throughout training, ultimately leading to larger (1.6-2.4x) number of Gaussians being added to fit the scene. However, de- spite this increased Gaussian count, the perceptual gap introduced by the approximation remains, indicating that the error cannot be fully corrected through densification alone.
4.5 BEAP ON/OFF STUDY
We further compare BEAP supervision against other widely used projection models on ScanNet++. In this experiment, we keep the projective-exact integral and PBF for Gaussian transmittance and association fixed, and vary only the ray-sampling distribution used for supervision (pinhole, equidis- tant, and BEAP). As shown in Tab. [ and Fig. K.5) our BEAP-based supervision outperforms all alternatives across every metric. Its more uniform sampling provides smoother and more consistent gradients, enabling better convergence on fine details.
The discrepancy becomes more pronounced under pinhole cameras. Under fixed resolution, well-balanced information distribution can matter more than simply using a larger FoV. Notably, when training in the perspective domain, the full FoV can even underperform the central region (see Tab. 6 rows 1-2) because large-FoV images (Fig. K.2, Left) have highly non-uniform angular sampling: informative content is concentrated at the center, while the less informative periphery occupies most of the image area and becomes disproportionately oversampled.
5 CONCLUSIONS
We presented 3DGEER, a complete, closed-form solution for projective-exact volumetric Gaussian rendering derived from first principles. To complement this formulation, introduced effi- cient PBF computation method for ray-particle association, enabling high-speed rendering while maintaining exactness in projective geometry. Additionally, we proposed the BEAP image repre- sentation to facilitate effective color supervision under wide-FoV camera models. Extensive experi- ments demonstrate that 3DGEER consistently outperforms state-of-the-art methods across multiple datasets, with particularly strong results in challenging wide-FoV scenarios.
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ETHICS STATEMENT
This work builds on publicly available datasets (ScanNet++, ZipNeRF, Aria, and MipNeRF) and does not involve any human subjects, private data, or personally identifiable information. The meth- ods developed are intended for udvancing 3D scene reconstruction and rendering research, with potential applications
XR, robotics and autonomous driving. We do not foresee direct harmful consequences, but we note that any misuse of high-fidelity 3D reconstruction techniques could raise privacy concerns. To mitigate this, we restrict our experiments to research datasets that are widely used in the vision community.
REPRODUCIBILITY STATEMENT
We make significant efforts ensure reproducibility. All mathematical derivations and proofs provided in the Appendix (Secs. B C and D). The datasets used in our experiments are publicly available (ScanNet++,, ZipNeRF. Aria, and MipNeRF). Details of preprocessing, training protocols, and evaluation metrics are described in Sec. G Our implementation will be released with training and evaluation scripts upon publication to further facilitate reproducibility.
REFERENCES
Jonathan T. Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin Bartin-Brualla, and Pratul P. Srinivasan. Mip-nerf: multiscale representation for anti-aliasing neural radiance fields. In IEEE International Conference on Computer Vision (ICCV), 2021.
Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. Zip-nerf: Anti-aliased grid-based neural radiance fields. In IEEE International Conference on Computer Vision, (ICCV), 2023.
Adam Celarek, George Kopanas, George Drettakis, Michael Wimmer, and Bernhard Kerbl. Does 3d gaussian splatting need accurate volumetric rendering? In Computer Graphics Forum (CGF), 2025.
David Charatan, Sizhe Lester Li, Andrea Tagliasacchi, and Vincent Sitzmann. pixelsplat: 3d gaus- sian splats from image pairs for scalable generalizable 3d reconstruction. In Proceedings of the IEEE/CVF conference on computer vision and patlern recognition (CVPR), pp. 19457-19467, 2024.
Jorge Condor, Sébastien Speierer, Lukas Bode, Aljaz Bozic. Simon Green, Piotr Didyk, and Adrián Jarabo. Don't 't splat your gaussians: Volumetric ray-traced primitives for modeling and rendering scattering and emissive media. ACM Transactions on Graphics (TOG), 2025.
Jonathan Courbon, Youcef Mezouar, Laurent Eckt, and Philippe Martinet. A generic fisheye camera model for robotic applications. In IEEE/RSJ International Conference on Intelligent Robots and Systems, (IROS), pp. 1683-1688, 2007.
Daniel Duckworth, Peter Hedman, Christian Reiser, Peter Zhizhin, Jean-François Thibert, Mario Lucic, Richard Szeliski, and Jonathan Barron. SMERF: streamable memory efficient radiance fields for real-time large-scene exploration. ACM Transactions on Graphics (TOG), 2024.
Zhongpai Gao, Benjamin Planche, Meng Zheng, Anwesa Choudhuri, Terrence Chen, and Ziyan Wu. 6dgs: Enhanced direction-aware aware gaussian splatting for volumetric rendering. In The Thirteenth International Conference on Learning Representations (ICLR), 2025.
Chun Gu, Xiaofei Wei, Zixuan Zeng, Yuxuan Yao, and Li Zhang. Irgs: Inter-reflective gaussian splatting with 2d gaussian ray tracing. In Proceedings of the Computer Vision and Pattern Recog- nition Conference (CVPR), pp. 10943-10952, 2025.
Yuliang Guo. Sparsh Garg, S. Mahdi H. Miangoleh, Xinyu Huang, and Liu Ren. Depth any camera: Zero-shot metric depth estimation from any camera. Conference on Computer Vision and Pattern Recognition (CVPR), 2025.
11

---

Published as a conference paper at ICLR 2026
Florian Hahlbohm, Fabian Friederichs, Tim Weyrich, Linus Franke, Moritz Kappel, Susana Castillo, Mare Stamminger, Martin Eisemann, and Marcus Magnor. Efficient perspective-cors correct 3d gaus- Sian splatting using hybrid transparency. In Computer Graphics Forum (CGF), 2025.
Alex Hanson, Allen Tu, Geng Lin, Vasu Singla, Matthias Zwicker, and Tom Goldstein. Speedy- splat: Fast 3d gaussian splatting with sparse pixels and sparse primitives. In Proceedings of the Computer Vision and Pattern Recognition Conference (CVPR), 2110//ppp11555155011111 21537-21546, 2025.
Qiqi Hou, Randall Rauwendaal, Zifeng Li, Hoang Le, Farzad Farhadzadeh, Fatih Porikli, Alexei Bourd, and Amir Said. Sort-free gaussian splatting via weighted sum rendering. In The Thirteenth International Conference Learning Representations (ICIR), 2025.
Binbin Huang, Zehao Yu, Anpei Chen, Andreas Geiger, and Shenghua Gao. 2d gaussian splatting for geometrically accurate radiance fields. In Andres Burbano, Denis Zorin, and Wojeiech Jarosz (eds.), ACM SIGGRAPH Conference Papers, 2024a.
Letian Huang, Jiayang Bai, Jie Guo, and Yanwen Guo. Gs++; Error analyzing and optimal gaussian splatting. CoRR, 2024b.
Juho Kannala and Sami S Brandt. A generic camera model and calibration method for conventional, sidelangle, and fish-eye lenses. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMII), 2006.
Bernhard Kerbl, Georgios Kopanas, Thomas Leimkühler, and George Drettakis. 3d gaussian splat- ting for real-time radiance field rendering. ACM Transactions Graphics (TOG), 2023.
Bernhard Kerbl, Andreas Meuleman, Georgios Kopanas, Michael Wimmer, Alexandre Lanvin, and George Drettakis. A hierarchical 3d gaussian representation for real-time rendering of very large datasets. ACM Transactions on Graphics (TOG), 131,, 2024.
Leonid Keselman and Martial Hebert. Approximate differentiable rendering with algebraic surfaces. In European Conference on Computer Vision (ECCV), 2022.
Shakiba Kheradmand, Daniel Rebain, Gopal Sharma, Weiwei Sun, Yang-Che Tseng, Hossam Isack, Abhishek Kar, Andrea Tagliasacchi, and Kwang Moo Yi. 3d gaussian splatting as markov chain monte carlo. Advances in Neural Information Processing Systems (NeurIPS), 37:80965-8000 80986, 2024.
Varun Ravi Kumar, Ciarán Eising, Christian Witt, and Senthil Kumar Yogamani. Surround-view fisheye camera perception for automated driving: Overview, survey & challenges. IEEE Trans- actions Intelligent Transportation Systems, 24(4):3638 00113333336699,00 2023.
Zimu Liao, Siyan Chen, Rong Fu, Yi Wang. Zhongling Su, Hao Luo, Li Ma, Linning Xu, Bo Dai, Hengjie Li, et al. Fisheye-gs: Lightweight and extensible gaussian splatting module for fisheye cameras. ECCV Workshop, 2024.
Changkun Liu, Shuai Chen, Yash Sanjay Bhalgat, Siyan HU, Ming Cheng, Zirui Wang, Vic- tor Adrian Prisacariu, and Tristan Braud. Gs-epr: Efficient camera pose refinement via 3d gaus- sian splatting. In The Thirteenth International Conference Learning Representations (ICLR), 2025.
Tao Lu, Mulin Yu, Linning Xu, Yuanbo Xiangli, Limin Wang. Dahua Lin, and Bo Dai. Scaffold-gs. gs:: Structured 3d gaussians for view- adw-tidative rendering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 20654-20664, 2024.
Zhaoyang Lv, Maurizio Monge, Ka Chen, Yufeng Zhu, Michael Goesele, Jakob Engel, Zhao Dong, and Richard Newcombe. Photoreal scene reconstruction from an egocentric device. In ACM SIGGRAPH Conference Papers, 2025.
Alexander Mai, Peter Hedman, George Kopanas, Dor Verbin, David Futschik. Qiangeng Xu, Falko Kuester, Jonathan T Barron, and Yinda Zhang. EVER: exact volumetric ellipsoid rendering for real-time view synthesis. IEEE International Conference Computer Vision (ICCV), 2025.
12

---

Published as a conference paper at ICLR 2026
Saswat Subhajyoti Mallick, Rahul Goel, Bernhard Kerbl, Markus Steinberger, Francisco Vicente Carrasco, and Fernando De La Torre. Taming 3dgs: High-quality radiance fields with limited resources. In SIGGRAPH Asia Conference Papers, 2024.
Xiaoyang Mao, Lichan Hong, and Arie Kaufman. Splatting of curvilinear volumes. In Proceedings Visualization '95, pp. 61-68. IEEE, 1995.
Christopher Mei and Patrick Rives. Single view point omnidirectional camera calibration from planar grids. In IEEE International Conference on Robotics and Automation (ICRA), 2007.
Nicolas Močnne-Locooz, Ashkan Mirzaei, Or Perel, Riccardo de Lutio, Janick Martinez Esturo, Gavriel State, Sanja Fidler, Nicholas Sharp. and Zan Gojeic. 3d gaussian ray tracing: Fast tracing of particle scenes. ACM Transactions on Graphics (TOG), 2024.
Hazem Rashed, Eslam Mohamed, Ganesh Sistu, Varun Ravi Kumar, Ciaran Eising, Ahmad El- Sallab, and Senthil Yogamani. Generalized object detection on fisheye cameras for autonomous driving: Dataset, representations and baseline. Proceedings of the IEEE/CVF Winter Confer- ence on Applications of Computer Vision, (WACV), pp. 2272-2280. 2021.
Liu Ren, Hanspeter Pfister, and Matthias Zwicker. Object space ewa surface splatting: A hardware accelerated approach to high quality point rendering. In Computer graphics forum (CGF), 2002.
Samuel Rota Bulò, Lorenzo Porzi, and Peter Kontschieder. Revising densification in gaussian splat- ting. In European Conference Computer Vision (ECCV), pp. 347-362. Springer, 2024.
Johannes Lutz Schönberger and Jan-Michael Frahm. cuncturr-fro-mmotion revisited. In Confer- ence on Computer Vision and Pattern Recognition (CVPR), 2016.
Ahmed Rida Sekkat, Yohan Dupuis, Varun Ravi Kumar, Hazem Rashed, Senthil Yogamani, Pascal Vasseur, and Paul Honeine. Synwoodscape: Synthetic surround-view fisheye camera dataset for autonomous driving. IEEE Robotics and Automation Letters (RAL), 7(3):8502-8509) 2022.
Chinmay Talegaonkar, Yash Belhe, Ravi Ramamoorthi, and Nicholas Antipa. Volumetrically con- sistent 3d gaussian rasterization. In Proceedings of the Computer Vision and Pattern Recognition Conference (CVPR), pp. 10953-10963, 2025.
Lee Westover. Footprint evaluation for volume rendering. In Proceedings of the 17th annual con- ference on Computer graphics and interactive techniques, pp. 367-376, 1990.
Qi Wu, Janick Martinez Esturo, Ashkan Mirzaei, Nicolas Moenne-Loccodd and Zan Gojeic. 3dgut: Enabling distorted cameras and secondary rays in gaussian splatting. Conference on Computer Vision and Pattern Recognition (CVPR), 2025.
Zhiwen Yan, Weng Fei Low, Yu Chen, and Gim Hee Lee. Multi-scale 3d gaussian splatting for anti- aliased rendering. In Proceedings the IEEE/CVF Conference Computer Vision and Pattern 2024.
Recognition, (CVPR), pp.
Chandan Yeshwanth, Yueh-Cheng Liu, Matthias Nießner, and Angela Dai. Scannet++: A high- fidelity dataset of 3d indoor scenes. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2023.
Senthil Yogamani, Ciarán Hughes, Jonathan Horgan, Ganesh Sistu, Padraig Varley, Derek O' O'Dea, Michal Uricár, Stefan Milz, Martin Simon, Karl Amende, et al. Woodscape: A multi-task, multi- camera fisheye dataset for autonomous driving. In Proceedings of the IEEF/CVF International Conference on Computer Vision (ICCV), pp. 9308-9318 2019.
Senthil Yogamani, David Unger, Venkatraman Narayanan, and Varun Ravi Kumar. Fisheyebevseg: Surround view fisheye cameras based bird's-eye view segmentation for autonomous driving. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (CVPR), pp. 1331-1334, 2024.
Zehao Yu, Anpei Chen, Binbin Huang, Torsten Sattler, and Andreas Geiger. Mip-splatting: Alias- free 3d gaussian splatting. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, (CVPR), pp. 19447-19456, 2024a.
13

---

Published as a conference paper at ICLR 2026
Zehao Yu, Torsten Sattler, and Andreas Geiger. Gaussian opacity fields: Efficient adaptive surface reconstruction in unbounded scenes. ACM Transactions on Graphics (TOG), 2024b.
Zheng Zhang, Wenbo Hu, Yixing Lao, Tong He, and Hengshuang Zhao. Pixel-gs: Density control with pixel-aware gradient for 3d gaussian splatting. In European Conference on Computer Vision (ECCV), pp. 326-342. Springer, 2024.
Zichao Zhang, Henri Rebecq, Christian Forster, and Davide Scaramuzza. Benefit of large field- of view cameras for visual odometry. 2016 IEEE International Conference on Robotics and Automation (ICRA), pp. 801-808. IEEE, 2016.
Hexu Zhao, Haoyang Weng, Daohan Lu, Ang Li, Jinyang Li, Aurojit Panda, and Saining Xie. On scaling up 3d gaussian splatting training. In The Thirteenth International Conference on Learning Representations (ICLR), 2025.
Matthias Zwicker, Hanspeter Pfister, Jeroen Van Baar, and Markus Gross. Ewa volume splatting. In Proceedings Visualization, 2001. VIS'O1., pp. 29-538. IEEE, 2001.
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CONTENTS
Introduction
Related Works
2.1
Splattin--Based Gaussian Rendering
2.2
Ray-Based Gaussian Rendering
3
Method
3.1
Rendering with Exact Projective Geometry
3.2
Ray-Particle Association with Exact Projective Geometry
3.3
Bipolar Equiangular Projection (BEAP)
Experiments
4.1
Scaling Gaussians vs. Projective Exactness in Wide-FoV Cameras
Comprehensive Evaluation on All-FoV Novel View Synthesis
Comprehensive Runtime Analysis
4.4
Local Affine Approximation vs. Projective-Exie Exact Integral
4.5
BEAP On/Off Study
10
5
Conclusions
10
a
Ethics Statement
Reproducibility Statement
11
LLM Usage Statement
17
Appendix: Exactness and Limitation
17
Appendix: Ray-Gaussian Transmittance with Projective Exactness
17
B. J Preliminarics And Notations
17
B.2 Differentiable Rendering with Projective Exactness
18
B.3 Ray Integral of Isotropic 3D Gaussian
19
Appendix: Cookbook of Gradient Computation
20
Chain Rule
20
Two-Stage Propagation
20
Gradients for Canonical Ray
................................
20
Gradients for Whitening Matrix
21
C.5
Gradients for Scaling and Rotation
21
6
Gradients for 3D Mean
21
Numerical Stability of Our Differentiable Framework
21
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D Appendix: Additional Details in Association
22
D.1 Particle Bounding Frustum (PBF) Solver
22
D.2 Details in Computing CSF-PBF Intersection
23
Relation to Prior Works
24
E
Appendix: BEAP Image Inverse Interpolation
25
Pinhole Camera Model
25
KB Fisheye Model
25
Appendix: High-level Algorithmic Summary
26
Appendix: Additional Implementation Details
26
Baselines
26
Implementation Details
26
Central and peripheral regions in ScanNet++ evaluation
27
Cross-camera generalization evaluation on ZipNeRF
27
H
Appendix: EWA Splatting Preliminaries
27
From camera coordinates to ray coordinates
27
Local afline approximation
................................
27
Splatting approximation error
Appendix: Advantage over Prior Tracing Methods
28
J
Appendix: Results on Egocentric Dataset
29
Appendix: Additional Material and Results
30
Additional Material.
30
Additional Results.
30
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LLM USAGE STATEMENT
We used a large language model (ChatGPT, OpenAT) as a writing assistant. Specifically, the LLM was employed to (i) improve the clarity and readability of the manuscript, (ii) suggest stylistic ed- its to figure captions and section transitions, and (iii) provide alternative phrasings for technical descriptions without altering the scientific content.
The LLM was involved in research ideation, algorithm design, data analysis, or experimental validation. All technical contributions, mathematical derivations, and results are the work of the authors. The authors take full responsibility for the content of this paper.
A APPENDIX: EXACTNESS AND LIMITATION
We clarify that our exactness refers to eliminating linear approximation in projective geometry dur- ing both the "rendering" and "ray-pan tray-particle association" procedure, where errors will not decrease as the number of Gaussians increases under our settings (see Sec. [.Ip. We do not claim exactness in the full physical volumetric sense. Our framework still involves approximations like ordering, self-attenuation, and overlapping which are our limitations.
Limitations, Our formulation integrates each Gaussian independently: hence it does not explicitly resolve self-occlusion, and ordering is handled through depth- sorting, As also shown in the supplementary videos, this may lead to occasional popping artifacts. To our knowledge, EVER (Mai et al., 2025) is the only method that analytically resolves ordering and self-occlusion by approximat- ing each Gaussian as a constant-density ellipsoid, though this comes with a substantial reduction in rendering speed.
Importantly, prior mny-tracing-basel Gaussian renderers such as Fuzzy Metaballs (Keselman & Hebert 2022) and subsequent methods including 3DGR/UT (Moënne- et al. [2024] Wu al. 2025) rely on the seaximum-response,
assumption for ray-Gaussian interaction. This assumption inherently ignores ordering and self-attenuation, yet has been shown to introduce negligible error in practice (see (Celarek et al. (2025) Fig. 9-11 / Sec. 7.1) regardless of the number of Gaussians. This explains that our depth- based ordering still can achieve the same practical accuracy as existing tn-tracing--ased Gaussian renderers.
Future work. While the practical impact of ordering and self-occlusion remains small in current Gaussian-based ray tracing, these factors fundamentally limit exact volumetric correct- ness. A promising future direction is incorporate analytic learnable self-attenuation mod- els els-potentially inspired by ellipsoidal integration (e.g.. EVER Mai et al. (2025)) but without incur- ring heavy runtime costs. Another important avenue is handling ordering more principled man- ner, which may further reduce the rare popping artifacts observed under extreme configurations. We believe these limitations are orthogonal to our contributions on projective exactness, and addressing them would make the framework even more robust for large-scale or high-dynamic scenes.
B APPENDIX: RAY-GAUSSIAN TRANSMITTANCE WITH PROJECTIVE EXACTNESS
B. PRELIMINARIES AND NOTATIONS
We follow the classic parameterization of 3D Gaussian primitives as introduced in prior work (Zwicker et al. 2001). Each 3D Gaussian distribution is defined by a mean position μ R and 3D covariance matrix Σ in the world coordinate system:
(x) = p 1/ exp (--(x- (-
1)
-1)). (x-)).
(B.I)
where we adopt the normalization constant p as V2T, and x E R denotes point in world space. This formulation differs from recent methods such 3DGS (Kerbl al.) 2023) and 3DGUT (Wul et al., 2025). which omit the determinant term |/ to simplify splatting or association during rendering. The form of covariance matrix obeys: Σ RSS. R given scaling matrix
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S diag(s) constructed from the scaling vector R&, and rotation matrix R SO(3) derived from a unit quaternion q E R.
To evaluate the transmittance contributed by single 3D Gaussian particle along a ray r(t) o ttdd, we have:
/
(o, d) = J,ER Gr. (o + id) dt,
(B.2)
where a is the opacity coefficient associated with the Gaussian.
Finally, considering set of 3D Gaussian particles sorted by depth contributing the rendering color of the given ray, the rendered color C' is computed as:
i-1 aI
C(o,d) = =Σ
Gs (o, 1.)T.
(B.3)
3=1
where (o, (4) derives the view-dependent color of the i-th Gaussian as seen from the optical center parameterized using spherical harmonics (SH) (Kerbl et al. 2023). Through minimizing the reconstruction loss between the rendered color and the observed color along each ray, the parameters 9, Jr, the coefficient as well as the SH coefficients, are jointly optimized via backpropagation.
B.2 DIFFERENTIABLE RENDERING WITH PROJECTIVE EXACTNESS
The core challenge in achieving projective-exact Gaussian rendering in ray marching formulation lies in obtaining a closed-form solution for integrating Gaussian density and color along a 3D ray. To address this, we first transform the ray- iny-relative to each Gaussian-ing a canonical coordinate system where the transmittance integral (see Eq. [B2)) remains consistent and the Gaussians become isotropic. We then derive the exact ray color from first principles.
Specifically, we define canonical coordinate system shared across all 3D Gaussian primitives. point u E R3 in this canonical space is mapped to world coordinates for each Gaussian via:
RS 0
u
H
(B.4)
From this transformation, an infinitesimal segment Au along a ray in the canonical frame corre- sponds a segment Ax = RSSu the world frame.
To ensure that the transmittance contributed by each Gaussian along the ray is preserved across coordinate systems, we require that the elementary transmittance (i.e., the product of density and segment length) remains invariant under the transformation (see Fig. 2 Righr):
GLo(u) |uu = Gx,n(x) |x
(B.5)
This constraint implies that the Jacobian determinant is absorbed into the measure. Note that
Ax
Au
|RS| = 1/2 We can thus express the corresponding canonical density as:
1 p (u) exp (-2uTu)
(B.6)
which is isotropic and shared across all primitives in canonical space.
This leads to an important conclusion that the exact closed-form solution for the transmittance inte- gral of each Gaussian along the ray r(t) can thus be obtained by computing the accumulated density along the transformed ray (t) = Ou + td, in the canonical isotropic Gaussian. Note that < is always negligible unless the optical center is within 3-sigma, then all 3D Gaussian rendering frameworks cull the Gaussian when optimization.
This admits closed-form solution in terms of the perpendicular distance from the ray the Gaus- sian center, as captured in Theorem
GLo(o. + id,) dt = a exp ( (P,s(o,)))))
T (o. = J
(B.7)
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where (o, d) denotes the perpendicular distance from the Gaussian center (i.e., the origin in the canonical space) to the transformed ray (t). Intuitively, this quantity also represents the minimal Mahalanobis distance from any points the ray the Gaussian in world space, measuring how close the ray passes by the particle distribution.
The remaining question is just about how compute the perpendicular distance from the canonical origin to the transformed ray. We derive the transformed ray's direction vector d, and moment vector m,
d = Rs- m. On X du.
(B.8)
where On RS- (o p) denotes the optical center in canonical space. Based on this, the square of D,, d) can be efficiently calculated as:
(D (D,.rood))3 m m
(B.9)
d d2
By substituting this distance square into Eq. B.7 and then the derived transmittance value into
Eq. B.3 we compute the accumulated color along the ray.
B.3 RAY INTEGRAL OF ISOTROPIC 3D GAUSSIAN
Theorem 1. Given ray and standard isotropic 3D Gaussian, the aay-Gaussian integral has closed-form proportional to:
exp (-2D²).
where D) is the perpendicular distance from the Gaussian center to the ray.
Proof. We first apply a translation to the coordinate system such that the Gaussian G centered at the origin, and the ray is translated accordingly. Next, we rotate the entire scene so that the ray direction is aligned with the z-axis. Due to the canslltion-invariann
the integral and the isotropy of the standard Gaussian, these transformations do not affect the result of the integral.
The translated standard isotropic 3D Gaussian is defined as:
1 u
GI,((u)
exp
(B.10)
P
which matches Eq. B.6 Let the z-axis aligned ray
To
0
ra(1)= =
O
(B.11)
the myy-Gussiinn integral becomes:
/ LER
ug
+
(_11)) 2
Gro((r(()) di exp
exp
2
IER
(B.12)
2m
exp
D2
p
where D + denotes the squared perpendicular distance from the ray to the Gaussian center in canonical space. Thus, the integral is proportional to exp D>)) as stated in theorem.
Note that the density function shared by 3D primitives in the canonical space (see Eq. B.6 follows the same formulation as in the isotropic case. Given the world-coordinate ray (o, d) as the input of D. its value also depends on the world-space Gaussian parameters μ. E, which are used to transform the ray into the canonical coordinate system. Therefore, for each Gaussian contributing the ray, we derive the transmittance as:
V cxp ( - (Dux(o,
(B.13)
which recovers Eq. B.7 when picking p 2t.
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C APPENDIX: COOKBOOK OF GRADIENT COMPUTATION
C.I CHAIN RULE
Recall that T aa is the ray- aa-Gaussian transmittance (Eg. B.7). where o is the opacity coefficient, and a is the ray- Gay-Gaas integral. We denote K = D as the minimal squared Mahalanobis distance from the ray to the Gaussian distribution,
the Gaussian scaling, and R as the rotation
matrix in the world space. Additionally, given Σ RSS R as the 3D covariance matrix of the Gaussian in the world space, we have W pea as the PCA whitening matrix satisfying:
W pca
= = =
!
(C.1)
where
5-1 diag
As illustrated in Fig. 2 we can apply the chain rule to find derivatives w.r.. the Gaussian parameters, e.g, scaling vector S, rotation quaternion q. and mean μr:
OT 0T
Ōx
aW pca
(C.2)
Os
dR
OW
ds
pca
OT 0T
ŌR
aw pca.
(C.3)
dq ÖK UW pea
dq
aT OT ŌR
(C.4)
0p 0x 0p
C.2 TWO-STAGE PROPAGATION
To reduce computational overhead shared propagation, we design our gradient computation fol- lowing the structured two-stage approach of 3DGS (Kerbl et al. 2023). comprising RenderCODA (RC) and P PreprocessCUD. (PC). The former performs ray-wise accumulation, efficiently ac- celerated by our Particle Bounding Frustum (PBF) association (see Section 3.2), while the latter operates with Gaussian-win wise complexity, leveraging view frustum culling for additional speedup.
According to Eq. B.7 can respectively write the gradient opacity coefficient and the shared partial derivative for Gaussian parameters in RC as:
0T
07
T
a,
(C.5)
Ök
2
RRC
RC
The remaining gradients to the Gaussian parameters can then be back-propagated from the squared distance, e.g., Us Öq 0p
0x 0x dn.
C.3 GRADIENTS FOR CANONICAL RAY
then backpropagate gradients to the canonical ray parameters: the origin Ou, direction d.,, and moment m., d,, as defined in Eq. [.8 in our paper. These quantities directly influence the Mahalanobis distance Eq. B.9 and are eritical for capturing canonical ray geometry.
8x
2 d d.
m.
(C.6)
0m,
ICC
0%
dR.
duy
(C.7)
do,
dm,
BC
2k
dk d, + o, x 0m.
(C.8)
0d,
d d2
RC
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C.4 GRADIENTS FOR WHITENING MATRIX
Recall that d, W pea and On pea 14), where o = -R t. denotes the world space optical center, and d denotes the ray direction. Once the gradients w.r.t the ray parameters obtained, we further propagate them to the PCA whitening matrix W pea via outer products with the world-space vectors:
Ōk
RenderCUDA:
d ådn
OW pea
Ōk
OK
PreprocessCUDA:
(o
1e)
(C.9)
OW
don
Ōk
Ōk
Overall:
OW pea
OW pca
OW pea PC
Here, the gradient backpropagated through d., is accumulated ray-wise in Rende while the gradient through o,, is computed Gaussian-wise in P Preprocess then summed to yield the complete gradient w.n.t. the PCA transformation matrix W pea.
These two components are
C.5 GRADIENTS FOR SCALING AND ROTATION
Thus, for scaling gradients
8x
@W Üs
we have:
DW
pea
aw
Wi
Ry
if
pca
(C.10)
Osk
otherwise
0x To derive gradients for rotation qak, we can write the whitening matrix as:
aw peu dq
given the quaternion form q = Qr + q.i + 933 +
8q
8W pea
1 1 93 g
(q9) + g.9%)
(g.9: g-
W pca =
7
$ 9:q; 9.9k)
1 of g
4
(g qk + 9rqs)
(C.11)
I G - q gf))
9999 + 9.9;)
(q/9s- 9-9()
As result, we derive the following end gradients for the quaternion q
0i
g1
98
aW
QE:
()
@i
aw poa
2
9i
0qr
Bq:
PC
41
()
ECC
20g
85
85
(C.12)
2q:
2qk
8W
81
()
8W
24k
8q;
P0
2q
I2C
80
83
83
83
C.6 GRADIENTS FOR 3D MEAN
Meanwhile, we compute end gradients w.r.f. the Gaussian 3D mean f as:
Ōk
WT Ōk
(C.13)
Dp
pea Do..
PC
C.7 NUMERICAL STABILITY OF OUR DIFFERENTIABLE FRAMEWORK
The choice of forward formulation, backward gradient computation, and intermediate cache vari- ables critically affects numerical stability. Prior approaches such as GOF (Yu et al., 2024b) suffer from instability in backward propagation and therefore rely additional 3D filtering heuristics to stabilize training. In contrast, our closed-form formulation. together with its exact gradient chain, avoids such instability altogether, eliminating the need for any auxiliary filtering.
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GOF
w/o 3D Filtering
GOF
Our Forwards & Backwards
Wi Filtering
Figure C.I: Artifacts without 3D filtering in GOE (Left) when degenerated Gaussians are not filtered in GOF, and (Right) the stable results obtained when applying our forward and backward formulations.
The key distinction from our gradient and rendering pipeline lies in that prior methods adopt an alternative expansion of m, m.. Ideally, disregarding numerical issues, the following two forms are geometrically equivalent:
m. m. = IIo.,//ldd,, || (o,, d.)²²
(C.14)
However, when 3D Gaussian has one axis scaled close to zero, the canonical-space vectors o,, and d, become nearly parallel. In this case, This makes the term |/o./| can even yield negative values and further lead to instable backward propagation as well.
d..)) - 1, while both |o..l and ||d, diverge. (o., d highly prone numerical overflow and, in practice,
Fig. C.I (Left) demostrates the numerical artifacts produced by GOF. When it does not apply addi- tional 3D filtering to remove degenerated Gaussians, severe artifacts appear. The snowflake noise arises from numerical overflow in Eq. C.14 where negative values are prematurely returned and extremely small positive values lead to amplified Gaussian intensity. Since the EWA approxima- tion with Jacobian and conic remains numerically stable, the artifacts mainly stem from inaccurate maximun-response computation, which also induces gridline patterns.
In contrast, our CUDA rasterizer implementation computes both the cross product and its gradients directly (Eq. B.9 (forward) and Eqs. C.6.C.9 (backward)), thereby fundamentally avoiding this numerical issue. As result, the forward pass artifact-free, while the backward pass yields accurate gradients. Fig. C.1 (Right) illustrates the results of applying our forward and backward formulations GOF; even without additional 3D filtering exclude degenerated Gaussians (i.e., flat disk- or needle Gaussians), our approach effectively eliminates the artifacts caused by numerical instability.
D APPENDIX: ADDITIONAL DETAILS IN ASSOCIATION
Section D.I provides our detailed derivation. Section D.2 provides implementation considerations required for stable PBF under large-FoV and omnidirectional cameras. Section D.3 clarifies the derivational relationships and explicitly states the conditions under which prior formulas coincide with ours.
D.1 PARTICLE BOUNDING FRUSTUM (PBF) SOLVER
Recall Eq. g we define P = (V E R4x4 as the plane transformation matrix from the view space the canonical space. Take the unknown bounds for tan 01, as example, we have:
Poo + cPuz
Pio cP12
gu P
(D.1)
P2o - cP22
P30 cP32
where P; denotes i-th row of the matrix, expressed as a x column vector, and P.ş further denotes j-th element. Recall Eq. 9. assuming we are computing the bounding box of the 1-standard-devianl contour ellipsoid of the Gaussian, ie.. À = 1, then we have the following quadratic equation:
((f PT) P2) d² ((f P.T) Pa) (f P.T) Po 0,
(D.2)
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where f = (1,1,1,-11 is a x 4 row vector. Here we denote the equation
T2ac 2Toze + Too 0,
(D.3)
where T a x 3 upper triangular matrix defined with Tij (5 P.) P3.
Similarly, for the bounds of Φ, we have = tan 61,2 satisfying:
Tz2c 2T22c Tn 0.
(D.4)
Interestingly, given a camera-view configuration [Re tz], we find that the parameter matrix T can be efficiently computed from the Gaussian's view-space covariance D. RRRSS R R and 3D mean mhc = Rof + to:
To
x (MoMe)
(D.5)
Thus, we can compute the angular bounds (defined by their tangent-space center and extent) from:
tan 01 + tan 02 tan Φ1 + tan Φ2
To2 T12
2
2
T22 To2
2/ 2 'T3
T22 Too
tan 01 tan 02ll
(D.6)
1222
2/ T12
tan Φ1 - tan dall
|Tzal
If the discriminant of either quadratic is negative-indicatiog that the camera's optical center lies inside the Gaussian ellipsoid and the angular bounds are undefined-we conservatively clamp the frustum angles using the camera's maximum field of view (FoV).
Note that this result is directly computed from the 3D view-space true covariance, in contrast to methods like EWA and UT, which rely intermediate conic approximations in screen space. Our PBF formulation offers both exactness and efficiency.
D.2 DETAILS IN COMPUTING CSF-PBF INTERSECTION
We uniformly sample spherical angles within the maximum field of view (FoV). The resulting x, y. and 2 coordinates -i.., the ray direction vectors d are projected onto unit sphere and then transformed into pixel space for inverse color interpolation (see Sec. Eb. These sampled rays are then grouped according to their corre- sponding CSFs for subsequent Gaussian asso- ciation.
Z
8
Ba
In most scenes using pinhole camera with limited FoV, the angular bounds in PBF typi- cally lie within [- 7 77. In such cases, since the tan function is monotonically increasing over this interval, the intersections between PBF and CSFs can be computed by directly comparing the tan values of their respective an- gular bounds. However, in wide FoV scenar- ios, even when the incidence angle formed by the Gaussian centers remains within [- ;, = the angular bounds used in PBF may still fall outside the monotonic region of the tan func- tion. In these cases, directly comparing tan val- may yield incorrect PBF-CSF intersections, such as mistakenly matching angular bounds as (0%, 01] (see Fig. D.I for an example).
# P
(0.0,
Figure D.I: Mirror Transformation of PBF An- gular Bounds. Four candidate mirrored angles 81,g, 0' "i.g, 02.2.. 6' are generated from a given pair of bounds. Given Gaussian centered at μa, the two closest to the corresponding Gaussian's 's mirrored angle Un, are used as PBF bounds for robust intersection with mirror-transoomee CSF partitions.
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To resolve the correct intersections, we introduce mirror transformation applied to the tangent values of both PBF angular bounds and CSF partitions. Inspired by the CMEI camera model (Mei & Rives 2007), we shift the optical center using mirror parameter ç, leading to the following transformation:
tan 0
sin 0 cos
tan 0g
(D.7)
+ E |
+ tan
{
where 0g represents the mirror-transformed angle of 0 on the æz-plane, using mirror parameter &. We set = throughout all experiments.
Given pair of tangent value [tan 81, tan 02] from PBF solver (See Equation. D.6., this trans- formation yields four candidate mirror angles, as illustrated in Fig. D.I. The Gaussian center, in contrast, corresponds to unique mirror-transformed angle O,.e- We exploit this property by select- ing the two candidate angles closest to On.e- These two then define the effective lower and upper bounds for PBF, i.e., [01.g: 02.gl. which are subsequently compared with CSF partitions to compute the correct intersections.
This trick ensures robust and consistent angular comparisons, even beyond the monotonic do- main of the tan function. As reported in our timing analysis (Tab. K.5 the pre-processing the mirror-angle-baat based PBF-CSF intersection computation and the canonical transformation for all Gaussians- takes less than 0.13 ms in total. This confirms the efficiency of our approach, even when applied to thousands of Gaussians across wide FoV scenarios.
D.3 RELATION TO PRIOR WORKS
Substituting the perspective screen space coordinates:
tan O,
V = tan Ø
into our formulation yields expressions identical to those in 2DGS (Huang et al., (002a) and HTGS (Hahlbohm et al., 2025). This is expected perspective alternative of our angular-domain formulation.
Concretely, 2DGS and HTGS start from a screen-space or tangent-plan plane parameterization and there- fore rely on assumptions that hold only for perspective (pinhole) cameras, e.g., geometric identities that arise when rays are implicitly tied to planar, z-parallel image plane. This assumption holds only for perspective cameras and does not extend to fisheye or omnidirectional systems, where the corresponding surface in screen space is curved. In contrast, derivation begins from camera- agnostic angular parameterization in 3D and directly parameterizes Camera Sub-Frustums (CSFs) and Particle Bounding Frustums (PBFs). These different starting assumptions change the known quantities and constraints available during derivation, and thus the mathematical conditions under which closed-form bounds and culling guarantees remain valid for generic cameras.
We' d like to emphasize that mathematics in itself is general tool, but deriving these bounds under weaker assumptions (i.e., without relying on screen-space structure) and validating them with com- prehensive experiments across both pinhole and wide-FOV camera families constitute an essential part of our contribution.
This difference in parameterization also affects the downstream computational pipeline. Gaussian association, tiling/duplication, and sorting depend on how Gaussians map to tiles or bins. Using an angular (CSF-based) domain yields a different bucketing structure compared to pinhole screen- space tiling. This is practically important for large-FoV scenes: under generic cameras a Gaus- sian's bound may appear partially "behind" the camera in screen coordinates (see Fig. D.I. caus- ing screen-space methods to produce mirrored or incorrect tangent-based bounds (dashed line in Fig. D.I). To robustly handle these cases, introduce mirror parameters in the association step (Section D.2) and incorporate the Gaussian center into sorting and correction cenetiion-oommentttt unnec- essary in perspective-only derivations.
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Imaging Plane
Imaging Sphere
(d.ar degs diee)
dcz
des
arcton
d
arcran
d..
Figure E.I: Spherical Angles Defining PBF and BEAP. A view-space direction d, is first nor- malized onto the optical sphere (green spade), followed by angular projections onto the 22 and ye planes. The orange point denotes the resulting coordinate (0, Φ) the unfolded BEAP imaging plane.
E APPENDIX: BEAP IMAGE INVERSE INTERPOLATION
To prepare training data, sample rays comprehensively cover the original image FoVs and use them as input to our CUDA renderer. Simultaneously, we interpolate the ground truth color for each input ray to ensure accurate and uniform supervision during training.
The projection sampled rays onto observed images begins with transforming the incidence angle projections 0 and into normalized ray direction vectors d = (z, y, expressed as:
sin cos ó T sin f cos Φ + cos sin² Φ + cos cos Φ
(E.1)
cos 0 sin ø
y sin 0 cos + cos sin cos 0 cos db
(E.2)
cos f cos Ø
(E.3)
v sin 0 cos² + cos sin? Φ cos" 0 cos
Several camera models are provided subsequently project the normalized rays (x, y, =) into pixel space (Ipp Up 1)
Pinhole Camera Model. For pinhole camera models, we first have tan and y tan Φ
Thus, we have:
p
fa
0
Cx
[aan 0 tan Φ
Up
fy
Cy
(E.4)
0
KB Fisheye Model. We have the incidence angle derived from its projections:
tan" + tan? &b and = arctan(r).
(E.5)
And we have the fisheye distorted angle:
Pa (01 k2 + kepp + kypp kapps).
(E.6)
As result, the rays are projected onto pixel space, allowing for inverse color interpolation from:
xp
Lapa
0
Cz
tann 0
yp
0
Cy
lan Φ
(E.7)
0
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F APPENDIX: HIGH-LEVEL ALGORITHMIC SUMMARY
We provide high-level overview of our volumetric rendering pipeline (See Alg. | and Alg. 2 For clarity, we follow the structure of 3DGS Kerbl et al. (2023) to illustrate difterences: newly introduced modules or replaced ones are highlighted in green, and significantly modified ones in yellow.
ALGORITHM GPU Software Asso. Render.
Function 3DGEER (Js, S, C, ra; V):
CullGaussian(f, V)
Frustum Culling Partition D Transform Sub-frustum Asso. Transform
F+ InitCSFs(w, h, K)
De: Me ViewspaeeCausiannsss.. V)
FID - PBF Intersecct u.
Wpa f- PCA PCA_ Whittninggssng
GID. FK + DuplicateeithkoossS W
pea: FDD)
Indices (Gaussian) & Keys (FrustumID+Deptt)) b Globally Depth Sort
Sort SyrtByyeys Fx) R -
> Init Canvas (w/ Rays)
for all CSFs f do
for all Rays i + do
x Gaunrustuu R, f)
Ma. du + Cano_ Ray W pea. He, i)
> Transform Mah-distance
D D.. Dist(m,,, d..)
[ f BlendInOrden(i, FR, Dp
b Alpha Blending
end end return I
ALGORITHM 2: 3DGEER Initialization and Training Framework w, h: width and height of the training images
K, V: intrinsics and extrinsic meta info.
Init Gaussian Attributes Position Scaling, Rotation > Color, Opacity
4 < SfM Points
q <- InitAttributess
0 <- InitCoefficients()
while not converged do
Fey Č BEAP(W, h, K)
View Rays / Colors
b Asso. & Render See Alg,
C S.
L Loss(C, Õ) AVL
Loss > Backprop & Step
s. C. o T then
μ. s. q. a Optimize(p,. q. c.
Kerbl et al.
2023
end
end
G APPENDIX: ADDITIONAL IMPLEMENTATION DETAILS
Baselines. We compare 3DGEER with leading volumetric particle rendering methods targeting
projective approximation errors in 3DGS (Kerbl et al., 2023). For large FoV fisheye inputs, we evaluate against FisheyeGS (Liao al. 2024), EVER (Mai et al., 2025), and 3DGUT (Wu et al. 2025). Under pinhole settings, we additionally compare with 3DGS and 3DGRT "idénne-Loccoos al. 2024
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Implementation Details. To ensure fair comparison. align implementation with the orig- inal 3DGS in terms of initialization and hyperparameters on the ScanNet++ and MipNeRF360 datasets. For ZipNeRF,
adopt the training configurations of EVER and SMERF (Duckworth et al. 2024) for all candidate methods. Similar to most ray-based frameworks, our method does not directly access gradients of mean2D control densification or pruning. Instead, we optimize using gradients of the view-space mean3D. Unless otherwise noted, all models are trained with consistent supervision strategies and architectural settings. For fisheye datasets (ScanNet++ and ZipNeRF), we precompute bijective grid mappings (Guo et al. 2025 and reproject all rendered outputs to the native KB- Fisheye image space (Kannala Brandt 2006) before evaluation. Experiments on Scan- Net++ and MipNeRF360 are conducted using full-resolution images. For ZipNeRF, due to the high frame count and extremely large fisheye resolution, we use the maximum resolution (see Tab. 3 that allows all baselines to run reliably on a local desktop setup.
Central and peripheral regions in ScanNet++ evaluation. follows the range:
The rays sampled in the central part
i E
[-
W
1.01
2fz 2fz
2fy 2
And the Peripheral part takes the rays sampled in remaining angle ranges. Here w, h indicate the width and height of the image and fa, fy are the focal lengths defined in the original KB camera model.
Cross-camers generalization evaluation on ZipNeRF. ZipNeRF provides two sets of data: fish- eye raw images and their corresponding undistorted frames, which are generated by applying dis- tortion correction and cropping the central regions. Although the frames are aligned on a per-frame basis, camera poses and point clouds were independently reconstructed using COLMAP (Schön- berger & Frahm 2016) for each set. Due to COLMAP's global optimization independently refining camera trajectories, applying single rigid transformation to align the two pose sets is insufficient and may introduce alignment errors. To ensure reliable cross-camera evaluation without introduc- ing errors from structure- structure-from-non from-motion (SfM), always use the same set of test-time camera poses as those used during training. Pross-camera experiments, we vary only the intrinsic parameters, while keeping the extrinsic poses fixed. This ensures fair comparisons without entangling pose inconsistencies arising from preprocessing.
H APPENDIX: EWA SPLATTING PRELIMINARIES
From camera coordinates to ray coordinates. As the coordinate definitions EWA Splatting Zwwicker et al., 200I), have the following mapping function from the camera (view) space points t to the ray space points for projective transformation in pinhole camera models, i.e., 7 t S x:
x
to/tz t
r(t)
(H.1)
[22]
[1 41, (2)
tt
Io/ll 1
1
= (x) =
I 21/1-122
(H.2)
12
1 -82
where 7 [[ 21, 1|
(H.3)
These mappings are not affine, which requires computationally expensive [Westvvrr ct al. 1995) or locally approximated solutions (Zwicker et al. 2001 Kerbl et al. 2023 to calculate integral in the ray space.
1990
Mao
Local affine approximation. EWA splatting by Zwicker et al. (200I) introduced local affine ap- proximation, denoted 7k. to keep the linear effects and the k-indexed reconstruction kernels as Gaussians in 2D screen space. It is defined by the first two terms of the Taylor expansion of r at the camera space point tg. then the ray space point x near around xς can be estimated by:
r() Xx + J (t tR),
(H.4)
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PSNNR 27.11
PSNK 21:30
PS 36 FPS: 322
FIS 341
PPS SDGRT
GT
JDGS
Ours
Figure 11: Qualitative Results on the Stump MMpp-NeRF300, 3DGEER outperforms both 3DGS- based (Gaussian Splatting) and 3DPRT-based (Particle Ray Tracing) methods on novel view synthe- sis benchmarks under (Left) MipNeRF360 (pinhole), while maintaining high runtime efficiency. Larger differences are visible in the colored boxes.
where J is the Jacobian matrix given by the partial derivatives of the point ty. For the k- indexed 3D Gaussian n in certain camera coordinate, using EWA local affine transformation, the approximated density function can be expressed as 3D Gaussian GZ, in the ray coordinate. For vx around X4. we have:
1 -- xx) T5p (x xs))
Gr,,, (x)
exp
(H.5)
(2T) & Ex|
where ME 2 (J
(H.6)
Integrating a 3D Gaussian Dx.xe along coordinate axis yields 2D Gaussian G
(2)
hence:
S
[
5 Ix (x) dæ
G(2) Sx (x),
(H.7)
where & 20 and the mean &g Ø1
Ik,0 and the variance matrix Eg is easily obtained [
from the 3 x matrix Σg by skipping the third row and column.
Splatting approximation error. Although we can not expect the density function ray space
still obey the Gaussian distribution when we apply general projective transformation the original view space, exact density can be derived through substitution via Eq. H.2
(t) Gr.n (T (x))
1
exp (r tx) tt)² 2 (r (x) -ta))
(H.8)
(2Tr)
$ Ex
which means that the approximation error can be estimated through Monte Carlo integration of the following equation:
/ (x)) GES.X (x) dæz.
Ex = ISE.
(H.9)
Note that da represents the dimension along the ray direction in the ray coordinate.
I APPENDIX: ADVANTAGE OVER PRIOR RAY-TRACING METHODS
Pure ray-tracing (e.g., 3DGRT) can ideally projective-exact, but in practice the BVH adopted in 3DGRT for intersection tests uses inscribed proxy icosahedron for faster speed, which is tight at vertices but face centers lie ~00 inward based on icosahedron's 's property. Proxy shapes may de- grade intersection accuracy and further give worse accumulated gradients during density control and eventually lead to incorrect geometry (e.g., missing leaves in Fig. I.I) Further, intersection tests for far, tiny, and cluttered Gaussians are limited by floating poating--ointt precision that causes noisy rendering. (E.g.. in Fig. 15 of the 3DGRT paper, native rendering in the leftmost block contains visible speckles in the tree and background areas.) Even 3DGUT with UT approximation outperforms 3DGRT on
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Living Room
Steakbouse
Ground Truth
FisheyeGS
Ground
FisheyeGS
3DGEER
Figure J.1: Qualitative Results on the Aria Dataset. Our method produces sharper floor textures
in indoor scenes and clearer distant tree details in outdoor scenes.
Table J.1:
Quantitative comparison on Aria. Higher PSNR/SSIM and lower LPIPS are better.
Method
Scene
PSNR1
SSIM1
LPIPS
livingroom steakhouse
(indoor)
36.398
0.933
0.226
3DGEER (Ours)
(outdoor)
30.662
0.881
0.246
livingroom
(indoor)
35.686
0.927
0.231
FisheyeGS
stcakhouse
(outdoor) 29.989
0.869
0.261
MipNeRF360. For validation,
3DGRT's ray-traaa ray-tracing and rendering engine and our 3DGEER rendering pipeline to render from the same pretrained MipNeRF360-stin
and compare on PSNR1/
SSIMY/ LPIPSJ:: 3DGRT- 25.79/ 0.745/ 0.264; ours- 26.83/ 0.774/ 0.246 (+1.04/ +0.029/ -0.018). 3DGEER builds projective-exact and efficient pipeline and further and optimizes with BEAP color supervision, achieving much faster speed and better accuracy (Tab.
APPENDIX: RESULTS ON EGOCENTRIC DATASET
Recent egocentric devices typically feature wide FoVs, but existing pipelines struggle to maintain reconstruction quality when using full-resolution images or when restricted to cropped views. The challenge is further compounded by low-light imaging conditions- pecially with small sensors and aperture-consttainne constrained lenses, exacerbates degradation.
limited light intake, es-
We evaluate on the Aria egocentric dataset (~ 110º FoV. circular shape). Our method achieves an average PSNR of 33.5dB, consistently surpassing FisheyeGS (32.8dB) across all metrics (Tab, (TT.. Qualitatively, our reconstructions preserve sharper details peripheral regions for the indoor scene (Fig. J.I. Left) and recover distant structures such as trees in the outdoor scene (Fig. J.I Right). The performance gain (+0..7 dB) lies between the improvements observed in pinhole datasets (e.g., MipNeRF360: +0.3dB; ZipNeRF-cross: +2.0dB) and in highly distorted wide-FoV datasets (e.g., ZcanNett+/ZiNNeRF 180 diagonal FoV: (..B), which aligns with Aria s moderate Fo and distortion level.
29

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Published as a conference paper at ICLR 2026
K APPENDIX: ADDITIONAL MATERIAL AND RESULTS
K.1 ADDITIONAL MATERIAL.
Code, video, and HTML page are provided in the Supplementary Material.
K.2 ADDITIONAL, RESULTS.
3DGS
FisheyeGG
3DGEER
Figure K.1: Out-of-Distribution (OOD) Region in Fig. 6 3DGS collapses to black due to the
numerical issue of tan 90 co, while method robustly renders views up to 180", sucessfully to reconstruct the green chair, where splat-based methods (e.g., FisheyeGS) fail. Similarly, in Fig. 6 (zoom-in), method also preserves structures like the corridor and fridge, table and door, window and cabinet.
Pinhole Extreme Focal Length
Fisheye Extreme Wide
Training
3DGS
3DGIEER
Training Time Fov
FisheyeGS
3DGEER
Bedroom
(Undastortad)
[Original Fov)
Gaus. Num:
Laboratory
(Undastorted)
[Original FoV)
Office Room
(Undistorted)
Num:
Gaus.Nun:
Figure K.2: Extreme-Fov Generalization. (Left) Comparison between method and 3DGS under extreme-FoV pinhole (170") views (trained on undistorted central regions). (Right) Zoom-out results of Fig. showing the complete circular 180 FoV, where method generalizes robustly to out-of-distribution regions.
Table K.I: MipNeRF360 Quantitative Results by Sequence. The overall results are shown in
Tab.
Method
Metric
Bicycle
Bonsai
Counter
Garrenn
Kitchen
Stump
Flowers
Room
Treehill
PSNR
25.63
32.23
29.05
27.60
31.78
26.83
21.83
32.09
22.83
Ours
SSIMT
0.774
0.946
0.910
0.860
0.931
0.774
0.946
0.637
LPIPS]
0.223
0.175
0.195
0.136
0.120
0.246
0.340
0.108
0.344
30

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Published as a conference paper at ICLR 2026
Bedroom (ffcbbdde1)
Office Room
Kitchen
Laboratory
(alla3aaa99)
(4ef7 5031e)
(1d003b0 7bd)
32.5
34
28
30.0
12
PSWR
PSW
29-
27.5
24
28
25.0
10
10a
10
0.90
SSM
SSIM
0.06
0.92
0.93
0.95
0.94
10a
10r
10a
0.14
0.24
0.22
LAIS
0.12
0.15
LPIPS
0.14
0.20
013
0.16
10"
30
10
10
Gaussian Number (log)
Gaussian Number (log)
Gaussian Number (log)
Gaussian Number (log)
HiheyeGS
#sheyeGS thres.
3DGEER
Figure K.3: PSNR SSIM LPIPS). Trends When Scaling-Up Gaussian Counts. Brute-force
scaling fails to close the gap in projective exactness.
Table K.2: ScanNet++ Quantitative Results by Sequence (Full FoV). The overall results are
shown in Tab. 22]
EVER
FisheyeGS PSNR
3DGUT SSIM
3DGEER (Ours)
Scene
PSNR
SSIM
LPIPS
SSIM LPIPS
PSNR
LPIPS
PSNR
SSIM LPIPS
Storage
28.24
0.916
0.164
24.74
0.932
0.146
29.26
0.934
0.147
29.18
0.940
0.141
Office Room
27.80
0.899
0.169
25.75
0.927
0.145
29.37
0.928
0.145
29.78
0.935
0.125
Bedroom
32.60
0.948
0.131
31.95
0.970
0.095
34.65
0.967
0.112
34.94
0.974
0.084
Kitchen
29.77
0.924
0.213
29.72
0.944
0.177
30.78
0.941
0.196
31.17
0.946
0.168
Laboratory
28.96
0.936
0.156
26.91
0.959
0.129
29.12
0.948
0.151
32.43
0.968
0.114
31

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Published as a conference paper at ICLR 2026
Id003507bd
1:7cboddel
feft
Figure K.4: Qualitative Results the ScanNet++ Dataset. Larger differences visible in the zoomed-in colored boxes, particularly in structural details and artifact regions. Our method deliv- ers superior peripheral reconstruction compared to FisheyeGS, exhibits fewer artifacts and sharper details than EVER, and produces crisper object boundaries and textures-such as wires, keyboards, and wooden floors-wher compared to 3DGUT.
3DGEER Pinhole)
(Tr. Equid.)
(Ours)
Figure K.5: Impact of BEAP supervision. Visual comparison ScanNet++ shows that BEAP's balanced angular sampling produces smoother gradients and better preservation geometric and texture details than pinhole equidistant sampling. (Tab. K.8 shows quantitative full stats).
32

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Table K.3: Quantitative Results on the ScanNet++.. PSNR SSIM LPIPS| Trends When Scaling- Up Gaussian Counts. Bedroom, Office Room, Kitchen, and Laboratory in order.
FisheyeGS
FisheyeGS w/ Low Thres.
3DGEER
Iter
PSNR
SSIM LPIPS
Gau
PSNRT SSIM LPIPS]
Gau
PSNR1
SSIM LPIPS
Gau!.
4k
28.67
0.955
0.127
512K
30.03
0.959
0.113
1914K
31.33
0.952
0.138
422K
7k
29.12
0.961
0.114
695K
30.61
0.965
0.101
2829K
32.87
0.963
0.114
543K
10k
29.36
0.964
0.107
827K
31,21
0.967
0.095
3489K
33.42
0.967
0.103
616K
20k
30.17
0.968
0.097
985K
31.68
0.971
0.085
4467K
34.59
0.973
0.087
664K
30k
30.36
0.969
0.095
985K
31.84
0.972
0.083
4535K
34.94
0.974
0.084
664K
FisheyeGS
FisheyeGS w/ Low Thres.
3DGEER
Iter
PSNR
SSIM LPIPS
Gau
PSNRT SSIM LPIPS! Gau
PSNR1
SSIM1 LPIPS|
Gau
4k
23.19
0.911
0.164
677K
23.72
0.916
0.151
2778K
27.84
0.915
0.167
480K
7k
23.33
0.916
0.157
867K
24.06
0.923
0.138
3936K
28.37
0.922
0.152
621K
10k
23.45
0.918
0.153
983K
24.31
0.925
0.133
4481K
28.54
0.925
0.145
668K
20k
23.51
0.921
0.149
1072K
27.07
0.932
0.127
5272K
29.49
0.933
0.128
732K
30k
25.75
0.927
0.145
1001K
27.50
0.935
0.125
5272K
29.78
0.935
0.125
732K
FisheyeGS
FisheyeGS w/ Low Thres.
3DGEER
Iter
PSNR
SSIM LPIPS
Gau
PSNRT SSIM LPIPS) Gau
PSNR1
SSIM, LPIPS,, Gau!.
4k
27.58
0.926
0.221
314K
28.12
0.929
0.211
1191K
28.11
0.918
0.239
292K
7k
28.47
0.935
0.203
485K
28.99
0.937
0.193
1997K
29.11
0.928
0.215
410K
10k
28.82
0.938
0.194
588K
29.29
0.941
0.181
2491K
29.73
0.935
0.198
476K
20k
29.60
0.944
0.179
760K
29.93
0.946
0.164
3563K
30.91
0.945
0.172
552K
30k
29.72
0.944
0.177
760K
29.97
0.945
0.162
3563K
31.17
0.946
0.168
552K
FisheyeGS
FisheyeGS w/ Low Thres.
3DGEER
Iter
PSNRT
SSIM LPIPS)
Gau.
PSNR1
SSIM LPIPS. Gau!.
PSNR
SSIM LPIPSJ
Gau!,
4k
23.27
0.940
0.152
602K
24.32
0.945
0.142
2429K
25.59
0.939
0.155
406K
7k
23.94
0.947
0.141
740K
25.36
0.953
0.131
3258K
27.27
0.952
0.133
524K
10k
24.12
0.950
0.137
788K
25.65
0.956
0.126
3569K
28.14
0.958
0.124
562K
20k
26.76
0.959
0.130
838K
27.77
0.961
0.120
4035K
30.39
0.967
0.108
582K
30k
26.91
0.959
0.129
838K
27.97
0.961
0.120
4035K
32.43
0.968
0.114
582K
Table K.4: Runtime Analysis vs. Other Methods. 3DGEER is compared with efficient splatting- based methods on MipNeRF360 and ScanNet++, with thorough comparison in stage-wise and total runtime.
Dataset
Method
Avg. Timings (ms)
Prep.
Dup.
Sort
Render
Total
MipNeRF366 360
3DGS
0.32
0.27
0.65
1.40
2.92
3DGEER (Ours)
0.37
0.17
0.27
2.10
3.06
ScanNet++
FisheyeGS
0.10
0.63
1.45
2.33
4.70
3DGEER (Ours)
0.13
0.26
0.59
2.89
3.98
33

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Published as a conference paper at ICLR 2026
Table K.5: Runtime Analysis: Asso. Method on ScanNet++, The efficiency of the PBF-based association is compared to the splatting-based EWA and UT with acceleration option SnugBox in- tegrated into 3DGEER pipeline.
3DGEER
Avg. Dup.
Timings (ms)
Avg. FPS f
Asso.
Prep.
Sort
Render
Total
EWA
0.13
1.17
3.27
12.69
17.84
56
+ SnugBox
0.60
1.42
6.22
8.68
UT
0.13
0.71
1.94
7.97
11.16
+ SnugBox
0.28
0.64
3.16
4.38
228
PBF (Ours)
0.26
0.59
2.89
3,98
251
Table K.6: Impact of PBF-CSF Association. The evaluation employs the artifact-sensitive LPIPS metric. To assess cross-model robustness, each variant is also tested using models trained with the other association strategies. Additionally, we report the memory cost during rendering along with the average number of Gaussian association per Tile, and the performance gap ALPIPS between
in-method and cross-method evaluations to quantify consistency and generalization.
Asso. Method Gaus. per Tile
Mem. (GB)
LPIPS
Cross-LPIPS |
ALPIPS (lc-2)
mcan
std
EWA
2203
1447
2.2
0.1250
0.1321
0.71
UT
1377
1058
1.4
0.1249
0.1302
0.53
PBF (Ours)
475
427
0.63
0.1245
0.1251
0.06
Table K.7: Training- Time Transmittance Func. Replacement. The overall results are shown in
Tab.
3DGS Splats PBF
FisheyeGS Splats PBF
3DGEER (Ours)
PSNR
Gaus, Num (k)
LPIPS
Num (k)
LPIPS
Storage
19.27
0.859
0.173
1293
24.75
0.933
0.150
781
29.18
0.940
0.141
429
Office
Bedroom
27.39
0.923
0.119
1542
31.96
0.970
0.096
1079
34.94
0.974
0.084
664
25.15
29.87
Laboratory
21.60
0.869
0.173
1379
27.03
0.962
0.131
823
32.43
0.968
0.114
582
Table K.8: Impact of Different Supervision Space. The overall results are shown in Tab.
3DGEER w/o BEAP
3DGEER w/o BEAP
3DGEER w/o BEAP
3DGEER (Ours)
Supervision Space
Central Perspective
Perspective
Equidistant
BEAP
Scene
PSNR SSIM LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM
LPIPS
Storage
28.42
0.931
0.143
19.74
0.834
0.302
29.03
0.938
0.142
29.18
0.940
0.141
Office Room
28.90
0.927
0.133
21.12
0.825
0.291
29.71
0.932
0.126
29.78
0.935
0.125
Bedroom
33.86
0.965
0.090
24.38
0.882
0.240
34.60
0.973
0.096
34.94
0.974
0.084
Kitchen
30.65
0.941
0.172
23.23
0.889
0.338
31.23
0.941
0.177
31.17
0.946
0.168
Laboratory
27.35
0.951
0.118
17.08
0.834
0.329
30.65
0.956
0.135
32.43
0.968
0.114
34

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Published as a conference paper at ICLR 2026
Table K.9: ZipNeRF Quantitative Results by Sequence. Alameda, Berlin, London, and NYC in
order. The overall results are shown in Tab.
Train
1/8 HB
1/8 FE
1/8
1/4PH
1/4PH
1/4
Test Data
1/8FE FE
P/1
1/8TE FE 1/4
1/8 FE 1/4 PII
1/8 FE 1/4 PII
1/8 FE 1/4 PII
1/8 FE
1/4 PII
SSIM
SSIM f
FisheyeGS
20.74
23.64
0.820
0.802
0.223
0.277
17.39
23.74
0.736
0.839
0.265
0.240
EVER
0.852
20.06
22.83
0.250
3DGUT
21.78
22.39
0.841
0.718
0.193
0.368
17.03
22,58
0.588
0.765
0.329
0.311
1/8
1/8
1/4
Test Data
1/8 FE 1/4 PH PSNR
1/8 FE
1/4 PH
1/8 FE 1/4 PH
1/8 FE 1/4 PH
1/8 FE 1/4 PH
1/8 FE 1/4 PH
Method
SSIM
LPIPS
PSNR
SSIM 1
LPIPS
Fisheye(IS
24.25
28.25
0.895
0.924
0,206
0.209
20.08
27.86
0.848
0.929
0.233
0.199
EVER
3DGUT
25.37
27.02
0.904
0.876
0.197
0.288
18.57
27.43
0.730
0.898
0.297
0.249
29.15
0.916
0.925
0.194
23.58
29.51
0.878
0.915
0.219
0.228
Data
1/8
1/8 FE 1/8 HE
1/8
1/4PH
1/4 1/1PH
1/4
Test Data
1/8PE FE 1/4 PH
1/4PH SSIM
1/8 FE LPIPS
1/4 PH
1/8 FE PSNR
1/4 PH
1/8 FE 1/4
1/88 HE 1/4 PH LPIPS
Method
PSNR
SSIM 1
Fisheye(3S
23.57
26.65
0.829
0.861
0.253
0.254
20.83
27.33
0.768
0.883
0.288
0.229
24.59
26.83
0.823
0.859
26.98
0.793
0.235
3DGUT
25.61
26.25
0.854
0.803
0,221
0.336
19.82
26.41
0.669
0.833
0.313
0.291
Ours
26.60
27.84
0.895
0.882
0.142
0.230
23.95
27.15
0.840
0.846
0.217
0.289
Data
1/8
1/8 FE 1/8 FE 1/4PH
1/8
1/4PH
1/4
1/4
Test Data
1/8 FE 1/4 PH
1/8 FE LPIPS
1/1 PH
1/8 FE 1/1 PH
1/8 FE
1/4
1/8 FE LPIPS
1/4 PH
Method
PSNR
SSIM
PSNR
SSIM 1
FisheyeGS8
24.15
27.21
0.890
0.885
0.162
0.215
19.41
27.50
0.811
0.907
0.203
0.183
27.71
0.931
25.22
28.27
0.192
3DGUT
26.31
26.69
0.917
0.817
0.121
0.302
19.00
27.42
0.659
0.866
0.263
0.228
Ours
28.26
28.49
0.922
0.906
0.097
0.190
24.52
28.55
0.874
0.885
0.168
0.220
35


Reviews:
[{"review_id": "CkSuEWovpo", "reviewer": "Reviewer_H5DA", "summary": "This paper presents a theoretically grounded and practically efficient framework for 3D Gaussian rendering under arbitrary camera models. It derives an analytical ray–Gaussian integral by transforming anisotropic Gaussians into a canonical isotropic space, yielding an exact closed-form transmittance that unifies differentiable rendering and projective geometry without approximation. A key innovation is the Particle Bounding Frustum, which computes exact angular bounds from each Gaussian’s covariance to perform direct frustum–ray association, replacing BVH traversal while preserving geometric correctness. The method further introduces a Bipolar Equiangular Projection that provides uniform ray parameterization and improves stability across wide-FoV distortions. Experimental results demonstrate the significant performance improvement upon existing works.", "questions": "1. The paper reports achieving higher quality with far fewer Gaussians, but it is unclear whether this reduction stems from better ray–particle association, regularization effects, or implementation details.\n\n2. How does the method scale to multi-million Gaussians in large scenes? Without BVH traversal, are there memory or parallelization bottlenecks at high particle counts?\n\n3. On standard pinhole datasets, does rendering quality keep improving with more Gaussians,\n\n4. Beyond numerical stability, what are the key algorithmic differences between 3DGEER and GOF in ray–Gaussian intersection and gradient computation?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Introduces an efficient PBF-based association strategy that achieves both geometric exactness and near real-time rendering, effectively bridging the gap between ray-tracing accuracy and splatting efficiency.\n2. The proposed framework generalizes naturally across pinhole, fisheye, and omnidirectional cameras through its unified angular-space formulation and the Bipolar Equiangular Projection.\n3. Demonstrates strong experimental performance and consistent generalization across diverse camera models, datasets, and fields of view, with results that are comprehensive and sufficiently support the paper’s claims.", "weaknesses": "1. An ablation study on the key components is missing.\n\n2. A more detailed discussion of the limitations is needed, for example explaining how sensitive the framework is to ordering effects as discussed in related works.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2043QENBb3", "reviewer": "Reviewer_uR7S", "summary": "The paper proposes, 3DGEER, a ray-based Gaussian rendering framework that achieves projective exactness with a closed-form integral of anisotropic Gaussians along rays via a canonical transform, eliminating projection linearization error. It introduces Particle Bounding Frustum (PBF) for exact, BVH-free ray–particle association and Bipolar Equiangular Projection (BEAP) for unified arbitrary-FoV sampling that accelerates association and improves quality. Experiments indicate that 3DGEER high FPS while preserving projective exactness across camera models, achieving consistent quality gains over prior exact-ray or splatting baselines.", "questions": "See weaknesses. I suggest to add the missing ablations to better understand how much improvement each contribution brings separately, and re-run some of the baselines on a consistent GPU for a coherent runtime comparison.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper proposes a projectively exact closed-form ray integral and an exact, BVH-free Particle Bounding Frustum (PBF) for ray–particle association, improving both accuracy and efficiency.\n- The paper proposes BEAP, which uniformly samples rays in angular space, to improve generalization to large FoVs and improve efficiency and rendering quality.\n- The method achieves SOTA performance compared to the baselines in both photometric metrics and FPS across camera models and generalizes to wider FoVs than those seen in training data", "weaknesses": "- Missing ablation for projective exactness: The paper does not include a controlled ablation that replaces the projective closed-form integral with 2D projective-space approximation (that is used in baselines) within the same pipeline. Such a study is crucial to learn how much improvement is brought specifically from the projective exact formulation (separate from PBF/BEAP)\n- Missing ablation for BEAP: The paper omits a controlled BEAP on/off study within the same pipeline and comparisons to alternative projections (e.g., perspective, equidistant, equal-area). A clear ablation should report quality and efficiency with/without BEAP, stratified by image region (full FoV, center vs. edge/periphery), to identify how much quantitative improvement is brought by BEAP (separate from the other 2 contributions).\n- In table 4, the runtime for some baselines are evaluated with different GPU models. It would be more clear if everything could be evaluated with a same GPU model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "G8sMLQMPjO", "reviewer": "Reviewer_SpuR", "summary": "This paper presents a new 3D Gaussian renderer designed for large field-of-view cameras, such as fish-eye cameras. It derives a closed-form expression for integrating the Gaussian density along a ray. To enhance efficiency, the paper introduces the Particle Bounding Frustum (PBF), which provides a tight bounding box to cull unnecessary ray-Gaussian intersections. \n\nUnlike pinhole cameras, where rays are typically sampled uniformly across a 2D plane, this paper proposes sampling rays uniformly from a sphere using Equiangular Projection. This approach allows for interpolation of the rays to produce the output image, further improving quality. \n\nExperiments conducted on both pinhole and fish-eye datasets demonstrate that the 3D Gaussian Efficient Ray Renderer (3DGEER) outperforms previous methods across all metrics. Additionally, it runs five times faster than existing projective exact ray-based baselines and generalizes well to wider fields of view that were not seen during training.", "questions": "See W1 in weakness.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Strong results.** The experimental results are impressive, achieving state-of-the-art performance across many large field-of-view (FOV) datasets. Additionally, there are significant improvements in memory efficiency and rendering speed compared to 3DGRT and EVER.\n\n2. **Good presentation.** The paper is well-structured and easy to read. The derivation of the formula is generally clear. However, some sections, such as the gradient flow in Figure 2, could be streamlined or moved to the appendix, as they do not seem essential for all readers.", "weaknesses": "**Limited discussion to existing work**.  The paper provides limited  discussion regarding existing work. While there are several derivations of the proposed techniques, including the closed-form expression for integrating Gaussian functions and the tight bounding box, these derivations have been adopted in prior research without clear reference in this paper. \n\nFor the exact projective geometry, Equation 5 in this paper is identical to Equation 15 in HTGS [1].  Similarly, the equations for calculating the tight bounding box (Equations 7 through 10) appear to be equivalent to Equations 8 through 10 in both 2DGS [2] (with the substitutions x = tan(θ) and y = tan(φ)). Both of these derivations originate from source [3]. However, these papers are not explicitly cited, and a more thorough discussion regarding their connections should be included. (The only reference seems to be in Section 2.2 (lines 139-142), which is not entirely accurate since it does not use BVH. )\n\nFurthermore, due to the similarities between these works, it is unclear how this new formula can eliminate the screen-space constraints present in HTGS and 2DGS that limit their field of view, as claimed in lines 285-286. This point requires additional justification.\n\n[1] Efficient Perspective-Correct 3D Gaussian Splatting Using Hybrid Transparency. Eurographics 2025.\n[2] 2D Gaussian Splatting for Geometrically Accurate Radiance Fields. SIGGRAPH'24.\n[3] A hardware architecture for surface splatting. ACM TOG 2007.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sk4564iuYT", "reviewer": "Reviewer_62Nx", "summary": "This paper investigates how Gaussians are projected within the 3D Gaussian Splatting framework and addresses the limitations of existing projection methods for general cameras, particularly those with large fields of view (FoVs) where standard approaches often fail. Specifically, it analyzes an approximation commonly used in the projection process and demonstrates that this approximation can negatively affect the accuracy of Gaussian projection under large FoVs.\n\nTo achieve general and accurate Gaussian projection, the paper makes three main contributions. First, it shows that the transmittance of a Gaussian depends solely on the distance between a ray and the Gaussian’s center. Second, it introduces a more precise yet computationally efficient formulation for determining Gaussian bounds. Third, it proposes Bipolar Equiangular Projection (BEAP), a novel technique for enhanced ray sampling that provides more uniform coverage and improves projection efficiency.\n\nThrough experiments on diverse datasets and camera configurations, the proposed method demonstrates notable improvements in representation quality and significantly reduces the number of Gaussians required. In contrast, prior methods relied on a much larger number of Gaussians to compensate for distortions in large FoVs.", "questions": "1. Very minor, but in Fig. K.2, the “laboratory” and “office” scenes appear to be mixed.", "limitations": "", "rating": 10, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "**Clear and well-structured presentation**: The paper is well-written and easy to follow. It clearly articulates the problems addressed, the proposed solutions, and their significance, supported by clear and informative figures.\n\n**Solid theoretical foundation**: The paper provides comprehensive mathematical formulations necessary to understand the proposed method, with additional derivations and supporting details presented in the appendix. It is also noteworthy that the paper explores alternative angular representations to overcome the limitations of conventional tan-based approaches (Appendix D.2).\n\n**Strong quantitative and qualitative results**: The proposed method demonstrates consistent performance improvements across diverse datasets and camera types, including both fisheye and pinhole cameras.", "weaknesses": "The contribution of the transmittance equation (Eq. 5) to the overall method could be articulated more clearly. In particular, the benefits of this formulation for general camera models are not fully evident, and the paper could better explain how this component integrates with and enhances the proposed framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)