GVKF: Gaussian Voxel Kernel Functions for Highly Efficient Surface Reconstruction in Open Scenes
Authors: Gaochao Song
Chong Cheng
Hao Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: 3dgs
Mesh
Sdf
Nerf
Surface Reconstruction
Primary area: machine_vision
TL;DR: High efficient surface reconstuction based on gaussian splatting

Abstract: In this paper we present a novel method for efficient and effective 3D surface reconstruction in open scenes. Existing Neural Radiance Fields (NeRF) based works typically require extensive training and rendering time due to the adopted implicit representations. 
In contrast, 3D Gaussian splatting (3DGS) uses an explicit and discrete representation, hence the reconstructed surface is built by the huge number of Gaussian primitives, which leads to excessive memory consumption and rough surface details in sparse Gaussian areas.
To address these issues, we propose Gaussian Voxel Kernel Functions (GVKF), which establish a continuous scene representation based on discrete 3DGS through kernel regression. The GVKF integrates fast 3DGS rasterization and highly effective scene implicit representations, achieving high-fidelity open scene surface reconstruction. Experiments on challenging scene datasets demonstrate the efficiency and effectiveness of our proposed GVKF, featuring with high reconstruction quality, real-time rendering speed, significant savings in storage and training memory consumption.

GVKF: Gaussian Voxel Kernel Functions for Highly Efficient Surface Reconstruction in Open Scenes
Gaochao Song"
Chong Cheng" Hao Wang
The Hong Kong University of Science and Technology (Guangzhou)
gesong40gmail. com
ccheng736@connect hkust-ga edu. haowangGhkust- edu.cn
Abstract
In this paper we present novel method for efficient and effective 3D surface reconstruction in open scenes. Existing Neural Radiance Fields (NeRF) based works typically require extensive training and rendering time due the adopted implicit representations. In contrast, 3D Gaussian splatting (3DGS) uses an explicit and discrete representation, hence the reconstructed surface built by the huge number of Gaussian primitives, which leads to excessive memory consumption and rough surface details in sparse Gaussian areas. To address these issues, we propose Gaussian Voxel Kernel Functions (GVKF), which establish continuous scene representation based on discrete 3DGS through kernel regression. The GVKF integrates fast 3DGS rasterization and highly effective scene implicit represen- tations, achieving challenging scene datasets demonstrate the efficiency and effectiveness of our proposed GVKF, featuring with high reconstruction quality, real-time rendering speed, significant savings storage and training memory consumption. Project page: https://3daggatoorld ..l.giithubb.. io/ to/gvkf/
open scene surface reconstruction. Experiments
Introduction
3D surface reconstruction in open scenes holds great significance various practical applications, such autonomous driving, virtual reality, urban planning and etc. However, achieving high-fidelity and efficient open scene reconstruction has been a longstanding challenge, due to the trade-off between the rendering quality and the required resources for optimization.
In pursuit of this goal, two predominant approaches are Neural Radiance Fields (NeRF) 27
I1 19 and 3D Gaussian Splatting (3DGS) [15 implicit representations typically require extensive training and rendering time, which limits the practical use in large-scale scene reconstruction [1 38 26 36). On the other hand, 3DGS (5) adopts explicit representations, which enables high-quality novel view synthesis while achieving real-time rendering. This makes 3DGS more feasible for efficient scene reconstruction in the applications such as autonomous driving and virtual reality.
4 21 based methods. On one hand, N
Recently, there are studies using 3DGS technology for novel view synthesis and surface reconstruction street scenes and urban environments reconstruct the 3D surfaces based on Gaussian points. However, it has been noted that overly large and sparse Gaussian points can significantly affect the geometric representations of the scene,
23). For instance, SuGaR [12] attempts to
Botth authors contributed equally to this research.
The corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Table 1: Comparison of 3DGS rendering and volume rendering methods.
Method
Math Expression
Pros
Cons
3DGS Rendering
Discrete summation
Fast rendering
High Mem consumption, Hard to fit 3D continuous surface
Volume Rendering
Continuous integration
Better 3D surface representation
Low rendering speed due con-
tinuous sampling
particularly in background areas. To overcome these challenges, the 2D Gaussian Splatting (2DGS) [4] proposes to use Gaussian surfaces as surfels to represent complex geometries [30, thereby improving the surface reconstruction quality. Particularly, 2DGS faces challenges when processing large-scale scenes, as it requires the explicit representation of a large number of Gaussian primitives, leading to significant GPU memory consumption. Therefore, 2DGS still exhibits limitations in novel view synthesis capabilities and the geometric representation of large-scat scgle scenes.
In Table [ we summarize the comparison of 3DGS rendering and volume rendering. To fully leverage the fast rendering advantages of Gaussian alpha blending while achieving effective implicit scene representation, we propose a novel Gaussian Voxel Kernel Functions (GVKF) method. Firstly, GVKF utilizes voxelization to implicitly represent 3DGS, managing the growth and pruning of Gaussian splats. This approach retains the expressive power of explicit Gaussian splats while enabling efficient management of these splats. Secondly, we carefully analyze the intrinsic connection between Gaussian splatting alpha blending rendering and traditional volume rendering from a mathematical perspective. We establish a through kernel regression. This makes it possible for discrete Gaussians to represent continuous scenes. By replacing the discrete opacity values in original 3DGS rendering pipeline (which can be viewed as collapsed kernel functions) with Gaussian kernel functions, we maintain the advantages of the original 3DGS alpha blending while optimizing the representation of continuous scenes. Moreover, we demonstrate that our proposed rendering method is mathematically consistent with traditional volume rendering. Thirdly, based on our constructed scene opacity representation, which is also known the scene opacity field, we derive the bidirectional mapping relationship between opacity and the scene surface. This enables direct mesh extraction for scene surface. In summary, our contributions are as follows:
method to represent continuous scene opacity density fields
We propose GVKF, an implicit continuous scene reconstruction method that integrates the effectiveness of implicit representation with the fast rasterization advantages of Gaussian Splatting, without the need for computationally intensive volume rendering.
Based on GVKE, we further propose implicit representation of the scene surface, achieving efficient and high-quality scene surface reconstruction.
Experiments demonstrate the usefulness GVKF in open scenes, showcasing high-quality surface reconstruction accuracy, real-time rendering speeds, and significant savings in storage and memory consumption.
2 Related Works
2.1 Novel View Synthesis
The introduction of Neural Radiance Fields (NeRF) 27 has significantly advanced the development of 3D reconstruction and novel view synthesis. NeRF employs volumetric rendering techniques to intricately simulate the geometric structure of scenes and neewponnt-dpeenden characteristics, thereby considerably enhancing the quality of image rendering. Following NeRF, variants such as Mip-NeRF [ and Zip-NeRF (2) have addressed the aliasing issues during rendering. Additionally, UC-NeRF 66, designed for outdoor scenes, enhances image consistency through color correction and pose refinement. InstantNGP 28 accelerates training and improves rendering efficiency by optimizing subvolume processing with grid pyramid techniques. Meanwhile, other feature grid-based scene representation methods 22 34 S 45) have been extensively explored to enhance the training capability and expressiveness of models. Recently, 3D Gaussian Splatting (3DGS) (15) effectively represents complex scenes using 3D Gaussian points, significantly boosting the efficiency of real-time high-resolution image rendering while maintaining rendering quality. Further research efforts like Scaffold-GS and Octree-GS [25] 32 have attempted more effective methods to organize and manage Gaussian points, which helps reduce memory usage and speed up training.

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Neural Gaussian Representation
Discrete-o-CConinuous Translation
Neural Scene Representation
Neural Opacity Field
Rasterization
Image
((x)
p(t)
Sparse Voxel
Kemel Regression
Φ
Implicit Surface
Mesh
Figure 1: Framework of Gaussian Voxel Kernel Functions (GVKF) for scene representation. In this framework, discrete Gaussian primitives represent continuous opacity density p(t) the ray via kernel regression. After slightly modifying the rasterization pipeline, the kernel function can be integrated into alpha blending rasterization without introducing dense points sampling. Additionally, we directly define the mapping relationship between the neural opacity field and the implicit surface.
2.2 Surface Reconstruction
Traditional isosurface extraction, relying on density thresholds, often struggles with fine details due to resolution and noise constraints. Recent studies propose
complex representation methods [48].
For instance, NeuS [37 uses MLP networks for occupancy grids or SDF, improving reconstruction
accuracy and noise reduction [29
43
20
7 42). Techniques like BakedSDF (44) translate the
optimization of NeRF or neural SDFs into 3D meshes, enhancing features through high- high-resolution grids but increasing computational load. NeuS2 39 introduces novel formula for second-oree -odder derivatives with multi-resolution hash encoding and CUDA-based MLP technology, significantly reducing training time. StreetSurf [[] optimizes SDF mappings in open scenes and decouples static and dynamic objects. Despite udvancements, NeRF-based methods still need optimization for processing speed and real-time rendering.
3DGS has gained attention for its high-quality scene reconstruction and rapid processing capabilities [5]. 3DGS uses multiple 3D Gaussian distributions with anisotropic covariance for precise control over scene attributes [49 [8]. This technology enhances surface reconstruction methods like SuGaR
12 which employs Poisson surface reconstruction for fast and accurate mesh extraction. However,
irregular Gaussian sphere distribution affects surface quality. To improve this, 2DGS 14 uses 2D Gaussian planes for better surface conformity and TSDF for accurate reconstruction, though may cause surface fragmentation. GOF [47] directly extracts surfaces using opacity thresholds and tetrahedral mesh extraction but limited by high VRAM requirements. GSDF (46) combines 3DGS with NeuS-like SDF branch for optimized rendering and reconstruction, increasing training time. Despite their potential, 3DGS-based methods face challenges like managing Gaussian points, high VRAM consumption, and degraded rendering quality.
3 Methods
As shown in Fig. I we first introduce the implicit neural 3DGS primitives representation based sparse voxel grid, which offers highly efficient storage management and the fitting power of neural networks. Secondly, rationale, we have analyzed its intrinsic connections with Gaussian alpha blending [15] and traditional volume rendering (27) from statistical analysis perspective. Finally, describe the relationship between the proposed continuous scene representation (a neural opacity field) and implicit surface, and derive explicit mapping function for mesh reconstruction.
present GVKF-based continuous scene representation,
explain its
3.1 Voxel Gaussian Representation
To achieve orderly 3DGS management while minimizing the explicit expression of them to save training storage consumption, we use a spatial sparse voxel grid manage Gaussian primitives. During the initialization phase, the sparse grid is generated from the downsampled SfM point clouds

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and dynamically grows or being eliminated during training. Each sparse grid is allowed to generate up m Gaussian primitives, and all these primitives are limited small range of space centered at the voxel grid.
Gaussian Generation For particular 3D Gaussian expression, five attributes required: p R (position), a E IR (opacity), R E 1 (rotation matrix), s E R (scaling), and e E R (color). Then, Gaussian (x) can be generated as:
-(z)= a. a (0=-p)² ((--1)
(I)
where Σ E: R3x3 is covariance matrix defined as Σ Rss" R" cs calculated via SH coefficients and camera direction. Different from traditional 3DGS[15] that treats them as explicit optimizable tensors, we decode them from feature vector F IR via several MLPs:
a MLP camera), R = MLPR
MLP, MLP. camera). (2)
For alpha and color MLPs, the view camera and feature vector are inputs, facilitating view- dependent fitting. Relative coordinates of Gaussians the parent voxel center are stored with F. compressing explicit Gaussian components and leveraging the MLP's fitting capacity. Gaussians are dynamically generated each iteration and recycled post-update, reducing memory usage.
Voxel Registration. To control Gaussian numbers in large open scene, we eschew the traditional adaptive density control strategy, adopting method inspired by caffold-Gaussinn [5] and Octree- Gaussian (3) The voxel registration is based on gradient accumulation. After each iteration, gradients from 3DGS are recorded and accumulated in their respective voxels, denoted as V. Voxels where exceeds a set threshold are subdivided into eight subvoxels increase grid resolution, continuing until the maximum depth is reached. Additionally, less frequently used voxels discarded after a specified period.
3.2 Neural Opacity Field of 3DGS
Since 3DGS rasterization rendering and traditional volume rendering share some overlapping con- cepts, in this section, we sort them out and introduce our method from statistical perspective while avoiding introducing redundant mathematical symbols.
Continuous Scene Description. We define p(t) [0, +00+ + [0, 1] as the opacity density function, which measures the probability of a ray encountering a particle at position t. We define T(t) [0, +oo] - [0, 1] as the transmission function, which measures the probability that a ray has not encountered any particles from its origin to point t. Considering the probability that a ray does not encounter any particles time step + dt, denoted as T(t + dt), it is evident that T(t dt) T(t) dt) Solving this differential equation, we obtain the relationship between T(t) and
p(():
' p(t)dt).
T(t) = exp(-
Therefore, we obtain the cumulative distribution function (CDF) of the probability that ray hits particle over the interval [0, t): ( = 1 Ttt), with the corresponding probability density function (PDF) being (L) = (() p(t). From the perspective of volume rendering. this PDF used as the probability of the appearance of color along the ray, ultimately taking the mathematical expectation of the color as the ray color:
B T(t) pl)) c(t)dt + T(B) Cbg.
C=/
(4)
The discrete formulation of volume rendering Eq. 4 iss::
N
i-1
C
T a4
- exlxxp((-))))) T. Il (1) j=1
(5)
where opacity a, represents the accumulated result in sampling interval à; of volume density di hence the value of N does not influence the result as long as as adapted enough. Based on the similar idea of volume rendering, the PDF 4'(t) T(t) p(t) can also be reasonably considered

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as the probability of the appearance of surface along the ray, where the place with the highest probability density is most likely to have surface. Correspondingly, on the CDF (t), this the place where the derivative is the largest. In this paper, we use the CDF (() to describe continuous scenes based on the camera rays, to facilitate integration with the 3DGS rasterization rendering pipeline. In section B.3 we will prove that under the 3DGS representation, the place where the
derivative of CDF is the largest is not actually the surface, so a method to locate the surface will be introduced.
Kernel Regression of 3DGS.
In implicit scene representation methods based on volume rendering, the continuous opacity density
Surface
Ray
function is directly predicted by a ML.((7). In approach, the continuous opacity density func- tion p(t) is fitted through kernel regression via discrete Gaussian primitives after a differentiable transformation: Gi y, 2) + (t,tt tx) from 3D to ID. The transformation consists of three steps: (I). The Gaussian primitives that the ray passes through are selected as kernel functions. (2). According to the Bay-Gaussian Intersection method [[]] the ray transformed into the local coordinate system of each 3DGS to obtain the 1D probability density alone the ray. Here, the peak of the ID probability density, denoted as f4, is defined
p(()
Rendering
p(t)
Opacity (SDGS)
ac-Ki(t-s) 4.)
p(t)
(GVVF)
as the Ray- May-Gaussian Intersection [10 16 (4), in- dicating that the 3DGS has the greatest influence at this point alone the ray. (3), To integrate with regularization methods 14 47]. we assume that each 3DGS fits the surface of the object. There- fore, after t., the probability density continues to remain at its maximum value, indicating that the object is solid. Without loss of generality, the opacity density p(t) on camera ray can be expressed as:
Figure 2: Comparison of Volume Rendering, 3D Gaussian Splatting with Alpha Blending, and GVKF Rendering.
N p(t) = Σ as Ki(t - -t), KG(t) (
[exp(-k: t2) t< <<0
(6)
1 0,
where N represents the number of activated kernel functions along the ray, and k; represents the summarized transform of Bay-Gaussian transform (See Appendix A. for details). a; represents the coefficient for each kernel function!!
Rendering. As for traditional 3DGS rasterization, the pixel color is rendered through alpha blending on N 3DGS being passed through by the ray:
C=
c a DDD a1
In this scenario, a; is constant value representing the opacity of Gaussians. This point-basse rendering is coherent with Eq. 5 with extremely sparse sampling points to simulate dense volume rendering. However, it is impossible to recover continuous opacity density alone the ray from such rendering equation, as illustrated in row-3 of Fig. 2 This is because the third row of the covariance matrix of 3DGS is discarded, and it is directly projected onto 2D plane to evaluate the impact on the opacity of points along the ray. From the perspective of Eq. 6 this means that along the ray, the influence range of all N kernel functions that intersect with the ray collapses to infinitesimally value, making it impossible to recover a continuous opacity density field. To solve this, Eq. 7 can be modified to:
N
i-1
G o2 (0,(0)
II (1 ag (c)
(
==1
Vks 'Specifically, a4 3, V where B is constant value representing opacity related to Gaussian primitives. "This rendering form firstly proposed by GOF (4
5

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This equation will not affect the goal of 3DGS rendering: to approximate traditional volume rendering using sparse sampling points. And it allows for broadening the collapsed kemnel functions to fit the continuous opacity function of the scene.
Scene Representation. Based the discussion at the beginning, the scene surface he described via CDF (t) on the ray in continuous way, which can be calculated like Eq. 8 (removing color);
i-1 a K,st 4.) II aj K2(t t,))
(t)
f=1
3.3 Implicit Surface Mapping
This implicit opacity field (denoted as neural opacity field since it is represented by neural Gaussians) measures the CDF of the probability that ray hits solid scene surface. In the next section, we introduce the mapping of (()) implicit surface.
*(w) (())
('()) [w))p(u)
A(W
We represent implicit surface with signed distance func- tion (SDF), denoted as function ((t) on the camera ray. To recover Dtt) of given (t) that is calculated from well trained 3DGS, we firstly study the reverse mapping prob-
lem: o: (t) (1)
14
Opacity Density Near the Surface. To ensure that the 3DGS aligns with the object's surface and thus reflects the object's shape, depth distortion regularization [14, 47) is introduced during the Gaussian training process. This encourages the distribution of 3DGS along the ray to ag- gregate together, causing the peak of the kernel functions to coincide with the object's surface. In the next discussion, the coordinate of object surface on the ray assumed as t* with D(t*) = 0.
Figure 3: Illustration of functions (u), &' (u),,,(())
Considering p(t) the interval / [0, (*), we have:
M
p(1) = p(c - D()) = Σ aj exp(-kn ((()))))D(()),, 020s0
(10)
Where M represents the number of Gaussian kernels concentrated the surface. To facilitate calculations, we convert the opacity density system to the SDF coordinate system, with t" as the origin and letting -tt),, illustrated in Fig. have:
('(u) T(u) p(u) exp(- S p(w)dw) p(u)
(11)
v"(n) = (() expl- 1"d ((w)dw) + p'(u) -xppe-pppp-- f p(w))dw)
= 11-p2(u) (1) + d'(")] exp(- f" p(w)dd)
(12)
where p(u) ~ NO00009) EM 2ां. which can be directly derived from the additive property of the normal distribution. Then letting h(u) = - -p²(u) + p'(u), we have:
h(u) =pp((u) + p/(u) =-((u)(p(u) 1 "
(13)
easy to prove that l(u)) crosses a unique zero point un from top to bottom on the u-axis, and Wp < 0. This means that the peak of ('(u) will appear before the surface, so it is not reasonable to simply determine the actual intersection point of the light ray with the surface by directly evaluating the peak of ('))). To locate the accurate surface, transcendental equation of is needed to be

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PSNR:
PSNR:
51
PSNB:
STOP
PSNEIIH
Ground Truth
Ours
StreetSurf
2DGS
Figure 5: Qualitative comparison novel view synthesis and surface reconstruction on the Waymo Open Dataset [35), with each subplot annotated with PSNR values to quantify image quality. Our method shows higher geometric precision and detail, validating its efficiency and superiority in processing open scenes, especially in geometric accuracy and detail reproduction.
solved to get uo:
u
M
p(u)=
p(u)
exp(
(14)
V2To
20²
2
impossible to directly get the analytical solution, however, numerical computation methods can be ap- plied to solve up. This may require some extra time for computation.
D(t)
Logistic 1, D(t) Inf o()
13/w
Mapping from Opacity Surface.
Based on the analysis above, we can always have ex- act number of uo via numerical computation method. However, it is hard to find out the inverse function of (u)) for directly building the mapping of (b(() D(t). For the balance surface smoothing while reducing the indelible error, represent mapping relationship of u > (u) via Logistic Function as follows:
4(1)
Figure 4: Illustration of opacity to SDF map- ping (Eq. T6p
(u) + exp(- (u(( vo))
(15)
where μ represents the smooth factor. We choose Logistic Function because of its formal is concise and shares similar shape of (uu). More importantly, it only has one inflection point (0, 0.5), which can be used to simulate the inflection point of Φ(u) after translation. Finally, represent implicit SDF function via Inverse function transformation of Logistic Function, as shown Fig.
D(t) In( (())
1)/H uo
(16)
7

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Ground Truth
Ours
SuGaR
2DGS
Figure 6: Qualitative comparison on the Tanks and Temples dataset [17 shows that our method excels in reconstructing complex backgrounds with high geometric granularity. In contrast, 2DGS often results fragmented backgrounds, while SuGaR displays uneven spherical shapes, affecting both visual and geometric quality.
Table 2: Quantitative evaluation of novel view synthesis and surface reconstruction on the Waymo Open Scene dataset (35). Using LiDAR data as ground truth, we calculated Chamfer Distance (C-D) values for reconstruction accuracy. Our method performs excellently in both novel view synthesis and surface reconstruction, outperforming other methods in Gaussian point usage, VRAM occupancy, and real-time rendering.
Method
PSNR
C-D 4
MB (Storage)
GB (GPU) +
FPS 1
Training Time +
NeuS
13.24
0.76
170
31
0.1
5
F²-NeRR
24.70
886.77
130
24
~ 0.1
0.8 h
StreetSurf
27.12
1.02
540
22
0.1
1.5
3DGS
27.99
3.57
230
23
63
0.75 h
SuGaR
23.71
3.08
228
33
56
1.5
2DGS
28.51
1.67
238
23
51
0.7 h
GVKF (Ours)
30.24
1.57
30
32
1.5
4 Experiments
4.1
Experimental Settings
To assess method's performance against baseline methods in open scenes, used three datasets. We first experimented with the Waymo Open Scene dataset (35), using three cameras scene from five available, each scene containing about 600 images. We employed LiDAR point clouds to evaluate reconstruction quality, although LiDAR data was not used as training input. We also tested the Tank and Temple dataset [17), which includes trajectories and ground truth for six selected scenes. Lastly, we evaluated the Mip-NeRF 360 dataset (I due to the absence of ground truth, our focus
Datasets.
novel view synthesis demonstrate our method's 's efficacy in this aspect.
Baselines. In terms surface reconstruction, we presented the results on the Waymo dataset [35] in tables and figures 5 comparing state-of- -f-the-art implicit methods (such as NeuS (37] F2-NeRF 38
StreetSurf [T]] and explicit methods (such as 3DGS [15], SuGaR [12, 2DGS 14 We utilized PSNR to evaluate the results of novel view synthesis and Chamfer distance to measure reconstruction accuracy, while also recording training time, VRAM usage, and the size of the Gaussian point files post-training. Additionally, as shown in Table and Figure conducted comparisons on the Tank

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Table 3: Quantitative evaluation on the Tanks and Temples dataset 17 using Fl scores and training time as metrics. Our method outperforms all existing explicit methods in F1 scores and is comparable to implicit methods in reconstruction accuracy, with significantly reduced training time. These results highlight our method's efficiency and accuracy. Comparation of concurrent work GOF 47
presented in Appendix A.4.
Implicit
Explicit
Method
NeuS
Geo-NeuS
Neuralangelo
SuGaR
3DGS
2DGS
Ours
Barn
0.29
0.33
0.70
0.14
0.13
0.36
0.40
Caterpillar
0.29
0,26
0.36
0.16
0.08
0.23
0.34
Courthouse
0.17
0.12
0.28
0.08
0.09
0.13
0.25
Ignatius
0.83
0.72
0.89
0.33
0.04
0.44
0.51
Meetingroom
0.24
0.20
0.32
0.15
0.01
0.16
0.23
Truck
0.45
0.45
0.48
0.26
0.19
0.26
0.40
Mean
0.38
0.35
0.50
0.19
0.09
0.30
0.36
Time
>24 h
>24 h
>24
>1 h
~5 min
~0 min
1.5
and Temple dataset with implicit methods (such as NeuS 377. Geo-NeuS [9]. Neuralangelo 200 and
explicit methods (such as 3DGS [T5], SuGaR [12), 2DGS [14] ). We used official scripts to evaluate
Fl scores. For novel view synthesis,
compared various advanced methods the Mip-NeRF 360
dataset, including NeRF [27), Deep Blending [3]. Instant NGP [28], MERF BI]. Mip-NeRF 360
[]], BakedSDD 44) 3DGS [15, SuGaR
[2]. and 2DGS 14 We evaluation metrics such as
PSNR, SSIM, and L.PISS.
Implementation Details Our method modifies the representation 3DGS and slightly adjusts the opacity weights in the rendering pipeline using Gaussian kernel functions. This ensures compatibility with other components of Gaus- sian rasterization rendering. Similarly, we em- ploy the same L1 loss and D-SSIM loss 3DGS to supervise color loss, and we use the same Gaussian regularization term 2DGS and GOF to promote aligmment between the Gaussians and the surface. After training, the SDF field of the scene can be directly extracted based mesh with the MC MM (33] algorithm. To export com- plete sky and background, the modified MT al-
Table 4: Quantitative evaluation on the Mip-NeRF 360 M outdoor scene dataset is presented. Since the dataset lacks ground truth for surface recon- struction, we assessed the results of novel view synthesis.
Method
PSNR f
SSIM 1
LPIPS
NeRF
21.46
0.458
0.515
Deep Blending
21.54
0.524
0.364
Instant NGP
22.90
0.566
0.371
MERF
23.19
0.616
0.343
Mip-NeRF 360
24.47
0.691
0.283
on Eq. 16 and exported
BakedSDF
22.47
0.585
0.349
3DGS
24.24
0.705
0.283
SuGaR
22.76
0.631
0.349
gorithm in GOF[7] is used.
2DGS
24.33
0.709
0.284
GVKF (Ours)
25.47
0.757
0.240
4.2 Analysis
Figure demonstrates Segureddemonstrattes bushes, and other objects. In contrast, the 2DGS method produced more holes and fragmentation, while the StreetSurf method lost some critical geometric features. The results Table 2 indicate that method surpasses other methods in terms of view synthesis and reconstruction accuracy, and it highlights restoration. The SuGaR method generated excessive
the superiority of our method in capturing detailed features of roadside houses,
requires fewer Gaussian points and VRAM for large-scale scene reconstructions. Figure 6
method's 's excellent performance in irregular protrusions, and 2DGS exhibited more fragmentation and floating debris. According to the results in Table 3, method outperforms all explicit methods and achieves comparable reconstruction results to implicit methods, while maintaining equivalent GPU time usage. The results in Table 4 confirm that our method leads in novel view synthesis across all compared methodologies.
4.3 Ablation Study
In this section, we evaluate the impact of varying voxel grid sizes on the neural Gaussians by conducting experiments with the Waymo datasets. We selected voxel sizes of 1, 0.1, 0.01, 0.001, as

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Table 6: Further ablation study on voxel Gaussian representation and SDF mapping. w/o voxel: We eliminate the using of MLPs and voxel grid, w/o sdf: we directly use linear assumption between opacity function and SDF function.
Ablation
PSNR
F1
Mem (GB)
Storage (MB)
Training Time Meshing Time
Ours
26.31
0.36
~ G
90 M
1.5 h
~ 15 min
w/o voxel
23.60 (-2.71)
0.39 (+0.03)
~ 16 G (x 1.6)
467 M (x 5.2)
~ 1.4 h
~ 15 min
w/o sdf
26.31
0.30 (-0.06)
9 G
90 M
~ 1.5 h
~15 min
presented in Table 5 When the voxel size is too large, the sparse neural Gaussians fail to learn the scene representation and return NaN errors. As the number of voxels increases, more Gaussians are generated for scene representation, thereby enhancing the quality of novel view synthesis. However, the improvements plateau when the voxel size is reduced to 0.001, which also requires more training time and becomes impractical. Therefore, we set the voxel size to 0.01 balance training time and rendering quality.
We further conducted ablation study on the Tanks and Temples dataset [17] to evaluate the
Table 5: Influence different voxel size.
Voxel Size Initial voxels Final voxels PSNR Time
impact of voxel representation and SDF map- ping. The results are presented in Tab. 6 It can be observed that utilizing voxel representation significantly improves the PSNR for NVS tasks and reduces memory consumption dramatically compared to naive 3DGS setup. Although there is slight decrease in the geometric quality of surface reconstruction, consider this trade-off acceptable.
2
0.1
80 k
110 k
29.34
1.2 h
0.01
~ 90 k
1100 k
30.24
1.5 h
0.001
100 k
1 1100 k
30.29
4
4.4 Limitation
Implicit methods, such as those based on NeRF 27 37 20). typically utilize global fitting approach for SDF, which allows them to fully leverage the universal approximation capabilities of MLPs. This is advantageous even in areas with sparse viewpoints. However, our current method employs local line-of-sight-basee SDF fitting, compromise made to adapt to the 3DGS rendering style. This means that regions not covered by the training viewpoints lack fitting capability, resulting in uneven surfaces.
In addition, While our method advances 3D surface reconstruction open scenes, it faces challenges with dynamic objects and the decoupling of distant and near views, sometimes misrepresenting the sky as surface enveloping the model. The lack of sufficient prior knowledge for optimizing complex scenes also poses limitations.
5 Conclusion
This paper introduces GVKF, combining Gaussian splatting's rapid rasterization with the efficiency of implicit expressions to enhance reconstruction quality and speed significantly. By employing voxelized implicit representation of 3DGS, GVKF retains the expressive power of explicit Gaussian maps while managing them effectively. We have explored the relationship between Gaussian splat- ting's ' alpha blending and traditional volume rendering, developing a GS-based method to represent continuous scene opacity density fields through kernel regression, addressing 3DGS's limitations in continuous scene representation.
Experimental results demonstrate GVKF's effectiveness in open scenes, showing notable improve- ments in reconstruction accuracy, real-time rendering speeds, and reductions in storage and memory usage. These advancements support applications in fields like autonomous driving and virtual reality, pushing forward surface reconstruction technology.
Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 62406267), Muangzhou-HKUSTCCC Joint Funding Program (Grant No. 20.200300300008), Education Bureau of Guangzhou Municipality, and Guangzhou Quwan Network Technology Co., Ltd.
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A Appendix Supplemental Material
A.1 Bay-Gaussian Intersection
Based on Eq. D in the main text, the influence of 3DGS in camera space on a one-dimensional ray can be expressed as follows:
(x) - p()
p(t) exp( (ot--)=- p))
(17)
Here, e represents the unit vector of the ray direction. This formula converts the three-dimensional influ- ence of 3DGS into one-dimensional function along specific camera ray, which is a one-dimensional Gaussian function. Fig. 7 demonstrates the rela- tionship of this transform. For ease of notation, we express it as:
exp(--k -ky- (t (t--1) 21)
&
pt) cxp(-k; (t)
(18)
where 4 denotes the point along the ray where 3DGS has the maximum impact, also known as the "ray- Gaussian intersection," which can be analytically given by:
Figure 7: Ray-Gaussian Intersection in local 3DGS coordinate.
4
(19)
More proof details can be found in "Approximate Differentiable Rendering with Algebraic Surfaces."
16
A.2 More Implementation Details
As demonstrated in the ablation experiments, to balance quality and speed, chose to downsample the initial Gaussian point cloud using a voxel size of 0.01. Within each voxel, the dimension of J is set to 32, and it stores the relative coordinates of 10 Gaussian points, indicating that the maximum number of Gaussians generated per voxel grid is 10. Gaussians with an opacity less than 0 will be hidden during each iteration. In each scene, all voxel grids share total of four MLPs, which decode different Gaussian attributes from the corresponding voxels.
Regarding voxel registration, the gradient threshold is empirically set to 2 10-4 meaning that voxel grids with an average gradient exceeding this value after each iteration will be subdivided using octree method. The maximum recursion depth is set to to control the number of Gaussians in the scene, ensuring it does not exceed a certain threshold. Voxel evaluation is performed every 500 iterations to determine which voxels should be subdivided reclaimed. For other settings, strive to remain consistent with the original 3DGS settings.
A.3 More Results
Our method focuses on the challenging task of open scene reconstruction. Here, we provide comprehensive quantitative comparison with other related methods on the Mip360 dataset, as shown in Table Additionally, we have included more experimental results the Mip360 and Tank and Temple datasets, as shown in Figures and For more qualitative results, please visit the project page.
Discussion on indoor scene. We observe that current methods based on 3DGS perform adequately for indoor scenes, where there is typically 360-degree viewpoint coverage. However, they underperform in outdoor scenes due to limited viewpoint coverage. Heuristic splitting and pruning strategies in original 3DGS tend to fit the training viewpoints rather than distributing evenly across the space. This leads to poorer novel view synthesis results in outdoor environments. As illustrated Fig. I0 without a voxel grid, heuristic Gaussian growth strategies result in uneven spatial distribution of GS, sometimes even creating holes. Conversely, using voxel grids to constrain Gaussians allows for efficient management of their spatial distribution, supporting better novel view synthesis.
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Ground Truth
Rendered Image
Depth Map
Normal Map
Figure Additional experimental results on the Mip-NeRF360 dataset [ From left to right: Ground Truth, Novel View Synthesis, Rendered Depth Map, and Normal Map.
Table 7: Quantitative evaluation the Mip-NeRF 360 1 All scene dataset is presented.
Outdoor Scene
Indoor Scene
Average
Method
PSNR
SSIM t LPIPS
PSNR
SSIM f LPIPS
PSNR
SSIM
LPIPS $
NeRF
21.46
0.458
0.515
26.84
0.79
0.37
23.85
0.61
0.45
Deep Blending
21.54
0.524
0.364
26.4
0.844
0.261
23.70
0.67
0.32
Instant NGP
22.9
0.566
0.371
29.15
0.88
0.216
25.68
0.72
0.30
MERF
23.19
0.616
0.343
27.8
0.855
0.271
25.24
0.31
MipNeRF360
24.47
0.691
0.283
31.72
0.917
0.18
27.69
0.79
0.24
BakedSDF
22.47
0.585
0.349
27.06
0.836
0.258
24.51
0.70
0.30
Mobile-NeRF
21.95
0.470
0.470
12.19
0.26
0.26
17.07
0.36
0.37
3DGS
24.24
0.705
0.283
30.99
0.926
0.199
27.24
0.80
0.25
SuGaR
22.76
0.631
0.349
29.44
0.911
0.216
25.73
0.29
2DGS
24.33
0.709
0.284
30.39
0.924
0.182
27.02
0.80
0,24
GVKF (Ours)
25.47
0.757
0.240
30
0.915
0.2
27.48
0.83
0.22
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Ground Truth
Rendered Image
Depth Map
Normal Map
Figure 9: Additional experimental results on the Tanks and Temples dataset 17 From left to right: Ground Truth, Novel View Synthesis, Rendered Depth Map, and Normal Map.
Reference
3DGS
Our
Figure 10: GVKF Gaussian point visualization compared to traditional Gaussian method.
Reference
Map
3D Gaussian Points
Figure 11: Failure case. The sparse view area with less Gaussians tends to appear uneven surface.
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Table 8: NVS and storage comparation to GOF on Mip-NeRF 360 Dataset .
Mip-NeRF 360
PSNR
SSIM
LPIPS 4 Storage 4
GOF GVKF (ours)
24.53
0.733 0.757
0.245
25.47
0.240
68 M
Table 9: Mesh quality comparation (F1-score) to GOF on Tanks and Temples Dataset.
TanT
Barn
Caterpillar Courthouse
Ignatius Meetingroom
Truck Mean
GOF
0.51
0.41 0.34
0.28 0.25
0.68 0.51
0.28
0.59
0.46
GVKF (ours)
0.40
0.23
0.40
0.36
Failure Case. As shown in Fig. T1 in areas with sparse viewpoint coverage, the distribution of 3DGS is sparse and irregular, which hinders the fitting of smooth planes. This sparsity compromises the integrity of the surface reconstruction, resulting in models that are geometrically inaccurate. Our method still struggles effectively address these issues.
A.4 Comparation to Gaussian Opacity Field
The similar rendering equation is firstly proposed by GOF [47]. while this work provides in-depth analysis of the relationship among this rendering strategy, volume rendering and Gaussian alpha blending. Different from GOR, our scene representation is implicit, addressing the common issue of high memory consumption faced by 3D Gaussian splatting. Additionally, developed mapping function from opacity to SDF to alleviate the influence of directly linear transform between these fields.
As shown in Tab. 8 GOF uses explicit Gaussian management, still faces high storage consumption issues, making training large scenes on single card challenging. Our method achieves better novel view synthesis results with less storage usage. However, as shown Tab. 9 our current implementation has some geometric precision gaps compared to GOF, the potential reasons may include:
GOF's s iterative optimization extraction method achieves more precise isosurfaces than (May require long meshing time ~ h)
Further adaptation of regularization term to voxel grids might be needed in our method to improve mesh quality.
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Reviews:
[{"review_id": "v8uYQxpzRp", "reviewer": "Reviewer_zurJ", "summary": "This paper introduces a method for 3D Gaussian reconstruction and surface extraction from the 3D Gaussians. The method can render scenes with low memory usage through a voxel organization. Additionally, the authors derived a continuous implicit field on top of the 3D Gaussians and extracted the surface on this implicit field. Overall, the results look good.", "questions": "(1) In Line 292, what is the unit of “1, 0.1, 0.01, …”? \n\n(2) Is there any visualization results of the voxels? I am curious about its quality.", "limitations": "As discussed by the authors in the appendix, the limitations include the inability to reconstruct dynamic scenes and the challenge of distinguishing distant regions without priors. These are interesting for future work to explore.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "(1) The method achieves good reconstruction quality (for both novel view synthesis and surface geometry) with low memory usage.", "weaknesses": "(1) Figure 1 is too simple and abstract, lacking intuitive demonstration.\n\n(2) I can somewhat understand what the authors want to do in Equation 5, but could the authors provide some figures for easier understanding?\n\n(3) Equation 6 seems incorrect. There appears to be a mixed use of 'i' and 'k'.\n\n(4) Equation 7 seems incorrect. What does the term 't' mean here? It is not defined or explained here.\n\n(5) Why do we need to convert to an SDF value? I do not understand the reasoning.\n\n(6) I cannot find how the authors convert this implicit field to a surface. Did you use Marching Cubes?\n\n(7) In Equation 15, it seems the final derived SDF is defined on the ray and varies with different view directions. However, to extract a surface, the SDF should not be view-dependent.\n\n(8) I find the writing style of the paper a bit tricky. The decrease in memory usage is inspired by ScaffoldedGS. Regarding the claimed surface reconstruction contribution (as another contribution), although the authors provide many mathematical equations, I struggle to grasp the intuition, and the motivation for these equations is not clearly provided.\n\nOverall, while the results appear promising, I find it difficult to understand how it works, especially considering that several equations are incorrect and lack clear motivations. To me, it feels more like a mathematical wrapper. The storyline is disjointed, lacking intuitive analysis.\n\nMinor issues:\nLine 53: We -> we\nLine 112: Sfm -> SfM", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Q1: Flowchart is too abstract**\n\nThank you for your suggestion. We have provided more figures in the rebuttal file to illustrate our pipeline.\n\n> **Q2: The explanation of Eq 5**\n\nAs shown in Fig 3 of the rebuttal file, the fourth row shows three 1D Gaussian kernel functions, where $t_i$ denotes the values of $t$ at the peaks of these functions. To represent the continuous function $\\rho(t)$, these kernels are linearly combined by $\\alpha_i$.  \n\n> **Q3: Mixed Use of 𝑖 and 𝑘 in Equation 6**\n\nWe apologize for any confusion caused by previous symbol misuse. Below, we clarify and correct the equations in conjunction with Fig 3 in the rebuttal file:\n\n**Equation 6** represents the discrete rendering of original 3DGS:\n$$C=\\sum_{i=1}^N c_i \\cdot \\alpha_i \\cdot \\mathcal{G_i}^{2D} \\prod_{j=1}^{i-1}(1-\\alpha_j \\cdot \\mathcal{G_j}^{2D})$$\nThe third row of Fig 3 (rebuttal file) illustrates Gaussian primitive distribution along the ray, showing the collapse of the kernels into a narrow area due to discarding the third row of the 3DGS covariance matrix.\n\n**Adaptation to Kernel Functions:** For 3DGS rendering, we substitute $\\mathcal{G_i}^{2D}$ with $\\mathcal{K_i}(0)$, leading to\n**Equation 7:** \n$$C=\\sum_{i=1}^N c_i \\cdot \\alpha_i \\cdot \\mathcal{K_i}(0) \\prod_{j=1}^{i-1}(1-\\alpha_j \\cdot \\mathcal{K_j}(0))$$\nThis change maintains fidelity to the original rendering equation, optimizing both $\\alpha_i$ and $\\mathcal{K_i}$ through RGB loss.\n\n**Representation of Scene Surfaces:** Based on the initial definition of $\\rho(t)$,\n**Equation 8 is:** \n$$\\Phi(t)=\\sum_{i=1}^{N}\\alpha_i \\cdot \\mathcal{K_i}(t-t_i) \\prod_{j=1}^{i-1}(1-\\alpha_j \\cdot \\mathcal{K_j}(t-t_j))$$\nDuring training, Equation 7 facilitates rendering without dense sampling. For opacity representation, Equation 8 computes a continuous scene representation along the ray.\n\n> **Q4: The definition of t in Equation 7 is unclear**\n\nPlease refer to Q3.\n\n> **Q5: The necessity of converting to SDF values**\n\nAs described in Equation 8, our implicit opacity field is defined on the camera ray. This means that to determine the \"absolute opacity\" at any point Q in space, we must traverse all training cameras to find the minimum value. This process is illustrated in Fig 6 of the rebuttal file, where camera rays may come from various directions. Since $\\Phi(t)$ is cumulative and increasing, the minimum value occurs when the corresponding camera ray experiences the least obstruction along its path.  \nOnce the \"absolute opacity\" of each point in space is established, we could extract a mesh by isolating an isosurface (e.g., $\\Phi(x)=0.3$). However, this approach does not comply with the linear interpolation assumption of the MC/MT algorithms and leads to artifacts, as shown in the Fig 4 of rebuttal file. Therefore, we defined a nonlinear mapping from opacity to SDF to mitigate this issue (Equation 15).  \nIn our experiments, we used the Marching Tetrahedra (MT) algorithm, although the Marching Cubes (MC) algorithm is also applicable.  \nFor a more in-depth analysis, please refer to the response @epfR Q-2.\n\n> **Q6,Q7:**\n\n Please refer to Q5.\n\n> **Q8: The writing style and surface reconstruction contributions are unclear**\n\nWe summarize the comparison of 3DGS rendering and volume rendering in the following table, hoping this can clarify our motivation in sec 3.2 of main text. Our goal is to find a new rendering form that combines them. \n| Method           | Math Expression     | Pros                   | Cons                                                 |\n|------------------|---------------------|------------------------|------------------------------------------------------|\n| 3DGS Rendering   | Discrete integration   | Fast rendering         | Hardly fit 3D surfaces due to discrete primitives      |\n| Volume Rendering | Continuous integration | Better 3D surface representation | Low rendering speed due to continuous sampling |\n\nAs we know, kernel regression is a non-parametric technique used to estimate the conditional expectation of a random variable. This offers flexibility and adaptability for modeling both continuous and discrete functions.\nTherefore, to  integrate the advantages and mitigate the disadvantages of the above two rendering techniques,\n\n(1) we propose to use multiple kernel functions, which are projected from 3DGS along the ray (see Fig 5 in rebuttal file),  to represent continuous 3D surface;\n\n(2) we take the maximum values of each kernel function to perform discrete numerical integration for rendering; \n\n(3) we use **mathematical equations to prove our proposed rendering strategy is equivalent to 3DGS rendering,** rather than a wrapper. This is illustrated in Fig 3 of rebuttal file.\n\nTo summarize, we propose a novel 3d surface reconstruction algorithm, which is represented by continuous function while has fast rendering speed. We hope the provided comparison diagrams in rebuttal file will help clarify our approach.\n\n> **Q9: The unit of voxel size in the ablation experiments**\n\nThe dimensions mentioned represent the edge lengths of the voxel grids. The initial 3DGS are constructed using sparse point clouds derived from Structure from Motion (SfM). To accommodate our voxel grid system, we have set up grids of various sizes to filter the sparse point clouds, which also serve as storage containers for the implicit 3DGS.\n\n> **Q10: The visualization effect of voxels**\n\nAs shown in the Fig 1 of rebuttal file, voxel grids exhibit a more regular spatial distribution compared to traditional 3DGS. This regularity aids in novel view synthesis and saves storage, representing a significant contribution of our work. For further details, please refer to the response @sv7m Q-1.", "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": "SjH7UeXTuc", "reviewer": "Reviewer_epfR", "summary": "The paper introduces Gaussian Voxel Kernel Functions (GVKF), a novel approach for efficient 3D surface reconstruction in open scenes, leveraging 3D Gaussian Splatting.\n\nThe approach aims to combine the strengths of both implicit representations (NeRFs, Neural SDFs) and explicit representations (2D/3D Gaussian Splatting) to achieve an accurate and fast reconstruction method with low VRAM requirements.\n\n**Details**\n\n1. *Compressing 3D Gaussians Using a Sparse Voxel Grid and MLPs*\n\nThe paper proposes optimizing neural 3D Gaussians stored in a sparse voxel grid, reminiscent of Scaffold-GS and Octree-GS. The voxel grid is initialized using the downsampled SfM point cloud output during camera calibration. Each voxel can generate a fixed number of Gaussians, limited to a small range around the voxel. Feature vectors are stored and optimized in the grid, alongside the positions of the Gaussians. To generate the rotation, scaling, opacity, and colors of a Gaussian bound to a voxel, the voxel feature is decoded by an MLP along with the position of the Gaussian. This helps compress the representation while leveraging the natural regularity and fitting power of MLPs. To control the number of Gaussians in the scene, the paper takes inspiration from Scaffold-GS, proposing to subdivide or prune voxels based on gradient accumulation in each voxel.\n\n2. *Defining a Neural Opacity Field with Gaussians to Render Images* \n\nFor rendering images, rather than using the vanilla 3DGS formula that relies solely on the opacity of the Gaussians for alpha-blending, the paper proposes explicitly computing the ray-Gaussian intersections and use them to convert Gaussian functions into small kernels. Using these kernels, the paper derives a continuous opacity density function defined along the rays. Similar to Gaussian Opacity Fields (GOF), the implicit opacity field defined with this approach allows for more geometry-accurate rendering and better alignment of the underlying neural Gaussians with the surface of the scene.\n\n3. *Extracting Accurate Surfaces from Neural Opacity Fields* \n\nThe paper provides mathematical descriptions of the relationship between the neural opacity field and the underlying surface of the scene, resulting in a formula that maps the opacity function to an SDF. A Marching Cubes algorithm can then be used to extract a surface mesh of the scene.\n\nThe authors conducted experiments on several challenging datasets, demonstrating that GVKF achieves high reconstruction quality, real-time rendering speeds, and efficient memory usage, outperforming existing techniques in the literature.", "questions": "1. How does the method perform without using a voxel grid? Is there a drop or improvement in rendering quality and surface reconstruction quality? What about the VRAM requirements?\n\n2. Why not apply Marching Cubes directly to the opacity function? Does the SDF really improve the quality of the reconstruction?", "limitations": "No limitations are provided in the main paper. I encourage the authors to move the limitations from the appendix to the main paper, as this is very important for further research.", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n\n2. I appreciate the detailed mathematical analysis of the proposed method.\n\n3. The blending formula proposed in the paper allows for more geometry-accurate rendering compared to vanilla 3DGS.\n\n4. The voxel structure proposed in the paper enhances resource efficiency (particularly in terms of VRAM usage) by compressing the 3D Gaussians.\n\n5. The paper includes extensive experiments on multiple datasets to validate the method. The qualitative results (reconstructed surface meshes) are convincing, and the quantitative comparisons effectively demonstrate that the approach achieves better geometric accuracy and rendering quality than several existing works.", "weaknesses": "1. The paper is extremely similar to Gaussian Opacity Fields (GOF). The rendering Equation 7 (describing the revisited alpha-blending using an increasing opacity function along rays) and opacity field Equation 8 are almost identical to Equations 8 and 9 from the GOF paper. The difference lies in the generation of Gaussians, as GOF does not use a sparse voxel grid with MLPs to decode Gaussian features. However, GOF achieves better results and shorter optimization times (see Table 1 of the GOF paper). Although the paper mentions GOF, it does not compare with it, even though the paper and code were available five weeks before the deadline. I understand that five weeks is a short period, so GOF could be considered concurrent work rather than preexisting work. However, the results from GOF raise important questions about the claims in the paper: GOF seems to perform better and faster using an almost identical approach, except that GOF does not use a sparse voxel grid representation but just a set of explicit Gaussians. Therefore, it seems the voxel grid is not needed for performance and may be useful only for compressing the representation. As no ablation in the paper compares the VRAM requirement and performance of the approach with or without a voxel grid of neural Gaussians, I would like to hear feedback from the authors about this: Does the voxel grid really help in decreasing the memory consumption of the approach? Doesn’t it just heavily decrease performance?\n\n2. Following my previous point, some important ablations are missing in the paper to better validate several important claims. Specifically, the benefits of the voxel grid are questionable when compared with Gaussian Opacity Fields. Moreover, it is unclear why deriving an SDF from the opacity function is needed for applying Marching Cubes; wouldn’t it be possible to apply MC directly on the opacity field? Does the SDF really improve the quality of the reconstruction?\n\n3. No limitations are provided in the main paper. I encourage the authors to move the limitations from the appendix to the main paper, as this is very important for further research.\n\n**Comment**\n\nI really like the main idea of the paper, as well as the overall approach. However, while being very similar to (but more complicated than) Gaussian Opacity Fields, GVKF achieves lower performance, which makes me question some claims in the paper. I believe the paper needs to better justify why each of its components is needed to support the different claims. I am very interested in hearing feedback from the authors about these points and am willing to increase my rating after more clarifications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Q1: How does the method perform without voxel grids? What is the impact on quality and VRAM?**  \n\nWe conducted an ablation study on the Tant dataset to evaluate the impact of voxel representation and SDF mapping. The results are presented in the table below: \n \n| Ablation |PSNR |F1 | Mem |   Stor | Training Time | Meshing Time |\n| :--- |:---: |:---: | :---: |  :---: |:---: | :---:|\n| GVKF | 26.31 | 0.36 | ~9 G | 90 M | ~1.5 h | ~15 min |\n| w/o voxel | 23.60 (-2.71) | 0.39 (+0.03) | ~16 G (x1.6) | 467 M (x5.2) | ~1.4 h | ~15 min |\n| w/o sdf | 26.31 | 0.30 (-0.06) | ~9 G | 90 M | ~1.5 h | ~15 min |\n\n **Observations:**\n\n- Utilizing voxel representation significantly improves the PSNR for NVS tasks and reduces memory consumption dramatically compared to the original 3DGS setup. Although there is a slight decrease in the geometric quality of surface reconstruction, we consider this trade-off acceptable.  \n \n **Further Analysis:**\n\n- The motivation behind introducing voxels is to serve as spatial anchors, providing a more regulated, structured distribution of 3DGS, as well as controlled growth and splitting strategies. This is crucial for reconstructing large outdoor scenes and indeed, the introduction of voxels has met our expectations by:\n         \n         1. Enhancing NVS performance (refer to Tables 2 and 4 in the main text).   \n         2. Reducing memory and storage requirements during training, enabling the training of large scenes on a single GPU.  \n \nHowever, the structured distribution of 3DGS does not significantly improve the results of mesh reconstruction as expected. We hypothesize that the current GS regularization techniques, which are designed specifically for explicit GS in 2DGS, are overly constrained by the voxel grid. The voxel grid restricts the movement of GS within a confined area, which is contrary to the expectation of regularization (to aggregate GS along the same ray path). Without proper regularization, achieving credible geometric quality is challenging. Therefore, we believe it is necessary to adapt the regularization terms to fit the voxel grid framework.\n\n> **Q2: Why not use Marching Cubes directly on the opacity function? Does converting to SDF improve quality?**  \n\nAs demonstrated in Table Q1 and illustrated in Fig 4 of the rebuttal file, applying MC directly to the opacity field is problematic. We have identified two main reasons for this issue:  \n1. **Linear Assumption of Opacity to SDF Mapping:** Directly using MC requires establishing an isosurface as the surface, which can be considered a linear mapping from opacity to SDF, e.g., SDF(opacity) = -opacity + 0.3 to extract opacity=0.3. This linear assumption does not strictly hold due to the nature of GS distribution. \n2. **Imprecision Due to Linear Interpolation:** Traditional MC relies on linear interpolation, which, when applied directly, results in inaccurate artifacts due to the non-linear properties of GS distributions, as depicted in Fig 4 of the rebuttal file.  \n\nTo address these challenges, we propose three viable approaches:  \n1. **Increasing MC Resolution:** This approach reduces the impact of non-linearity but leads to significant storage overhead, especially in open scenes. \n2. **Iterative Optimization to Approximate the Isosurface:** However, this method incurs substantial computational costs and the optimization algorithm is highly customized, limiting its generality. \n3. **Finding and Applying the Appropriate Inverse Function as Mentioned in Our Work:** This method incurs almost no additional computational cost and is universally applicable.  \n\nOur contemporaneous work, GOF, has adopted the second approach. For further analysis, please refer to the response @6M18.\n\n> **Q3: Limitations are only in the appendix. Why not include them in the main text?**  \n\n Thanks for your suggestion, we will add more analysis of limitations in main text. \n\n> **Q4: GOF results suggest voxel grids might not be necessary for performance. Is the voxel grid needed for memory reduction?**  \n\nPlease refer to Q1.\n\n> **Q5: Why derive SDF from opacity instead of using MC directly? Does SDF really enhance reconstruction quality?** \n \nPlease refer to Q2.", "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": "DgwJ03OYLw", "reviewer": "Reviewer_sv7m", "summary": "The paper presents interesting combintation of implicit and explicit representation that achieves efficient and high-fidelity open-scene reconstruction. Basiclaly, they combine the sparse voxel representation attached with per-voxel Gaussian splatting representation and proposes formulation for 3D surface reconstruction and volume rendering.", "questions": "- Even though the advantages of the proposed work will be prominent with open scenes, I think the formulation can be applied to general scenes, including objects. How will the performance gap (speed, quality, memory, etc.) be different in different settings? The results indicate that the performance is better for outdoor scenes and comparable to other works for indoor scenes (Table 6). Do you have any intuition what constraints reflect the performance gaps, and why do you emphasize reconstructing open scenes?\n\n- line 121 function -> feature?\n- Under Equation 5, $k_i$ is not clearly defined. Can you put definition of the ray-gaussian transfom?\n- In equations 6, 7, and 8, is the index $i$ the same as $k$?\n- Please make the embedded fonts in Figure 2 larger.\n- I do not fully follow the derivation in Equation 11. Why do you put a square $-\\rho^2(u)$?\n- Figure 6 is hard to interprete. What are the main differences?", "limitations": "- The limitations are addressed in apppendix. I think the reconstruction of sky are intentionally blocked in Figure 4. I would appreciate such gap to be revealed - we can interpret the results as a trade-off to achieve more complete scene without holes.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The formulation is mathematically sound.  The paper is well-written. The experiments are thorough and informative. Especially the geometric quality in Figure 5 achieves highly detailed and smooth results compared to baseline.", "weaknesses": "The geometric reconstruction is inferior to Neuralangelo, according to Table 3. Can you elaborate more on this? For example, the qualitative results are performed only against the explicit methods. I think the current method is strong enough, but the current paper does not clearly indicate when it fails, which makes the process appear weaker.\n\nMinor comments\n- Fonts in figure 2 can be larger.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Q1: The geometric reconstruction is inferior to Neuralangelo as shown in Table 3. Can you elaborate more on this?**  \n\nImplicit methods, such as those based on NeRF, typically utilize a global fitting approach for SDF, which allows them to fully leverage the universal approximation capabilities of MLPs. This is advantageous even in areas with sparse viewpoints. However, our current method employs a local line-of-sight-based SDF fitting, a compromise made to adapt to the 3DGS rendering style. This means that regions not covered by the training viewpoints lack fitting capability, resulting in uneven surfaces, as shown in Fig 2 in the rebuttal file. In areas with sparse viewpoint coverage, the distribution of 3DGS is sparse, which hinders the fitting of smooth planes.  \n\nDespite this, it is important to note that our method offers more practical advantages compared to implicit representations. For instance, Neuralangelo still relies on computationally intensive volumetric rendering, which results in longer training times (over 24 hours, see Table 3 in the main text). In contrast, our method can be trained in less than 1.5 hours, benefiting significantly from the integration with 3DGS.  \n\nWe believe that upgrading our current line-of-sight-based implicit representation to a global implicit representation, while avoiding computationally intensive volumetric rendering, is a promising direction. We plan to pursue this as future work.\n\n> **Q2: Fonts in Figure 2 can be larger.**  \n\nWe will increase the font size in Figure 2 for better readability.\n\n> **Q3: How will the performance (speed, quality, memory, etc.) differ for general scenes versus open scenes? Why performance gaps？**  \n\nWe observe that current methods based on 3DGS perform adequately for indoor scenes, where there is typically 360-degree viewpoint coverage. However, they underperform in outdoor scenes due to limited viewpoint coverage. Heuristic splitting and pruning strategies in original 3DGS tend to fit the training viewpoints rather than distributing evenly across the space. This leads to poorer novel view synthesis results in outdoor environments. As illustrated in Fig 1 of rebuttal file, without a voxel grid, heuristic GS growth strategies result in uneven spatial distribution of GS, sometimes even creating holes. Conversely, using voxel grids to constrain GS allows for efficient management of their spatial distribution, supporting better novel view synthesis.  \n\nTherefore, while our method shows significant improvement in NVS performance for outdoor scenes, the improvement is not as pronounced for indoor scenes. Despite this, the implicit representation using GS consistently saves space across both indoor and outdoor settings:  \n\n| Method | Outdoor | Indoor | Avg | \n|--------|---------|--------|-----| \n| GOF (3DGS-based)   | 1045 M     | 254 M    | 649 M | \n| GVKF   | 91 M    | 45 M   | 68 M |\n\n> **Q4: Should \"function\" be \"feature\" on line 121?**  \n\nThanks for correct.\n\n> **Q5: Under Equation 5, what is the definition of the ray-Gaussian transform?**  \n\nBased on Equation 1 in the main text, the influence of 3DGS $\\mathcal{G^{3D}_i}$ in camera space on a one-dimensional ray can be expressed as follows: \n$$ \\rho(t) = \\exp(-\\frac{1}{2}(vt-p)\\Sigma^{-1}(vt-p)) $$ \nHere, $v$ represents the unit vector of the ray direction. This formula converts the three-dimensional influence of 3DGS into a one-dimensional function along a specific camera ray, which is a one-dimensional Gaussian function. Fig 5 in rebuttal file demonstrates the relationship of this transform. For ease of notation, we express it as: \n$$ \\rho(t) = \\exp(-k_i \\cdot (t-t_i)^2) $$ \nwhere $t_i$ denotes the point along the ray where 3DGS has the maximum impact, also known as the \"ray-gaussian intersection,\" which can be analytically given by: \n$$ t_i = \\frac{p^T \\Sigma^{-1} v}{v^T \\Sigma^{-1} v} $$  \n\nFor further reference, see: \n*Approximate Differentiable Rendering with Algebraic Surfaces. (ECCV 22)*\n\n\n> **Q6: In equations 6, 7, and 8, is the index $i$ the same as $k$ ?**  \n\nWe appreciate the feedback. Based on the response by @zurJ Q3, we have corrected the formula and added figure to aid your understanding.\n\n> **Q7: Why is there a square in Equation 11?**  \n\nPlease note that $\\mathcal{T}^\\prime(u) = -\\rho(u) \\mathcal{T}(u)$, as shown in lines L144-L145, is due to the exponential term containing an integral.\n\n> **Q8: Figure 6 is hard to interpret. What are the main differences?**  \n\nIn Figure 6 of the main paper, we demonstrate the influence of different initial voxel grid resolutions (0.1 to 0.01, 0.001) on the final reconstruction quality. While a finer initial grid (v=0.001) produces higher quality results (see Tab 5), it also results in overly dense distribution of GS points, leading to reduced training speed. As discussed in Section 4.3, the initial voxel grid resolution of v=0.01 yields similar quality in novel view synthesis compared to v=0.001 but with shorter training times. Therefore, we ultimately selected v=0.01 as the initial voxel grid resolution.\n\n> **Q9: The limitations are in the appendix. Can you reveal gaps such as sky reconstruction in Figure 4?**  \n\nAs you mentioned, our method is designed to reconstruct complete scenes without holes, including distant views and sky, which can be easily cropped or retained as needed. We appreciate the suggestion and will move the limitations section to the main text. Additionally, we will provide more comprehensive scene reconstruction comparisons.", "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": "BY3N7PFfza", "reviewer": "Reviewer_6M18", "summary": "This paper presents Gaussian Voxel Kernel Functions (GVKF) for 3D surface reconstruction. The authors establish a continuous signed distance function derived from discrete 3D Gaussians, achieving high-fidelity open-scene surface reconstruction. They claim that the proposed method has high reconstruction quality, real-time rendering speeds, and significant memory efficiency in both runtime and training time. However, these claims lack sufficient discussion or validation throughout the paper.", "questions": "1. In Eq (7), how is the integration over the variable $t$ performed? From my understanding, Eq (6) can be used for rasterization by indexing over $i$, but $t$ is a continuous variable along the ray in Eq (7). Do you need sample points on a ray?\n\n2. How do you transform from Eq. (9) to Eq. (11)? Does $\\rho(t)$ relate to $\\rho(u)$, and if so, how?\n\n3. In L35, the paper states that 2DGS requires a large number of Gaussian primitives, leading to significant GPU memory consumption. This seems to be a common issue in GS-based methods. How is this solved in the paper?", "limitations": "The paper does not include a limitation section.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The voxel kernel function is a relatively new concept and appears interesting.\n\n2. The evaluation is extensive. The paper conducts experiments on up to three datasets. The PSNR is high and achieves state-of-the-art results on the MipNeRF360 dataset.\n\n3. The paper includes an ablation study to analyze the impact of voxel size.", "weaknesses": "1. In the abstract and contributions section, the paper claims significant memory savings. However, there is no discussion or validation related to memory consumption throughout the paper.\n\n2. The statements in Table 1 lack justification and contradict the experimental results. It lists that implicit methods are good at rendering quality while explicit methods like SuGaR and 2DGS are not, which contradicts the results in Table 2. Also, stating that implicit methods have low mesh quality is inaccurate, as shown by state-of-the-art methods like Neuralangelo in Table 3.\n\n3. The overall idea is very similar to GOF, which also establishes a transformation from 3D Gaussians to an opacity function. The regularizations are also the same as GOF, borrowed from 2DGS. The paper also adopts MC and MT for mesh extraction. Despite this, the paper does not compare to GOF or discuss it, even though the authors seem aware of it.\n\n4. The mathematical formulation is relatively difficult to understand due to multiple misalignments in definitions. It also lacks sufficient detail on how one equation leads to another.\n\n5. There are insufficient implementation details, making reproduction difficult. It is unclear if neural networks are used, as Figure 1 mentions a \"Neural\" Gaussian representation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Q1: Discussion or validation of memory consumption.**  \n\nThe quantitative comparison of memory usage is already presented in Table 2 of the main text. Compared to explicit GS methods (3DGS, 2DGS), our approach significantly reduces training memory consumption and storage, thanks to the introduction of voxel grids (Section 3.1):  \n\n1. Attributes of 3DGS such as opacity, color, rotation, and scaling are encoded into voxel properties and decoded by several global MLPs, reducing storage requirements. \n\n2. Voxel grids provide a more structured and clearer GS spatial distribution, saving the number of GS primitives, as shown in Fig 1 in rebuttal file. \n\n3. Compared to the heuristic growth and pruning strategy of the original 3DGS, our gradient accumulation-based voxel registration strategy controls the growth of 3DGS more effectively, avoiding wasteful distribution of GS in cluttered spaces.  \n\nAdditionally, we appreciate the suggestion and will include voxel grid ablation studies. Please refer to the response @epfR Q1 for further details.\n\n> **Q2: Statements in Table 1.**  \n\nWe apologize for any confusion caused by the table and will replace it with the table in @zurJ Q8. Hope this can help to clarify the question occoured on both implicit/explicit methods, as well as our motivation.\n\n> **Q3: Comparison to GOF.**  \n\n**Why not compare?**\n\nOur work and the GOF study were developed concurrently. Since GOF was not published at the time of our submission, we did not include it in our quantitative comparison.\n\n**Comparison with GOF.**\n\nOur work is similar to GOF in terms of deriving from ray-gaussian intersections. However, our scene representation is implicit, addressing the common issue of high memory consumption faced by 3DGS. Additionally, we have developed a mapping from opacity to SDF to adapt to general MC and MT algorithms, a consideration lacking in GOF.\n\n**Our Pros:**\n\n1. As shown in the table, GOF uses explicit GS management, still faces high storage consumption issues, making training large scenes on a single card challenging. Our method achieves better novel view synthesis results with less VRAM usage.\n| Mip 360 | PSNR | SSIM | LPIPS | Storage  |\n| :--- | :---: | :---: |  :---: | :---:  |\n| GOF | 24.53 | 0.733 |  0.245 |  649 M  |\n| GVKF(ours) | **25.47** | **0.757** | **0.240** | **68 M**  |\n2. GOF requires a long mesh refinement process (sometimes exceeding training time), although it achieves high geometric accuracy, this process is not general for typical MC/MT algorithms. However, our nonlinear mapping can mitigate this issue.\n| Tant | F1 | Meshing Time |\n| :--- | :---: |  :---: | \n| GOF w/ refine | 0.46  |  ~2 h | \n| GOF w/o refine | 0.34 | ~15 min |\n| GVKF(ours) | 0.36   | ~15 min  |\n\n3. GOF lacks in-depth mathematical analysis, which might confuse volumetric rendering with modified GS rendering. We revisited this rendering approach, providing solid mathematical backing.\n\n**Our Cons:**\n\nWhile we acknowledge that our current implementation has some geometric precision gaps compared to GOF, the potential reasons include: \n1. GOF's iterative optimization extraction method achieves more precise isosurfaces.\n2. As analyzed in @epfR Q1, further adaptation of regularization term to voxel grids might be needed. We leave it as our future work.\n\n> **Q4: Derivation between equations.**  \n\nFig 3 in the rebuttal file demonstrates our difference of volume rendering and 3DGS rendering, corresponding to Eq 6-8.  For Eq 9, we show that the opacity $\\rho(t)$ near surface, linearly combined by concentrated kernel functions, can be regarded as a new gaussian distribution. Eq 10-13 to analyze its property after integral (one inflection point, which is extremely similar to Logistic function). Hence we simulate  the nonlinear inverse mapping of opacity to SDF via Logistic function (Eq 14-15).\n\n> **Q5: If neural networks are used**  \n\n3DGS are encoded into one-dimensional vectors and stored in voxel grids for implicit representation. We employ four global MLPs (as described in Equation 2 in the main text) to dynamically decode the necessary attributes of 3DGS during the rendering process.\n\n> **Q6: Question of Eq 7**  \n\nWe apologize for any confusion caused by our oversight. The correct form of Equation 7 should be:\n$$\nC = \\sum_{i=1}^N c_i \\cdot \\alpha_i \\cdot \\mathcal{K_i}(0) \\prod_{j=1}^{i-1} (1 - \\alpha_j \\cdot \\mathcal{K_j}(0))\n$$\n We can index the peak of the kernel function for rasterization, which is consistent with the form of origin 3DGS rendering (Eq 6).\n\n\n> **Q7: How do you transform from Eq. (9) to Eq. (11)? Does $\\rho(t)$ relate to $\\rho(u)$, and if so, how?**  \n\nPlease note that both $t$ and $u$ represent points along the ray, but they differ in terms of the origin of coordinates. For $t$, the origin is at the camera center, which complicates the analysis of the surface. Therefore, we let $u$ represent a point along the ray, with the origin at the scene surface, as shown in Fig 2 in the main text. From the perspective of the camera, the origin in Fig 2 is at $t_i$.\n\nIn Equation 9, due to regularization during the training process, 3DGS near the surface aggregate together to form a new Gaussian distribution $\\rho(t)$. However, from the perspective of $t$, the axis of symmetry of $\\rho(t)$ is not at the origin. Therefore, we align the axis of symmetry with the origin to obtain $\\rho(u)$, as shown in Fig 2. This can be understood as $\\rho(t)$ undergoing a simple translation to become $\\rho(u)$; the analysis in Equation 10 is entirely based on $\\rho(u)$.\n\n> **Q8: How to save Gaussian primitives ?**  \n\nPlease refer to Q1.\n\n> **Q9: About limitation section.**  \n\nThank you for the suggestion. We will move the limitations section from the appendix to the main text.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers,

Please see the attached PDF page, which includes additional experimental results and formula illustrations, to help clarify our approach. We are deeply grateful for the constructive feedback provided by all reviewers, which has significantly helped improve our paper. We are pleased to receive recognition from reviewers for our paper's:

1. Convincing evaluations and good reconstruction quality (Reviewers zurJ, epfR, sv7m, 6M18).
2. Lower resource consumption (Reviewers zurJ, epfR).
3. Appreciation of our mathematical analysis (Reviewers epfR, sv7m, 6M18).

We have been diligently working to improve the paper in various aspects to address your critiques. Below, we summarize the changes made in the updated draft:
1. Corrected symbol misuse in Equations 6, 7, and 8.
2. Added more intuitive explanations for Equations 5, 6, and 7, with illustrations to aid understanding.
3. Included ablation studies and visualizations for voxel and SDF mapping.
4. Enhanced descriptions of research motivation and implementation details.
5. Moved the limitations section from the appendix to the main text and expanded the explanations.
6. Added comparisons and discussions with GOF.

We will add the following content in the updated appendix:
1. More displays of failure cases.
2. Explanations of the effects in indoor scenes.
3. More detailed derivations (ray-gaussian transform) and illustrations.

Please see our reviewer-specific feedback for more information.

Meta-review: Initially, there were issues with the writing of the paper. The rebuttal was very clear and informative. Reviewer zurJ maintained a borderline reject rating, while the other three reviewers upgraded their ratings to 2x weak accept and 1x borderline accept and agree that there is an interesting contribution and strong results. There was a big discussion on the similarity to GOF. As it was published less than 8 weeks before, it is concurrent work and should not be held against this submission.

Therefore, the decision is to accept the paper. This decision was also discussed with the SAC. The authors are requested to integrate all the feedback to improve the writing of the paper.
Decision: Accept (poster)