GVKF: Gaussian Voxel Kernel Functions for Highly Efficient Surface Reconstruction in Open Scenes

Q1: Discussion or validation of memory consumption.

The 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):

1. Attributes of 3DGS such as opacity, color, rotation, and scaling are encoded into voxel properties and decoded by several global MLPs, reducing storage requirements.

2. 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.

3. 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.

Additionally, we appreciate the suggestion and will include voxel grid ablation studies. Please refer to the response @epfR Q1 for further details.

Q2: Statements in Table 1.

We 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.

Q3: Comparison to GOF.

Why not compare?

Our 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.

Comparison with GOF.

Our 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.

Our Pros:

1. 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. | Mip 360 | PSNR | SSIM | LPIPS | Storage | | :--- | :---: | :---: | :---: | :---: | | GOF | 24.53 | 0.733 | 0.245 | 649 M | | GVKF(ours) | 25.47 | 0.757 | 0.240 | 68 M |

2. 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. | Tant | F1 | Meshing Time | | :--- | :---: | :---: | | GOF w/ refine | 0.46 | ~2 h | | GOF w/o refine | 0.34 | ~15 min | | GVKF(ours) | 0.36 | ~15 min |

3. GOF lacks in-depth mathematical analysis, which might confuse volumetric rendering with modified GS rendering. We revisited this rendering approach, providing solid mathematical backing.

Our Cons:

While we acknowledge that our current implementation has some geometric precision gaps compared to GOF, the potential reasons include:

1. GOF's iterative optimization extraction method achieves more precise isosurfaces.
2. As analyzed in @epfR Q1, further adaptation of regularization term to voxel grids might be needed. We leave it as our future work.

Q4: Derivation between equations.

Fig 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).

Q5: If neural networks are used

3DGS 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.

Q6: Question of Eq 7

We apologize for any confusion caused by our oversight. The correct form of Equation 7 should be:

We can index the peak of the kernel function for rasterization, which is consistent with the form of origin 3DGS rendering (Eq 6).

Q7: How do you transform from Eq. (9) to Eq. (11)? Does $\rho(t)$ relate to $\rho(u)$, and if so, how?

Please 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 $t_i$.

In 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 $\rho(u)$.

Q8: How to save Gaussian primitives ?

Please refer to Q1.

Q9: About limitation section.

Thank you for the suggestion. We will move the limitations section from the appendix to the main text.

Thank you for your response. I would like to share some of my thoughts on the current draft. I have noticed that the paper bears a strong resemblance to Gaussian Opacity Fields (GOF). For instance, as pointed out by Reviewer epfR, Eq. 7 and Eq. 8 closely resemble Eq. 8 and Eq. 9 from the GOF paper. Additionally, the paper incorporates the same regularization as 2DGS (and mentions GOF in L268–L270). Furthermore, the paper adopts MC/MT for meshing, though it lacks sufficient details to ascertain whether the MC/MT implementation is the same as GOF. Based on the table in Q3, it appears that the current meshing implementation is almost identical to GOF w/o refinement (MT is a main contribution from GOF). These observations raise questions about whether the current method was developed concurrently and independently.

Although the method achieves better PSNR with significantly fewer parameters in MipNeRF 360, the author should provide more emphasis on how this gain is achieved, such as the significantly higher PSNR (25.47) achieved with 10x fewer parameters. It should also be carefully checked that there is no test-set pollution.

Moreover, there are several important details missing, such as the settings of MLP and meshing, making reproducibility difficult. Additionally, the presentation is poor, including numerous errors in symbols and tables in the initial submission, which complicates the evaluation process. Despite the author's rebuttal commitment to major revisions, I believe that if a submission requires numerous revisions and further confirmation by the reviewer, it would be better suited for resubmission. Consequently, I maintain my rating for rejecting the submission.

Additional Questions:

If I understand correctly, then the correct form of Equation 7 is $C = \sum_{i}^{N}c_i \cdot \alpha_i \prod_{j=1}^{i-1}(1-\alpha_j)$ since $K(0) = 1$? If so, how is it consistent with the form of origin 3DGS rendering (Eq 6)? Also, should Equation 8 also be corrected?

After correcting some symbols, I have re-evaluated the submission and would like to share further thoughts on the current formulation. However, some concerns remain that make this paper difficult to accept in its current form.

1. Kernel Function for Ray Intersections: If a ray intersects with the 3D Gaussian, the kernel function should be expressed as $K_i(t)=G^{2d}\exp⁡(−k_i⋅t^2)$ instead of $K_i(t) = \exp(-k_i \cdot t^2)$. This discrepancy needs clarification. Can the authors explain why the 2D Gaussian factor was omitted, and how this impacts the accuracy of the formulation?

2. Volume Rendering Formulation (NeRF vs. 3DGS): The paper attempts to establish a volume rendering approach using both a density-based (NeRF) formulation and an alpha-based (3DGS) formulation in Eq. 6 and Eq. 7. However, I believe this approach is inaccurate. Referring to Fig. 3 in the rebuttal PDF, if the Gaussian points are treated as sampling points for a continuous version of volume rendering, then the sampling interval between these points should not be omitted in the quadrature of volume rendering. The current formulation suggests that the kernel intersecting with the ray collapses to an infinitesimally small value, but this does not justify omitting the $\Delta t$ factor. Therefore, the transformation from Eq. 4 to Eq. 7 seems problematic. Could the authors clarify the rationale behind this omission and how it affects the overall rendering process?

3. Density Along the Ray: If the density along the ray is defined as $\rho(t) = \sum_{i}K_i(t - t_i)$, should the second sampling point be represented as $\rho_2(t) = K_1(t_2 - t_1) + K_2(t_2 - t_2)$ instead of just $K_2(t_2−t_2)$? The current formulation seems to overlook the contribution of preceding Gaussian kernels when calculating the density at subsequent points. Could the authors provide an explanation for this simplification and its implications on the accuracy of the density calculation?

Q1: The geometric reconstruction is inferior to Neuralangelo as shown in Table 3. Can you elaborate more on this?

Implicit 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.

Despite 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.

We 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.

Q2: Fonts in Figure 2 can be larger.

We will increase the font size in Figure 2 for better readability.

Q3: How will the performance (speed, quality, memory, etc.) differ for general scenes versus open scenes? Why performance gaps？

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 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.

Therefore, 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:

Q4: Should "function" be "feature" on line 121?

Thanks for correct.

Q5: Under Equation 5, what is the definition of the ray-Gaussian transform?

Based 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: $\rho(t) = \exp(-\frac{1}{2}(vt-p)\Sigma^{-1}(vt-p))$ Here, $v$ 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: $\rho(t) = \exp(-k_i \cdot (t-t_i)^2)$ where $t_i$ 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: $t_i = \frac{p^T \Sigma^{-1} v}{v^T \Sigma^{-1} v}$

For further reference, see: Approximate Differentiable Rendering with Algebraic Surfaces. (ECCV 22)

Q6: In equations 6, 7, and 8, is the index $i$ the same as $k$ ?

We appreciate the feedback. Based on the response by @zurJ Q3, we have corrected the formula and added figure to aid your understanding.

Q7: Why is there a square in Equation 11?

Please 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.

Q8: Figure 6 is hard to interpret. What are the main differences?

In 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.

Q9: The limitations are in the appendix. Can you reveal gaps such as sky reconstruction in Figure 4?

As 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.

Q1: Flowchart is too abstract

Thank you for your suggestion. We have provided more figures in the rebuttal file to illustrate our pipeline.

Q2: The explanation of Eq 5

As 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 $\alpha_i$.

Q3: Mixed Use of 𝑖 and 𝑘 in Equation 6

We 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:

Equation 6 represents the discrete rendering of original 3DGS: $C=\sum_{i=1}^N c_i \cdot \alpha_i \cdot \mathcal{G_i}^{2D} \prod_{j=1}^{i-1}(1-\alpha_j \cdot \mathcal{G_j}^{2D})$ The 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.

Adaptation to Kernel Functions: For 3DGS rendering, we substitute $\mathcal{G_i}^{2D}$ with $\mathcal{K_i}(0)$, leading to Equation 7: $C=\sum_{i=1}^N c_i \cdot \alpha_i \cdot \mathcal{K_i}(0) \prod_{j=1}^{i-1}(1-\alpha_j \cdot \mathcal{K_j}(0))$ This change maintains fidelity to the original rendering equation, optimizing both $\alpha_i$ and $\mathcal{K_i}$ through RGB loss.

Representation of Scene Surfaces: Based on the initial definition of $\rho(t)$, Equation 8 is: $\Phi(t)=\sum_{i=1}^{N}\alpha_i \cdot \mathcal{K_i}(t-t_i) \prod_{j=1}^{i-1}(1-\alpha_j \cdot \mathcal{K_j}(t-t_j))$ During training, Equation 7 facilitates rendering without dense sampling. For opacity representation, Equation 8 computes a continuous scene representation along the ray.

Q4: The definition of t in Equation 7 is unclear

Please refer to Q3.

Q5: The necessity of converting to SDF values

As 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.
Once 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).
In our experiments, we used the Marching Tetrahedra (MT) algorithm, although the Marching Cubes (MC) algorithm is also applicable.
For a more in-depth analysis, please refer to the response @epfR Q-2.

Q6,Q7:

Please refer to Q5.

Q8: The writing style and surface reconstruction contributions are unclear

We 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.

As 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. Therefore, to integrate the advantages and mitigate the disadvantages of the above two rendering techniques,

(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;

(2) we take the maximum values of each kernel function to perform discrete numerical integration for rendering;

(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.

To 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.

Q9: The unit of voxel size in the ablation experiments

The 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.

Q10: The visualization effect of voxels

As 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.

Discussion-Q1: Memory reduction contribution

The memory reduction is resulted from our proposed voxel grids in Section 1. The voxels serve as spatial anchors, providing a more regulated, structured distribution of 3DGS. They also control the growth and splitting strategies.

1. The attributes of 3DGS such as opacity, color, rotation and scaling are encoded into voxel properties and decoded by several global MLPs when rendering, as shown in equation 2 in the main text. This implicit representation greatly saves the memory consumption of 3D gaussian primitives.
2. Voxel grids provide more structured and clearer 3DGS distributions in space. While in original 3DGS, the primitives are uncontrollable. As shown in Fig. 1 of our submitted rebuttal file.
3. Our gradient-accumulation-based voxel registration strategy controls the growth of 3D gaussian primitives effectively, compared to the heuristic growth and pruning strategy of the original 3DGS.

In addition, the Tab. 2 of main text also demonstrates the efficiency of our method in memory reduction. More in-depth analysis can be found in @epfR Q1, @sv7m Q3

Discussion-Q2: "Taking the maximum values of each kernel function to perform discrete numerical integration for rendering" is not a novel contribution.

Currently, it appears that direct integration between 3DGS and NeRF is not feasible. If we use 3DGS, it cannot represent 3d continuous surface (e.g. SDF function). If we use 3d continuous surface representation, currently it must be optimized through volume rendering.

As we mentioned in our response to Q8, our goal is to combine the advantages of 3DGS rendering and the continuous representation in volume rendering. This requires us to solve two problems:

1. How can we represent the continuous opacity function $\rho(t)$ on a ray using discrete gaussian primitives?
2. How can we optimize this continuous opacity function $\rho(t)$ without using volume rendering?

Our contribution lies in finding a method to solve these two issues:

Firstly, we consider the effects of 3D Gaussian primitives on the ray as 1D Gaussian kernel functions. Hence, the weighted sum of $\alpha_i$ of each gaussian primitive can represent the continuous opacity function $\rho(t)$ on the ray. As shown in equation 5 in the main text. This is the defination of kernel regression.

Secondly, to optimize the continuous function $\rho(t)$，we take the maximum values of each kernel function (including the coefficient $\alpha_i$) to perform discrete numerical integration for rendering, which is equivalent to 3DGS alpha blending.

Please be clarified "equivalent to Gaussian splatting alpha blending" is NOT a trivial conclusion, but demonstrates our proposed method to optimize this continuous function makes sense and is mathematically correct. This indicates we propose a NEW solution, instead of volume rendering, to optimize continuous 3D surface.

Discussion-Q3: Why authors refer to the method as 'kernel regression.' Kernel regression is not equivalent to 'differentiable rendering with Gaussian kernels.'

It is notable that the concept of kernel regression is used to represent continuous functions, rather than to depict the rendering process. Our aim is to represent complex continuous function $\rho(t)$ using a simple set of Gaussian kernel functions. This aligns with the process of kernel regression: for each spatial point, opacity can be calculated through the weighted sum of $\alpha_i$, where the weights come from Gaussian kernel functions.

Discussion-Q4: Lack of analysis explaining why a 'kernel regression' is necessary and how it enhances performance.

As mentioned in Discussion-Q3, kernel regression is the concept for describing how continuous function $\rho(t)$ formed along the ray. In terms of the reason of improvement, our performance benefits from continuous representation.

In the original 3DGS-based methods, such as 2DGS, discrete gaussian primitives are employed to fit surfaces, and mesh extraction relies on TSDF method. This approach may lead to holes in areas with sparse gaussian primitives or occlusions. The produced meshes are often overly smooth, which hinders the representation of fine details. In contrast, with our continuous representation of kernel regression, we obtain more details on the surface and have less holes, as shown in Fig. 4 and Fig. 5 of main text as well as our project page.

Discussion-Summary

We highly value your feedback on our writing style and are dedicated to enhancing the clarity and flow of our paper. We sincerely hope the reviewers will recognize our efforts and overall contributions. Thank you once again for your valuable time and constructive feedback.

First, I want to express my appreciation for your efforts in clarifying the equations and improving the overall clarity of the description. However, I still have concerns regarding the positioning and motivation of the paper.

• My initial question regarding the claimed memory reduction contribution in the abstract remains unaddressed.
• The statement, 'We take the maximum values of each kernel function to perform discrete numerical integration for rendering,' seems to lead to a somewhat trivial conclusion, which is 'equivalent to Gaussian splatting alpha blending.' I struggle to see this as a novel contribution.
• I am also confused as to why the authors refer to the method as 'kernel regression.' Kernel regression is not equivalent to 'differentiable rendering with Gaussian kernels.'
• Furthermore, I personally feel that the experiments are disconnected from the overall narrative of the paper. While the experiments demonstrate that the proposed method outperforms the state-of-the-art (SoTAs), there is a lack of analysis explaining why a 'kernel regression' is necessary and how it enhances performance.

However, I would not insist on rejecting the paper. The performance is excellent, and the experimental results are strong. My concern is primarily with the writing style, which I personally do not favor.

As a result, I am inclined to maintain my original score.

Q1: How does the method perform without voxel grids? What is the impact on quality and VRAM?

We 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:

Observations:

• 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.

Further Analysis:

• 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:

     1. Enhancing NVS performance (refer to Tables 2 and 4 in the main text).   
     2. Reducing memory and storage requirements during training, enabling the training of large scenes on a single GPU.  
  

However, 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.

Q2: Why not use Marching Cubes directly on the opacity function? Does converting to SDF improve quality?

As 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:

1. 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.
2. 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.

To address these challenges, we propose three viable approaches:

1. Increasing MC Resolution: This approach reduces the impact of non-linearity but leads to significant storage overhead, especially in open scenes.
2. Iterative Optimization to Approximate the Isosurface: However, this method incurs substantial computational costs and the optimization algorithm is highly customized, limiting its generality.
3. Finding and Applying the Appropriate Inverse Function as Mentioned in Our Work: This method incurs almost no additional computational cost and is universally applicable.

Our contemporaneous work, GOF, has adopted the second approach. For further analysis, please refer to the response @6M18.

Q3: Limitations are only in the appendix. Why not include them in the main text?

Thanks for your suggestion, we will add more analysis of limitations in main text.

Q4: GOF results suggest voxel grids might not be necessary for performance. Is the voxel grid needed for memory reduction?

Please refer to Q1.

Q5: Why derive SDF from opacity instead of using MC directly? Does SDF really enhance reconstruction quality?

Please refer to Q2.