GVKF: Gaussian Voxel Kernel Functions for Highly Efficient Surface Reconstruction in Open Scenes
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.
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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.
Strengths
1. The voxel kernel function is a relatively new concept and appears interesting. 2. 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. 3. 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. 2. 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. 3. 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. 4. 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. 5. There are insufficient implementation details, making reproduction difficult. It is unclear if neural networks are used, as Figure 1 mentions a "Neural" Gaussian representation.
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? 2. How do you transform from Eq. (9) to Eq. (11)? Does $\rho(t)$ relate to $\rho(u)$, and if so, how? 3. 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?
Rating
5
Confidence
2
Soundness
3
Presentation
2
Contribution
3
Limitations
The paper does not include a limitation section.
Response
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?
> Summarize Thank you for your comments; they will greatly assist us in our future work. As part of our concurrent efforts, we acknowledge that GOF has achieved higher geometric quality in their rendering implementation than we have, and we recognize that this rendering approach was proposed earlier than ours. We will clarify in our paper that equations 5-8 are the main contributions of GOF. However, as pointed out in Q3, while this newly developed rendering method produces attractive results, it is still imperfect: the original GOF paper lacks a mathematical basis and requires extensive post-processing. We hope our analysis can better help readers understand this innovative rendering method and the improvements we have made in Section 3.3. Thank you for your suggestions regarding Section 3.1; we will add more implementation details and provide a more in-depth analysis. We apologize again for the misuse of mathematical formulas in our initial submission, but we assure readers that the revised formulas are correct, and we will thoroughly address your questions. > Answer to Question The equation 7 can be simplified as: $$ C=\sum_{i=1}^N=c_i \cdot \alpha_i \cdot \prod_{j=1}^{i-1} (1-\alpha_j) $$ Here we keep $\mathcal{K_i}(0)$ for the coherence with equation 5,8. This formulation also appears in the equation 3 of original 3DGS paper. Both equation 7 and equation 6 are valid, the difference is whether consider $\alpha_i$ as constant value when ray-gaussian intersection changes. For perfect aligning with equation 6, the equation 7 could be $$ C=\sum_{i=1}^N c_i \cdot \beta_i \frac{\sqrt{k_i}}{\sqrt{\pi}} \cdot \mathcal{K_i}(0) \prod_{j=1}^{i-1}(1-\beta_j \frac{\sqrt{k_j}}{\sqrt{\pi}} \cdot \mathcal{K_j}(0)) $$ In this scenario, $\beta_i$ is a constant value, and does not change when ray-gaussian intersection changes. More details can be found in next comment.
Additional comments
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?
Answer to more Math questions
Thanks for your in-depth analysis, this helps us a lot for the revision. > Q-1 Kernel Function for Ray Intersections In general, the Gaussian kernel function does not take into account the coefficient in front of the exponential, like $e^{-\gamma t^2}$. This means that with different values of $\gamma$, the Gaussian kernel function is not a strict normal distribution. For simplicity, we have retained the general form of the kernel function in our Equation 5. When considering ray gaussian intersection along the ray, the probability density influence of the Gaussian kernel on the ray is a strict normal distribution function $\frac{\sqrt{k_i}}{\sqrt{\pi}}e^{-k_it^2}$. In addition, the Gaussian kernel itself should also have a constant opacity $\beta_i$. To be precise, we have $\alpha_i=\beta_i*\frac{\sqrt{k_i}}{\sqrt{\pi}}$. From the perspective of yours, the missing $G^{2d}$ should be $\frac{\sqrt{k_i}}{\sqrt{\pi}}$, it is already included in $\alpha_i$. > Q-2 Volume Rendering Formulation (NeRF vs. 3DGS) I can mainly understand your question. To clarify, please do not confuse our opacity $\rho(t)$ and volume density $\sigma(t)$ in NeRF. In the main text, we do not use any symbol or concept like $\sigma(t)$. Firstly, we would like to clarify some basic concept for better explanation: **Volume rendering (NeRF)** $$ C=\sum_{i=1}^N T_i(1-\exp(-\sigma_i\delta_i))c_i, \quad T_i=\exp(-\sum_{j=1}^{i-1}\sigma_j\delta_j) $$ which can be written as: $$ C=\sum_{i=1}^{N}T_i \cdot \alpha_i \cdot c_i, \quad \alpha_i=(1-\exp(-\sigma_i\delta_i)), \quad T_i=\prod_{j=1}^{i-1}(1-\alpha_i) $$ These are illustrated in equation 1 and equation 2 of original 3DGS paper. **3DGS rendering** This is the rendering of 3DGS in original 3DGS paper: $$ C=\sum_{i=1}^N=c_i \cdot \alpha_i \cdot \prod_{j=1}^{i-1} (1-\alpha_j) $$ **Relationship of NeRF and 3DGS rendering** As we can see, the only difference between them is: $N$ in volume rendering is larger. Hence the goal of 3DGS and volume rendering is consistent and the difference is sampling resolution. This is also the fundation of 3DGS rendering equation in original 3DGS paper. We also illustrated this in Fig 3 of rebuttal PDF file. **Please note** 1. The opacity $\alpha_i$ is the function of volume density $\sigma_i$ and sampling interval $\delta_i$ in volume rendering, which is the **accumulated result in a sampling interval** 2. In NeRF, dense sampling is used for better record $\sigma$, rather than opacity $\alpha$ 3. The concept of volume density $\sigma$ does not appear in 3DGS and our paper, and rebuttal PDF file. We directly using guassian primitives to represent opacity $\alpha$, rather than volume density $\sigma$ 4. Our goal is to find continuous opacity $\alpha$ along the ray, represented as $\rho(t)$, rather than volume density $\sigma(t)$ **Answer your question** - *"Gaussian points are treated as sampling points for a continuous version of volume rendering"* This is right, and as analysed above, this is the fundation of 3DGS rendering equation. - *"The sampling interval between these points should not be omitted in the quadrature of volume rendering"* This is not right, $\alpha_i$ already records the accumulated influence to the ray in a short or long sampling interval, hence $\alpha_i$ should not multiply any $\delta$ again; - *"Why omitting the $\Delta t$ factor"* If we use kernel function to represent continuous volume density $\sigma(t)$, rather than $\rho(t)$ in our paper, the $\Delta t$ should be included to calculate opacity. However, this will make things trouble. > Q3 Density Along the Ray **Contribution of preceding gaussians** In the equation 7, the contribution of preceding gaussian kernels are reflected in the term $\prod_{j=1}^{i-1}(1-\alpha_j \cdot \mathcal{K_j}(0))$. Hence any kernel before should not be included in current point. **Correct sampling point** Based on equation 5, the second sampling point should be like $K_1+K_2+...+K_N$, the linear combination of all kernel functions, rather than $K_2$ or $K_1+K_2$. **Optimizing kernel fucntions** Based on Equation 5, our continuous opacity function $\rho(t)$ is defined on linear combination of kernel functions, hence the best way to optimize it is to directly optimize the kernel function and linear coefficients, rather than the sampling point represented by all of the kernels. As shown in Equation 7, this way considers every kernel's contribution and the influence of preceding kernels, which is coherent with volume rendering euqation and 3DGS rendering equation in Q2
Response
Thank you for the detailed explanation. While I appreciate the effort, I find that I still struggle to fully grasp some of the equations. I would like to raise the score and lower my confidence level.
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.
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. Minor comments - Fonts in figure 2 can be larger.
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? - line 121 function -> feature? - Under Equation 5, $k_i$ is not clearly defined. Can you put definition of the ray-gaussian transfom? - In equations 6, 7, and 8, is the index $i$ the same as $k$? - Please make the embedded fonts in Figure 2 larger. - I do not fully follow the derivation in Equation 11. Why do you put a square $-\rho^2(u)$? - Figure 6 is hard to interprete. What are the main differences?
Rating
6
Confidence
3
Soundness
3
Presentation
3
Contribution
3
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.
Summary
The paper introduces Gaussian Voxel Kernel Functions (GVKF), a novel approach for efficient 3D surface reconstruction in open scenes, leveraging 3D Gaussian Splatting. The 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. **Details** 1. *Compressing 3D Gaussians Using a Sparse Voxel Grid and MLPs* The 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. 2. *Defining a Neural Opacity Field with Gaussians to Render Images* For 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. 3. *Extracting Accurate Surfaces from Neural Opacity Fields* The 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. The 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.
Strengths
1. The paper is well-written and easy to follow. 2. I appreciate the detailed mathematical analysis of the proposed method. 3. The blending formula proposed in the paper allows for more geometry-accurate rendering compared to vanilla 3DGS. 4. The voxel structure proposed in the paper enhances resource efficiency (particularly in terms of VRAM usage) by compressing the 3D Gaussians. 5. 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? 2. 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? 3. 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. **Comment** I 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.
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? 2. Why not apply Marching Cubes directly to the opacity function? Does the SDF really improve the quality of the reconstruction?
Rating
6
Confidence
5
Soundness
4
Presentation
4
Contribution
3
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.
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.
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. (2) I can somewhat understand what the authors want to do in Equation 5, but could the authors provide some figures for easier understanding? (3) Equation 6 seems incorrect. There appears to be a mixed use of 'i' and 'k'. (4) Equation 7 seems incorrect. What does the term 't' mean here? It is not defined or explained here. (5) Why do we need to convert to an SDF value? I do not understand the reasoning. (6) I cannot find how the authors convert this implicit field to a surface. Did you use Marching Cubes? (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. (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. Overall, 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. Minor issues: Line 53: We -> we Line 112: Sfm -> SfM
Questions
(1) In Line 292, what is the unit of “1, 0.1, 0.01, …”? (2) Is there any visualization results of the voxels? I am curious about its quality.
Rating
4
Confidence
4
Soundness
2
Presentation
1
Contribution
2
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.
Discussion
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.
Reply to Discussion
> 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.
I read the rebuttal and other reviews. My questions are all answered, which I truly appreciate.
I would like to thank the authors for the detailed rebuttal and very clear answers to my questions. I recommend acceptance for the paper, as I think it brings interesting contributions to the field of mesh reconstruction using 3DGS-based approaches. However, although I appreciate the detailed mathematical framework presented in the paper to support its claims, I think the proposed approach overlaps considerably with the existing work Gaussian Opacity Fields (GOF). Consequently, I decide not to raise my rating, but maintain a Weak Accept.
Decision
Accept (poster)