Title: Effective Rank Analysis and Regularization for Enhanced 3D Gaussian Splatting
Venue: NeurIPS 2024
Accepted: yes
Keywords: 3D reconstruction

3D Gaussian Splatting

NeRF

Surface reconstruction

3DGS regularization
Abstract: 3D reconstruction from multi-view images is one of the fundamental challenges in computer vision and graphics. Recently, 3D Gaussian Splatting (3DGS) has emerged as a promising technique capable of real-time rendering with high-quality 3D reconstruction. This method utilizes 3D Gaussian representation and tile-based splatting techniques, bypassing the expensive neural field querying. Despite its potential, 3DGS encounters challenges such as needle-like artifacts, suboptimal geometries, and inaccurate normals caused by the Gaussians converging into anisotropic shapes with one dominant variance. We propose using the effective rank analysis to examine the shape statistics of 3D Gaussian primitives, and identify the Gaussians indeed converge into needle-like shapes with the effective rank 1. To address this, we introduce the effective rank as a regularization, which constrains the structure of the Gaussians. Our new regularization method enhances normal and geometry reconstruction while reducing needle-like artifacts. The approach can be integrated as an add-on module to other 3DGS variants, improving their quality without compromising visual fidelity. The project page is available at https://junhahyung.github.io/erankgs.github.io/.

Effective Rank Analysis and Regularization for Enhanced 3D Gaussian Splatting

Abstract

3D reconstruction from multi-view images is one of the fundamental challenges in computer vision and graphics. Recently, 3D Gaussian Splatting (3DGS) has emerged as a promising technique capable of real-time rendering with high-quality 3D reconstruction. This method utilizes 3D Gaussian representation and tile-based splatting techniques, bypassing the expensive neural field querying. Despite its potential, 3DGS encounters challenges such as needle-like artifacts, suboptimal geometries, and inaccurate normals caused by the Gaussians converging into anisotropic shapes with one dominant variance. We propose using the effective rank analysis to examine the shape statistics of 3D Gaussian primitives, and identify the Gaussians indeed converge into needle-like shapes with the effective rank 1. To address this, we introduce the effective rank as a regularization, which constrains the structure of the Gaussians. Our new regularization method enhances normal and geometry reconstruction while reducing needle-like artifacts. The approach can be integrated as an add-on module to other 3DGS variants, improving their quality without compromising visual fidelity. The project page is available at https://junhahyung.github.io/erankgs.github.io/ .

1 Introduction

Creating 3D models from multiple images is a central challenge in computer vision and graphics. Neural Radiance Fields (NeRF) [20] have revolutionized this area by demonstrating remarkable capabilities in novel view synthesis through implicit neural fields and differentiable rendering techniques. Despite their impressive 3D reconstruction quality, the training and rendering processes of NeRFbased methods are computationally intensive, posing significant challenges for real-time applications. To improve training and rendering efficiency, various acceleration techniques, such as baking with shell [12, 32] and grid representations [5, 21], have been introduced. While these solutions enhance efficiency to some extent, there are still limitations for real-time interactive scenarios.

Recently, 3D Gaussian Splatting (3DGS) has emerged as a promising technique capable of realtime rendering with high-quality results. This method utilizes 3D Gaussian representations and tile-based splatting techniques instead of expensive neural field querying, making it feasible to apply the technique in practical applications. This opens up new possibilities in areas that require faster rendering, such as virtual and augmented reality, gaming, and real-time avatars.

However, despite its potential, 3DGS encounters several challenges in terms of geometry reconstruction, including noisy rendering results with needle-like artifacts, especially in novel and extreme views far from the training images. These issues stem from the primitive-based nature of 3DGS, where individual primitives lack geometric constraints.

(a)

(b)

(c)

For accurate geometry reconstruction, it is well known that the density field should be concentrated near the surface [30]. To this end, previous efforts, such as SuGaR [10], have focused on regularizing the 3D Gaussians to be flatter, i.e. , regularizing the primitives into anisotropic Gaussians with one of its variance very small. Similarly, 2DGS [13] utilizes 2D Gaussians instead of 3D Gaussians to force this effect.

While the flatness of Gaussians is necessary for proper alignment with the surface, we argue that flatness alone is insufficient for accurately representing surface geometry. Gaussians should also avoid being needle-like or highly anisotropic, where one variance dominates the others. Needle-like Gaussians hinder accurate reconstruction as they cover only a negligible portion of the surface and produce spiky artifacts. Instead, disk-like Gaussians are preferable as they cover meaningful areas and contribute effectively to surface reconstruction.

However, existing methods fail to adequately differentiate between disk-like and needle-like Gaussians, as both can exhibit one scale that is near or exactly zero. Empirically, we find that most Gaussians converge into anisotropic forms, becoming needle-like with small scales along two axes due to the absence of appropriate regularization mechanisms.

To directly examine the shape statistics (whether their geometries are disk-like or needle-like) of 3D Gaussian primitives and understand their structural changes during training, we first propose performing the effective rank analysis on the covariance matrices of Gaussians. The effective rank [25], which is a real-valued and differentiable extension of the integer rank, can be used to monitor the training dynamics and structural transformations of Gaussian primitives. Indeed, our analysis reveals that the effective ranks of Gaussians approach an effective rank of 1 ( erank -1), resulting in needle-like shapes in 3DGS and other methods, such as SuGaR [10] and 2DGS [13].

Additionally, we propose using the effective rank as a regularization term to constrain the structure of the Gaussians. The differentiable nature of effective rank, with its concave logarithmic term providing stable gradients, makes it directly applicable to continuous optimization problems. Our new regularization method enhances normal and geometry reconstruction while reducing needle-like

SuGaR

3DGS

2DGS

Baselines

Baselines + erank reg.

Iteration 1

Iteration 14000

Iteration 30000

25000

25000

25000

15000

15000

15000

5000

5000

5000

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

25000

25000

25000

15000

15000

15000

5000

5000

5000

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

25000

25000

25000

15000

15000

15000

5000

5000

5000

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

artifacts, particularly in novel view scenarios. Furthermore, our effective rank regularization can be applied as an add-on module to other 3DGS variants, improving their quality.

The main contributions of our work are as follows:

We are firstly analyzing the dynamics of Gaussian primitive structures using the effective rank in the optimizing process, discovering that Gaussians converge into anisotropic forms with one dominant variance.

We propose an effective rank regularization method that alleviates needle-like artifacts in 3DGS rendering and improves geometric reconstruction.

Our approach is an add-on module that can be integrated with other 3DGS variants, and demonstrate that our method enhances 3D geometry reconstruction without compromising visual quality.

2 Related work

Novel view synthesis NeRFs [20] have revolutionized photo-realistic rendering from novel viewpoints by introducing a neural implicit representation of 3D scenes. This approach uses high-frequency positional encoding and differentiable volume rendering to achieve unprecedented realism. Enhancements to NeRF address challenges like anti-aliasing [1, 3], parameterizing unbounded scenes [2, 37], and training from in-the-wild images [19, 8, 29] through probabilistic transience modeling. Further

erank 3

erank 2

erank 1.65

erank 1.25

erank 1.08

improvements reduce training time and enhance rendering quality by incorporating low-rank tensor components [5].

Other research efforts have aimed for real-time rendering using alternative implicit models that do not rely on MLPs. Notable examples include sparse voxel grids [9] and multi-resolution hash encoding [21]. Despite these advancements, ray tracing methods are inherently slower than rasterization. To address this, 3DGS [16] introduced a point-based rasterization technique for real-time, high-fidelity view synthesis. Inspired by EWA Volume Splatting [39], 3DGS uses a fully differentiable pipeline, representing 3D scenes with 3D Gaussians and performing volume splatting to known camera poses for rasterization.

Surface reconstruction Surface reconstruction is a critical area in computer vision and graphics, aiming to recreate 3D shapes and structures from 2D images or other data forms. Among recent innovations, NeuS [30] leverages volume rendering and signed distance functions (SDF) for highfidelity reconstructions. NeuS2 [31] significantly improves training speed and extends modeling capacity to dynamic scenes. UNISURF [22] integrates implicit surface models and radiance fields for both surface and volume rendering. VolSDF [33] models volume density as a function of geometry, achieving high-quality geometry reconstructions. Neuralangelo [18] uses multi-resolution hash grids and neural surface rendering to recover detailed structures. BakedSDF [34] introduces a hybrid neural volume-surface representation optimized for mesh extraction.

3D Gaussian Splatting Recent advancements in 3DGS have further propelled surface reconstruction. NeuSG [6] refines surface details using 3DGS and neural implicit models. SuGaR [10] focuses on mesh extraction with SDF-based regularization and Poisson reconstruction. 2DGS [13] collapses 3D volumes into 2D Gaussian disks for view-consistent geometry and detailed mesh reconstruction. GaussianShader [15] enhances rendering quality in reflective surfaces using a shading function on 3D Gaussians. GOF [36] utilizes ray-Gaussian intersection for density estimation and geometric regularization. GIR [28] employs 3D Gaussians for inverse rendering, enabling accurate estimation of material properties, illumination, and geometry. These advancements showcase the potential of 3DGS for high-speed, detailed, and versatile surface reconstructions.

3 Preliminaries

3.1 3D Gaussian splatting

3DGS [16] represents a scene with a set of learnable 3D Gaussian primitives {G k | k = 1 , · · · , K } , where each 3D Gaussian G k consists of mean µ k ∈ R 3 × 1 , covariance Σ k ∈ R 3 × 3 , point opacity α k ∈ [0 , 1] and view-dependent color c k in spherical harmonics. Covariance matrix Σ k = R k S k S ⊤ k R ⊤ k is positive semi-definite, where S k = diag( s k ) is a scaling matrix, s k = ( s k 1 ; s k 2 ; s k 3 ) ∈ R 3 × 1 is a scale parameter, and R k ∈ R 3 × 3 is a rotation matrix parameterized by a quaternion. A 3D Gaussian primitive can be represented in 3D space as:

The primitives are then rasterized via differentiable volume splatting. Specifically, a 3D Gaussian is projected to 2D screen space as Σ ′ k = JW Σ k W ⊤ J ⊤ , where W is a world-to-camera transform and J is the Jacobian of the affine approximation of the projection matrix [39]. The covariance and mean

of the projected Gaussian G 2 k D ( x ) are then obtained by removing the third column and row of Σ ′ k and simply projecting µ k to screen space, respectively. Finally, the Gaussians are alpha-blended in the order of depth as:

where u is a screen space coordinate. The rendered images are supervised with photometric loss L for 3D primitive optimization similar to NeRF [20].

As Gaussians are initialized by sparse SfM points, Adaptive Density Control (ADC) is designed for densification during optimization. Specifically, ADC subsamples and splits Gaussians that satisfy the condition:

where P and p i denote a set of pixel indices and the i -th pixel, respectively, and τ is a predefined threshold. The intuition behind Eq. 3 is that regions not yet well reconstructed exhibit large viewspace positional gradients. This occurs because the optimization process attempts to move the Gaussians to correct these areas, so densifying such Gaussians can effectively increase expressibility.

3.2 Effective rank

Consider a real-valued non-all-zero M × N matrix A . The singular value decomposition (SVD) of A can be expressed as A = UDV , where U and V are unitary matrices of sizes M × M and N × N respectively, and D is a diagonal matrix of size M × N containing the real positive singular values in descending order:

where L = min { M,N } . The singular value distribution is then defined as

where σ = ( σ 1 , σ 2 , · · · , σ L ) T , and ∥ · ∥ 1 denotes ℓ 1 -norm.

Definition 1 (Effective rank) . The effective rank [25] of the matrix A is concisely defined as erank ( A ) = exp { H ( q 1 , q 2 , · · · , q L ) } , where H ( q 1 , q 2 , · · · , q L ) is the Shannon entropy given by

4 Method

In Section 4.1, we introduce the effective rank analysis to inspect the geometries of Gaussians of 3DGS and its variants, shedding light on their underlying structures. Based on the findings from our effective rank analysis, we propose a novel effective rank regularization method in Section 4.2.

4.1 Effective rank analysis of 3D Gaussians

We propose to analyze the effective rank to investigate the structural dynamics of individual 3D Gaussians by calculating the effective rank of the covariance matrix of the Gaussians. The covariance matrix of the 3D Guassians is defined as Σ k = R k S k S T k R T k , and the diagonal matrix after SVD is D = S k S T k , with real positive singular values in a descending order as follows:

where we omit subscript k of s k for brevity.

Accordingly, we can derive the effective rank of a 3D Gaussian G k with the covariance matrix Σ k . The entropy term is H ( G k ) := H ( q 1 , q 2 , q 3 ) := -∑ 3 i =1 q i log q i , with

and the effective rank of a 3D Gaussian G k with the covariance matrix Σ k is defined as follows:

The effective rank, being a differentiable extension of an integer rank, is a suitable tool for geometric analysis of 3D Gaussians since it jointly considers all of the scale parameters and can identify the relative scales of the three axes. The advantage of effective rank becomes more apparent when compared to recent works that only analyze individual or pair-wise variances of the 3D Gaussians [15]. Such approaches do not fully represent the geometry of Gaussians, potentially leading to planar and needle-like Gaussians being categorized together. For better understanding, we visualize the effective ranks of a sphere and 2D disks in Fig. 3.

With the distinct advantage of our approach, we can differentiate between needle-like Gaussians, which have effective ranks close to 1, and planar disk-like Gaussians. To reconstruct a scene with an accurate surface, we need Gaussians that represent a plane that aligns and concentrates well with the surface [30]. Ideally, 3D Gaussians with erank ( G k ) ≈ 2 are preferred, but Gaussians with an effective rank smaller than 2 are also required for representing thin and elongated objects and patterns. However, the needle-like Gaussians with erank ( G k ) ≈ 1 are undesirable because they account for a negligible region of the surface and produce degenerate results in novel views.

The first row of Fig. 2 (green graph) shows the effective rank histogram for 3DGS during training. As the model converges, the number of 3D Gaussians with erank ( G k ) ≈ 1 increases, indicating overfitting without improvements in PSNR and Chamfer distance metrics (metrics are provided in the Appendix A.5, Table 5). This indicates that the majority of "flat" Gaussians (singular values close to 0 ) are actually needle-like (erank ( G k ) ≈ 1 ), rather than disk-like (erank ( G k ) ≈ 2 ). It is also interesting to note that 3DGS naturally forms a small mode at erank ( G k ) = 2 , indicating an observed preference that can be further strengthened with our regularization.

Despite having different geometric constraints on the Gaussians, SuGaR [10] (the second row in Fig. 2) and 2DGS [13] (the third row in Fig. 2) also exhibit a similar tendency to have a large amount of needle-like Gaussians with a single dominant variance along an axis. Notice that all Gaussians in 2DGS start with an effective rank of exactly 2 , but the majority still fail to remain disk-shaped and instead become needle-like 2D Gaussians.

4.2 Optimization

The real-valued and differentiable nature of the effective rank allows us to utilize it as a regularization objective to impose structural constraints on 3D Gaussians. Specifically, our goal is to keep the effective rank of 3D Gaussians below 2, thereby promoting planar shapes, while penalizing Gaussians with an effective rank close to 1 to minimize needle-like artifacts. Although disk-like Gaussians with erank ( G k ) ≈ 2 are preferred, shapes with erank ( G k ) < 2 are also essential for representing complex geometries. We propose an effective rank regularization term that increases exponentially as the effective rank nears 1, strongly penalizing such Gaussians:

where ϵ = 1 × 10 -5 ensures numerical stability, and s 3 is the smallest scale parameter of G k . The regularization effectively constrains the effective rank of Gaussian primitives when added to the baselines, as shown in the purple graphs of Fig. 2. Also, the regularization is scheduled to be applied from 7000-iteration, adhering to the coarse-to-fine training paradigm, which enables stable training upon early iterations with erank ( G k ) > 2 Gaussians.

ADC algorithm We adopt the revised version of the densification algorithm presented in [4, 36], which densifies Gaussians based on the summation of norms instead of the norm of the summation in

Eq. 3 (further details in Appendix A.4). This change is particularly important for our regularization method. Disk-like Gaussians, unlike needle-like ones, often fail to satisfy the splitting criterion set by Eq. 3 because their axes with larger variances produce smaller gradient signals per pixel. Moreover, since disk-like Gaussians typically cover more pixel space, unaligned signals tend to cancel each other out. In contrast, the revised densification algorithm facilitates the splitting of disk-like Gaussians. Notably, due to their superior ability to reconstruct surfaces compared to needle-like Gaussians, our method requires approximately 10% fewer Gaussians than the baseline [16].

5 Experiments

We evaluate the effective rank regularization, comparing its performance as an add-on to baseline models. Additionally, we analyze the contributions of different components of the method.

5.1 Implementation

The regularization hyperparameter λ erank = 0 . 01 is used for all training. For other components belonging to the baselines, we use the same settings as described in the corresponding papers. All experiments are conducted on a Tesla V100 GPU. For mesh extraction, the truncated signed distance function (TSDF) fusion with Open3D [38] is used, with details in the Appendix A.3.

5.2 Comparison

Dataset We evaluate our model on the DTU [14] and Mip-NeRF360 [2] datasets. The DTU dataset consists of 15 forward-facing bounded scenes with a resolution of 1600 × 1200 . Following prior standards [13, 36], we downsample the images to a resolution of 800 × 600 . The DTU dataset is used for evaluating both geometry reconstruction (using Chamfer distance) and novel view synthesis. The Mip-NeRF360 dataset comprises 9 indoor and outdoor scenes with images at a resolution of 1600 × 1050 and is used exclusively for novel view synthesis evaluation. For novel view synthesis, the images are split into training and test sets, while the entire set of images is used for geometry reconstruction. COLMAP [26, 27] is used to initialize point clouds for the baselines.

Baselines Our method is applicable to other baselines as an add-on term. Therefore, we compare baselines with and without our regularization. We choose SuGaR, 2DGS, and GOF as our baselines, works that focus on better geometry reconstruction, along with the original 3DGS. All of the experiments are performed with the proposed setting of the original paper.

Geometry reconstruction Table 1 presents the quantitative results of geometry reconstruction on the DTU dataset. We report the Chamfer distance for each scene, along with the mean Chamfer

SuGaR

2DGS

GOF

GOF+erank reg.

G.T.

3DGS needles

GOF

GOF + erank reg.

distance and mean PSNR. The '+e' symbol indicates the addition of the effective rank regularization (with fixed densification) to the baseline methods.

The results show that methods enhanced with our add-on term outperform the baselines. Notably, applying our regularization to 3DGS (3DGS+e) results in a significant improvement in geometry reconstruction, demonstrating the effectiveness of the regularization. This supports our hypothesis that reducing needle-like Gaussians and achieving flatness as in Fig. 2 improves performance. Additionally, the figure shows that SuGaR contains both needle-like and non-planar Gaussians with effective ranks greater than 2. By attaining flatness and removing spikes through effective rank regularization, we achieve a substantial performance gain for SuGaR (SuGaR+e).

GOF and 2DGS already incorporate well-designed regularization terms, such as depth distortion loss [13, 2], to align Gaussians with surfaces and enhance geometry reconstruction. Furthermore, 2DGS explicitly uses 2D Gaussians as their primitive, inherently achieving planarity. Nonetheless, our method prevents Gaussians from converging into needles in both approaches (and enforces flatness in GOF), resulting in performance gains.

Fig. 4 shows mesh reconstruction results, where baseline methods often exhibit empty holes in the reconstructed meshes. Our regularization term enforces disk-like Gaussians, reducing such holes and proving advantageous for surface reconstruction.

Fig. 5 and the first row of Fig. 1 display normal reconstruction results. In Fig. 5, the resulting image from GOF shows spiky artifacts and a hollow surface on the pear. Similarly to the mesh results, needle-like Gaussians often fail to cover the entire area, leaving empty holes or transparent regions, resulting in hollow or incomplete reconstructions. The effective rank regularization mitigates these noisy artifacts, leading to a more accurate reconstruction of the underlying geometry.

Novel view synthesis Since 3D reconstruction from 2D images is an ill-posed problem, Gaussians tend to overfit to the training views, converging into needle-like shapes and causing spiky artifacts in test views, as shown in Fig. 1 (b), (c), and Fig.6. For better understanding, we visualize Gaussians with erank ( G k ) < 1 . 02 (scale ratio of approximately 20:1 or larger) in red. Our method mitigates overfitting and the resulting artifacts by enforcing structural priors on the Gaussians.

3DGS needles

3DGS

3DGS + erank reg.

3DGS ( PSNR: 25.1 / 1.4 GB)

3DGS + erank reg. (PSNR: 25.5 / 1.2 GB)

Furthermore, as seen in Fig. 7, our method adaptively preserves some elongated Gaussians when necessary, allowing the representation of slender structures. The results indicate that while 3DGS heavily relies on needle-like Gaussians to represent the scene, our method limits their use to only when required, leading to improved novel view synthesis performance.

We also provide quantitative results in Table 1, where we report the average PSNR for the DTU dataset. Results for Mip-NeRF360 are reported in Table 3 in the Appendix A.5. While many geometry regularization techniques degrade visual quality, our method does not exhibit this trade-off and actually shows slight improvements by properly constraining the shape of the Gaussians.

Efficiency As shown in Fig. 7 and Table 4, the efficacy of disk-like Gaussians in 3D reconstruction, compared to needle-like Gaussians, leads to a better memory footprint. The average storage usage for the DTU and Mip-NeRF360 datasets is reported in Table 4 (Appendix A.5).

5.3 Ablations

Our method comprises two key components: (a) the fixed densification (ADC) algorithm and (b) the effective rank regularization. We performed an ablation study on these components to observe their performance gains compared to the naive 3DGS method. Table 2 shows the Chamfer distance and PSNR measured on the DTU dataset. The results indicate that both components contribute to performance gains in geometry reconstruction and novel view synthesis tasks. Additionally, incorporating techniques such as depth distortion loss [13, 36] can further enhance the best-performing model (row +a+b+c). These techniques are discussed in the Appendix A.2.

6 Conclusion

Limitations Our regularization term constrains individual Gaussians but does not account for the local and global structure of the scene. Thus, it may be beneficial to pair our method with structure-aware regularizations, such as the depth distortion loss [13], which considers the Gaussians

along the ray collectively. Another limitation is the manual selection of the hyperparameter λ erank . While our chosen hyperparameter works well for the scenes used in our evaluation, it may not be optimal for extreme scenes dominated by thin objects and structures.

Acknowledgments and Disclosure of Funding

Junha Hyung and Susung Hong conducted this work during the internship at NAVER AI Lab. The NAVER Smart Machine Learning (NSML) platform [17] had been used for experiments. This work was supported by KAIST-NAVER hypercreative AI center. This work was partly supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIT) (No.RS-2019-II190075 Artificial Intelligence Graduate School Program (KAIST)). This work was also partly supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2022R1A2B5B02001913).

References

J. T. Barron, B. Mildenhall, M. Tancik, P. Hedman, R. Martin-Brualla, and P. P. Srinivasan. Mip-nerf: A multiscale representation for anti-aliasing neural radiance fields. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 5855-5864, 2021.

J. T. Barron, B. Mildenhall, D. Verbin, P. P. Srinivasan, and P. Hedman. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 5470-5479, 2022.

J. T. Barron, B. Mildenhall, D. Verbin, P. P. Srinivasan, and P. Hedman. Zip-nerf: Anti-aliased grid-based neural radiance fields. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 19697-19705, 2023.

S. R. Bulò, L. Porzi, and P. Kontschieder. Revising densification in gaussian splatting. arXiv preprint arXiv:2404.06109 , 2024.

A. Chen, Z. Xu, A. Geiger, J. Yu, and H. Su. Tensorf: Tensorial radiance fields. In European Conference on Computer Vision , pages 333-350. Springer, 2022.

H. Chen, C. Li, and G. H. Lee. Neusg: Neural implicit surface reconstruction with 3d gaussian splatting guidance. arXiv preprint arXiv:2312.00846 , 2023.

Z. Chen, T. Funkhouser, P. Hedman, and A. Tagliasacchi. Mobilenerf: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 16569-16578, 2023.

C. Dudai, M. Alper, H. Bezalel, R. Hanocka, I. Lang, and H. Averbuch-Elor. Halo-nerf: Learning geometryguided semantics for exploring unconstrained photo collections. In Computer Graphics Forum , page e15006. Wiley Online Library, 2024.

S. Fridovich-Keil, A. Yu, M. Tancik, Q. Chen, B. Recht, and A. Kanazawa. Plenoxels: Radiance fields without neural networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 5501-5510, 2022.

A. Guédon and V. Lepetit. Sugar: Surface-aligned gaussian splatting for efficient 3d mesh reconstruction and high-quality mesh rendering. arXiv preprint arXiv:2311.12775 , 2023.

P. Hedman, J. Philip, T. Price, J.-M. Frahm, G. Drettakis, and G. Brostow. Deep blending for free-viewpoint image-based rendering. ACM Transactions on Graphics (ToG) , 37(6):1-15, 2018.

P. Hedman, P. P. Srinivasan, B. Mildenhall, J. T. Barron, and P. Debevec. Baking neural radiance fields for real-time view synthesis. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 5875-5884, 2021.

B. Huang, Z. Yu, A. Chen, A. Geiger, and S. Gao. 2d gaussian splatting for geometrically accurate radiance fields. arXiv preprint arXiv:2403.17888 , 2024.

R. Jensen, A. Dahl, G. Vogiatzis, E. Tola, and H. Aanæs. Large scale multi-view stereopsis evaluation. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 406-413, 2014.

Y. Jiang, J. Tu, Y . Liu, X. Gao, X. Long, W. Wang, and Y. Ma. Gaussianshader: 3d gaussian splatting with shading functions for reflective surfaces. arXiv preprint arXiv:2311.17977 , 2023.

B. Kerbl, G. Kopanas, T. Leimkühler, and G. Drettakis. 3d gaussian splatting for real-time radiance field rendering. ACM Transactions on Graphics , 42(4):1-14, 2023.

H. Kim, M. Kim, D. Seo, J. Kim, H. Park, S. Park, H. Jo, K. Kim, Y. Yang, Y. Kim, et al. Nsml: Meet the mlaas platform with a real-world case study. arXiv preprint arXiv:1810.09957 , 2018.

Z. Li, T. Müller, A. Evans, R. H. Taylor, M. Unberath, M.-Y. Liu, and C.-H. Lin. Neuralangelo: Highfidelity neural surface reconstruction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 8456-8465, 2023.

R. Martin-Brualla, N. Radwan, M. S. Sajjadi, J. T. Barron, A. Dosovitskiy, and D. Duckworth. Nerf in the wild: Neural radiance fields for unconstrained photo collections. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 7210-7219, 2021.

B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, and R. Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM , 65(1):99-106, 2021.

T. Müller, A. Evans, C. Schied, and A. Keller. Instant neural graphics primitives with a multiresolution hash encoding. ACM transactions on graphics (TOG) , 41(4):1-15, 2022.

M. Oechsle, S. Peng, and A. Geiger. Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 5589-5599, 2021.

C. Reiser, S. Garbin, P. P. Srinivasan, D. Verbin, R. Szeliski, B. Mildenhall, J. T. Barron, P. Hedman, and A. Geiger. Binary opacity grids: Capturing fine geometric detail for mesh-based view synthesis. arXiv preprint arXiv:2402.12377 , 2024.

C. Reiser, R. Szeliski, D. Verbin, P. Srinivasan, B. Mildenhall, A. Geiger, J. Barron, and P. Hedman. Merf: Memory-efficient radiance fields for real-time view synthesis in unbounded scenes. ACM Transactions on Graphics (TOG) , 42(4):1-12, 2023.

O. Roy and M. Vetterli. The effective rank: A measure of effective dimensionality. In 2007 15th European signal processing conference , pages 606-610. IEEE, 2007.

J. L. Schönberger and J.-M. Frahm. Structure-from-motion revisited. In Conference on Computer Vision and Pattern Recognition (CVPR) , 2016.

J. L. Schönberger, E. Zheng, M. Pollefeys, and J.-M. Frahm. Pixelwise view selection for unstructured multi-view stereo. In European Conference on Computer Vision (ECCV) , 2016.

Y. Shi, Y. Wu, C. Wu, X. Liu, C. Zhao, H. Feng, J. Liu, L. Zhang, J. Zhang, B. Zhou, et al. Gir: 3d gaussian inverse rendering for relightable scene factorization. arXiv preprint arXiv:2312.05133 , 2023.

J. Sun, X. Chen, Q. Wang, Z. Li, H. Averbuch-Elor, X. Zhou, and N. Snavely. Neural 3d reconstruction in the wild. In ACM SIGGRAPH 2022 conference proceedings , pages 1-9, 2022.

P. Wang, L. Liu, Y. Liu, C. Theobalt, T. Komura, and W. Wang. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. arXiv preprint arXiv:2106.10689 , 2021.

Y. Wang, Q. Han, M. Habermann, K. Daniilidis, C. Theobalt, and L. Liu. Neus2: Fast learning of neural implicit surfaces for multi-view reconstruction. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 3295-3306, 2023.

Z. Wang, T. Shen, M. Nimier-David, N. Sharp, J. Gao, A. Keller, S. Fidler, T. Müller, and Z. Gojcic. Adaptive shells for efficient neural radiance field rendering. arXiv preprint arXiv:2311.10091 , 2023.

L. Yariv, J. Gu, Y. Kasten, and Y. Lipman. Volume rendering of neural implicit surfaces. Advances in Neural Information Processing Systems , 34:4805-4815, 2021.

L. Yariv, P. Hedman, C. Reiser, D. Verbin, P. P. Srinivasan, R. Szeliski, J. T. Barron, and B. Mildenhall. Bakedsdf: Meshing neural sdfs for real-time view synthesis. In ACM SIGGRAPH 2023 Conference Proceedings , pages 1-9, 2023.

Z. Yu, A. Chen, B. Huang, T. Sattler, and A. Geiger. Mip-splatting: Alias-free 3d gaussian splatting. arXiv preprint arXiv:2311.16493 , 2023.

Z. Yu, T. Sattler, and A. Geiger. Gaussian opacity fields: Efficient and compact surface reconstruction in unbounded scenes. arXiv preprint arXiv:2404.10772 , 2024.

K. Zhang, G. Riegler, N. Snavely, and V. Koltun. Nerf++: Analyzing and improving neural radiance fields. arXiv preprint arXiv:2010.07492 , 2020.

Q.-Y. Zhou, J. Park, and V. Koltun. Open3d: A modern library for 3d data processing. arXiv preprint arXiv:1801.09847 , 2018.

M. Zwicker, H. Pfister, J. Van Baar, and M. Gross. Ewa volume splatting. In Proceedings Visualization, 2001. VIS'01. , pages 29-538. IEEE, 2001.

A Appendix / supplemental material

A.1 Broader impact

The broader impact of our work on 3D reconstruction lies in its potential to advance various fields such as virtual and augmented reality, medical imaging, and digital content creation by enabling more efficient and high-quality 3D model generation. However, as with any advanced technology, it also presents potential risks and avenues for misuse. For instance, enhanced 3D reconstruction techniques could be exploited to create deepfakes or unauthorized reproductions of proprietary designs, posing ethical and legal challenges. To mitigate these risks, we propose implementing strict usage guidelines to ensure the integrity and rightful use of 3D models. We aim to maximize the positive impact of our research while minimizing potential negative consequences.

A.2 Additional regularization

For rendering normals, we add other regularization terms, such as depth distortion loss [13] and normal regularization, as proposed in [13, 36]. (We do not utilize these regularization terms for evaluating effective rank regularization as an add-on module in Tab. 1.) The depth distortion loss, which concentrates splats on a surface and mitigates floater artifacts, is given as

where ω i = α i G i ( x ) ∏ i -1 k =1 (1 -α k G k ( x ))) for the blending weight and z i is the depth of the intersection point of the i -th Gaussian, and i, j are indexes over Gaussians contributing to a certain ray.

The normal regularization minimizes the difference between the rendered normal map ¯ n of the splats and the gradient normals ˆ n derived from the rendered depth map,

which locally aligns the 3D Gaussians with the actual surfaces. Since the effective rank regularization does not account for the local and global structure of the scene, it is beneficial to pair our method with these structure-aware regularizations.

A.3 Mesh extraction

We utilize the Truncated Signed Distance Function (TSDF) fusion for mesh extraction. The algorithm encodes the distance of any point in the voxel grid to the nearest surface, with the distance being truncated to a maximum value to limit the influence of faraway points. The sign of the distance function indicates whether the point is inside (negative) or outside (positive) the object. Multiple TSDFs are combined from different viewpoints to create a more accurate and complete 3D reconstruction, forming a coherent and comprehensive 3D model. The Marching Cubes algorithm is then used for triangulation.

A.4 ADC fix

We adopt the revised version of the densification algorithm presented in [4, 36], which densifies Gaussians based on the summation of the norm instead of the norm of the summation in Eq. 3:

As discussed in the main paper, this approach is crucial to our regularization because disk-like Gaussians typically cover more screen space and receive gradient signals from various pixels, which can cancel out when summed. The revised algorithm ensures the effective splitting of Gaussians with our regularization. However, due to the efficiency of disk-like Gaussians in surface reconstruction, our method requires about 10% fewer Gaussians compared to the baseline [16].

Table 4: Storage usage of our method, along with Chamfer distance, PSNR, and optimization time.

A.5 Additional quantitative results

We report novel view synthesis results on the Mip-NeRF360 dataset in Table 3. The results show that our add-on regularization term improves the visual quality of 3DGS in terms of PSNR, SSIM, and LPIPS. Also, the method even shows comparable or slightly better performance compared to the NeRF variants with slow and computationally intensive rendering.

We report the training time of our method in Table 4. The training time for 3DGS on the DTU [14] dataset averages 11.2 minutes per scene. Adding the effective rank regularization with the densification fix incurs no overhead since the additional computation is compensated with a reduced number of Gaussians. Total training time is on average 11.1 minutes for the DTU dataset and 40 minutes for the Mip-NeRF360 dataset, on a single V100 GPU, reported in Table 4.

Also, with a reduced number of Gaussians, our method requires less memory and storage for scene representation, as in Table 4. While being more compact, our method outperforms baselines in terms of Chamfer distance and PSNR.

Table 5 demonstrates Chamfer distance and PSNR changes during the course of training, for the baselines shown in Fig. 2. Results are reported for the scene 37 of the DTU dataset. Needle-like Gaussians increase, but the performance plateaus, indicating overfitting. Additionally, different Gaussian structures with similar metrics suggest the heterogeneous nature of Gaussians in 3DGS and its variants. Also, the reported 'Number of needles' corresponds to Gaussians with an effective rank smaller than 1.04. The results suggest that our regularization term effectively minimizes the number of needles without a visual quality trade-off.

We present per scene PSNR on the DTU dataset in Table 6. The mean PSNR is already shown in Table 1 and Table 2 of the main paper.

A.6 Cause of needle-like Gaussians

While not directly related to our methodology, we investigate some reasons for the convergence of 3D Gaussians into anisotropic Gaussians with one dominant variance.

First, the scale of the 3D Gaussians is not properly constrained due to the dilation operation, which adds a small constant to screen space Gaussians [16] to ensure a minimum scale, as noted in Mip-

Splatting [35]. Combined with the inherent implicit shrinkage bias of 3DGS [16, 35], this results in the underestimation of the scale parameters during the optimization process.

Second, the densification along the longer axis does not occur effectively since the longer axes, or the axes with large variance, have smaller gradients. When Gaussians move in the direction of the shorter axis, pixel values change abruptly. In contrast, there are only small changes in pixel values when moving along the longer axis. Specifically, when ∂ p i ∂ u aligns with the direction of the longest axis, the gradient values are typically small. Consequently, the norm of the final gradient often falls below the densification threshold ∥ ∂L ∂ x ∥ 2 < τ x , preventing effective densification. We visualize ∂ G k ∂ x in arrows in Fig. 8 (a), which is proportional to ∂ p i ∂ u , for better understanding. Therefore, the splats are biased towards adjusting their scale parameters (Fig. 8 (b)) rather than splitting along the longer axis, converging into needle-like Gaussians.

Third, scale parameters are kept the same after splitting, so needles are not shortened after densification.

It will be interesting future work to delve deeper into these reasons and address the problem with other approaches.

A.7 Additional qualitative results

We present a normal rendering of our method results. Fig. 9 are results of the scene 122, with the depth distortion and normal regularization loss used together. Fig. 10 shows the results of scene 55. Fig. 11 shows the rendering results of the Mip-NeRF360 dataset of our method. We visualize the Gaussians with an effective rank smaller than 1.02 in red. The effective rank regularization is adaptive to the scene, reducing the number of needle-like Gaussians, while effectively representing the required regions.

(a)

(b)

(x)

(o)

Gradientofa2DGaussian

1.05

0.90

0.75

0.60

0.45

0.30

0.15

0.00

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Abstract and introduction has main claims and overall explanation of the paper included.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We include limitations and discussion.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [NA]

Justification:

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We fully disclose information including hyperparameters and the code base.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We release the code and the data is publicly available.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Yes, we provide all the details necessary.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We provide mean and standard deviation of the results.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We specify the GPU and amount of time it takes for training, along with memory footage.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: Yes our paper conform to NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We discuss the societal impacts in the Appendix.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [Yes]

Justification: We discuss such information in the Appendix.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We are the original owners of code and the models.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: We provide code and the model.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification:

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification:

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a new regularization for Gaussian splatting (GS) method that increases the Shannon entropy of the scale parameters (it was named erank as while rendering the scale is presented as a diagonal matrix, and the aforementioned calculation turns into the effective rank), to better reconstruct 3d mesh out of the GS point cloud. The paper argues that a majority of the gaussians in previous methods are close to erank of 1, which ruins the 3d reconstruction because of needle-like gaussians that effectively add noise to the reconstructions. Given that regularization as an add-ons to other methods it improves the 3D reconstruction, as shown in the experiments on DTU dataset. Additionally, it shows in the appendix that the image rendering abilities of the GS is not impaired because of the added regularization.\n\nSTRENGTHS:\nThe paper dives into the attributes of the GS and impressively recognizes an issue relating to the GS ability to reconstruct 3d mesh. The authors exhibit an analysis of the amount of 1-ranked gaussians that strengthen their claim, and showed improved results on DTU dataset.\n\nWEAKNESSES:\nThere are several concerns regarding this paper, first it should include more benchmarks for 3d reconstruction, for example Tanks and Temples dataset which was used as benchmark in many other works in the field, including the works that was cited in the paper. Secondly, there are two works that the authors did not mention [1,2] and should be compared to in the DTU experiment. The first [1] has the same idea as 2DGS and was already published. It might be that their regularization solves part of the issue of the GS that this paper presents. The second [2], does not use the gaussian splatting directly for reconstruction and instead renders stereo images for depth estimate. Thus, it does not share the reconstruction error because of the needle-like gaussians. Comparing both of them will strengthen this paper. Lastly, there is no adequate justification for the erank regularization in comparison to other possibilities, e.g. linear, parabolic, or exponential loss on the size of the smallest magnitude scale. Finally, an ablation study showing why erank-regularization is superior compared to other losses is missing. \n[1] Dai, P., et al. \"High-quality surface reconstruction using gaussian surfels.\" arXiv preprint arXiv:2404.17774 (2024).\n[2] Wolf, Y., et al.. \"Surface Reconstruction from Gaussian Splatting via Novel Stereo Views.\" arXiv preprint arXiv:2404.01810 (2024).\n\nQUESTIONS:\nsee weaknesses\n\nLIMITATIONS:\nsee weaknesses", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n3D Gaussian Splatting is a remarkable technique in novel view synthesis. However, it usually degenerates into noodle-like shapes which sometimes bring visual artifacts and inaccurate geometry. This paper analyzes the phenomenon and proposes an effective rank loss to regularize the Gaussians. The technique seems to be a plug-and-play module for 3DGS variants. Extensive experiments suggest the effectiveness.\n\nSTRENGTHS:\n1. The observation and analysis are strong and the solution of using effective rank is simple yet effective.\n2. The results show clear improvements over representative works including 3DGS, SuGaR, and 2DGS, which cover a broad range of scene representations.\n3. The paper is well-written and easy to follow. I have no issue with the written. Most of the related works are adequately discussed. \n4. The technical details are sufficient and I believe an expert in 3DGS can easily implement it.\n\nWEAKNESSES:\n1. There is no supplemental video, which is essential for nvs tasks. \n2. The improvement of e-rank for Novel view synthesis is limited. It would be beneficial to highlight how this could improve novel view synthesis.\n3. Although the paper analyzes that the noodle-like artifacts are due to the training bias, they can also be created to represent high-frequency details. The paper should highlight the PSNR in Figure.1 to give readers a sense that such regularization indeed improves novel view synthesis.\n4. It would be beneficial to add the number of Gaussians since enforcing the Gaussian to 2D from 1D should reduce the number of Gaussians in representing a scene.\n\nQUESTIONS:\n1. How the normal is constructed for 3D Gaussian and how it is visualized?\n2. What is the effect of densification on geometry reconstruction in 3DGS? In Table 1. Does 3DGS+e indicate with ERANK only or plus densification? It should be clearly ablated where the performance comes from.\n\nLIMITATIONS:\nOverall, the paper did a good job with minor issues required to clarify. The limitation of tuning the hyperparameter is also discussed and the potential negative effect is reasonable. \n\n**minors:**\n\nThe normal visualization seems not to be normalized witn [0,1]", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper performs a statistical analysis on 3DGS for its effective rank distribution of the learned Gaussians. It claims that most of the Gaussian learned are close to rank 1 effectively, giving needle-like artifacts in novel view synthesis and reconstruction. Hence, this paper proposes a regularization loss to discourage low range Gaussians from forming, showing improved reconstruction results.\n\nSTRENGTHS:\n**Motivation**\n* This paper starts with a clear and strong motivation, where the Gaussian learning with 3DGS creates needle-like artifacts. This reduces both the reconstruction and novel view synthesis quality.\n\n**Method**\n* This paper proposes an intuitive and effective method of regularizing the effective rank of the Gaussians, to regularize on the shape of individual Gaussian. The experiment results also demonstrate the effectiveness of the proposed method on various baselines, with considerable improvement in reconstruction results and minor improvement in novel view synthesis results.\n\nWEAKNESSES:\nI do not doubt the motivation and effectiveness of the proposed method. In fact, needle-like artifacts have been a well-known issue for 3DGS in the community. However, I feel reluctant to accept this paper due to the following reasons:\n1. Although the phenomenon is well known by many, the root cause of this issue is not clear. This paper tries to explain the cause in its appendix, and gives two hypotheses: a) incorrect dilation observed by MipGaussian, b) Gaussians are more likely to densify along their short axis instead of their long axis. Based on this, I would like to point out the following issues:\n- This part should be put into the main paper instead of the statistical analysis. The causes of the low-rank Gaussians would be much more valuable than reiterating the phenomenon that many people already know.\n- This cause analysis needs to be more rigorous. The version provided in the appendix is more of a hypothesis than an analysis. \n- If it is because of the dilation error, is it already fixed by MipGaussian? What is the rank distribution like for MipGaussian? Can the proposed method improve the MipGaussian performance as well? If yes, then something else is still causing the issue. If not, why shouldn't MipGaussian be used instead of the proposed method?\n- If it is because of the densification bias, does this problem still occur without densification at all? In fact, I have personally observed many cases of needle-like artifacts that are so long that extend across the entire image. In this case, the rendering loss should be enough to prevent this from happening without anything to do with the densification. But it is not happening, why? Is there something wrong with the rendering loss supervision we have with 3DGS? I am not saying that the densification bias is not a cause, but it requires more evidence and experiment results to be proved.\n\n2. Continuing from the previous point, the method proposed does not address the cause hypothesized. It is not fair to say that regularization methods are just bad, but there have been even simpler regularization methods used by other papers (e.g. PhyGaussian). The most common method is to regularize the ratio between the long axis and the short axis of the 3DGS. It takes only a few lines of code and works reasonably well. This paper does not mention this simpler regularization and does not compare with it as a baseline. However, I think it is very important to justify the more complicated method proposed in this paper, with either rigorous analysis as well as empirical results (preferably both). \n\nIn summary, I believe that this paper lacks a rigorous analysis of the cause of the low-rank Gaussian formation, and the proposed method lacks novelty and performance comparison against the simple naive regularization used in other papers.\n\nQUESTIONS:\nTo summarize my opinions mentioned in the weakness section into questions, it would be:\n1. what is really the cause for the low-rank Gaussian formation and how does the proposed method address it?\n2. why is the proposed method better than the simple regularization on the ratio of the long/short gaussian axis?\n\nLIMITATIONS:\nThe paper does not have potential negative societal impact and there is no significant limitation of the method that needs to be addressed other than the ones mentioned in the previous sections.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper identifies a common problem in 3D Gaussian splatting where Gaussians converge to needle-like shapes, with its variance mostly contained in one axis. These needle-shaped Gaussians can cause artifacts and inaccurate surface reconstruction. \nTo quantify this phenomenon, the paper uses the concept of effective rank, a continuous generalization of the rank of a matrix.  During scene reconstruction, while 3D gaussians may begin with an effective rank of 3, they tend to converge to an effective rank of 1 after several thousand iterations of training.  Based on effective rank, the paper proposes a regularization method to reduce needle-shaped Gaussians. On the DTU dataset, the authors show that 3DGS with rank regularization causes better chamfer distance and PSNR metrics.\n\nSTRENGTHS:\nThe paper introduces a method to prevent needle-like Gaussians in 3DGS, an important challenge in view synthesis, and presents a clear way to quantify the problem. The authors propose a new regularization term that is simple and can be used alongside other Gaussian splat variants such as 2DGS and SuGaR. On the DTU dataset, the authors report better chamfer distance and PSNR metrics than prior work.\n\nOverall I found the paper very interesting and its claims well justified. Because of the widespread popularity of Gaussian Splatting, the method can be useful to both researchers and 3D designers.\n\nWEAKNESSES:\nWhile the paper introduces a new way to quantify needle-like Gaussians, this is not the first paper to propose a regularization term for them.  PhysGaussian [A] introduces an anisotropic regularizer to prevent skinny Gaussians. This regularization term has already been implemented in Nerfstudio’s Splatfacto method [B]. This seems like the most relevant and important comparison to the proposed method, and is completely missing from the paper. This missing comparison is the main reasoning behind my score. \n\nIn addition, I wish that the paper did a deeper investigation into what causes spikey Gaussians. I found the discussion in Appendix A.4. very fascinating, and think it could be very valuable to the community if elaborated on. Maybe a simple experiment validating the hypothesis could strengthen the paper. \n\nMinor Comments: \nI found that Fig. 1 was hard to parse. I think it would help to include labels for the figure’s images. \n\n[A] https://arxiv.org/pdf/2311.12198\n[B] https://docs.nerf.studio/nerfology/methods/splat.html\n\nQUESTIONS:\nWhat happens if you do not regularize erank(G) < 2? Do Gaussians with erank(G) > 2 hurt reconstruction quality? \n\nWhat are x and o in Fig. 8?\n\nLIMITATIONS:\nYes the authors have discussed the limitations and potential societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}