R$^2$-Gaussian: Rectifying Radiative Gaussian Splatting for Tomographic Reconstruction

3D Gaussian splatting (3DGS) has shown promising results in image rendering and surface reconstruction. However, its potential in volumetric reconstruction tasks, such as X-ray computed tomography, remains under-explored. This paper introduces R$^2$-Gaussian, the first 3DGS-based framework for sparse-view tomographic reconstruction. By carefully deriving X-ray rasterization functions, we discover a previously unknown integration bias in the standard 3DGS formulation, which hampers accurate volume retrieval. To address this issue, we propose a novel rectification technique via refactoring the projection from 3D to 2D Gaussians. Our new method presents three key innovations: (1) introducing tailored Gaussian kernels, (2) extending rasterization to X-ray imaging, and (3) developing a CUDA-based differentiable voxelizer. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art approaches in accuracy and efficiency. Crucially, it delivers high-quality results in 4 minutes, which is 12$\times$ faster than NeRF-based methods and on par with traditional algorithms. Code and models are available on the project page https://github.com/Ruyi-Zha/r2_gaussian.

Paper

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Peer review

Reviewer DGKr5/10 · confidence 4/52024-07-08

Summary

The paper introduces R2-Gaussian, a framework for tomographic reconstruction using 3D Gaussian splatting (3DGS). This framework aims to address the limitations of traditional 3DGS in volumetric reconstruction, specifically for tasks like X-ray computed tomography (CT).

Strengths

- Identifying and addressing the integration bias in standard 3DGS formulation for volumetric reconstruction - The proposed R2-Gaussian framework is developed with tailored Gaussian kernels, rectified projection techniques, and a CUDA-based differentiable voxelizer. - The paper provides simulated X-ray validation, comparing the proposed method against state-of-the-art techniques

Weaknesses

- The proposed method shows results in experimental settings. However, its performance in real-world clinical or industrial scenarios is not thoroughly examined. For instance, real X-ray images are given to reconstruct the CT scan. - I am not sure if 75, 50, and 25 views of X-rays are considered sparse-view reconstruction and if it is practical to have 75, 50, or 25 views of X-rays for CT reconstruction. - In the proposed method, the kernel formulation removes view-dependent color. However, it cannot model the scattering effect in X-ray. - In Fig. 7, it is unclear how to get X-3DGS slices and what "queried from three views" means in detail. Why not implement the voxelization on X-3DGS (or vanilla 3DGS) for a fair comparison?

Questions

- After the voxelization from Gaussians, can the final density volume be compatible with traditional CT scans and be viewed in CT software?

Rating

5

Confidence

4

Soundness

2

Presentation

2

Contribution

2

Limitations

Yes.

Reviewer P8A37/10 · confidence 4/52024-07-08

Summary

The paper aims to achieve high tomographic reconstruction performance with a limited number of views in a time-efficient manner. To this end, the paper modifies 3DGS for X-ray projection by adjusting the rendering equation, correcting 2D projection errors, and using voxelizers for regularization. The experiments demonstrate the efficacy of the proposed method.

Strengths

First of all, the paper is well written. The problem in the current tomographic representation is clearly stated, and the authors' objectives are sufficiently addressed. The overall structure is easy to understand, and the ablation study covers most of the arising questions.

Weaknesses

One critical weakness of this paper is the existence of prior work using 3DGS for X-ray projection. Although the paper seems structurally well-written, the paper seems less novel due to the presence of X-Gaussian. X-Gaussian, which has been accepted to ECCV 2024, employs a similar method. More importantly, X-Gaussian achieved a PSNR of 43 in Human Organ reconstructions, whereas this paper (R2 Gaussian) achieved a PSNR of 36, despite slightly different experimental settings. If the authors provide persuasive explanation on the novelty of this paper, I am willing to raise my score. Additionally, it would be helpful to augment the related works and baseline models. For example, C^2RV (CVPR 2024) is another recent tomographic representation model.

Questions

In the SAX-NeRF paper, the performance gap between SAX-NeRF and NAF is significant. However, in the R^2 Gaussian paper, the performance gap between them is not significant. Could you elaborate on why this happens?

Rating

7

Confidence

4

Soundness

4

Presentation

4

Contribution

3

Limitations

The paper clearly states the limitations and the potential societal impact of the work.

Reviewer P8A32024-08-08

I appreciate the authors for their detailed rebuttal. It resolved most of the critical concerns. Therefore, as long as the authors incorporate the explanations from the rebuttal into the final version, I will raise my score accordingly.

Authorsrebuttal2024-08-09

Thank you for your positive feedback and for the effort you put into reviewing our rebuttal. We are glad that our explanations have addressed most of your concerns. We will ensure that these clarifications are fully integrated into the final version of our paper.

Reviewer w7Ci7/10 · confidence 5/52024-07-10

Summary

### Motivation - The authors propose to adapt 3D Gaussian Splatting (3DGS) to sparse-view tomographic reconstruction, i.e., to recover a radiodensity 3D volume from a small set of X-ray images and corresponding sensor information. This is relevant for various clinical and industrial applications. ### Contributions - Their _R2-Gaussian_ model iterates over existing 3DGS solutions tuned for XR/CT imaging, proposing a 3DGS initialization scheme better suited for tomographic reconstruction. - The authors also correct an integration bias in 3DGS (meant for faster image inference but causing ambiguities in volumetric reconstruction) and provide the corresponding CUDA patch. - The proposed _R2-Gaussian_ includes other adaptations (custom densification parameters, voxel-based regularization, simplified isotropic kernels), resulting in an end-to-end CT reconstruction system. ### Results - The authors provide extensive qualitative evaluation and quantitatively compare to other NeRF-based and traditional CT reconstruction methods, showing that their method provides a better trade-off between volume accuracy and reconstruction time. ### Relevance - Effort in applying 3DGS to XR/CT imaging has grown the past year [6, 27, 39], as 3DGS appears to be a well-suited representation for such applications (due to its compactness, fast convergence, etc.). This work is a meaningful iteration and could benefit the community.

Strengths

_(somewhat ordered from most to least important)_ ### S1. Convincing Comparative Evaluation and Qualitative Results - The quantitative comparison to state-of-the-art CT reconstruction methods appear convincing, with the proposed solution demonstrating a better trade-off between volume accuracy and reconstruction time. - The authors provide a lot of meaningful qualitative results, to illustrate theoretical contributions, to highlight the benefits of their method, but also to showcase its limitations. This makes reading this paper the more interesting. - An ablation study w.r.t. some of the key contributions and w.r.t. some hyperparameters is also provided. - Though limited in number (15 volumes), the authors evaluate on different categories (animal, vegatal, and synthetic targets). ### S2. Iterative yet Meaningful Contributions Towards 3DGS for XR/CT - The discussion w.r.t. the integration bias in vanilla 3DGS is interesting, and the technical solution brought by the authors appear valuable to the community. As mentioned in the paper, their corrected CUDA implementation could benefit other 3DGS works targeting volumetric reconstruction. - The authors propose an initialization scheme dedicated to volumetric tomography, as usual 3DGS initialization techniques (e.g., SfM) are not applicable here. This is a relevant contribution, properly described and evaluated (qualitative + quantitative evaluation). - The proposed system, tackling xrays-to-CT reconstruction in an end-to-end differentiable manner, is indeed novel. Existing 3DGS models [6, 27, 39] for XR/CT imaging rather focus on digitally-reconstructed-radiograph (DRR) novel-view synthesis (NVS) rather than CT reconstruction. ### S3. Sound Theory and Reproducibility - Background theory is well described by the authors, and the scientific/technical insight of the authors w.r.t. the identified integration bias could benefit the community. - The authors provide their model implementation, which appears sound and well-structured (note that I did not try to run the code, but had a look at key files). ### S4. Detailed Discussion of Limitations and Contributions - The authors put significant effort in discussing and illustrating some of their method's limitations (needle-like artifacts inherent to 3DGS, varying convergence time, limited extrapolation ability, etc.), as well as summarizing its impact (possible clinical/industrial applications, benefit of their CUDA code to the CV community, etc.) in Appendices G and H. ### S5. Well-Written and Illustrated Paper - Overall, the paper is nicely structured, written, and illustrated. E.g., Figures 2-4 are helpful to understand the methodology at a glance.

Weaknesses

_(somewhat ordered from most to least important)_ ### W1. Lack of Consideration for Real-World Noise and Anisotropic Effects - The authors claim that "X-ray attenuation depends only on isotropic density" [L140] to justify their radiodensity-based model, but this is not entirely correct. While most models generating digitally reconstructed radiographs (DRRs) from CT volumes indeed consider x-ray attenuation as an isotropic phenomenon, this is a approximation. Some x-ray transport effects, such as Compton scattering, are actually anisotropic (but because CT volumes do not inherently contain the material information necessary to accurately simulate Compton scattering, DRR models ignore the anisotropy part) [a]. However, the authors do claim that they trained their model on DRRs generated using TIGRE [5] configured to simulate Compton scattering [L217-224]. Does it mean that the authors preprocessed the CT volumes to replace the attenuation values by material information (e.g., mapping HU values to a set of predefined materials)? More information on the data generation process would be helpful here. If indeed the model was trained on DRRs containing anisotropic residual noise (c.f. Compton effect's residual impact on XR imaging), but the proposed algorithm itself only consider isotropic attenuation, how does it impact the results? E.g., it could be interesting to generate 2 sets of input DRRs (one generated with the approximated/simplified attenuation model and one more realistic) and compare the final accuracy of the reconstructed CT volumes. - The fact that the method is only applied to synthetic inputs (DRRs generated by TIGRE) rather than real, usually noisier, X-ray images is also problematic. The paper would benefit from a real-world evaluation, or at least a discussion on why it was not performed. ### W2. Lack of References/Comparisons to SOTA on XR-3DGS - The authors mention some of the existing 3DGS solutions applied to CT/XR imaging (e.g., X-Gaussian [6], GaSpCT [39], Li et al.'s model [27]) [L89-94] but do not perform any form of comparison with those. The lack of qualitative/quantitative comparison is fair (the authors argue that these models "cannot generate CT models" [L91-92], which is somewhat true ; though a comparison on the XR-NVS task could have made this paper stronger). But I would argue that the authors should have better referenced some of these works in the Methodology. E.g., even if less formalized, radiative Gaussians are already presented in X-Gaussian [6] ; and even though it is performed in the pixel domain rather than voxel one, GaSpCT [39] already proposes a total variation loss to regularize their XR-3DGS. While I still believe that the proposed work is a valuable iteration over these works (by better formalizing and adapting the Gaussian properties and rasterization to CT data), I think the authors should be more transparent w.r.t. the SOTA. - References w.r.t. total variation (TV) theory are also missing, making it hard to contextualize the scope of the authors' contribution w.r.t. the objective function. ### W3. Somewhat Overstated Contributions - The CUDA implementation of the radiodensity voxelization seems to be more a technical feat rather than a scientific contribution. While possibly valuable to the community, I do not see any novelty in this module (maybe not as GPU-optimized, but differentiable point-cloud-to-voxel-grid tools already exist, e.g., in PyToch3D). - Changes to the adaptive control are unclear/minor, according to the authors' descriptions [L210-214] (i.e., changing the size threshold w.r.t. pruning large Gaussians, editing the density of cloned/split Gaussians). - The positive impact of the TV regularization does not appear that statistically significant (+0.32dB for PSNR, +0.009 for SSIM, +1m33s for convergence). Maybe some qualitative results could help grasp its contribution? ### W4. Limited Dataset Size - The quantitative evaluation is performed on only 15 samples, even though varied. Larger CT datasets are available, e.g. CTPelvic1K [b]. ### W5. Limited Impact of Integration Bias Correction (?) - The authors claim that "this integration bias, though having a negligible impact on imaging rendering, leads to significant inconsistency in density retrieval" [L184-186], but they only provide qualitative imaging results (Fig. 6) to justify their un-biasing contribution. Additional results could better contextualize the corresponding claims. ---- ### Additional Reference: [a] Gao, Zhongpai, et al. "DDGS-CT: Direction-Disentangled Gaussian Splatting for Realistic Volume Rendering." arXiv preprint arXiv:2406.02518 (2024). [b] Liu, Pengbo, et al. "Deep learning to segment pelvic bones: large-scale CT datasets and baseline models." International Journal of Computer Assisted Radiology and Surgery 16 (2021): 749-756.

Questions

_see **Weaknesses** for key questions/suggestions._ ### Q1. Typo? - [L223] Do the authors mean "Compton scatter" rather than "ponton scatter"?

Rating

7

Confidence

5

Soundness

3

Presentation

4

Contribution

3

Limitations

Some limitations and societal impacts are discussed in detail (see **S4** above).

Reviewer w7Ci2024-08-08

I thank the authors for their thorough response, as well as my fellow reviewers for their insightful comments. I appreciate the author's effort to address my (overall minor) concerns and questions, and I lean towards maintaining my current score (_accept_). I do hope that, were the paper accepted, the authors would account for the reviewers' remarks, as summarized by the authors in their global response, e.g.: - **Including results on real-world data** c.f. _DGKr_ and _w7Ci_ (me). These new experiments gathered by the authors demonstrate the real-world applicability of their work (c.f. Tab. 1 and Fig. 1 of rebuttal PDF). - **Better discussing/referencing existing XR-GS works** c.f. _P8A3_ and _w7Ci_ (me). The results and discussion provided in the authors' response would benefit the readers, by better contextualizing their work. I would also suggest: - **Clarifying the isotropic simplification at the core of some claims/contributions**. E.g., the authors' claims that "[X-Gaussian] _uses alpha-blending, which contradicts the unordered nature of X-ray imaging_" [response] and that "_we can individually integrate each 3D Gaussian to rasterize an X-ray projection_" [L162 + Equation 5] are only correct in the context of the isotropic simplification of X-ray imaging. I.e., if actual physics effects, such as Compton scattering, were to be considered, then the ordering of the Gaussians would matter (see preprint [i] for contributions to XR-GS orthogonal to R$^2$-Gaussian's, as well as [ii, iii] w.r.t. why ordering matters in GS-based scattering simulation). I do agree with the authors that most CT reconstruction models adopt the isotropic simplification; and, therefore, that their rasterization simplification is legitimate. However, readers should be more explicitly made aware of the basis of the authors' claims (isotropic approximation [L100, L140, L162]). - **Clarifying the benefits of proposed GS voxelizer** compared to existing solutions, e.g., PyTorch3D PC-to-voxel solution, which is also CUDA-based and differentiable but may require a few tweaks to work on Gaussians. - **Including discussed references**, e.g. [c] (TV). ----- #### Reference: [i] Gao, Zhongpai, et al. "DDGS-CT: Direction-Disentangled Gaussian Splatting for Realistic Volume Rendering." arXiv preprint arXiv:2406.02518 (2024). [ii] Zhou, Yang, Songyin Wu, and Ling-Qi Yan. "Unified Gaussian Primitives for Scene Representation and Rendering." arXiv preprint arXiv:2406.09733 (2024). [iii] Condor, Jorge, et al. "Volumetric Primitives for Modeling and Rendering Scattering and Emissive Media." arXiv preprint arXiv:2405.15425 (2024).

Authorsrebuttal2024-08-09

Thank you for your careful review of our rebuttal. Your detailed comments are crucially helpful in improving our paper, and we sincerely appreciate your support. We will incorporate the reviewers' remarks into the final version. We will also thoroughly address the points you raised, especially clarifying the isotropic simplification and the benefits of our proposed voxelizer.

Reviewer sKx68/10 · confidence 5/52024-07-12

Summary

This paper presents a 3D reconstruction method for sparse-view computed tomography using 3D Gaussian Splatting. The core contribution is the reformulation of the volumetric rendering equation to include view-independent central density estimation. Additionally, the paper introduces a differentiable voxelizer that converts a set of 3D Gaussians into a voxel grid of densities, proving effective in computed tomography tasks.

Strengths

- The paper reveals the view-dependent integration bias in 3D Gaussian Splatting (3DGS), which, to my knowledge, has not previously been reported in the community. This discovery may have a high impact to computer vision. - While this is not the first paper to apply 3DGS to computed tomography, it is the first to accurately reconstruct 3D volumes with an image formation tailored specifically for this task. - The proposed method is well evaluated and compared against other baseline methods. - Exposition is clear. The paper really reads well.

Weaknesses

I don’t see any particular weakness of the paper.

Questions

I have a couple of questions. The paper focuses on sparse-view computed tomography. I am curious about how the method would perform in dense-view scenarios. In industrial CT scanning, capturing a few thousand projections for high-quality microscale geometric details is not uncommon. With a sufficient number of projections, the FDK algorithm typically performs well. How does the proposed method compare to FDK when a large number of projections are used? At what point might it start to underperform, if at all? Would it still outperform in dense-view scenarios? Additionally, what are the implications of correcting the integration bias in image-based 3D reconstruction tasks? Would this correction lead to improved 3D reconstructions as well? If not, why?

Rating

8

Confidence

5

Soundness

4

Presentation

4

Contribution

4

Limitations

The only limitation I can think of is the scope of this work; computed tomography represents a relatively niche area in the fields of machine learning and computer vision.

Reviewer DGKr2024-08-10

Thanks for the response. The authors have addressed my concerns. I will raise my score accordingly.

Authorsrebuttal2024-08-11

Thank you for your valuable feedback and the time you took to review our paper. We’re pleased that our response addressed your concerns. We will ensure these clarifications are incorporated into the final version of the paper.

Program Chairsdecision2024-09-25

Decision

Accept (poster)

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