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CoCoGaussian: Leveraging Circle of Confusion for Gaussian Splatting from Defocused Images

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Pith's one-line read CoCoGaussian reconstructs sharp 3D scenes from defocused images by modeling each Gaussian's blur as a circle of confusion whose diameter comes from depth and a learned aperture scalar.

desk verdict Solid engineering that transfers DoF-NeRF's circle-of-confusion model into 3DGS with a useful adaptive scaling; the physical depth model is approximate, but the paper is honest about it and the results are strong. read the letter →

arxiv 2412.16028 v2 pith:GLUQRQE3 submitted 2024-12-20 cs.CV

Jungho Lee , Suhwan Cho , Taeoh Kim , Ho-Deok Jang , Minhyeok Lee , Geonho Cha , Dongyoon Wee , Dogyoon Lee
show 1 more author
Sangyoun Lee
This is my paper · ORCID
classification cs.CV
keywords defocusblurcircleofconfusion3DGaussianSplattingnovelviewsynthesisdepthfieldblinddeblurringradiancefieldsapertureestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

CoCoGaussian claims that defocus blur, usually considered an obstacle to 3D reconstruction, can itself serve as the training signal. For each 3D Gaussian the paper computes a circle-of-confusion diameter from that Gaussian's depth and a learned aperture scalar, then spawns additional Gaussians that reproduce the blur pattern during training. Given only defocused multi-view images, the model renders the blurry training views and, at inference, drops the blur Gaussians to render sharp novel views. If the claim holds, all-in-focus captures are no longer required for 3D scene reconstruction, and the same trained model can refocus or change depth of field after capture.

What carries the argument

The central machinery is the generation of CoC Gaussian sets: a small MLP $h_\theta$ maps the camera position and base Gaussian parameters to the aperture scalar $K$, M direction vectors $d_m$, M scaling factors $\beta_m \in (0,1]$, and scale/quaternion deltas. The load-bearing identity is $\sigma(\mu_B) \approx K |1/d(\mu_B) - 1/d_F|$, which ties each Gaussian's depth to the diameter of its blur disk, and the offset rule $\mu_{\text{CoC};m} = \mu_B + (\sigma(\mu_B)/2)\,\beta_m d_m$, which places the spawned Gaussians inside that disk. This is what lets the model render defocus during training and simply omit it at inference.

What would settle it

Render a synthetic scene containing a planar target tilted about 45 degrees to the optical axis with a known thin-lens aperture, train CoCoGaussian on the resulting defocused views, and compare the recovered CoC diameters of off-axis Gaussians with the ground-truth lens equation: a systematic mismatch that grows with radial distance would confirm that the Euclidean-depth proxy, rather than the learned parameters, is driving the error.

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Extended reading notes

Core claim

The central claim is that a defocused image can be decomposed into a sharp base set of 3D Gaussians plus M sets of circle-of-confusion Gaussians whose placement is dictated by photographic defocus geometry rather than by free-form kernel learning. For each base Gaussian with mean $\mu_B$, the CoC diameter is approximated as $\sigma(\mu_B) \approx K |1/d(\mu_B) - 1/d_F|$, where $d(\mu_B)$ is the camera-to-Gaussian depth, $d_F$ is a learnable focus-plane distance per image, and $K = f \times D$ is a learnable aperture scalar. Each CoC Gaussian mean is an offset $\mu_{\text{CoC}} = \mu_B + (\sigma(\mu_B)/2)\,\beta_m d_m$ with learned direction vectors and scaling factors, and the final defocused image is a softmax-weighted sum of images rendered from the base and CoC Gaussians. At inference the CoC Gaussians are set aside, so sharp novel views, aperture control, and focus-plane control all emerge from a model trained only on defocused images.

Load-bearing premise

The straight-line distance from the camera center to each Gaussian center is treated as the optical depth that sets the circle of confusion, so scene points away from the optical axis get blur sizes from a proxy that can be systematically wrong, and a single learned aperture scalar cannot correct that off-axis bias.

Editorial extensions

If this is right

  • Rendering with only the base Gaussian set yields sharp images, turning the method into a render-time deblurring tool.
  • Changing $K$ at inference produces shallower or deeper depth of field, and changing $d_F$ refocuses the image, without retraining.
  • Because the CoC formula also describes sub-pixel blur, the model works on all-in-focus images and reports higher quality than plain 3DGS on the NeRF-LLFF benchmark.
  • On the Deblur-NeRF synthetic and real datasets and the DoF-NeRF real dataset, the paper reports higher quantitative scores than the compared ray-tracing and rasterization baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The depth used in the CoC formula is Euclidean distance from the camera center to the Gaussian mean, which overestimates true optical depth for off-axis points; replacing it with ray-Gaussian intersection depth would be a direct testable extension.
  • Because $\beta_m$ is bounded above by 1, the adaptive mechanism can shrink an overestimated CoC but cannot grow an underestimated one, so a two-sided scaling or a depth regularizer is a natural next step.
  • The learned per-image focus plane suggests a post-hoc focal-stack application: refocused renders could be validated against images actually captured at different focus distances.
  • The same CoC-Gaussian spawning strategy could be combined with motion-blur kernels to handle scenes with both defocus and camera shake.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper proposes CoCoGaussian, a 3D Gaussian Splatting method that reconstructs a sharp 3D scene from multi-view defocused images. For each base Gaussian, a Circle of Confusion (CoC) diameter is computed from the Euclidean distance between the camera center and the Gaussian mean, a learned aperture scalar K, and a per-image learned focus distance dF (Eq. 7). The method generates M additional 'CoC Gaussian' sets by offsetting the base Gaussian means along learned unit directions, with offsets scaled by the CoC diameter and learnable factors beta_m in (0,1] (Eq. 9). The base and CoC Gaussians are rasterized into M+1 images, which are combined by a CNN-computed pixel-wise weighted sum. At inference, rendering only the base Gaussians is claimed to produce sharp novel views, and K and dF can be adjusted for depth-of-field and focus-plane customization. Experiments on Deblur-NeRF synthetic and real datasets, DoF-NeRF real dataset, and NeRF-LLFF show state-of-the-art or competitive metrics, with ablations demonstrating the contribution of each component.

Significance. If the claims hold, CoCoGaussian is a practically valuable contribution: it extends 3DGS to defocused inputs while retaining real-time rendering, and it provides an interpretable, physically motivated mechanism for depth-of-field control. The paper is strong in its breadth of comparisons, reproduction of baseline codes, and explicit ablations of each proposed module. The central idea of generating CoC-shaped Gaussian copies from a per-Gaussian depth and a learned aperture is novel in the 3DGS literature and outperforms previous blur-agnostic Gaussian methods. The main risk is that the physical depth used in the CoC formula is not the axial depth required by the thin-lens model, which weakens the claim that the blur is 'accurately' modeled by the physical CoC; the empirical gains may partly come from the learned adaptive components rather than the physics.

major comments (1)
  1. [Sec. 4.2, Eq. (7); Sec. 4.3, Eq. (9); Sec. 6] The depth d(µ_B) in Eq. (7) is the Euclidean distance between the camera center x_cam and the Gaussian mean µ_B. In the thin-lens model underlying Eq. (6), the relevant object distance is the axial distance along the optical axis, not the radial distance. For a point at angular offset θ from the optical axis, radial distance overestimates axial distance by a factor of 1/cos θ, causing the CoC diameter to be systematically overestimated for points behind the focus surface and underestimated for points in front of it. Because the scaling factors β_m are constrained to (0,1] in Eq. (9), the model can shrink overestimated CoCs but cannot enlarge underestimated ones; the limitation discussion in Sec. 6 acknowledges only the over-estimation case. The learned direction vectors and CNN weights could in principle compensate, but then the sharp base-Gaussian output is produced by learned corrections rather than by the physical CoC model. Please either replace d(µ_B) with the axial depth in the camera coordinate frame, or provide a quantitative analysis of the angular bias on the tested datasets showing it is negligible.
minor comments (5)
  1. [Sec. 1] There is a typo in the sentence 'reduce dependence on estmiated depth'; 'estmiated' should be 'estimated'.
  2. [Sec. 4.3] The text says 'However, theses offsets ∆µCoC presents two potential issues'; 'theses' should be 'these' and the verb should agree.
  3. [Appendix Sec. 12] The sentence 'as demonstrated in Tab. 5 of the main paper' refers to the NeRF-LLFF results, which are in Table 4 of the main paper; the table number is incorrect.
  4. [Appendix Table 9] The reference numbers in the table header are inconsistent with the main text: 'DP-NeRF [18]' should be [11], 'Deblurring 3D-GS [25]' should be [10], and 'BAGS [11]' should be [26].
  5. [Appendix Sec. 10] In the sentence 'the C ISCO and C ORAL scenes have higher luminance', the scene names are broken by spaces; this appears to be a formatting artifact.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CoCoGaussian fits a structured CoC forward model to defocused views and validates sharp rendering against external all-in-focus ground truth.

full rationale

The derivation chain is not circular. Eq. (7) is the standard thin-lens CoC approximation, cited to Hecht and to DoF-NeRF, and while K, dF, beta, direction vectors, and CNN weights are learned from defocused training images, they are parameters of a forward blur model fitted by the reconstruction objective, not predictions derived from the target quantity. The sharp output is obtained by rendering only the base Gaussians (Appendix 13: 'CoCoGaussian renders sharp images using only the GB') and is evaluated on external all-in-focus ground truth from the DoF-NeRF and Deblur-NeRF benchmarks, so the sharp-view result is not a renamed training loss or a re-statement of the input. The self-citations (e.g., [10], [12], [26]) are implementation borrowings, such as weighted-sum blending and Gaussian perturbation, and are not load-bearing for the central reconstruction claim. The Euclidean-depth proxy and the one-sided beta limitation acknowledged in Sec. 6 are physical/correctness risks rather than circular reductions; nothing in the paper equates the CoC diameter to the fitted blur by definition or imports a conclusion solely from an author-overlapping citation.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The central claim depends on several learned quantities: K, dF, beta, direction vectors, and covariance adjustments are all optimized against defocused training images, so the 'physical' component is partly fitted. The depth proxy and the discrete-Gaussian PSF are explicit modeling choices rather than measured camera properties.

free parameters (6)
  • K (aperture parameter f times D) = learned, value not reported
    Replaces physical focal length and aperture diameter in Eq. 7; fitted to reproduce the defocus blur in training images rather than measured from the camera.
  • dF (focus plane distance) = learned per input image, initialized from SfM average depth
    Determines which depths are in focus in Eq. 7; fit per image because each defocused image can have a different focus plane.
  • beta_m (CoC scaling factors) = learned in (0, 1]
    Scales CoC Gaussian offsets in Eq. 9; added to handle unreliable depth on reflective and refractive surfaces.
  • d_m (CoC direction unit vectors) = learned via h_theta, M by N by 3
    Set the 3D directions of CoC Gaussian offsets in Eqs. 8 and 9; replace fixed circular directions with learned ones.
  • delta_s and delta_q (CoC covariance adjustments) = learned in [1, 1.1]
    Scale and rotate CoC Gaussians relative to base Gaussians in Eqs. 10 and 11, following Deblurring 3DGS.
  • M (number of CoC Gaussian sets) = 5
    Chosen by hand; ablations in Appendix Table 5 show M=5 is a reasonable setting but PSNR and SSIM fluctuate with M.
assumptions (4)
  • domain assumption The circle of confusion diameter follows Eq. 6 from Hecht with a single aperture scalar K, ignoring lens aberrations.
    Used in Sec. 4.2 to derive Eq. 7; assumes an ideal thin lens and a single effective aperture.
  • ad hoc to paper For each base Gaussian, the depth relevant to defocus is the Euclidean distance between camera center x_cam and Gaussian mean mu_B.
    Invoked in Sec. 4.2 and Fig. 2; overestimates axial depth for off-axis points and is not corrected because K is a scalar.
  • ad hoc to paper Defocus blur can be represented by M discrete 3D Gaussian copies plus a CNN-weighted sum of rasterized images.
    Introduced in Secs. 4.3 and 4.4; no proof that this spans real defocus PSFs, and the ablation shows M matters.
  • ad hoc to paper After training, rendering only the base Gaussians gives the sharp scene.
    Used at inference and in Appendix Sec. 13; the base set is never supervised by sharp images, so this is an inductive assumption validated only by benchmarks.
invented entities (1)
  • CoC Gaussians (G_CoC)
    purpose: Simulate the circle of confusion by rendering M additional copies of each base Gaussian with offset means, adjusted scales, and rotations, then blending with the base image.
    A modeling construct defined in Secs. 4.2 and 4.3; it makes no externally falsifiable prediction outside the paper, and its correctness is judged only by end-to-end rendering metrics.

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Cite this review

Pith. "Pith review of CoCoGaussian: Leveraging Circle of Confusion for Gaussian Splatting from Defocused Images." pith.science (2026). https://pith.science/paper/GLUQRQE3

@misc{pith2026241216028,
  author       = {Pith},
  title        = {Pith review of: CoCoGaussian: Leveraging Circle of Confusion for Gaussian Splatting from Defocused Images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLUQRQE3}},
  note         = {Machine review of arXiv:2412.16028}
}
read the original abstract

3D Gaussian Splatting (3DGS) has attracted significant attention for its high-quality novel view rendering, inspiring research to address real-world challenges. While conventional methods depend on sharp images for accurate scene reconstruction, real-world scenarios are often affected by defocus blur due to finite depth of field, making it essential to account for realistic 3D scene representation. In this study, we propose CoCoGaussian, a Circle of Confusion-aware Gaussian Splatting that enables precise 3D scene representation using only defocused images. CoCoGaussian addresses the challenge of defocus blur by modeling the Circle of Confusion (CoC) through a physically grounded approach based on the principles of photographic defocus. Exploiting 3D Gaussians, we compute the CoC diameter from depth and learnable aperture information, generating multiple Gaussians to precisely capture the CoC shape. Furthermore, we introduce a learnable scaling factor to enhance robustness and provide more flexibility in handling unreliable depth in scenes with reflective or refractive surfaces. Experiments on both synthetic and real-world datasets demonstrate that CoCoGaussian achieves state-of-the-art performance across multiple benchmarks.

Figures

Figures reproduced from arXiv: 2412.16028 by the authors.

Figure 1
Figure 1. We propose CoCoGaussian, a novel framework for 3D scene reconstruction from defocused images. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. We take the camera position xcam and the base Gaussian parameters µB, sB, and qB as inputs to the MLP hθ, which produces five outputs in total. (a) We set the depth d(µB) as the Euclidean distance between xcam and µB. Using the output K from hθ, the d(µB), and the learnable focus plane dF , we apply Eq. (7) to determine the CoC diameter. The diameter, combined with the outputs β and d from hθ, is then used in Eq. (9… view at source ↗
Figure 3
Figure 3. Qualitative comparison on the Deblur-NeRF and DoF-NeRF datasets. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Visualization of Aperture parameter and Focus Plane Customization. The top row of images decreases the aperture parameter [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: (a) CoC sizes based on the position of object relative to the focus plane, and (b) CoC sizes based on the aperture size. CISCO CORAL SAUSAGE Defocused Images Sharp Images Defocused Images Sharp Images Defocused Images Sharp Images Luminance: 0.4275 Luminance: 0.5329 Lu…
Figure 6
Figure 6. Figure 6: The Luminance Difference between Defocused and Sharp Images. All the luminance values are normalized ranging between 0 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Qualitative Ablation of CoC Scaling Factor. The figures [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Gaussian Positions of Defocused Images. The black and gray dots indicate base and CoC Gaussians, respectively. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Gaussian Positions of All-in-Focus Images. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.