3D Gaussian Splatting for Real-Time Radiance Field Rendering
1. Paper overview
In one sentence: A scene-specific set of anisotropic Gaussians makes high-quality radiance fields fast to render, at the cost of substantial memory and approximate visibility. e01e02e07e09e13
| At a glance | What to know |
|---|---|
| Research problem | Source description The problem is reconstructing and interactively viewing complete captured scenes without the rendering cost of volumetric ray marching. Sparse structure-from-motion (SfM) points provide an initial scaffold, but the representation must acquire missing detail, remove misplaced geometry and represent directional appearance. Success is measured against held-out photographs, not physical scene interaction. e02e08 |
| Core mechanism | |
| A key reported result | Novel-view synthesis on Mip-NeRF360: Ours-30K: 27.21; 0.815; 0.214; 41m33s; 134 FPS; 734 MB. PSNR (dB) ↑; SSIM ↑; LPIPS ↓; training time; rendering FPS; stored parameters. Nine real scenes; every eighth photograph held out; native input resolutions. Ours uses an A6000 and 30K optimization iterations. Mip-NeRF360: 27.69; 0.792; 0.237; 48h; 0.06 FPS; 8.6 MB. Its three quality metrics are copied from the original publication. INGP-Big: 25.59; 0.699; 0.331; 7m30s; 9.43 FPS; 48 MB. Quality is metric-dependent: Ours improves SSIM and LPIPS but has lower PSNR than Mip-NeRF360. The 48h figure converts 12h on four A100s into single-GPU-equivalent time; it is not matched-A6000 wall time. e08e09 |
| Reading caution | Source description Poorly observed regions can produce elongated or splotchy Gaussians. Large splats can pop because of guard-band rejection and abrupt depth-order changes. Regularization and antialiasing are proposed remedies, not evaluated solutions. e13 |
Core contributions
Figure 2. The same explicit scene representation participates in image formation and image-based optimization. Original paper, p. 5 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Follow the black arrows from left to right: SfM points initialize the Gaussian set, and a camera supplies the projection used by the tile rasterizer. The image at the right closes the training loop through the blue gradient arrows. The lower branch uses optimization signals to adapt the Gaussian population, then returns the revised set to rendering. Cross-checking Figure 2 with Sections 3–5 and Algorithm 1 shows that density control includes discrete cloning, splitting and removal. The blue arrow into that branch denotes gradient-informed decisions; it does not establish differentiability through the creation or deletion of a primitive. e02e04e05e06e07
What it supports. The method's efficiency depends on a representation that can both be optimized from photographs and directly rasterized. Geometry, opacity and directional color live in the Gaussians themselves. Once a scene is fitted, novel-view rendering follows the forward projection/rasterization path without running the training loop.
Where the evidence stops. The diagram summarizes scene fitting, not a neural-network architecture or action policy. Image supervision can admit incorrect geometry, and the figure alone says nothing about unseen-region accuracy.
2. Motivation
2.1 The problem and the proposed response
The problem is reconstructing and interactively viewing complete captured scenes without the rendering cost of volumetric ray marching. Sparse structure-from-motion (SfM) points provide an initial scaffold, but the representation must acquire missing detail, remove misplaced geometry and represent directional appearance. Success is measured against held-out photographs, not physical scene interaction. e02e08
2.2 What this reading follows
Imagine fitting a captured scene with translucent, oriented blobs whose shapes and colors can change. Project those blobs into a camera, blend their contributions and compare the image with a real photograph. That is the core loop of 3D Gaussian Splatting. Its distinctive move is to optimize both the blobs and their population while using a renderer designed for GPU efficiency. This reading follows the supplied 2023 arXiv v1: how the representation learns, why splitting and gradients matter, and what the reported speed actually buys. The result is a static-scene rendering method; its image-quality evidence does not establish dynamic prediction or robot control. e01e02e07e09e13
3. Research context
We place the paper in the collection through its world–action interface. The catalog labels and the reading’s assessment are shown separately.
| Catalog dimension | Recorded classification |
|---|---|
| Major category | Foundational work |
| Architecture | Not applicable |
| Prediction paradigm | Not applicable |
| Quadrant | Not applicable |
This table preserves the labels recorded at reading time. The current major category is Components of WAMs. View the current classification.
3.1 Evidence-based assessment
Supports the recorded classification
The foundational 3D-representation category is supported by the explicit static-scene radiance field. Architecture, prediction paradigm and quadrant are correctly not applicable to the world-action taxonomy: there is no action variable, temporal dynamics model, inverse dynamics or joint future/action predictor. This is a positive scope determination rather than an unverified classification. e02e07
This is the collection’s architectural analysis, not a new related-work survey. Benchmark comparisons and their protocols appear in Section 6.
4. Problem formulation
4.1 Inputs and outputs
| Inputs | Outputs |
|---|---|
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4.2 Equations and their role
5. Method in detail
5.1 From a world-space ellipsoid to a supervised pixel
A Gaussian begins as an explicit object in the reconstructed scene, carrying position, opacity, shape and directional color. Its covariance is constructed from independent scales and rotation, which avoids unconstrained updates to an invalid covariance matrix. For a training camera, the viewing transform and projection Jacobian map this covariance into a two-dimensional footprint. Evaluating that footprint at a pixel gives a contribution weighted by the learned opacity; ordered compositing then attenuates colors behind earlier splats. The resulting image is compared with the captured photograph through L1 and D-SSIM losses. Gradients therefore adjust the scene attributes that produced the image. Changing the camera after fitting changes the projection and SH appearance evaluation, while the optimized scene parameters remain fixed for rendering. e02e03e04e05e07
Figure 4. Clone to expand coverage; split to refine an oversized primitive. Original paper, p. 6 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Read each row left to right. The black outline is the schematic geometry, and the green regions represent Gaussian coverage. In the upper row, a small Gaussian covers too little of the shape, so cloning supplies another primitive that optimization can move. In the lower row, one broad Gaussian spans detail that it cannot express well; splitting creates smaller components. The arrows and row labels agree with Section 5.2 and Algorithm 1: first require a sufficiently large position-gradient signal, then split the large-scale case and clone the smaller-scale case. The figure illustrates the intended behavior rather than showing a measured reconstruction trajectory. e06e11
What it supports. Density control changes the model's capacity locally. A parameter update can move or reshape an existing blob, but cannot alone add independent components. Cloning and splitting let the optimization allocate those components where image gradients indicate remaining reconstruction error.
Where the evidence stops. The diagram omits numerical thresholds and pruning. It does not prove either operation is optimal, and a larger population can itself improve quality; the ablation must be read with that capacity change in mind.
5.2 Why fitting also changes the number of Gaussians
Ordinary gradient updates can reshape a primitive, but they cannot give one Gaussian independently controllable pieces. The paper alternates those updates with population edits driven by large view-space position gradients. A small Gaussian can be cloned to cover missing structure; a large one can be split so smaller components refine the approximation. Pruning and opacity resets oppose uncontrolled growth and remove primitives that no longer help. Algorithm 1 makes the branch direction explicit: the large-scale case splits, and the other case clones. Reader interpretation: this is adaptive allocation of representational capacity as well as geometric fitting. Table 3's splitting ablation supports its practical importance, but because removing splitting also changes the population trajectory, a count-matched comparison would better isolate the placement rule itself. e05e06e11
5.3 Why the renderer is fast, and where the approximation enters
The renderer groups work before shading individual pixels. Gaussians are duplicated across the tiles they overlap and sorted using tile ID together with depth. Each tile can then load shared batches and blend front-to-back, rather than sorting a separate list for every pixel. The backward pass reuses the sorted lists in reverse and avoids a fixed cap on contributing primitives; this matters because truncating gradients at ten points damages the reported ablation results. Alpha thresholds still skip negligible contributions and terminate saturated pixels. The speed therefore comes from organized GPU work and an explicit representation, with a visibility approximation from shared ordering. When a large Gaussian changes order abruptly, the limitations section identifies popping as a possible consequence. The system renders images; it does not decide camera motions or execute actions. e02e07e11e13e16
5.4 Training and inference
During training
This is per-scene optimization, with Adam named in Algorithm 1. Each iteration samples a training view, renders it, compares against its photograph and updates Gaussian attributes. Opacity uses a sigmoid, scales an exponential, and initial isotropic axes use mean distance to the three nearest points; position learning rates decay exponentially. e05
Optimization begins at four-times-smaller image resolution and upsamples at iterations 250 and 500. Only the zero-order SH component is optimized initially; one band is added every 1,000 iterations until four bands are present. The paper motivates this schedule by missing angular coverage. e15
During inference
A requested camera determines projection, ordering and SH-based directional color for the fitted scene. Rendering needs no training photograph, loss evaluation or density refinement. The output is an image, with camera navigation supplied externally; no action extraction, feedback controller or future-state rollout is defined. e02e07
Appendix C skips blending updates below alpha 1/255, clamps alpha above at 0.99, and stops before accumulated opacity would exceed 0.9999. Thus an unlimited gradient depth means no fixed count cap, not that every scene Gaussian receives a gradient. e16e07
5.5 Implementation flow
- Represent geometry and appearance
Initialize Gaussians at SfM points. Store a position, opacity, three scales, a rotation quaternion and spherical-harmonic (SH) color coefficients. Scale and normalized rotation parameterize valid covariance; no surface normals or learned rendering network are required. e02e04e05
- Project and composite
Transform each Gaussian covariance into image space using the affine approximation of perspective projection. Its projected Gaussian footprint, multiplied by learned opacity, supplies the pixel-dependent blending weight. Depth-ordered alpha compositing accumulates color. e03e04
- Adapt representational capacity
After warm-up, densify every 100 iterations when average view-space position-gradient magnitude exceeds 0.0002. Clone small Gaussians; split large ones into two, sampling positions from the parent and dividing scale by 1.6. Prune nearly transparent or excessively large Gaussians; reset opacity near zero every 3,000 iterations. e06
- Render with tiles
Cull, project and duplicate splats into overlapping 16×16-pixel tiles. A single radix sort uses tile ID and depth; each tile traverses its list front-to-back. This avoids per-pixel sorting but makes visibility approximate. Backpropagation traverses the reused lists in reverse without a fixed contributor-count cap. e07
6. Experiments & results
3D Gaussian Splatting fits an explicit radiance field to calibrated photographs of a static scene. Anisotropic Gaussians, adaptive splitting/cloning and a differentiable tile rasterizer jointly enable fast novel-view synthesis. Its main tradeoff is high rendering speed with competitive image quality at substantial scene-memory cost; it learns neither actions nor temporal dynamics (e02, e09, e13).
6.1 Read the original evidence
Table 1. Fast display comes with metric-dependent quality and a larger stored scene. Original paper, p. 8 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Read within one dataset block before moving across datasets. SSIM and PSNR increase with quality, whereas LPIPS decreases. Compare Ours-7K with Ours-30K to see the effect of continued optimization; compare Ours-30K with M-NeRF360 for the quality benchmark and with INGP-Big for a fast baseline. The retained dagger note identifies inherited Mip-NeRF360 quality scores. FPS is measured rendering performance, while Mem stores optimized parameters rather than peak training memory. Sections 7.1–7.2 specify the viewer, every-eighth-image test split and differing native resolutions. Consequently the three blocks should not be treated as one fixed-resolution throughput benchmark. e08e09e14e15
What it supports. On Mip-NeRF360, Ours-30K reports 134 FPS with SSIM 0.815 and LPIPS 0.214, compared with 0.06 FPS, 0.792 and 0.237 for M-NeRF360. Its PSNR is lower, 27.21 versus 27.69 dB, and its parameter storage is larger, 734 versus 8.6 MB. The practical gain is fast display with competitive, not uniformly superior, quality.
Where the evidence stops. Page 9 footnote 2 converts 12 hours on four A100s into the baseline's 48-hour figure; Ours uses an A6000. The timing comparison is not a controlled same-hardware wall-clock experiment. Appendix D separately reports rerun baseline quality.
Figure 12. A fast renderer still inherits the ambiguity of poorly observed regions. Original paper, p. 11 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. The caption identifies the left panel as Mip-NeRF360 and the right as the proposed method. This is a DrJohnson view with little overlap with the training views. Inspect the near wall and dark foreground furniture, where the two representations produce different kinds of error. There is no ground-truth panel in this figure, so it should be read as an illustrated failure case rather than a scored comparison. Section 7.4 links weak observation coverage with artifacts and separately discusses elongated Gaussians, culling and depth-order changes. A single still image cannot distinguish all of those mechanisms. e13
What it supports. Good aggregate held-out-view scores do not guarantee reliable images from poorly constrained viewpoints. The figure supports the authors' stated coverage limitation: explicit anisotropic primitives can fit observed appearances well yet yield visible defects where the photographs leave scene structure ambiguous.
Where the evidence stops. Neither method has ground truth shown here, and a still crop cannot demonstrate temporal popping. Proposed regularization, antialiasing and improved culling remain future remedies in this source, not validated fixes for this image.
6.2 Results and evaluation conditions
| Task & protocol | Reported result | Comparison & interpretation |
|---|---|---|
| Novel-view synthesis on Mip-NeRF360 Nine real scenes; every eighth photograph held out; native input resolutions. Ours uses an A6000 and 30K optimization iterations. | Ours-30K: 27.21; 0.815; 0.214; 41m33s; 134 FPS; 734 MB. PSNR (dB) ↑; SSIM ↑; LPIPS ↓; training time; rendering FPS; stored parameters | Mip-NeRF360: 27.69; 0.792; 0.237; 48h; 0.06 FPS; 8.6 MB. Its three quality metrics are copied from the original publication. INGP-Big: 25.59; 0.699; 0.331; 7m30s; 9.43 FPS; 48 MB. Quality is metric-dependent: Ours improves SSIM and LPIPS but has lower PSNR than Mip-NeRF360. The 48h figure converts 12h on four A100s into single-GPU-equivalent time; it is not matched-A6000 wall time. e08e09 |
| Novel-view synthesis on Tanks&Temples Truck and Train; every eighth photograph held out, native resolutions; A6000, 30K iterations. | Ours-30K: 23.14; 0.841; 0.183; 154 FPS. PSNR (dB) ↑; SSIM ↑; LPIPS ↓; rendering FPS | Mip-NeRF360: 22.22; 0.759; 0.257; 0.14 FPS. All three aggregate image metrics favor Ours on these two scenes. This is a limited scene subset, not a claim about every scene in the dataset. e08e09 |
| Novel-view synthesis on Deep Blending DrJohnson and Playroom; every eighth photograph held out, native resolutions; A6000, 30K iterations. | Ours-30K: 29.41; 0.903; 0.243; 137 FPS. PSNR (dB) ↑; SSIM ↑; LPIPS ↓; rendering FPS | Mip-NeRF360: 29.40; 0.901; 0.245; 0.09 FPS. Aggregate quality differences are small. Table 8 separately shows DrJohnson PSNR favoring Mip-NeRF360, so aggregate parity does not imply a uniform per-scene win. e08e09e14 |
| Novel-view synthesis on Synthetic NeRF Eight bounded Blender scenes, exact cameras, white background; 100K random initial Gaussians; 30K iterations. Competing scores are taken from their respective papers. | Ours-30K: 33.32. Mean PSNR (dB) ↑ | Point-NeRF: 33.30; INGP-Base: 33.18; Mip-NeRF: 33.09; Plenoxels: 31.76. Random initialization can work in this well-constrained bounded setting. The tiny mean advantage over Point-NeRF lacks reported uncertainty and does not establish a statistically reliable improvement. e10 |
| Component ablation on Truck, Garden and Bicycle 5K and 30K iterations; high-resolution inputs manually downsampled for this experiment. | Full: 26.05; Limited-BW: 19.19; Random Init: 20.42; No-Split: 23.90; No-SH: 25.35; No-Clone: 25.91; Isotropic: 25.23. Mean PSNR (dB) ↑ at 30K | Limited-BW permits gradients only for the ten front-most points. Other rows remove the named choice; the full model is the reference. Gradient truncation and removing splitting cause large losses in this setup. The different preprocessing prevents direct substitution of these values for the main benchmark; no repeated-run uncertainty is reported. e11 |
6.3 Ablations and diagnostic examples
Read component removals and qualitative examples within their stated evaluation conditions.
Table 3. The strongest losses come from restricting gradients and weakening scene refinement. Original paper, p. 9 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Start with the Full row and compare one removed choice at a time, keeping the scene and iteration column fixed. Average-30K provides a compact summary, but the individual columns reveal exceptions: No-Clone scores 24.82 on Truck against Full's 24.81. Limited-BW caps gradients at the ten front-most points, Random Init removes the SfM scaffold, and Isotropic replaces the full covariance shape with one radius. The caption on the inspected page states that inputs were manually downsampled from higher-resolution originals to reduce artifacts. Those preparation differences mean this table is a controlled ablation setting, not a replacement for the main benchmark scores. e11e12
What it supports. At 30K, mean PSNR falls from 26.05 to 19.19 with Limited-BW and to 23.90 without splitting. Isotropic Gaussians reach 25.23 and removing SH reaches 25.35. The results support retaining gradients through depth and enough geometric flexibility, while the small No-Clone gap requires a more measured interpretation.
Where the evidence stops. Only three scenes are included, with no repeated-run uncertainty. Disabling population changes also alters representation size and optimization trajectory, so this table does not isolate a count-matched benefit of splitting or cloning.
Figure 10. Independent scales and rotation help splats align with fine image structure. Original paper, p. 11 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Each group repeats the same order: Ground Truth, Full, Isotropic. Compare the separated leaves in Ficus, the narrow structures around the distant car and the wheel boundary. The isotropic option can optimize one radius but cannot independently stretch and orient its Gaussian footprint as the full covariance model does. Read this alongside Equation (6), where rotation and axis scales determine covariance, and the scalar-radius ablation described on page 10. Figure 10's caption adds an essential control: the Ficus example restricts both configurations to at most 5K Gaussians for illustration. That special budget is part of this visual comparison. e04e11e12
What it supports. The examples show how anisotropic primitives preserve thin structures that become diffuse under the illustrated isotropic configuration. Table 3 provides a complementary quantitative result: mean 30K PSNR is 26.05 for the full model and 25.23 for Isotropic across its three real-scene ablations.
Where the evidence stops. The Ficus image uses a special count cap, not the standard synthetic benchmark configuration. These selected views demonstrate appearance differences; they do not directly measure surface accuracy or establish superiority at every memory budget.
7. Analysis & limitations
7.1 What the evidence leaves open
Poorly observed regions can produce elongated or splotchy Gaussians. Large splats can pop because of guard-band rejection and abrupt depth-order changes. Regularization and antialiasing are proposed remedies, not evaluated solutions. e13
Peak training memory can exceed 20 GB. Rendering needs the full scene model plus another 30–500 MB for rasterization, depending on scene and resolution. Very large scenes may require a reduced position learning rate. e13
Novel-view image fidelity does not verify metric geometry, dynamic prediction or executable control. Comparisons also mix inherited and rerun baselines; Appendix D reports rerun Mip-NeRF360 averages separately. e02e08e14
7.2 Questions for discussion
- How much of splitting's gain survives a matched Gaussian-count budget?
- Can a stricter visibility calculation reduce popping without losing the reported rendering advantage?
8. Reproducibility audit
8.1 Requirements and known gaps
A reproduction needs calibrated images/SfM points, the stated test split and image preparation, PyTorch, custom CUDA rasterization, CUB sorting and the SIBR viewer for the reported FPS protocol. The source advertises code/data, but availability alone is not evidence of a working reproduction. e08e15
The PDF supplies loss weights, several refinement and stability constants, algorithms and covariance derivatives, but does not enumerate a complete executable configuration: exact learning-rate values, all pruning/size thresholds, software versions and random seeds remain unspecified. These must be resolved before calling a run faithful. e04e05e06e15e16
8.2 Proposed reproduction checks
The following checks are proposals motivated by the paper. They have not been run as part of this reading.
Check 1: Separate useful splitting from simply adding more primitives
Reader-proposed experiment, not performed: reproduce the Full, No-Split and No-Clone comparisons on Truck, Garden and Bicycle at 5K and 30K using the ablation's manually downsampled inputs. Fix cameras, held-out images, loss, SH schedule and random seeds. Record PSNR, LPIPS, Gaussian count, peak memory and wall time. Add a count-matched control that supplies the same number of new primitives as Full through a predeclared alternative placement rule. Repeat with three seeds. A consistent Full advantage at matched count would support its placement/refinement logic; disappearance of the advantage would suggest that extra capacity explains much of the original difference. e05e06e08e11e15
Check 2: Check gradients before attributing errors to visibility sorting
Reader-proposed experiment, not performed: use a small synthetic collection of overlapping anisotropic Gaussians and a fixed camera, comparing the full reverse pass against automatic differentiation or finite differences of the same forward compositing expression. Hold sorting and active threshold decisions fixed during the local check; test opacity, scale, rotation and position derivatives. Then compare full gradients with a ten-contributor cap at shallow and deep overlap, keeping the forward image identical. Record relative gradient error, optimization loss and runtime. Agreement for the full pass but increasing error under the cap would support the claimed depth-complexity mechanism. Test order swaps and alpha-threshold crossings separately, because those discontinuities should not be confused with an incorrect derivative inside a fixed branch. e03e04e05e07e11e16
8.3 Reading coverage
Visual audit: All 14 supplied PDF pages were rendered and actually viewed, including the title/version/affiliations, Equations (1)–(11), Figures 1–12, Tables 1–9 and Algorithms 1–2. All six final crops were individually viewed for readability and complete labels. Figure 2's operation/gradient arrows and Figure 4's clone/split rows were checked against Sections 3–6 and Algorithm 1; no claim-relevant branch conflict was observed. Page 9's timing footnote and ablation preprocessing, page 10's gradient-cap definition, and page 14's numerical thresholds and per-scene results were visually checked. The short Table 1 caption is retained because it contains the required dagger provenance note. Supplemental videos, code and external project material were outside the supplied reading.
PDF pages inspected for this edition: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. Appendix coverage: reviewed.
Original figures and tables remain the work of the source’s authors. Extractions preserve their scientific content; any HTML wrapper layout is disclosed with each figure. The surrounding reading notes are our own.
Text reading scope & known omissions
- Abstract; 1 Introduction
- 2 Related Work, including 2.1–2.3
- 3 Overview
- 4 Differentiable 3D Gaussian Splatting
- 5 Optimization with Adaptive Density Control, including 5.1–5.2
- 6 Fast Differentiable Rasterizer for Gaussians
- 7 Implementation, Results and Evaluation, including 7.1–7.4
- 8 Discussion and Conclusions; Acknowledgments; References
- Appendix A: Details of Gradient Computation
- Appendix B: Optimization and Densification Algorithm
- Appendix C: Details of the Rasterizer
- Appendix D: Per-Scene Error Metrics
Outside the original text pass
- Text extraction does not reconstruct figure images; inspect the retained PDF for figures and equation/table layout.
- Separate supplemental material availability has not been fully verified.
- The supplied extraction limitation was addressed by inspecting all 14 original PDF pages and six final figure/table crops. All seven supplied text chunks were read individually.
- Identity/version note: the title, all four authors and DOI match the catalog. The inspected artifact explicitly identifies arXiv:2308.04079v1, 8 August 2023. Its ACM reference block and running pagination use Article 1, whereas the catalog BibTeX records 139:1–139:14. No publisher edition or later revision was supplied for comparison; equivalence beyond the verified identity is not asserted.
- The separately referenced supplemental videos, project website, source code and datasets were not inspected. No experiments were reproduced.
The visual audit above records the subsequent illustrated pass.
8.4 Traceable evidence
e01PDF p. 1, title/author block, arXiv margin and ACM Reference Format
The observed title matches the catalog; authors are Bernhard Kerbl, Georgios Kopanas, Thomas Leimkühler and George Drettakis. Kerbl and Kopanas contributed equally. The artifact says arXiv:2308.04079v1, 8 August 2023, and DOI 10.1145/3592433; its ACM reference uses Article 1.
Go to primary source ↓e02PDF pp. 2 and 4, introduction contributions and Section 3; p. 5, Figure 2 and caption
Static photographs, calibrated cameras and sparse SfM points are converted into a scene-specific set of 3D Gaussians with optimized position, covariance, opacity and SH appearance. Figure 2 distinguishes operation flow from returning gradients.
Go to primary source ↓e03PDF p. 3, Section 2.3, Equations (1)–(3), especially Eq. (3) and following definitions
Ordered point alpha blending uses projected Gaussian weights multiplied by learned opacity and attenuates later colors by preceding transparency; the paper compares this image-formation model to volumetric compositing.
Go to primary source ↓e04PDF p. 4, Section 4, Equations (5)–(6); p. 13, Appendix A, Equations (8)–(11)
Covariance is parameterized by scale and normalized quaternion rotation, projected using a viewing transform and projective Jacobian. Appendix A derives scale and rotation gradients; normals are not required.
Go to primary source ↓e05PDF p. 5, Section 5.1 and Eq. (7); p. 13, Appendix B, Algorithm 1
Image loss combines L1 and D-SSIM with lambda 0.2. Sigmoid opacity, exponential scales, three-neighbor initialization and position-only exponential learning-rate decay are stated. Algorithm 1 samples a view, renders, evaluates loss and applies Adam.
Go to primary source ↓e06PDF pp. 5–6, Section 5.2 and Figure 4; p. 13, Algorithm 1 refinement branches
Densification occurs every 100 iterations after warm-up for average view-space position-gradient magnitude above 0.0002. Small Gaussians are cloned and large ones split into two at scale divisor 1.6. Opacity resets every 3,000 iterations; low-opacity and oversized Gaussians are pruned. Algorithm 1's greater-than size branch selects splitting.
Go to primary source ↓e07PDF p. 6, Section 6; pp. 13–14, Appendix C and Algorithm 2
Rasterization culls and projects Gaussians, duplicates them across 16×16 tiles and sorts by tile/depth keys before front-to-back compositing. There is no further per-pixel sorting. The reverse pass reuses lists without a hard contributor-count limit.
Go to primary source ↓e08PDF p. 8, Section 7.2 Real-World Scenes; p. 9, continuation and footnote 2
Evaluation covers 13 real scenes with every eighth photo held out and native input resolutions. An A6000 is used except for Mip-NeRF360; its runs used four A100 GPUs for 12 hours, reported as 48 single-GPU-equivalent hours. Image-quality sources and timing hardware are therefore not uniform.
Go to primary source ↓e09PDF p. 8, Table 1, all dataset groups and Ours-7K/Ours-30K, M-NeRF360 and INGP-Big rows
Table 1 reports quality, training time, FPS and stored-parameter memory. At 30K, Ours reaches Mip-NeRF360 27.21/0.815/0.214 at 134 FPS, Tanks&Temples 23.14/0.841/0.183 at 154 FPS and Deep Blending 29.41/0.903/0.243 at 137 FPS (PSNR/SSIM/LPIPS). Daggers identify inherited Mip-NeRF360 quality metrics.
Go to primary source ↓e10PDF p. 8, Table 2 including caption; p. 9, Section 7.2 Synthetic Bounded Scenes
Synthetic NeRF uses 100K random initial Gaussians, exact cameras, white background and 30K iterations. Mean PSNR is 33.32 versus Point-NeRF 33.30 and INGP-Base 33.18; competing scores are inherited from their papers.
Go to primary source ↓e11PDF p. 9, Table 3 and caption, Section 7.3; p. 10, gradient-depth ablation and Figures 7–9
Ablations use separately downsampled inputs. Average-30K PSNR is Full 26.05, Limited-BW 19.19, Random Init 20.42, No-Split 23.90, No-SH 25.35, No-Clone 25.91 and Isotropic 25.23. Limited-BW caps gradient recipients at ten; Table 3 contains no uncertainty intervals.
Go to primary source ↓e12PDF p. 10, Section 7.3 Anisotropic Covariance; p. 11, Figure 10 and caption
The isotropic ablation optimizes one radius for all axes. Figure 10 contrasts ground truth, full and isotropic representations; the Ficus illustration restricts both configurations to at most 5K Gaussians.
Go to primary source ↓e13PDF pp. 10–11, Section 7.4; p. 11, Figures 11–12 and captions
The authors identify unobserved-region artifacts, elongated Gaussians and popping from culling/order changes. They report peak training memory above 20 GB and extra rasterizer memory of 30–500 MB. Figure 12 compares Mip-NeRF360 left and Ours right on a DrJohnson view with little training overlap.
Go to primary source ↓e14PDF p. 14, Appendix D and Tables 4–9, especially Table 8 Dr Johnson column
The paper's own Mip-NeRF360 reruns average 27.58 PSNR, 0.790 SSIM and 0.240 LPIPS on that dataset, distinct from copied values. DrJohnson PSNR is 29.140 for Mip-NeRF360 and 28.766 for Ours-30K.
Go to primary source ↓e15PDF p. 8, Section 7.1 Implementation and Optimization Details
The system uses Python/PyTorch, custom CUDA, CUB sorting and SIBR for measured frame rates. Resolution increases at 250 and 500 iterations; SH bands are introduced every 1,000 iterations up to four. A code/data URL is advertised, without version-pinned environment specifications in this section.
Go to primary source ↓e16PDF p. 14, Appendix C Numerical stability
The text specifies skipping blending updates below alpha 1/255, clamping alpha to at most 0.99, and stopping before accumulated opacity exceeds 0.9999.
Go to primary source ↓8.5 Primary sources
3D Gaussian Splatting for Real-Time Radiance Field Rendering ↗
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