LinPrim: Linear Primitives for Differentiable Volumetric Rendering
Authors: Nicolas von Lützow
Matthias Nießner
Venue: NeurIPS 2025 poster ()
Keywords: Novel View Synthesis
Real-Time Rendering
Scene Representations
Primary area: applications

Abstract: Volumetric rendering has become central to modern novel view synthesis methods, which use differentiable rendering to optimize 3D scene representations directly from observed views. While many recent works build on NeRF or 3D Gaussians, we explore an alternative volumetric scene representation. More specifically, we introduce two new scene representations based on linear primitives - octahedra and tetrahedra - both of which define homogeneous volumes bounded by triangular faces. To optimize these primitives, we present a differentiable rasterizer that runs efficiently on GPUs, allowing end-to-end gradient-based optimization while maintaining real-time rendering capabilities. Through experiments on real-world datasets, we demonstrate comparable performance to state-of-the-art volumetric methods while requiring fewer primitives to achieve similar reconstruction fidelity. Our findings deepen the understanding of 3D representations by providing insights into the fidelity and performance characteristics of transparent polyhedra and suggest that adopting novel primitives can expand the available design space.

LinPrim: Linear Primitives for Differentiable Volumetric Rendering
Nicolas von Lützow Technical University of Munich nicolas. von- wwwelluettttmdeec
Matthias Nießner
Technical University of Munich
niessnerStun.de
Reconstructed Primitives
View Synthesis
Figure 1: We present LinPrim, new take novel view synthesis, that leverages linear primitives octahedra and tetrahedra for differentiable volumetric rendering in order to facilitate 3D scene reconstruction. To this end, we propose differentiable rendering pipeline in conjunction with real-time capable rasterizer tailored to such primitives, thus achieving interactive frame rates for novel-view rendering and showcasing the potential of polyhedral primitives in NVS workflows.
Abstract
Volumetric rendering has become central to modern novel view synthesis meth- ods, which use differentiable rendering to optimize 3D scene representations directly from observed views. While many recent works build NeRF [18] or 3D Gaussians [13], we explore an alternative volumetric scene representa- tion. More specifically, introduce two scene representations based on linear primitivees Primitives-octahedrra octahedra and tetrahedra- both which define homogeneous volumes bounded by triangular faces. To optimize these primitives, present differentiable rasterizer that runs efficiently on GPUs, allowing end-to-end gradient- based optimization while maintaining real-time rendering capabilities. Through experiments on real-world datasets, we demonstrate comparable performance to state-of-the-at volumetric methods while requiring fewer primitives achieve similar reconstruction fidelity. Our findings deepen the understanding of 3D repre- sentations by providing insights into the fidelity and performance characteristics of transparent polyhedra and suggest that adopting novel primitives can expand the available design space.
Website: httpe://riccolavvollleeaaa http::/wwww...iti.uuiuubeee..e github.io//iiPrriii..
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Introduction
Novel View Synthesis (NVS) enables a holistic understanding of 3D scenes given only 2D observations. This capability is essential to range of applications, including VR/AR, robotics, and autonomous driving. The success of state-of-the-an NVS approaches depends heavily on the chosen scene representations and the processes used to render them. In addition to visual quality, the representations have inherent characteristics that largely impact the ability to work with the reconstructed scenes in downstream applications. Explicit representations have proven particularly desirable due to their easily interpretable and manipulable structures.
Approaches in NVS utilize wide range of scene representations that differ significantly from the conventional Trangle-mesh-bre central to standard 3D content creation, their direct optimization in NVS tasks remains challenging [21, 6]. Instead, it is essential to explore the extensive design space of possible representations to assess how they impact fidelity, performance, and stability in NVS settings.
pipelines used in most 3D applications. While triangle meshes are
The strongest visual quality NVS applications is currently achieved by approaches relying on Neural Radiance Fields (NeRF) [18, 3. 1, 2]. Although the results are impressive, NeRFs encode an implicit function, making them suboptimal for downstream applications. For instance, editing or scene animation require elements to be easily identified, accessed, manipulated, which is inherently difficult with NeRF representations [31, 22]. Additionally, the corresponding volumetric rendering process is computationally expensive, often making real-time applications infeasible. More recently, 3D Gaussian Splatting (3DGS) [13] introduced a real-time capable, explicit representation that also achieves strong visual performance. The 3DGS representation consists of 3D Gaussian kernels, which are optimized on known views to adjust their position and features. As evidenced by the large body of derivative works relying on 3DGS [30, 12, 17, 5, 25], the explicit nature of the representation simplifies integration into other pipelines and settings.
Motivated by the scientific curiosity of how simple, bounded primitives can be used to reconstruct real-world scenes, we explore an alternative to 3DGS by employing transparent polyhe- dra as scene representation. Specifically, we introduce two novel scene representations based on octahedron and tetrahedron primitives. Each primitive characterized by a compact set of features that describe its position, vertices, opacity, and view-dependent appearance. These primi- tives are bounded by triangular faces and of homogeneous density, which makes them an intuitive representation that can be easily understood and modified.
We build on the 3DGS pipeline to accommodate the new primitives. Our method first constructs poly- hedra from a concise set of features in a fully differentiable manner. During rendering, we then rely on simple ray-triangle intersections to determine the opacity of each primitive and successively blend them. As result, we can backpropagate image-space errors through the intersection calculations to distribute gradients onto the geometric features of each polyhedron. The optimization remains analogous to 3DGS, yet the bounded, triangular geometry of our primitives expands the applicability of differentiable rendering pipelines. Finally, we analyze the photometric reconstruction quality and performance of our approaches on real-world scenes. We are able to show comparable results while retaining real-time rendering speeds.
To sum up, our contributions are as follows:
We introduce two novel scene representations hased on transparent octahedron and tetrahe- dron primitives.
We derive gradient-based optimization processes for the representations to adjust the primi- tives and population based on known views.
We demonstrate that differentiable, GPU-based rendering pipeline is real-time capable and produces high-fidelity scene reconstructions on real-world datasets.
2 Related Works
Volumetric Rendering of Radiance Fields A major innovation in NVS has been the development of volumetric rendering approaches that represent scenes as continuous radiance fields. The seminal Neural Radiance Fields [18] demonstrated how to learn a 5D function, mapping 3D coordinates

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and viewing directions to color and density, using neural network. The rendering process used differentiable, enabling the optimization of the representation given a known set of input views. Since its introduction, many subsequent works have improved on the initial approach. In particular, Mip-NeRF [1, 2] and Zip-NeRF [3] have drastically improved the visual quality by introducing multi-scale representations, anti-aliasing, and learned compression techniques. Instant-NGP [20] takes a different approach by accelerating both training and inference times via a multiresolution hash-based encoding strategy, allowing near real-time reconstruction and rendering.
Despite these advancements, NeRF-based methods remain primarily implicit, often resulting in comparatively high rendering and optimization costs. Their implicit nature can limit usability in downstream tasks. Since geometry and appearance are learned as continuous fields, no explicit surface representation is available by default. Using WeRF-based representations in downstream applications thus relies on specialized approaches expensive post-processing [27, 26].
Differentiable Point-based Rendering Point-based rendering has traditionally been seen as an alternative to mesh- based pipelines due to its simplicity in handling geometry and varying levels of surface detail [32, 10]. More recently, differentiable point-based methods have emerged, enabling end-to-end optimization of both geometry and appearance from image supervision [23, 28, 16).
The seminal work of 3D Gaussian Splatting [13] replaces traditional point primitives with elliptical Gaussian kernels. By optimizing the position and shape of the kernels from known views, they can create an explicit 3D reconstruction from known views. Additionally, each Gaussian can be efficiently "splatted" onto the screen space, allowing for real-time rendering performance. Building on the foundation of 3D Gaussian Splatting, Mip-Splatting [30] tackles aliasing by controlling the maximum frequency of the splats, effectively reducing flickering and dilation artifacts. This is achieved by filtering and scaling Gaussian primitives to better match the level of detail required at different viewing distances, preventing excessively sharp or overlapping splats.
Other works replace the Gaussian distributions with other primitives to achieve improved performance. Beyond Gaussians [4), replaces Gaussian kernels with linear kernels to achieve sharper and more precise results in high-frequency regions. Compared to our work, however, the primitives remain elliptical and non-homogeneous in density. In contrast, EVER [17] uses homogeneous density ellipsoids to replace alpha-blending with an exact rendering process, which avoids popping artifacts. 3D Convex Splatting [11] employs 3D smooth convexes, completely detaching it from the Gaussian context. Their primitives offer improved flexibility and reconstruction quality.
Although these methods offer notable improvements, we instead aim to explore novel representations that rely on simple distinct primitives. While all of the above methods introduce novel primitives or rendering processes, they share the general optimization process using known views and differentiable rendering. As such, we aim to foster innovation by exploring foundational primitives to improve the general understanding of the scene representations used in NVS settings. For this, our work focuses on transparent polyhedra bounded by triangular faces, keeping compatibility with conventional tools while maintaining high fidelity and rendering speed.
3 Method
3.1 Linear Rendering Primitives
We introduce two representations based primarily on polyhedron primitives. For each primitive, set of features that can accurately describe its shape needs to be defined. In our case, these include geometric parameters, position, opacity, and color.
Octahedra. Importantly, there is a wide range of possible octahedra of which we only model a subset. We specifically ignore cases with non-triangular faces, which are difficult to handle and can lead to unpredictable numbers of vertices and edges. Moreover, to prevent degenerate cases, we limit the degrees of freedom by restricting vertex positions to lie on the coordinate axes relative to the center. To increase expressiveness, we instead optimize a rotation quaternion for each primitive, which ensures the shape be oriented arbitrarily in space.
We enforce symmetry by choosing a single distance to describe opposing corners. Hence, each pair of opposite vertices shares the same distance from the center, effecttivly reducing the complexity
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Preprocessing
Tiling & Global Sort
3D Reconstruction
Rasterization
Adjust Population
Adjust Primitives
SfM points
Figure 2: Overview of our method. Primitives constructed from SfM points. During rendering, primitives are preprocessed according to the camera pose and sorted front-to-back, Afterward, the list traversed and the individual color contributions blended for each pixel. Through comparison with a known view, the primitives' features and the population are adjusted based on gradient flow.
of shape parameters. Aside from improving stability, this ensures that the center is always the true geometric center of the octahedron, simplifying downstream tasks such as bounding volume calculations and ray space approximation.
Tetrahedra. Although we focus primarily on octahedra, the same principles can be extended to accommodate many types of homogeneous volumes. As particular example, demonstrate the usage of tetrahedron primitives to reconstruct scenes, Contrary to octahedra, even regular tetrahedra not symmetric along
and thus require adjustments the stored features. In particular, define four basis vectors, which describe directions from the center on which corners are located. All basis vectors are equally spaced such that they can represent regular tetrahedra. We then once again optimize for the distance between the center and each corner, keeping the process largely the same. Contrary to octahedra, this means that the optimized position can differ from the geometric center of the primitive.
Memory Requirements. In total, omitting color, each primitive is fully described by its center, rotation, corner distances, and opacity. For octahedra, this yields 11 floats in total, matching the parameter budget of standard Gaussian kernel. Due to the absence of symmetry in our formulation, tetrahedra require 12 floats instead.
To describe colors and their view-dependent appearance, we utilize Spherical Harmonics (SH) coefficients, following the same approach described in 3DGS. Each primitive thus also stores set of 48 SH coefficients. which aim capture complex reflectance properties under varying angles of illumination and view. Because each coefficient is stored as float, this set of SH parameters accounts for the majority of the memory used per primitive by a significant margin compared the geometric parameters. Thus, reducing memory consumption, in practice, hinges largely on either lowering the SH degree or decreasing the total number of primitives in the scene.
3.2 Rendering Process
We use two-stage rendering process, beginning with per-primitive preprocessing and followed by per-pixel rasterization. This approach enables efficient parallelization by allowing many primitives to be processed in parallel before the final compositing stage.
Preprocessing. During preprocessing, we construct each primitive from the aforementioned features (i.e. the position, distances, and rotation). Given the camera pose, then transform the primitives into

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the camera's 's coordinate system and calculate the siew-dependen color based on the viewing direction. Afterward, we project these primitives into the 3D affine ray space approximation introduced in EWA Splatting [32, 33). Because small primitives are needed to achieve High-fidelity reconstructions, and because the approximation error grows with increasing distance from the center points, we found that the resulting error remains negligible while still reducing overall performance overhead (see Appendix D). Lastly, we compute each primitive's projected bounding box in screen space to later derive the pixels from which it is visible.
Global Sorting and Tiling. Following preprocessing, we leverage the tiling and global sorting techniques from 3DGS to map primitives to their corresponding pixels efficiently. We tile the screen into regions and sort primitives within each tile according to approximate front-to-back order.
Rasterization. During rasterization, each pixel ray either intersects a given primitive twice or not all, since the introduced primitives convex. To find intersections, we the intersection algorithm (MTIA) [19], which is fully differentiable and thus usable in our traddeet-bbsee optimization. If there depends on the distance between the intersection points. As our reconstruction is scale ambiguous, we must normalize the obtained intersection distances to ensure consistent appearance across different scene sizes. We do this by determining the smallest distance d of the primitives and then scaling all distance measurements accordingly in manner similar to EVER [17]. Lee given the opacity of primitive a, we define the density of an octahedron as:
intersections, the primitive is not visible; if there are two, the opacity
log(1 0.99 a)
a(a) 2 min(dg, (d..ddd..... dz)
We did experiment with optimizing for the unnormalized densities directly instead of the intermediate value of the opacity, but found no performance gains in practice, while decreasing the interpretability of the feature values.
Compositing, Finally, we aggregate the color contributions of all visible primitives via alpha blending in front-to-bace order. Because we maintain sorted list of primitives for each tile, we can stop blending once cumulative opacity reaches a predefined threshold (i.e. 0.999), improving performance without noticeably affecting quality. While this approach simplifies behavior when primitives overlap. the end result is a high-fidelity rendering that faithfully reproduces fine geometry while being efficient enough for real-time applications, shown in Appendix A.
3.3 Anti-Aliasing
In the context of Gaussian kernels, Mip-Splatting [30] introduced two antialiasing methods for Gaussian primitives: a 3D smoothing filter and a 2D Mip filter. In conjunction, they are able to control the maximum frequency of primitives, mitigating aliasing and dilation artifacts.
We adopt their 3D smoothing strategy to limit how small primitives can become based on visibility from the training views. As result, each primitive is visible from at least one pixel of training view. However, applying their 2D Mip filter approach is not possible for our primitives because they rely on modifying the screen space Gaussian distribution. Instead, for each primitive, we identify vertices that lie on the bounding box used for tiling and shift them outward, parallel to the screen. This effectively enlarges the bounding box footprint and increases the minimum size at which the primitive appears. Moreover, because the fall-off within each primitive is still treated as linear, expanding the distance from the center raises the opacity along the primitive as it is evaluated over a larger extent. Conceptually, this approach acts as a coarse approximation that serves to band-limit the high frequencies that would otherwise cause visible aliasing. Although this approach is not fully mathematically equivalent to 2D Mip filter, it captures its function and empirically mirrors its behavior. We evaluate the effectiveness of the filters in Appendix C.
3.4 Optimization
We employ differentiable rendering to optimize our primitives' features from known views. For this, we use the loss formulation of 3DGS [13], consisting of L.1 and SSIM terms. The gradients are then optimized using ADAM [15] and backpropagated through the entire rendering pipeline, effectively distributing the updates across the primitive features.
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Rasterization Backpropagation. As first step, backpropagation through the blending process yields gradients with respect to each primitive's alpha and color. When these gradients are propagated further, the alpha gradient indicates whether the primitive should be more or less transparent and, consequently, whether the intersections should move closer together or farther apart along the pixel's ray. By following the resulting intersection gradients, we can adjust each primitive's ' center and vertices accordingly to achieve the desired changes. This corresponds to backpropagation through the MTIA, where the intersection depth is influenced by the position of each of the corners of the triangle. turn, the positions of the corners depend on the position of the ray space center and the offsets to it. Combining the above, by aggregating over the corners and both intersections, we propagate gradients from alpha to the geometric features. Further details are in Appendix E.
Preprocessing paoppaaangn By this point, the gradients describe the loss w.r.t. the features that were originally passed to the rasterization step. While they already describe desired changes to the position of the center and corners, they do so in ray space. As consequence, propagation through the preprocessing first involves reversing the projection and view transformations, yielding world space gradients. The final center gradients are combination of the propagated ray space center gradients and the impact of the position on the ray space approximation and view-depent nee-deepndeent color. Finally, the gradient flow w.r.t. to the corners splits when undoing the rotation, onto quaternion and distance features, respectively. As our scenes are scale ambiguous, we use the maximum distance between training cameras approximate their size and adjust the learning rates for position and distances accordingly.
3.5 Population Control
While the above optimization process can effectively optimize set of primitives, it cannot create or remove them required. In addition to an initialization scheme, this raises the need for additional processes to control the population of primitives. While generally adapt the processes introduced by 3DGS, they require adjustments to accommodate the novel primitives.
Initialization. We initialize a primitive for each Structure-from-Moicce (SfM) [24] point using its position and color. Every primitive constructed with equal distance features to all corners, using the distance between the cor- responding SfM point and its closest neighbor starting point. While Gaussian kernels are circular when initialized and thus unaffected by rotations, our kernels not. Thus, initialize the primitives with a uniformly random rotation quaternion create a more homogeneous spa- tial coverage.
Figure 3: Visualization of our scene representa- tion. We show all octahedra with an opacity of at least 0.25 on an optimized Mip-NeRF 360 [2] Bicycle scene reconstruction.
Adaptive Population Control. Efficient pop- ulation control requires processes to prune unde- sired primitives and create ones that can improve fidelity. For pruning, primitives are removed if they are too transparent, too large relative to the scene extent, or take up too much screen space of a known view. This necessitates defining the size of primitive to determine these criteria. For octahedra, we define their size their longest axis, i.e. twice the size of the longest distance. For tetrahedra, determining their largest possible depth is more involved, and we instead approximate using 2 times the distance to the furthest corner.
Creating new primitives is realized as either cloning splitting existing ones. For this, primitives that have large viecespace positional gradients are chosen, with smaller ones being cloned and larger ones being split based on the size of the primitives, as described above, and the size of the scene. Cloning a primitive creates a new primitive with the same features but without the gradient momentum, which directly applicable our representations.
Splitting, on the other hand, usually involves creating two new primitives with smaller sizes and placing them by sampling from the original Gaussian as PDF. To adjust this for octahedra, since the

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distances are aligned with the coordinate axes and symmetric, we sample the new positions from PDF with standard deviations set to the corresponding distance and apply rotation afterwards. This way. primitives are more likely to spread along the longer dimensions of the octahedra. For tetrahedra, this is not trivially possible, and we instead set all standard deviations to half the largest distance. sample result of the resulting population of octahedra can be seen in Figure 3.
LinPrim-MCMC. We aim to keep many factors apart from the primitives themselves fixed between our approach and the Gaussian baseline. As such, our population control paradigms closely mirror the behavior of 3DGS. Nonetheless, we demonstrate the fact that LinPrim remains compatible with advancements made on Gaussian kernels and analyze the impact of population control by implementing separate population control technique from GS-MCMC [14] for our octahedron primitives. They rethink population control as drawing samples from the distribution of the scene, which not only improves performance but also allows precise control over the final population size.
We again adopt their approach as close as possible to minimize undesired performance impacts. In this case, we essentially act as if our primitives are Gaussians with the same features. We account for the differences in magnitude between octahedron distances and Gaussian standard deviations by scaling down distances by a factor of 2.6. After this scaling, the size of the supposed Gaussians more closely resembles the primitives they imitate. related concept was explored as Distribution Alignment in Beyond Gaussians [4].
Experiments
We evaluate our results on Mip-NeRF 360 [2] and ScanNet++ v2 [29]. For Mip-NeRF 360, we use all 9 available scenes, while for ScanNet++, we use only the first five NVS scenes with available test views. Mip-NeRF 360 primarily features forward-facing captures centered around a single object region of interest, whereas ScanNet++ consists of indoor scenes with more diverse camera trajectories and test views located farther from the training coverage. We train and test on the official splits provided by each dataset and maintain consistent parameters across all evaluated scenes. All approaches are trained for 30k iterations.
Following prior work, we use Mip-NeRF 360 outdoor scenes at quarter resolution and indoor scenes half resolution. For ScanNet++ scenes, we use the official toolkit to undistort the images and use them at full resolution. All approaches are trained, rendered, and evaluated at the same resolution.
While large body of work has proposed improvements 3DGS, we primarily focus the original formulation, the modifications introduced by Mip-Splatting, and the convex primitives from 3D Convex Splatring (3DCS) [11], as these are most closely aligned with the foundational changes introduced in our method. It is important to note that, although most evaluated approaches share an equal number of parameters per primitive, each 3D convex in 3DCS is defined by six points, resulting in higher overall parameter count and making direct comparison of parameter budgets less straightforward.
4.1 Reconstruction Quality
We measure reconstruction quality using the standard image metrics PSNR, SSIM, and LPIPS. On the ScanNet++ v2 dataset, our method consistently achieves scores close to 3DGS, Mip-Splatting and 3DCS despite using significantly fewer primitives, as visible in Table 1. We also compare against GS-MCMC [14], which enables explicit control over the primitive population, and demonstrate that our primitives can effectively leverage the same population hcuristics, achieving improved average similar trend of
performance at equal population counts. For Mip-NeRF 360 scenes, comparable performance but with a more compact representation, as can be seen in Table 2. These results align with other works that suggest that the number of used Gaussians can be drastically reduced without a negative impact on performance through the introduction of more advanced population control paradigms [8, 9].
Figure 4 illustrates that our method faithfully reconstructs many challenging regions while sometimes showing slightly lower overall fidelity. These outcomes stem, in part, from the bounded nature of our primitives, which enforce sharper, more "binary" decisions undereonstrained areas. In practice, this can lead to better depth estimates and crisper reconstructions in reflective transparent
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Ground Truth
3DGS
Mip-Splattin Splatting
Ours
Figure 4: Qualitative Results on test views from ScanNet++ v2 scenes [29].
regions-such glass distant backgrounds- also result in more conspicuous edges rarely observed surfaces like ceilings (see Appendices and F). Overall, we find our approach to work the best in smaller, more densely captured scenes, it retains strong geometric and visual clarity without overly densifying frequently observed regions. Further results on Mip-NeRF 360 scenes can be seen in Appendix H.
4.2 Tetrahedra Evaluation
In Table we evaluate our tetrahedron approach using the same datasets, parameter constraints, and optimization framework described previously. By simply swapping octahedra for tetrahedra, we demonstrate that our method generalizes well to polyhedra with different geometric constraints and can still produce photorealistic reconstructions. Quantitatively, our tetrahedron approach performs similarly to octahedra on the ScanNet++ scenes and struggles on Mip-NeRF 360, highlighting the benefits of the increased symmetry and stability inherent pipeline was designed primarily around octahedra, these results confirm that it is not narrowly tailored to one specific primitive type. Nonetheless, tetrahedra could likely benefit from specialized adaptive population control and projection techniques that fully consider their asymmetric geometry, potentially closing the performance gap currently challenging scenes. For instance, tuning how we split tetrahedra could yield further gains accuracy.
octahedra. Although our existing
Notably, tetrahedra and octahedra both converge to similar final primitive counts. On average, obtained 259k tetrahedra for ScanNet++ scenes and 2.20M for Mip-NeRF 360, matching the counts of octahedra used in our experiments reasonably well. This suggests that switching between these

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Table I: Quantitative Results ScanNet++ v2 scenes [29]. We limit the MCMC-based approaches use as many primitives as LinPrim and 3DGS do on average. Our method achieves comparable performance while using much fewer primitives under default population control, and slightly higher quality when combined with MCMC-based densification. Scene identifiers are shortened for readability.
39136d
5a269b
dc263d
08bbbd
fb564c
Mean
Prim.
3DGS [13]
27.70
23.68
23.44
22.44
23.21
24.09
738k
Mip-Splatting [30]
27.73
23.64
23.45
22.57
23.22
24.12
977k
3DCS [11]
27.59
24.19
22.92
22.78
23,84
24.26
440k
LinPrim
27.56
23.84
22.67
22.31
23.80
24.04
255k
27.90
23.79
23.78
23.66
23.37
24.50
255k
GS-MCMC [14]
27.70
23.40
23.80
23,44
22.28
24.12
738k
27.62
24.03
23.33
22.98
24.07
24.41
255k
LinPrim + MCMC
27.74
24.13
23.57
23.33
23.97
24.55
738k
Table 2: Quantitative Results Mip-NeRF 360 scenes [2). We limit the MCMC-based approaches
to use as many primitives as 3DGS do on average. 3DCS [11] uses two different sets of hyperpa- rameters for outdoor and indoor scenes.
PSNR
SSIM
LPIPS
Primitives
3DGS [13]
27.43
0.813
0.217
3.32M
Mip-Splattin [30]
27.79
0.827
0.203
4.17M
3DCS [11] +
27.22
0.801
0.208
1.02M
LinPrim
26.63
0.803
0.221
1.79M
GS-MCMC [14]
28.09
0.836
0.187
3.32M
LinPrim + MCMC
27.04
0.812
0.211
3.32M
primitive types does not inherently impose significant memory or computational overhead. Future work on refining geometric constraints, feature parameterization, and splitting heuristics could unlock even better performance for tetrahedron- based approaches.
Table 3: Comparison of the performance of our Octahedron and Tetrahedron approaches on Scan- Net++ v2 [29] and Mip-NeRF 360 [2] scenes. Both approaches produce comparable results on ScanNet++ scenes, but our Octahedron approach reaches higher reconstruction quality Mip-NeRF 360 scenes.
ScanNet++ SSIM
Mip-NeRF 360
PSNR
LPIPS
PSNR
SSIM
LPIPS
Octahedron
24.04
0.849
0.281
26.63
0.803
0.221
Tetrahedron
24.05
0.848
0.302
25.96
0.790
0.247
4.3 Limitations
While our method demonstrates comparable results and easily understood and modified, is not without drawbacks. First, the bounded nature of our primitives can introduce hard edges in poorly observed regions, resulting in more "segment-like" artifacts under limited view coverage. In these regions, the ability of Gaussian kernels smoothly transition into each other leads to more visually pleasing results, even if the reconstruction is not necessarily better. Second, while our adapted population control and MCMC-based updates modulate primitive populations well, these processes were originally designed for Gaussian primitives and are therefore not yet fully optimal for polyhedral representations. Future work could explore more specialized strategies for splitting, cloning, and sampling that better exploit the geometric structure of polyhedra. Finally, the rendering efficiency of our current implementation remains below that of the optimized Gaussian- rasterizer (see Appendix A). Reducing the computational overhead of polygon-bar based rasterization by leveraging

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existing optimized tools and pipelines for triangle- scalability and practicality of our approach.
triangle-based based rendering would substantially enhance the
Broader Impact. Our work provides a new approach to novel view synthesis, which extends the design space of 3D scene representations. It has the potential improve the quality and performance of 3D reconstruction and downstream applications, benefiting research in graphics, vision, robotics, and related fields. We do not anticipate any adverse environmental, societal, or ethical effects.
5 Conclusion
We introduced a novel framework for volumetric scene reconstruction based on linear primitives, specifically octahedra and tetrahedra. Our formulation jointly optimizes geometry and appearance through a compact set of shape parameters in fully differentiable manner. Exploiting the inherent symmetries of octahedra enables stable, high-fidelity reconstructions while maintaining low memory footprint. In addition, our experiments with tetrahedra show that the concept generalizes naturally to other triangle-ban volumes, highlighting flexibility. Despite using significantly fewer primitives, method achieves competitive performance across challenging datasets, demonstrating its effec- tiveness for real-world applications. At the same time, the bounded nature of our primitives promotes more distinct structural representations, resulting in crisper reconstructions in highly view dew-dependent areas and a more accurate geometric understanding, all without increasing sensitivity to initialization (see Appendices B and F).
Promising future work could focus on further leveraging the inherent properties of our primitives. For instance, their triangular faces closely resemble traditional meshes, suggesting a natural bridge between primitive-based and mesh-based representations. Developing efficient and accurate methods to binarize primitive opacity [7] could enable direct mesh optimization from images. Moreover, while the rendering enhancements introduced by EVER [17] require substantial modifications for Gaussian -asee methods, our primitives naturally support intersection tests with homogeneous vol- umes, eliminating the need for explicit ray tracing and allowing exact blending without additional computational overhead. Leveraging existing triangle-mesh rendering pipelines could further ac- celerate our approach, substantially improving efficiency while removing the need for ray-space approximations (see Appendices A and D). Beyond static reconstruction, exploring downstream applications of our ciimitiesssuse as dynamic or deformable scene representations coceenatttions-ccnttttttet interesting direction for future research.
constitutes an
To summarize, we hope our linear primitives open up new possibilities for representing scenes in NVS applications and foster the creation of novel scene representations.
This work was supported by the ERC Consolidator Grant Gen3D (101171131) and the German Research Foundation (DFG) Research Unit "Learning and Simulation in Visual Computing". We thank Angela Dai for the video voice-over.
Acknowledgements.
References
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A Rendering Performance and Efficiency
Rendering Speed. We measure rendering speed the average time required to render each train and test of a scene. All timings are recorded on single RTX 3090 GPU the same resolutions used for optimization, ensuring a fair comparison across all methods. As shown in Table 4, all evaluated approaches achieve real-time performance, demonstrating that our primitives can be rendered interactively. Our octahedron-baa Mip-NeRF 360. Mcanwhile, our tetrahedron variant achieves 175fps on ScanNet++ and 66fps on Mip-NeRE 360, making it about twice fast as the octahedron approach due to having only half the faces to process per primitive.
method reaches 68fps on ScanNet++ and 29fps on
Rendering Efficiency and Trade-offs. To further assess efficiency, we optimize vari- ants of the same scene with different prim- itive counts and measure rendering speed across these populations (Table 5). While primitive subsampling would simplify com- parisons, it fails to reflect realistic optimiza- tion behavior, as smaller populations are compensated by larger primitives. Render- ing speed depends mainly on the number of visible primitives rather than total popu- lation: notably, GS-MCMC renders slower than 3DGS despite using roughly one third as many primitives. We can estimate a performance equilibrium of octahedra and Gaussians from Table 5 around five Gaussians per octahedron, around 1.6 faces per Gaussian, providing starting point for future optimizations.
Table 4: Quantitative Results the rendering speed of tested approaches. We report the average time to render every train and test view ScanNet++ v2 and Mip-NeRF 360 scenes.
ScanNet++
Mip-NeRF 360
3DGS [13]
5.7ms
8.8ms
Mip-Splatting [30]
11.0ms
20.3ms
LinPrim
14.6ms
34.6ms
LinPrim Tetrahedron
5.7ms
15.2ms
Table Comparison of the rendering speed varying primitive populations on optimized versions of ScanNet++ scene 39F36da05b. We subsample the initial SfM points to 1/3 for the 50k and 1/2 for the 100k version since we cannot remove primitives when using MCMC densification.
Primitives
Rendering Speed
3DGS [13]
795k
4.94ms (202fps)
GS-MCMC [30]
255k
5.60ms (179fps)
50k t
5.59ms (179fps)
LinPrim + MCMC
100k +
8.55ms (117fps)
250k
15,87ms (63fps)
Runtime Profiling and Memory Usage We profile both LinPrim and 3DGS under default and MCMMCccnntrolld populations (Table 6) analyze runtime and memory behavior. The majority of computation time occurs during rendering, where per-tile lists are traversed to accumulate color contributions. In this stage, 3DGS only requires evaluating the function value of Gaussian, while LinPrim calculates intersections between the pixel ray and the primitives faces. Although LinPrim processes fewer primitives per pixel, this stage is currently less optimized. Given the prevalence of ray-triangle intersection in modern graphics pipelines, integrating hardware-accellratte or existing optimized kernels offers clear potential for further speedups.
Table 6: Performance and storage analysis on ScanNet++ scene 39/36da05b. We report average per-frame times for preprocessing, sorting and tiling, and rendering, as well as the total number of optimized primitives and resulting file sizes.
Preprocessing
Sort+Tile
Rendering
Primitives
File Size
3DGS [13] GS-MCMC 255k [14]
0.29ms
0.66ms
3.99ms
795k
193MB
0. 1mss
1.85ms
3.57ms
255k
62MB
LinPrim
0. 19ms
0.60ms
12.02ms
328k
81MB
LinPrim + MCMC
0. 19ms
1. 1.08ms
14.60ms
250k
62MB
LinPrim Tetrahedron
0. 0.20ms
0.90ms
5.27ms
325k
81MB
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We evaluate the memory demands of the methods by analyzing how many primitives are kept in memory during each part the rendering process and present the mean values on the tested views in Table 7. We also include the values when limiting either method to a similar number of total primitives (250k) by leveraging the corresponding MCMC-based approach.
As can be seen, the number of primitives in the view frustum is generally similar for both primitive types when the scene's population is similar. However, looking the total length of the per-tile sorted lists clearly shows that the bounding boxes of LinPrim are generally much smaller, as they intersect fewer tiles. This trend is further exemplified by the total number of primitives that are iterated over. Due to the distinct boundaries of linear primitives, we can further differentiate between primitives that were only iterated over and those that were intersected and contributed to the output image. In total, linear primitives create an image with only around 5-10% of the intersections of Gaussians.
Table Analysis of the number of primitives in memory and in the tile-list, as well as the count of total primitives iterated over and intersected by pixel rays. Output images are of size 1752 x 1168 with patch size 16 16, and values are evaluated on ScanNet++ v2 scene 39f36da05b.
Frustum
Tile-List
Iterated
Intersected
3DGS [13]
327k
5.0M
937M
937M
GS-MCMC [14]
113k
5,5M
1.3B
1.3B
LinPrim LinPrim + MCMC
139k
1.4M
275M
52M
102k
1.7M
373M
95M
B Depth and Surface Reconstruction
Depth Accuracy. We evaluate geometric accuracy by comparing predicted depth maps from our method and 3DGS against ground-truth depths across all images of ScanNet++ v2 scenes. For each pixel, the predicted depth is defined as the distance to the first primitive along the viewing ray where cumulative opacity exceeds 0.5. Ground-truth depths are derived from the official ScanNet++ meshes, and metrics are computed only at pixels with valid predictions and ground-truth values. Both methods achieve nearly complete coverage (3DGS: 99.97%, LinPrim: 99.72%) and yield visually convincing results, as shown in Figure 5. Quantitatively, LinPrim attains slightly lower L1 errors (Table 8), indicating more accurate average depth estimates, while the similar L2 errors suggest higher presence of occasional outliers.
Table 8: Pixel-wise depth errors on five ScanNet++ scenes. We report per-scene LI and L2 distances between predicted and ground-truth depth maps, computed over all valid pixels.
39f36da05b
5a269ba6fe
dc263dfbf0
08bbbdec3d
fb564c935d
L1
L2
L1
L2
L1
L2
L1
L2
L1
L2
3DGS
0.155
0.268
0.135
0.209
0.185
0.313
0.186
0.288
0.125
0.215
LinPrim
0.142
0.263
0.114
0.202
0.166
0.318
0.154
0.285
0.121
0.238
Surface Reconstruction. To further evaluate the 3D consistency of the scene geometries, reconstruct meshes from predicted depth maps using TSDF-Fusion with a 5 cm voxel size, followed by Marching Cubes. We then compute the Chamfer Distance ground-truth meshes by uniformly sampling 5,000 surface points within the GT mesh boundaries. Results are summarized in Table 9, Both approaches produce coherent surfaces, with LinPrim obtaining closer matches to the GT average.
Table 9: Surface reconstruction accuracy measured via Chamfer distance between meshes recon- structed from predicted depth maps on five ScanNet++ scenes.
39/36da05b
5a269ba6fe
dc263dfbf0
08bbbdec3d
fb564c935d
Average
3DGS
0.126
0.092
0.096
0.088
0.093
0.099
LinPrim
0.070
0.052
0.123
0.092
0.060
0.079
21

---

Ground Truth
3DGS
Ours
Figure 5: Qualitative depth comparison ScanNet++ v2 scenes. Predicted depth maps from LinPrim and 3DGS are shown alongside depths rendered from ground truth meshes.
C Anti-Aliasing
Filter Ablation. We analyze the effect of our proposed anti-aliasing filters on reconstruction quality for representative scenes from Mip-NeRF 360 and ScanNet++ v2 (Table 10). Enabling either filter individually improves quantitative performance, most notably PSNR, while LPIPS remains largely cechanged-indicattin that perceptual structure is preserved even pixel-level similarity improves. Combining both filters yields the strongest overall gains, confirming their complementary nature. Importantly, neither filter significantly alters the number of primitives, which indicates that these techniques refine visual fidelity without complicating the underlying geometry or increasing memory overhead.
Table 10: Ablation showing the impact of the filters used in our anti-aliasing efforts. Combining both filters reduces the reconstruction error.
Filters
Bonsai SSIM
dc263dfbf0
2D
3D
PSNR
LPIPS
PSNR
SSIM
LPIPS
30.84
0.929
0.196
22.46
0.870
0.267
30.96
0.931
0.196
22.59
0.870
0.266
30.97
0.930
0.195
22.66
0.871
0.266
31.21
0.936
0.195
22.67
0.871
0.266
Zoom-Out Evaluation. To assess robustness under scale changes, we evaluate single-scale training progressively smaller rendering scales, following the Mip-Splatting evaluation [30].
As shown in Table 11, anti-aliasing consistently improves performance across zoom-out factors, even though absolute gains modest. These results show that our filtering strategy generalizes beyond the training scale and maintains visual quality in downsampled views.
D Ray Space Approximation
We analyze the impact the ray-space approximation [32, 33], which our method employs to simplify the rasterization. We use MCMC- densification [14] to minimize differences in performance caused by population control, and since the gradient magnitudes differ, which would require re-tuning hyperparameters. Furthermore, observed in our evaluations, as this should maximize average primitive size, and consequently the influence of the ray-space approximation, which is most accurate for small primitives.
conduct the experiments at the smallest primitive count (255k) we
22

---

Table 11: Ablation of zoom-out performance on NeRF synthetic scenes. Results are PSNR values on test images. Approaches are optimized full resolution and evaluated on smaller resolutions simulating zoom-outs.
Chair
Full Res.
1/2 Res.
1/4 Res.
1/8 Res.
Avg.
No AA
35.67
34.39
30.29
27.17
31.88
LinPrim
35.70
34.47
30.42
27.28
31.97
Drums
Full Res.
1/2 Res.
Res.
1/8 Res.
Avg.
No AA
25.96 25.98
26.42
25.34
23.14
25.22
LinPrim
26.44
25.43
23.31
25.29
Ficus
Full Res.
1/2 Res.
1/4 Res.
1/8 Res.
Avg.
No AA
34.49
34.38
31.02
26.17
31.52
LinPrim
34.53
34.39
31.14
26.45
31.63
As shown in Table 12, disabling the approximation yields minor increases in reconstruction quality. However, simple implementation incurs considerable runtime costs from removing the approxi- mation: on the ScanNet++ scene 39136da05b, the average rendering time increases from 15.87ms (with ray space) to 25,15ms (without ray space). This increase comes solely from the rasterization step, which increases from 14. 14.60ms to 28. 28.34ms. more efficient implementation of the rendering could likely achieve the improved quality without the drastic impact performance by leveraging efficient
or optimized kernels.
Table 12: Ablation of the influence the ray space approximation [32, 33] on ScanNet++ v2 scenes [29]. Since the version without the ray space does not use our 2D filter, we also give results without it. Scene identifiers are shortened for readability.
39f36d
5a269b
dc263d
08bbbd
fb564c
PSNR
SSIM
LPIPS
LinPrim MCMC
27.62
24.03
23.33
22.98
24.07
24.41
0.858
0.272
no 2D Filter no Ray Space
27.60
24.01
23.52
22.95
23.99
24.42
0.857
0.272
27.62
24.14
23.64
23.01
24.07
24.50
0.857
0.272
E Gradient Computation
In the following we provide the gradient computation of the intersection depths i w.r.t to the ray space corners ray that span triangle. After passing through the blending process, gradients are propagated onto the intersection depths following the opacity calculation:
oi, i2) 1- exp(-o(a) (i2 - i.)).
From here, we derive the impact on either intersection and aggregate afterward. Given the barycentric coordinates (1 u U. w) of the intersection, the resulting partial derivatives are:
Zu
Py
Tay
may
8y
my
Øi
Bv
By
0,
1.2
2.2
Ov ray
8
V
0,
ray
By
8u
1.5
ray
2.3
ray
(3)
1
u
0i
9y
0.
1,3
1.
my
+
2,
0v ray
dv
191
V
0.5
ray
By
1,5
Tay
2,5
(4)
au
Oy
0i
8V 3,3
1,2
2.2
ray
8u
ray
(5)
Ov
dv
7,9
1.5
By
2.2
23

---

In the case of octahedra, since the vertices are either in positive or negative direction from the center, the gradient onto the actual ray space feature can be negated, which we omit for the sake of readability.
To obtain the derivatives of the barycentric coordinates w.r.t. to the triangle corners, we backpropagate through the MTIA [19]. Below, we give the results for corner vo, the formulas for Vı and V2 analogous. Notably, the barycentric coordinates are invariant to the depth and as such have only two non-zero components. Let r denote pixel ray coordinates and d' the determinant, then:
= ((220= r4 Ovu
Ou
V2 V1.2 v2,xz ))
V1.8
dr 1 = ivo à (()
( V1.y Vi, -x
V2.
V1.1
2,2
V1.2
V2,0
(6)
Gradient flow towards the ray space center follows the same process as to that of the corners. Since they are just matrix multiplications, undoing ray space and view-space transformations is straightforward. Finally, once the gradients have been mapped back into world space, we propagate each corner's gradient into the primitive's s distance and rotation parameters.
F Robustness and Failure Analyses
Sensitivity to Initialization. We evaluate the robustness of LinPrim and 3DGS to initialization perturbations by modifying the structure-from-mottee The ablation is conducted on the first ScanNet++ scene 39f36da05b, where both methods achieve comparable baseline performance. We test three variants: (i) Noisy Points, where Gaussian noise with standard deviation of 5% of the maximum camera distance (about 14 cm) is added to each SIM point; (ii) Half Points, where half of the SfM points are randomly removed; and (iii) Half-Size, where initial primitive sizes are halved, reducing spatial coverage. As shown in Table 13, all of the above methods reduce performance across all metrics and for either approach. Noticeably, reducing the coverage through smaller initial primitives hurts 3DGS's ' performance more than ours. This suggests that even though LinPrim boundaries are distinct and intersections are fewer, relevant visual cues can be transferred onto the primitives without issue.
(SfM) input used for primitive initialization.
Table 13: Sensitivity to initialization. We evaluate robustness to initialization ScanNet+ scene 39f36da05b by comparing reconstructions from perturbed training setups.
3DGS SSIM
LinPrim SSIM
PSNR
LPIPS
PSNR
LPIPS
Base
27.70
0.859
0.211
27.56
0.857
0.225
Noisy Points
27.61
0.858
0.217
27.54
0.855
0.228
Half Points
27.69
0.859
0.214
27.51
0.854
0.232
Half-Size
27.25
0.849
0.235
27.44
0.856
0.227
Per-Class Reconstruction and Failure Cases. To better understand where each approach excels or struggles, we compute per-class PSNR ScanNet++ v2 using semantic renderings from the first three scenes. Results for a selection of classes are shown in Table 14. In absolute terms, the failure cases of our approach were the classes of Storage Cabiner and Computer Tower, in which LinPrim achieved the lowest relative performance compared to 3DGS. On the contrary, the classes that exhibit the strongest view-dependent appearance in the scenes windows (+0.41), doors (+1.59), and monitors (+2.77) are better reconstructed by LinPrim. There further seem to be types of geometry and appearance LinPrim is better able capture, e.g., whiteboards (+1.95), blinds (+0.64), and heaters (+0.96). These results suggest that linear and Gaussian primitives capture complementary appearance and geometry properties, and that hybrid representations combining both may alleviate such edge-case failures.
Effeet of Spherical Harmonics Degree. We evaluate the impact of the spherical harmonics degree on performance for LinPrim and 3DGS by optimizing scenes for either approach with reduced maximal degrees and comparing test-view performance. We show results in Table 15 and omit SSIM and LPIPS results for brevity reasons, and since they are less impacted by the changes to spherical harmonics than PSNR.
Higher SH degrees do not always improve reconstruction quality and can even harm performance in less observed regions by overfitting appearance while masking geometric inaccuracies. This trend
24

---

Table 14: Per-class reconstruction quality on the first three ScanNet++ scenes. We report PSNR for 3DGS and LinPrim across selected semantic categories, along with relative differences and pixel counts per class.
3DGS
LinPrim
Relative
Pixels
Wall (Top 1)
33.74
34.76
+1.02
2.7M
Floor (Top 2)
27.80
25.80
-2.00
1.1M
Table (Top 3)
32.65
32.88
+0.22
646k
Storage Cabinet
36.50
32.22
-4.28
283k
Monitor
26.50
29.27
+2.77
223k
Window
23.36
23.77
+0.41
110k
Computer Tower
34.93
23.41
-11.52
15k
more evident on ScanNet++, where test views are farther from the training distribution than on Mip-NeRF 360. In our ablation, we find that 3DGS generally benefits more from spherical harmonics than LinPrim. As seen in Table 14, LinPrim nonetheless reaches higher visual quality in more view-dependent areas. In conjunction, this can likely be attributed to better understanding of the underlying surfaces and geometry instead of an affinity to spherical harmonics.
Table 15: Ablation on the impact of reduced SH-degrees on the PSNR on three ScanNet++ and two Mip-NeRF 360 scenes, and compared the degree 3 baselines.
ScanNet++ 5a269ba6fe
Mip-NeRF 360 Bicycle
SH-degree
39f36da05b
dc263dfbf0
Room
3DGS
2
-0.38
-0.55
-0.58
-0.64
-0.76
1
-0.37
-0.41
-0.64
-0.79
-0.88
0
-0.29
-0.37
-0.41
-1.01
-1.00
LinPrim
2
+0.05
+0.21
-0.22
-0.18
-0.09
1
+0.01
+0.15
-0.22
-0.47
-0.39
0
-0.01
+0.10
-0.33
-0.89
-0.58
G Experimental Setting
In this section, give specific values for the hyperparameters used and the experimental setting to ensure the reproducibility of our results. Experiments were performed using either a single RTX A6000 RTX 3090, rendering speed was evaluated using the same RTX 3090 setup for all approaches.
We use the following learning rates for both our Octahedron and Tetrahedron approaches, the position learning rates and the corresponding schedule are consistent with 3DGS [13]. Specific values were chosen empirically.
Table 16:
The learning rates used for our Octahedron and Tetrahedron approaches. f The distance learning rate is scaled by the maximum distance between known camera pose and the mean camera pose to ensure consistent behavior even scale-ambiguous settings.
Learning Rate
Color
2.5 x 10 3
SH Coefficients
1.25 10-4
Opacity
2.5 x 10-2
Rotation
1 x 1
Distance
2.6
10-4
Other notable paradigms and parameters concerning the initialization and population of primitives follows:
25

---

Primitives are initialized with distances equal to the nearest SIM point, clamped between 10 and 0.5. Opacities are to 0.1 and rotations are sampled uniformly random.
The population is adjusted every 250 iterations. Primitives with position gradients larger than 1.5 x 10-4 are considered for densification, and ones with opacity smaller than 0.025 removed. Similarly, primitives with size larger than 40% of the scene or 20 pixels removed.
During densification, primitives smaller than 1% of the scene are duplicated. Larger ones are replaced with two newer ones with relative size of 1.2 1 For octahedra, the position is chosen normally at random with a standard deviation aligned with, and equal to, the distances in each dimension. For tetrahedra, sample normally with standard deviations equal to half of the largest distance.
We do not propagate gradients through the density normalization factor (see Equation 1).
For our anti-aliasing efforts, set the kernel size of the 2D filter 0.1. For octahedra, the ray space vector is moved by half that amount, as the adjustment is mirrored on both sides. The 3D filter remains consistent with Mip-Splatting.
H
Additional Results
In Figure 6, we show qualitative results of our representation Mip-NeRF 360 scenes. In Table 17, we show the average PSNR, SSIM, and LPIPS metrics on the used ScanNet++ v2 scenes.
Table 17:
Quantitative Results on ScanNet+ v2 scenes [29]. We limit the MCMC-based approaches to use as many primitives as LinPrim and 3DGS do on average. The presented results are the average over the five scenes considered during our evaluation.
PSNR
SSIM
LPIPS
Primitives
3DGS [13]
24.09
0.853
0.263
738k
Mip-Splatting [30]
24.12
0.852
0.261
977k
3DCS [11]
24.26
0.848
0.273
440k
LinPrim
24.04
0.849
0,281
255k
24.50
0.862
0.256
255k
GS-MCMC [14]
24.12
0.861
0.251
738k
24.41
0.858
0.272
255k
LinPrim + MCMC
24.55
0.860
0.263
738k
I
Existing Assets
3DGS [13]: Code can be accessed under https://github com//graphdeco-nriaa deussian-splatting and uses custom License also found in the repository. For experiments and code, we use commit d9fad7b.
Mip-Splatting [30]: Code can be accessed under https://github com/ wttp://wwwwiiiiiiiisssesseeeeeee
and follows the 3DGS license. For our ex- periments and code, we use commit dda02ab.
3DGS-MCMC [14]: Code be accessed under https: com/ubc-vision/ 3dgs-memc and follows the 3DGS license. For our experiments and code, we use commit 7b4fc9f.
Mip-NeRF 360 Data [1]: Available from ftamttttp////oabbrorooottt/ppppp and does not provide license terms.
ScanNet++ v2 Data [29]: Available from https://kald::: tum. de/ scannetpp/ and uses custom license also found on the website.
26

---

Ground Truth
3DGS
Mip- Sppattplt
Ours
Figure Qualitative Results on test views from Mip-NeRF 360 scenes [2].
27


Reviews:
[{"review_id": "YGkDkM5c97", "reviewer": "Reviewer_Pgum", "summary": "The paper introduces LinPrim, a differentiable 3D representation for volumetric-rendering-based reconstruction that replaces Gaussian kernels with bounded, linear polyhedral primitives: axis-aligned octahedra and tetrahedra.\n\nEach primitive is described by 11–12 floats for geometry + 48 SH coefficients for color and is rasterized with a custom GPU kernel that (i) projects the polyhedron into the ray space used by EWA splatting, (ii) computes differentiable ray-triangle intersections, and (iii) composites in front-to-back order (with simple anti-aliasing heuristics).\n\nAs evaluated on ScanNet++ and Mip-NeRF 360 dataset, the proposed method achieves comparable (slightly off) quality while using 4x fewer primitives compared with 3DGS.", "questions": "- Complexity trade-off. Given octahedra are slower but fewer than Gaussians, where is the break-even point in primitive count vs. faces for equal FPS?\n- Surface extraction. Is this representation friendly for surface extraction? If so, how does accuracy and performance looks like? I also assume there is better algorithm for iso-surface extraction compared with marching cubes on this representation.\n- SH bandwidth. Did you experiment with degree-1 or learned MLP colour decoders? How sensitive is quality to SH order?\n- Dynamic scenes. Your rasterizer is well suited to convex, bounded volumes—does it extend to per-frame primitive motion (e.g. for dynamic NeRFs) without severe overhead?\n- Edge-case failures. Besides ceilings, which scene structures (thin wires, foliage, specular objects) break most often, and can hybrid primitives (polyhedra + Gaussians) alleviate that?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- It is always interesting and worth exploring to bring polyhedra-based primitives into differentiable rendering with splatting. \n- The representation is compact with efficiency improvement over 3DGS. Octahedra need only 11 float params (same budget as a Gaussian) yet deliver comparable PSNR/SSIM with one quarter the primitive count on ScanNet++.\n- Fully differentiable rasterizer introduced by analytical ray-triangle gradients (derived in the appendix) enable end-to-end training without resorting to volumetric sampling.\n- The paper compares octahedra vs. tetrahedra, tests anti-aliasing filters, and re-uses publicly available 3DGS/Mip-Splatting code with identical training schedules for fair comparisons.\n\nWeaknesses:\n- Runtime still heavier than 3DGS. Octahedra render at ~14.6 ms per 1 K×1 K view vs. 5.7 ms for 3DGS on the same hardware; the face count largely offsets the smaller population.\n- It is always a downside for polygon-based primitives to represent hard edges when coverage is sparse, whereas Gaussians fade gracefully.\n- Limited evaluation scope. It would be great and more comprehensive to have more evaluations especially on the success and failure cases of the proposed representation, which would unveil unique characteristics brought by this representation compared with 3DGS. Testing on more different scenes would be beneficial, such as large-scale scenes, indoor scenes with lots of surfaces with Manhattan assumptions, and scenes with reflections, etc. \n- Anti-aliasing is heuristic. The 2D bbox inflation filter is an ad-hoc substitute for Mip-Splatting’s analytic filter and might not generalize to zoom-out scenarios.", "was_revised": true, "final_justification": "All of my concerns have been properly addressed. Please include those additional discussions and evaluations in the revised version of the paper. I vote for acceptance.", "rebuttal": "We thank the reviewer for the thorough and insightful review. We address the individual points and questions below:\n\n**Complexity Trade-off.** To estimate the break-even point between the rendering speed of our octahedron approach and 3DGS, we optimize several variations of the same scene with different primitive counts and track the average rendering speed in Table 1\\. Please refer to our reply to reviewer *Z51C*  for more details on how primitive counts were limited.\n\nWhile subsampling primitives for evaluating rendering speed would allow us to more easily test rendering speeds at different primitive counts, it would not accurately reflect an optimized scene since the optimization will compensate for smaller populations with larger primitives, which in turn increase rendering times. Additionally, rendering speed is largely dependent on the number of visible primitives, not the population, as clearly demonstrated by 3DGS-MCMC taking longer to render while having only around a third the population size of 3DGS.\n\nWhile an exact measure for relative rendering efficiency is difficult, when optimizing with the same MCMC heuristics, the ratio seems to be rendering 5 Gaussians for every octahedron or rendering 1.6 faces for each Gaussian. We kindly refer to Table 6 and Table 8 and the accompanying comments for additional insights on rendering efficiency. We will add this discussion and evaluation to the final manuscript.\n\nTable 1: Comparison of the rendering speed of ScanNet++ scene 39f36da05b at different primitive populations. All experiments were run on an RTX 3090\\. *\\*We subsample the initial SfM points to 1/3 for the 50k and 1/2 for the 100k version since we cannot remove primitives when using MCMC densification.*\n\n|  | \\# Primitives | Rendering Speed |\n| :---- | :---- | :---- |\n| **3DGS** | 795k | **4.94ms (202fps)** |\n| **3DGS-MCMC** | 255k | 5.60ms (179fps) |\n| **LinPrim-MCMC** | **50k\\*** | 5.59ms (179fps) |\n|  | 100k\\* | 8.55ms (117fps) |\n|  | 250k | 15.87ms (63fps) |\n\n**Surface Extraction.** We kindly refer to our comment to reviewer *EARy*, where we analyze the quality of depth maps and surface reconstructions. In summary, we find that our method exhibits slightly stronger geometric reconstruction than the 3DGS baseline.\n\nWe agree with the reviewer that our triangle-based primitives are a more natural fit for direct iso-surface extraction, and exploring specialized mesh-extraction algorithms is a promising direction for future work. As shown by prior works (e.g., 2D Gaussian Splatting \\[SIGGRAPH 2024\\]), there is often a trade-off between geometric accuracy and visual fidelity in reconstructed scenes. We argue that LinPrim’s explicit triangular representation aligns more closely with the goal of producing high-quality meshes, and that incorporating additional optimization constraints \\- such as opacity requirements or encouraging face-normal alignment \\- could further improve surface quality, possibly without or with only little sacrifice in terms of appearance. Investigating these surface-aware regularizers may unlock both sharper geometry and maintained rendering performance.\n\n**SH Bandwidth.** We evaluate the impact of the spherical harmonics degree on performance for LinPrim and 3DGS by optimizing scenes for either approach with reduced maximal degrees and comparing test-view performance. We show results in Table 2 and omit SSIM and LPIPS results for brevity reasons, and since they are less impacted by the changes to spherical harmonics than PSNR.\n\nNoticeably, higher SH-degrees don’t necessarily improve the quality of reconstructions for unobserved regions, and can instead hurt performance by obfuscating underlying geometry and overfitting on known views. We can observe this by comparing performance on ScanNet++, where test views are more out-of-distribution, to the results on Mip-NeRF 360\\.\n\nIn our ablation, we find that 3DGS generally benefits more from spherical harmonics than LinPrim. As seen in Table 3, LinPrim nonetheless reaches higher visual quality in more view-dependent areas. In conjunction, this can likely be attributed to a better understanding of the underlying surfaces and geometry instead of an affinity to spherical harmonics. We will add a more expansive analysis and discussion in the revised manuscript.\n  \nTable 2: Ablation on the impact of reduced SH-degrees on the PSNR on three ScanNet++ and two Mip-NeRF 360 scenes, and compared to the degree 3 baselines.\n\n| *PSNR* | SH-degree | ScanNet++ |  |  | Mip-NeRF 360 |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  |  | **39f36da05b** | **5a269ba6fe** | **dc263dfbf0** | **Bicycle** | **Room** |\n| **3DGS** | **2** | \\-0.38 | \\-0.55 | \\-0.58 | \\-0.64 | \\-0.76 |\n|  | **1** | \\-0.37 | \\-0.41 | \\-0.64 | \\-0.79 | \\-0.88 |\n|  | **0** | \\-0.29 | \\-0.37 | \\-0.41 | \\-1.01 | \\-1.00 |\n| **LinPrim** | **2** | **\\+0.05** | **\\+0.21** | **\\-0.22** | **\\-0.18** | **\\-0.09** |\n|  | **1** | \\+0.01 | \\+0.15 | **\\-0.22** | \\-0.47 | \\-0.39 |\n|  | **0** | \\-0.01 | \\+0.10 | \\-0.33 | \\-0.89 | \\-0.58 |\n\n**Dynamic Scenes.** We expect our rasterizer to be applicable to object motion and dynamic scenes, following approaches introduced to Gaussians. As an example, 4D Gaussian Splatting \\[CVPR 2024\\] leverages a Spatial-Temporal Encoder to derive primitive deformations based on the timestep and could similarly be used to extend our work. In the context of applying motion to an existing scene, the explicit nature of our representation would allow us to select and move primitives that make up an object. In theory, the bounded nature of our primitives could aid motion by more distinctly separating individual appearances, and it would be interesting to see the performance in practice. We will discuss it as future work in our revised manuscript.\n\n**Edge-case Failures.** To provide a deeper understanding of which regions and types of objects are well-suited to our approach and which ones are reconstructed better by 3DGS, we created semantic images for ScanNet++ renderings and calculated the per-class PSNR for both approaches. We again consider the first three ScanNet++ scenes, but since coverage is not required as in the depth case, we just use test images. We present the top 3 categories, some select results, and specific failure cases we identified in Table 3\\.\n\nTable 3: Comparison of visual errors on specific classes when evaluating on three ScanNet++ scenes. \n\n| *PSNR* | 3DGS | LinPrim | Relative Performance | \\#Pixels |\n| :---- | :---- | :---- | :---- | :---- |\n| **Wall (Top 1\\)** | 33.74 | **34.76** | \\+1.02 | 2.7M |\n| **Floor (Top 2\\)** | **27.80** | 25.80 | \\-2.00 | 1.1M |\n| **Table (Top 3\\)** | 32.65 | **32.88** | \\+0.22 | 646k |\n| **Storage Cabinet** | **36.50** | 32.22 | \\-4.28 | 283k |\n| **Monitor** | 26.50 | **29.27** | \\+2.77 | 223k |\n| **Window** | 23.36 | **23.77** | \\+0.41 | 110k |\n| **Computer Tower** | **34.93** | 23.41 | \\-11.52 | 15k |\n\nIn absolute terms, the failure cases of our approach were the classes of “Storage Cabinet” and “Computer Tower,” in which LinPrim achieved the lowest relative performance compared to 3DGS. On the contrary, the classes that exhibit the strongest view-dependent appearance in the scenes \\- windows (+0.41), doors (+1.59), and monitors (+2.77) \\- are better reconstructed by LinPrim. There further seem to be types of geometry and appearance LinPrim is better able to capture, e.g., whiteboards (+1.95), blinds (+0.64), and heaters (+0.96). \n\n*Values in brackets describe the relative performance of LinPrim compared to 3DGS as measured by PSNR.*\n\nThe results suggest that linear and Gaussian primitives are better able to reconstruct different types of appearance and geometry. Following this, we believe the hybrid primitives proposed by the reviewer could indeed alleviate edge-case failures of either primitive type. Exploring how to bridge the gap between the types of primitive and manage their representations and populations is an interesting avenue for future work. We will add a discussion in the revised paper.\n\n**Anti-Aliasing.** To evaluate the zoom-out capabilities of the proposed anti-aliasing approach, we evaluate the visual quality of single-scale training at multiple smaller scales following Mip-Splatting in Table 4\\. While the improvements over the baseline are minor, they are consistent and generalize to zoom-out scenarios. We will add this ablation to our existing discussion on the anti-aliasing approach.\n\nTable 4: Evaluation of zoom-out performance on NeRF synthetic scenes. Results are PSNR values on test set images. Approaches are optimized at full resolution and evaluated on smaller resolutions simulating zoom-outs.\n\n| Chair | Full Res. | 1/2 Res. | 1/4 Res. | 1/8 Res. | Avg. |\n| :---- | :---- | :---- | :---- | :---- | :---- |\n| **No AA** | 35.67 | 34.39 | 30.29 | 27.17 | 31.88 |\n| **LinPrim** | **35.70** | **34.47** | **30.42** | **27.28** | **31.97** |\n\n| Drums | Full Res. | 1/2 Res. | 1/4 Res. | 1/8 Res. | Avg. |\n| :---- | :---- | :---- | :---- | :---- | :---- |\n| **No AA** | 25.96 | 26.42 | 25.34 | 23.14 | 25.22 |\n| **LinPrim** | **25.98** | **26.44** | **25.43** | **23.31** | **25.29** |\n\n| Ficus | Full Res. | 1/2 Res. | 1/4 Res. | 1/8 Res. | Avg. |\n| :---- | :---- | :---- | :---- | :---- | :---- |\n| **No AA** | 34.49 | 34.38 | 31.02 | 26.17 | 31.52 |\n| **LinPrim** | **34.53** | **34.39** | **31.14** | **26.45** | **31.63** |"}, {"review_id": "lOXkge0rXD", "reviewer": "Reviewer_HtCm", "summary": "This manuscript proposes to use linear primitives for 3D scene reconstruction and novel view synthesis. Inspired by 3D Gaussian Splatting, this work designs a differentiable rasterization pipeline and a population control strategy for the linear primitives, enabling gradient-based optimization. Compared to vanilla 3D Gaussian Splatting, the proposed method achieves comparable rendering quality with fewer primitives, while exhibiting qualitatively superior fidelity in some challenging cases, such as sharper geometric edges and mitigation of over-densification (e.g., the table edge and carpet in Fig.4).", "questions": "1.\tThe manuscript does not clearly explain how and why EWA splatting is applied on linear primitives. L149 mentions a trade-off between approximation error and computational overhead, it is better to provide a quantitative comparison.\n\n2.\tWhy is opacity maintained by each primitive and use it to derive density, rather than directly learning a density attribute?\n\n3.\tAlthough 3DGS also have a truncated range, its AABB is somewhat over-estimated. However, the proposed linear primitive seems incapable of receiving gradients from non-intersecting pixels. Could this property impose stricter requirements on initialization?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1.\tThe proposed method can achieve close rendering quality with significantly fewer primitives. In some challenging cases, as shown in the qualitative comparisons, it exhibits unique advantages over previous arts.\n\n2.\tThe proposed linear primitive holds strong potential to inspire future exploration. Compared to 3DGS, it allows for more precise define of ray-primitive intersection, while easier to equipping topological structuring, and more like the triangle meshes commonly used in graphics pipelines.\n\nWeaknesses：\n\n1.\tAs shown in Table 4, the advantage of using fewer primitives does not transform into rendering efficiency (although it can already achieve interactive frame rates on standard benchmarks). In addition, since L321 claims the advantage of low memory footprint, it is suggested to compare the memory usage quantitatively to better illustrate the compactness of the representation.\n\n2.\tThe benefits of the proposed linear primitives over the vanilla 3D Gaussians are not convincingly described. Now its potential is far from fully unleashed, and the demonstrated advantages appear circumstantial rather than fundamental or indispensable.", "was_revised": true, "final_justification": "Most of my concerns have been resolved by the rebuttal. Taking into account both the results presented in the submission and responses, I think the proposed primitive constitutes a non-trivial improvement over vanilla 3DGS. Therefore, I decide to raise my rating to accept.", "rebuttal": "We thank the reviewer for the precise and perceptive analysis and comments. We address the comments below:\n\n**How and why is EWA splatting applied?** We employ EWA splatting by using the ray-space approximation to cut down per-primitive work during rasterization. Although our primitives remain full 3D objects, projecting them into ray space lets us treat each triangle as an effectively 2D shape for intersection and gradient computations. Concretely, this simplifies intersection calculations as the ray directions are always orthogonal to the image plane and have only one nonzero component. This effectively means that testing a ray against a triangle reduces to a 2D problem. Additionally, this applies similarly to the gradient calculations, where position gradients collapse to two dimensions until transformed out of ray space.\n\nWe agree that an ablation study comparing the rendering speed and quality, each with and without the ray-space approximation, would help clarify the trade-offs and will include the corresponding quantitative results and discussion in the final manuscript.\n\n**Optimizing Densities.** Despite the representations being largely equivalent in our use case, optimizing a density directly is certainly plausible and might help performance by avoiding unnecessary transformations. We will run an ablation on this to see if there are positive effects in practice. \n\nFor the submission, we used opacities for two main reasons: \n\n1. They are bounded, and we thus assume them to be more numerically stable during optimization.  \n2. Compared to densities, opacities are more easily interpretable, making it easier to understand distributions of primitives or contributions to final appearance.\n\n**Sensitivity to Initialization.** To evaluate the sensitivity of our approach and 3DGS to initialization, we apply changes to the initialization scheme and evaluate the relative performance impact. We perform this ablation on the first ScanNet++ scene (39f36da05b), on which both approaches performed similarly well. \n\nThese are the specifics of the changes made:\n\n1. Noisy Points: We add noise to the position of the SfM points used for initialization. For this, we sample random offsets in each dimension with a standard deviation of 5% of the maximum distance between cameras, which is around 14cm for the scene.  \n2. Half \\#Points: We randomly drop half of the SfM points used for initialization.  \n3. Half-Size Primitive Initialization: The initial size of the primitives in either approach depends on the distance between neighbouring SfM points. For this test, we reduce the initial size of primitives by half, leading to sparse initial coverage of the 3D space.\n\nAs can be seen in Table 5, all of the above methods reduce performance across all metrics and for either approach. Noticeably, reducing the coverage through smaller initial primitives hurts 3DGS's performance more than ours. This suggests that even though LinPrim boundaries are distinct and intersections are fewer, relevant visual cues can be transferred onto the primitives. In general, while the scope of the experiment is limited, we believe it showcases that our approach is not more dependent on a strong initialization and will include the results in the revised manuscript.\n\nTable 5: Ablation on the impact of worse initialization schemes on the final reconstruction quality on ScanNet++ scene 39f36da05b.\n\n|  | 3DGS |  |  | LinPrim |  |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | **SSIM** | **PSNR** | **LPIPS** | **SSIM** | **PSNR** | **LPIPS** |\n| **Base** | **0.859** | **27.70** | **0.211** | **0.857** | **27.56** | **0.225** |\n| **Noisy Points** | 0.858 | 27.61 | 0.217 | 0.855 | 27.54 | 0.228 |\n| **Half \\#Points** | **0.859** | 27.69 | 0.214 | 0.854 | 27.51 | 0.232 |\n| **Half-Size Primitive Init.** | 0.849 | 27.25 | 0.235 | 0.856 | 27.44 | 0.227 |\n\n**Memory Usage and Bounding Boxes.** We evaluate the memory demands of the methods by analyzing how many primitives are kept in memory during each part of the rendering process. For this, we run the evaluation on the first ScanNet++ scene (39f36da05b) and present the mean values on the tested views in Table 6\\. We also include the values when limiting either method to a similar number of total primitives by leveraging the corresponding MCMC-based approach as introduced in our reply to reviewer *Z51C*.\n\nAs can be seen, the number of primitives in the view frustum is generally similar for both primitive types when the scene's population is similar. However, looking at the total length of the per-tile sorted lists clearly shows that the bounding boxes of LinPrim are generally much smaller, as they intersect fewer tiles. \n\nThis trend is further exemplified by the total number of primitives that are iterated over. Due to the distinct boundaries of linear primitives, we can further differentiate between primitives that were only iterated over and those that were intersected and contributed to the output image. In total, linear primitives create an image with only around 5-10% of the intersections of Gaussians.\n\nAs Table 5 shows that this does not translate to issues with gradient flow, it suggests that the large majority of intersections in 3DGS only contribute minor effects to the output image and similarly do not enable significant gradient flow. We believe these values present a significant opportunity for future work to optimize the rendering process and leverage the concise nature of our representation. And will include the findings and discuss opportunities in our revised manuscript.\n\nTable 6: Analysis of the number of primitives in memory and in the tile-list, as well as the count of total primitives iterated over and intersected by pixel rays. Output images are of size 1752x1168 with patch size 16x16, and values are evaluated on ScanNet++ scene 39f36da05b.\n\n|  | Frustum | Tile-List | Iterated | Intersected |\n| :---- | :---- | :---- | :---- | :---- |\n| **3DGS** | 327k | 5.0M | 937M | 937M |\n| **GS-MCMC 255k** | 113k | 5.5M | 1.3B | 1.3B |\n| **LinPrim** | 139k | **1.4M** | **275M** | **52M** |\n| **LinPrim-MCMC 250k** | **102k** | 1.7M | 373M | 95M |\n\n**Benefits of Linear Primitives over 3DGS.** We kindly refer to our reply to reviewer *EARy* about geometry and surface reconstruction and reviewer *Pgum* about semantic-wise visual quality. In summary, we show that our primitives provide improvements in geometric reconstruction and are better able to reconstruct view-dependent appearance when compared to the Gaussian baseline."}, {"review_id": "N4WdVIQlXR", "reviewer": "Reviewer_EARy", "summary": "This paper introduces LinPrim, a novel explicit scene representation for novel-view synthesis that replaces Gaussian kernels with transparent linear polyhedra. LinPrim achieves competitive reconstruction quality on ScanNet++ and Mip-NeRF 360, while using significantly fewer primitives than 3DGS.", "questions": "See the weaknesses.", "limitations": "yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The method provides a complete and systematic solution, including full forward and backward derivations for triangle-ray intersections, seamless integration into the 3DGS training loop, and full preservation of end-to-end differentiability.\n2. This approach enables a marked decrease in attribute complexity while preserving high-fidelity rendering performance.\n\nWeaknesses\n1. Recent work like 3D Convex Splatting [CVPR 2025] also propose modifications to Gaussian primitives. A more in-depth comparison and analysis with these methods would strengthen the paper.\n2. Although the method employs fewer primitives, its training and inference times are still slower than those of 3DGS. The potential reasons, including per-triangle computation costs or overdraw effects, are not thoroughly investigated.\n3. Despite emphasizing a substantial reduction in primitive count, the paper omits quantitative measurements of total storage requirements, making the practical memory efficiency unclear.\n4. It would be interesting to explore whether this representation provides enhanced geometric fidelity relative to 3DGS, such as in depth maps or mesh reconstruction.", "was_revised": true, "final_justification": "The author provided a detailed and persuasive response. I think this article has good potential and encourage the author to apply the representation to other fields (such as 4D reconstruction, Feed-forward generation, etc.). I maintain my score to accept.", "rebuttal": "We thank the reviewer for their positive feedback and insightful questions. We address the reviewer’s questions below:\n\n**Comparison to 3DCS.** We will add a full quantitative and qualitative comparison of 3D Convex Splatting to our evaluation and discuss the approach. For the sake of brevity, we include only a partial comparison of the approaches on the first three ScanNet++ scenes in Table 7\\. Keeping in mind that 3DCS was not tuned on this dataset, we are able to achieve comparable performance. Including densification paradigms via Markov Chain Monte Carlo (as further described in our comment to reviewer *Z51C*), we are able to slightly outperform their results. In the revised paper, we will extend this comparison to all ScanNet++ and Mip-NeRF 360 scenes and include additional visualizations.\n\nTable 7: Quantitative evaluation on three ScanNet++ scenes.\n\n|  | 39f36da05b |  |  | 5a269ba6fe |  |  | dc263dfbf0 |  |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | **SSIM** | **PSNR** | **LPIPS** | **SSIM** | **PSNR** | **LPIPS** | **SSIM** | **PSNR** | **LPIPS** |\n| **3DGS** | 0.859 | **27.701**  | **0.211** | 0.829 | 23.678 | 0.290 | 0.879 | 23.440 | **0.246**  |\n| **3DCS** | 0.857 | 27.624 | 0.212 | 0.828 | **24.219** | 0.290 | 0.868 | 22.766 | 0.264 |\n| **LinPrim** | 0.857 | 27.557 | 0.225 | 0.825 | 23.845 | 0.310  | 0.871  | 22.668 | 0.266 |\n| **LinPrim-MCMC** | **0.862** | 27.666 | 0.212 | **0.838** | 24.127 | **0.284** | **0.883** | **23.611** | 0.247 |\n\n**Performance.** To provide deeper insights into our method’s runtime characteristics, we profiled the evaluation of our method and 3DGS. We evaluate the methods as-is, as well as using the MCMC population control to ensure similar populations for both approaches, as can be seen in Table 8\\.\n\nOur results show that the large majority of the cost is incurred in the rendering stage, where per-tile lists are traversed to aggregate color contributions. In this stage, 3DGS only requires evaluating the function value of a Gaussian, while LinPrim calculates intersections between the pixel ray and the primitives faces. As a consequence, even though LinPrim traverses fewer primitives per pixel than 3DGS (as shown in Table 6), the current implementation is simply less optimized. Since ray-triangle intersections are commonly required, we believe leveraging existing optimized implementations for this use presents a significant opportunity for runtime improvements and will discuss the findings in the revised manuscript.\n\n**Storage Requirements.** As shown in Table 8, our representation not only uses fewer primitives in general it also employs smaller bounding boxes and consequently holds far fewer primitives in memory at any given time. Concerning total storage requirements, the actual disk size of an optimized scene is almost solely reliant on the number of primitives. Looking at the file sizes of the optimized scenes, we find that given unrestricted optimizations, the LinPrim scene is around 1/3 of the 3DGS size.\n\nTable 8: Analysis of the runtime of individual steps in the rendering pipeline and corresponding primitive counts and file sizes on the ScanNet++ scene 39f36da05b.\n\n|  | Preprocessing | Sorting & Tiling | Rendering | \\#Primitives | File Size |\n| :---- | :---- | :---- | :---- | :---- | :---- |\n| **3DGS** | 0.29ms | 0.66ms | 3.99ms | 795k | 193MB |\n| **GS-MCMC 255k** | **0.18ms** | 1.85ms | **3.57ms** | 255k | **62MB** |\n| **LinPrim** | 0.19ms | **0.60ms** | 12.02ms | 328k | 81MB |\n| **LinPrim-MCMC 250k** | 0.19ms | 1.08ms | 14.60ms | **250k** | **62MB** |\n| **LinPrim-Tetrahedra** | 0.20ms | 0.90ms | 5.27ms | 325k | 81MB |\n\n**Depth Maps and Mesh Reconstruction.** We evaluate the accuracy of depth maps predicted by our method and by 3DGS on all images from the first three ScanNet++ scenes. For each pixel, the predicted depth is defined as the depth of the first primitive along the ray where cumulative opacity exceeds 0.5. The resulting depth maps look convincing for both approaches.\n\nGround‑truth depths are extracted from the ScanNet++ meshes via the official toolkit. We compute error metrics only at pixels with valid ground‑truth and prediction values \\- pixels missing either are excluded. Both methods achieve similarly high coverage, with 3DGS covering 99.97 % of pixels and LinPrim covering 99.79 %. Results can be seen in Table 9\\.\n\nTable 9: Pixel-wise depth map errors on three ScanNet++ scenes.\n\n|  | 39f36da05b |  | 5a269ba6fe |  | dc263dfbf0 |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | *L1* | *L2* | *L1* | *L2* | *L1* | *L2* |\n| **3DGS** | 0.1553 ± 0.1124 | 0.2678 ± 0.1519 | 0.1352 ± 0.0749 | 0.2086 ± 0.0944 | 0.1854 ± 0.0511 | **0.3126** ± 0.0905 |\n| **LinPrim** | **0.1415** ± 0.1079 | **0.2634** ± 0.1469 | **0.1143** ± 0.0790 | **0.2022** ± 0.1081 | **0.1662** ± 0.0408 | 0.3183 ± 0.0935 |\n\nTo assess the 3D consistency and extract surfaces from the predicted depth maps, we apply TSDF-Fusion with a voxel size of 5cm and apply Marching Cubes to reconstruct meshes from the images. We then compute the Chamfer Distance to the ground-truth meshes from ScanNet++ by uniformly sampling 5000 surface points, as can be seen in Table 10\\.\n\nTable 10: Chamfer distances between mesh meshes reconstructed on three ScanNet++ scenes by fusing predicted depth maps.\n\n| *Chamfer Distance* | 39f36da05b | 5a269ba6fe | dc263dfbf0 |\n| :---- | :---- | :---- | :---- |\n| **3DGS** | 0.1264 | 0.0924 | **0.0959** |\n| **LinPrim** | **0.0700** | **0.0518** | 0.1232 |\n\nThe results indicate that LinPrim produces slightly more accurate depth estimates than 3DGS, suggesting better geometric reconstruction. The Chamfer Distance evaluation shows that LinPrim achieves better 3D reconstruction in two out of three scenes. This indicates that while depth predictions are generally of higher quality, exact results can vary based on scene characteristics. We will add qualitative and quantitative results and discuss them in our revised manuscript."}, {"review_id": "9R9W8qxFHW", "reviewer": "Reviewer_Z51C", "summary": "This paper proposes two new primitives to perform volumetric rendering, namely octahedra and tetrahedra, and adapt the existing 3D Gaussian Splatting (3DGS) framework to select a suitable number of primitives and to optimize the rendering process efficiently. The method is evaluated on two datasets, showing comparable performance to competitive baselines while, in some cases, reducing the number of primitives required.", "questions": "- the population control technique seems to be potentially complex, what is the impact on the training time and stability of the overall approach?\n- the comparison between using tetrahedra and octahedra could be improved, what are the advantages and disadvantages of one compared to the other based also on the empirical evidences?\n- why are the baselines in Table 1 and 2 different? What is the performance of those not reported in one of the two tables?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n- the method is evaluated on multiple datasets and ablation studies are presented\n- visualizations aid the explanation of the method\n\n**Weaknesses:**\n- the paper is overall difficult to follow: several concepts are not introduced (e.g., primitives, alpha blending, rasterization, quaternions, Structure-from-Motion) and the method section goes into detail too quickly (e.g., octahedra and tetrahedra are not introduced in Sec. 3.1, which only presents their technical challenges for the problem at hand) \n- the contribution appears limited: the paper adapts the existing 3DGS framework to accommodate 2 new primitives, without significantly diverging from it. The numerical results also do not show a significant enough performance improvement to justify this as a relevant contribution (e.g., on ScanNet++, the performance and number of primitives is very similar to GS-MCMC)\n- the experimental section is overall not clear: the two datasets are not introduced (what do they contain, what is the specific task) and some choices appear arbitrary:\n\t- why are the baselines for the two datasets different?\n\t- what are the metrics in Table 1?", "was_revised": true, "final_justification": "I thank the authors for their effort in answering my questions.\n\nI have also studied the other reviews and respective answers and realized that most of my concerns (especially related to lack of details and clarity) are definitely not shared by other reviews. This is likely due to my limited understanding of the paper and my limited background which gave me a hard time judging the work.\n\nI am thus willing to increase my rating above the acceptance threshold.", "rebuttal": "We thank the reviewer for the insightful comments. We address the questions and concerns below:\n\n**Population Control.** Our aim was to keep as many factors apart from the primitives themselves fixed when comparing to 3DGS to make the evaluation as fair as possible, and focus on the underlying primitive type. As such, our population control paradigms closely mirror the behavior of 3DGS, with changes necessitated by the change of primitive. \n\nTo analyze the impact of the population control on performance, we implement a completely separate population control technique from the 3D-MCMC paper \\[NeurIPS 2024\\], which rethinks population control as drawing samples from the distribution of the scene. This not only improves performance but also allows precise control over the final population size. We once again adopt their approach as close as possible to minimize undesired performance impacts. In this case, we essentially act as if our primitives are Gaussian-like with standard deviations equal to the linear distances divided by 2.6. This value was chosen as it results in a close match in appearance between the original shape and its Gaussian substitute.\n\nAs can be seen in Table 11, this approach is able to consistently improve performance when evaluating on the first three ScanNet++ scenes. We further believe that this illustrates how approaches made as improvements to 3DGS can be similarly applied to LinPrim. We will add these results to the revised manuscript.\n\nTable 11: Ablation of the performance of MCMC-based densification on linear primitives with a fixed number of 250k and 1M primitives in the scene on the first three ScanNet++ scenes.\n\n|  | 39f36da05b |  |  | 5a269ba6fe |  |  | dc263dfbf0 |  |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS |\n| **LinPrim** | 0.857 | 27.56 | 0.225 | 0.825 | 23.85 | 0.310 | 0.871 | 22.67 | 0.266 |\n| **LinPrim-MCMC 250k** | 0.857 | 27.65 | 0.227 | 0.834 | 24.05 | 0.299 | 0.881 | 23.35 | 0.256 |\n| **LinPrim-MCMC 1M** | **0.862** | **27.67** | **0.212** | **0.838** | **24.13** | **0.284** | **0.883** | **23.61** | **0.247** |\n\n**Comparison between Octahedra and Tetrahedra.** We will include the results of our experiments on comparing octahedra and tetrahedra-based approaches in the final manuscript. For purposes of readability, we include only the results on three ScanNet++ and Mip-NeRF 360 scenes in Table 12\\.\n\nBased on the quantitative and qualitative results, we find that octahedra and tetrahedra generally show similar visual characteristics in equally populated areas of a scene. Differences arise when considering the population control paradigms. They are required to enhance the initial coverage of the scene and enable higher fidelity in well-observed regions. Using the symmetry of the octahedra primitives allows us to mirror established approaches for Gaussian primitives to place new primitives which is not directly possible for tetrahedra. \n\nNonetheless, we believe that tetrahedra promise a large improvement in runtime efficiency (see Table 8\\) when compared to octahedra due to their reduced face count, with the current drawbacks largely concerning them being less related to established approaches. As such, applying novel or well-suited population control approaches to tetrahedra offers a promising avenue for future work.\n\nTable 12: Comparison of the performance of LinPrim’s octahedron and tetrahedron-based approaches on three ScanNet++ and three Mip-NeRF 360 scenes.\n\n|  | 39f36da05b |  |  | 5a269ba6fe |  |  | dc263dfbf0 |  |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS |\n| **Octahedra** | **0.857** | **27.56** | **0.225** | 0.825 | **23.85** | **0.310** | **0.871** | 22.67 | 0.266 |\n| **Tetrahedra** | 0.853 | **27.56** | 0.250 | **0.828** | 23.71 | 0.335 | 0.871 | **22.94** | **0.286** |\n\n|  | Bicycle |  |  | Treehill |  |  | Room |  |  |\n| :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- |\n|  | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS | SSIM | PSNR | LPIPS |\n| **Octahedra** | **0.739** | **24.76** | **0.236** | **0.617** | **21.80** | **0.346** | **0.913** | **29.88** | **0.216** |\n| **Tetrahedra** | 0.722 | 24.29 | 0.275 | 0.612 | 21.41 | 0.361 | 0.899 | 29.05 | 0.240 |\n\n**Used Baselines.** We thank the reviewer for his feedback and will update our experiments section to make it clearer and more readable. Keeping space limitations in mind, the baselines were chosen following the availability of results and relevance to our work. We will ensure that a more complete overview of results and introduction of the datasets will be available in the final manuscript."}]

Author rebuttal:

Meta-review: This paper introduces LinPrim, a new differentiable representation for volumetric rendering that uses polyhedral primitives, such as octahedra and tetrahedra, as an alternative to the Gaussians in 3DGS.
During the review process, reviewers identified several strengths. They noted the representation is compact, delivering high-quality results with significantly fewer primitives than 3DGS. The new differentiable rasterizer, the solid experimental setup, and the potential for future work were also highlighted as key positives.
However, the reviewers also raised a few concerns about the initial submission:
1. A lack of runtime improvement despite the reduction in primitive count.
2. A limited evaluation, with a need for more comparisons on different scenes and against recent methods (e.g., 3D Convex Splatting).
3. Missing evaluations for anti-aliasing (in zoom-out cases) and for depth/mesh quality.
4. An unclear discussion of the key benefits of the proposed representation over 3DGS.
5. A lack of clarity on the practical memory efficiency.

The authors' detailed rebuttal and the follow-up discussion successfully addressed most of these points, leading all reviewers to be positive about the paper. I agree with the reviewers' final recommendation and recommend this paper for acceptance. I expect the authors will incorporate the new results and discussions from the review period into the final version.
Decision: Accept (poster)