Convex Splatting: Efficient Differentiable Rendering

• Convex splatting is a rendering method that replaces Gaussian splats with bounded convex primitives, enhancing edge fidelity and mesh extractability.
• It encompasses formulations like 3D convex, triangle, tetrahedron splatting, and order-independent weighted sum rendering for varied scene representations.
• These techniques optimize differentiable rendering through smoothing, explicit geometry, and sorting-free compositing to achieve robust, real-time performance.

Convex splatting denotes a family of differentiable rendering and scene-representation methods that replace or reinterpret Gaussian splats through convex primitives or convex weighted aggregation. In this literature, the term encompasses at least three distinct but related constructions: explicit convex primitives such as smooth convex polytopes in 3D Convex Splatting, projected convex polygons such as triangle splats, and structured volumetric convex cells such as tetrahedra; it also includes an order-independent rendering view in which per-pixel colors are formed by normalized convex combinations rather than non-commutative alpha compositing (Held et al., 2024, Held et al., 25 May 2025, Gu et al., 2024, Hou et al., 2024). Across these variants, the central objective is to retain the efficiency of splat-based rendering while improving geometric boundedness, edge fidelity, mesh extractability, or rendering robustness.

1. Definition and conceptual scope

Convex splatting is motivated by limitations of 3D Gaussian Splatting. In the explicit-primitive setting, Gaussian splats are described as inherently soft, rounded distributions lacking physical boundaries; they struggle to capture hard edges and flat surfaces without increasing primitive count, and they can disperse around target surfaces unless constrained by hand-crafted regularizers (Held et al., 2024). Convex primitives instead provide bounded support, interpretable interiors and boundaries, and tighter alignment with piecewise planar or polyhedral scene structure.

The notion is instantiated differently across recent work. "3D Convex Splatting: Radiance Field Rendering with 3D Smooth Convexes" defines a scene as a composition of 3D smooth convexes derived from half-space intersections and smoothed by log-sum-exp (Held et al., 2024). "Triangle Splatting for Real-Time Radiance Field Rendering" treats each triangle as a convex support with a differentiable window function defined over its projected 2D footprint (Held et al., 25 May 2025). "Tetrahedron Splatting for 3D Generation" uses tetrahedra embedded in a deformable tetrahedral grid, where opacity is derived from an SDF and composited front-to-back (Gu et al., 2024). "Sort-free Gaussian Splatting via Weighted Sum Rendering" uses the phrase conceptually: the renderer replaces ordered alpha blending with order-independent normalized weighted sums, so the final color at a pixel is a convex combination of fragment colors (Hou et al., 2024).

A useful distinction is therefore between primitive-level convex splatting and compositing-level convex splatting. The former concerns the geometry of each splat, while the latter concerns the algebra of how splat contributions are aggregated. This suggests that convex splatting is better understood as a design space than as a single algorithm.

2. Primitive formulations

The principal formulations differ in whether convexity is encoded by half-spaces, polygonal support, tetrahedral cells, or normalized accumulation.

Method	Primitive or support	Defining mechanism
3D Convex Splatting	3D smooth convexes	Half-space intersections with log-sum-exp smoothing
Triangle Splatting	Projected triangles	Maximum of three edge half-spaces plus normalized window
Tetrahedron Splatting	Tetrahedra in a grid	Linearly interpolated SDF inside each tetrahedron
Weighted Sum Rendering	Convex color mixture	Order-independent normalized weighted sums

In 3D Convex Splatting, a convex polyhedron is defined by $J$ planes $H_j$ with outward normal $\mathbf{n}_j$ and offset $d_j$, using signed plane distances

$L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$

The non-smooth support is

$$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$

and the differentiable surrogate is

$$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$

with smoothness parameter $\delta>0$. The indicator is

$$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$

with sharpness $\sigma>0$, so that the implicit shape is $H_j$0 and the boundary is $H_j$1 (Held et al., 2024).

Triangle Splatting defines each triangle by vertices $H_j$2, projected by

$H_j$3

Its image-space SDF is

$H_j$4

and its normalized coverage kernel is

$H_j$5

where $H_j$6 is the incenter. The method explicitly contrasts this with a sigmoid-based alternative and argues that the normalized kernel keeps support strictly confined to the triangle footprint (Held et al., 25 May 2025).

Tetrahedron Splatting uses a tetrahedral grid $H_j$7 in which each vertex carries an SDF value. Inside tetrahedron $H_j$8, the SDF is linearly interpolated by barycentric coordinates:

$H_j$9

The gradient is constant inside each tetrahedron, and the SDF zero set can be extracted directly through Marching Tetrahedra (Gu et al., 2024).

Weighted Sum Rendering does not alter Gaussian primitive geometry, but it reformulates rendering so that pixel color is

$\mathbf{n}_j$0

with

$\mathbf{n}_j$1

The normalized weights define a convex combination when all terms are non-negative, hence the claim that the renderer realizes a form of convex splatting at the aggregation stage (Hou et al., 2024).

3. Projection, coverage, and compositing

A common feature of explicit convex splatting methods is that rendering operates through projected 2D convex support rather than through explicit 3D hull rasterization. In 3D Convex Splatting, each convex is parameterized by a point set $\mathbf{n}_j$2, the points are projected with a pinhole camera model, and the 2D convex hull of the projected points is computed by the Graham Scan algorithm (Held et al., 2024). The method does not explicitly build the 3D hull during rendering; convexity is instead constructed in image space. For a 2D point $\mathbf{n}_j$3, the line-based smooth indicator becomes

$\mathbf{n}_j$4

where $\mathbf{n}_j$5 is the distance to camera. The scaling of $\mathbf{n}_j$6 and $\mathbf{n}_j$7 by $\mathbf{n}_j$8 is reported as crucial in ablations (Held et al., 2024).

Triangle Splatting also renders via the projected convex footprint, but the support is exactly the triangle. The method emphasizes depth-consistent softening: if a triangle is scaled by $\mathbf{n}_j$9, then

$d_j$0

so the normalized window remains invariant to projection scale (Held et al., 25 May 2025). This is presented as an alternative to the manual depth scaling required by 3D Convex Splatting.

Compositing in 3D Convex Splatting and Triangle Splatting follows transmittance-based front-to-back blending. In 3D Convex Splatting, if $d_j$1 for the $d_j$2-th convex, then

$d_j$3

with $d_j$4 given by spherical harmonics color coefficients (Held et al., 2024). Triangle Splatting uses the same front-to-back accumulation, treating $d_j$5 as opacity and setting per-pixel opacity to $d_j$6 (Held et al., 25 May 2025).

Tetrahedron Splatting derives opacity from SDF samples along a ray rather than from an explicit indicator kernel. Using NeuS-style conversion with inverse width $d_j$7,

$d_j$8

and accumulation is again

$d_j$9

for PBR color or analogous quantities for opacity, normal, and depth maps (Gu et al., 2024).

Weighted Sum Rendering departs from this model by replacing order-sensitive alpha blending with commutative weighted sums. The paper presents this as a way to remove sorting while preserving competitive image quality, boundedness, and temporal stability (Hou et al., 2024). A plausible implication is that convex splatting can be interpreted not only as a geometric prior but also as a rendering algebra.

4. Differentiability and optimization

Differentiability is central to all convex-splatting variants, but it is achieved through different smoothing or subgradient mechanisms.

In 3D Convex Splatting, log-sum-exp smoothing yields a differentiable approximation of the maximum over half-spaces, and gradients propagate through point positions $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$0, smoothness $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$1, sharpness $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$2, opacity $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$3, and SH color. The paper gives

$L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$4

and

$L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$5

with camera projection and convex hull steps implemented in custom CUDA kernels and treated as differentiable (Held et al., 2024). Initialization uses Structure-from-Motion seeds and a Fibonacci sphere, with initial radius $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$6 the mean distance to the three nearest neighbors, initial $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$7, $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$8, and opacity $L_j(\mathbf{p})=\mathbf{n}_j\cdot \mathbf{p}+d_j.$9 (Held et al., 2024).

Its training objective combines photometric and structural terms with a compactness regularizer:

$$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$0

with $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$1 and $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$2. Densification begins after 500 iterations, repeats every 200 iterations, stops after 9,000 iterations, and splits a convex into $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$3 new convexes when the sharpness $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$4 loss exceeds $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$5 (Held et al., 2024).

Triangle Splatting uses the actual $$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$6 in the SDF rather than a smooth surrogate. The paper states that gradients are well-defined almost everywhere inside the convex and at edges via subgradients. Chain-rule expressions are given through projection, SDF, window function, and compositing, for example

$$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$7

Its loss combines photometric and perceptual terms with regularizers:

$$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$8

Densification is midpoint subdivision, so each triangle splits into four, and it occurs every 500 iterations until iteration 25,000, increasing shapes by 30% each step (Held et al., 25 May 2025).

Tetrahedron Splatting backpropagates through barycentric interpolation, NeuS opacity, and front-to-back blending. It adds eikonal and normal consistency regularization:

$$\tilde{\phi}(\mathbf{p})=\max_{j=1,\dots,J} L_j(\mathbf{p}),$$9

and combines these with SDS-based objectives in generation settings (Gu et al., 2024). The method explicitly emphasizes that it can be trained without mesh extraction during optimization.

Weighted Sum Rendering is differentiable through the quotient

$$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$0

with autograd used in practice. It introduces learnable depth weighting functions such as EXP-WSR,

$$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$1

and LC-WSR,

$$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$2

which are optimized jointly with Gaussian parameters (Hou et al., 2024).

5. Systems design and computational properties

Convex splatting is closely tied to rasterization design. 3D Convex Splatting implements a tile-based CUDA rasterizer inspired by 3DGS, projecting points, computing 2D convex hulls, evaluating indicators per pixel, sorting convexes by increasing distance from camera to primitive center, and compositing via alpha blending (Held et al., 2024). With $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$3 and SH degree 3, the representation uses 69 parameters per convex versus 59 per Gaussian, but the paper argues that fewer convexes are typically needed in structured scenes (Held et al., 2024).

Triangle Splatting is also tile-based and emphasizes compatibility with standard graphics stacks and GPU hardware. Because each triangle requires signed-distance evaluation for only three lines per pixel, the paper states that it is cheaper per pixel than 3D Convex Splatting, which requires six lines (Held et al., 25 May 2025). The optimized triangle soup is said to be compatible with any mesh-based renderer, and the paper reports rendering at over 2,400 FPS at $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$4 resolution for the Garden scene using an off-the-shelf mesh renderer (Held et al., 25 May 2025).

Tetrahedron Splatting uses tile-based CUDA kernels with GPU radix sort and a short per-pixel resorting window, reported as $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$5 at grid resolution 256. The algorithm includes frustum culling, tile replication, sorting by tile ID and average vertex depth, per-pixel face coverage tests, perspective-correct barycentrics, SDF sampling, NeuS opacity evaluation, and accumulation (Gu et al., 2024). Its main target is not conventional novel-view synthesis alone but 3D generation pipelines requiring stable optimization and precise mesh extraction.

Weighted Sum Rendering is designed explicitly to remove sorting overhead. Because its numerator and denominator are simple sums, the accumulation is commutative and associative, so fragments can be processed in any order without sorting (Hou et al., 2024). The method is implemented with hardware blending and an extra normalization pass, and on a mobile device GPU it achieves on average $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$6 faster rendering (Hou et al., 2024). This addresses a different systems bottleneck from the one targeted by explicit convex primitives.

These system-level differences reflect a broader trade-off. Explicit convex support generally improves geometric boundedness and edge behavior, while sort-free convex aggregation improves order-independence and deployment on resource-constrained hardware. This suggests that future systems could combine both directions.

6. Empirical behavior, advantages, and limitations

The reported empirical record of convex splatting is heterogeneous because the methods target different tasks and datasets.

For 3D Convex Splatting, evaluation is reported on Mip-NeRF360, Tanks and Temples, and Deep Blending. On Tanks and Temples, 3DCS reports LPIPS $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$7, PSNR $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$8, SSIM $$\phi(\mathbf{p})=\log\!\left(\sum_{j=1}^{J}\exp\!\big(\delta\,L_j(\mathbf{p})\big)\right),$$9, training time $\delta>0$0 minutes, FPS $\delta>0$1, and memory $\delta>0$2 MB, compared with 3DGS at LPIPS $\delta>0$3, PSNR $\delta>0$4, SSIM $\delta>0$5, training time $\delta>0$6 minutes, FPS $\delta>0$7, and memory $\delta>0$8 MB (Held et al., 2024). The abstract states an improvement of up to $\delta>0$9 in PSNR and $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$0 in LPIPS over 3DGS. On indoor Mip-NeRF360 scenes, it reports LPIPS $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$1, PSNR $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$2, and SSIM $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$3, surpassing Mip-NeRF360 in SSIM and LPIPS and improving over 3DGS (Held et al., 2024). The same paper notes that outdoor performance is comparable to 3DGS and that PSNR or SSIM may sometimes favor 3DGS on unstructured vegetation.

Triangle Splatting reports on Mip-NeRF360 and Tanks and Temples. On average Mip-NeRF360, it reports LPIPS $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$4 and FPS $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$5, and states that it improves over 3DGS and 3DCS by $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$6 and $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$7 respectively on LPIPS; on Tanks and Temples it reports LPIPS $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$8, PSNR $$I(\mathbf{p})=\mathrm{Sigmoid}\!\big(-\sigma\,\phi(\mathbf{p})\big),$$9, SSIM $\sigma>0$0, and FPS $\sigma>0$1 (Held et al., 25 May 2025). The paper also states that it achieves higher perceptual quality than Zip-NeRF on indoor scenes and faster rendering than 3DCS by $\sigma>0$2 and faster training by $\sigma>0$3 (Held et al., 25 May 2025).

Weighted Sum Rendering reports nearly identical or slightly better quality than 3DGS on the standard benchmarks while eliminating sorting. On Mip-NeRF360, LC-WSR reports PSNR $\sigma>0$4 versus $\sigma>0$5 for 3DGS, SSIM $\sigma>0$6 versus $\sigma>0$7, and LPIPS $\sigma>0$8 versus $\sigma>0$9; on Tanks and Temples, LC-WSR reports PSNR $H_j$00 versus $H_j$01, SSIM $H_j$02 versus $H_j$03, and LPIPS $H_j$04 versus $H_j$05 (Hou et al., 2024). It also reports that memory is about $H_j$06 of that of 3DGS-Graphics across scenes (Hou et al., 2024).

Tetrahedron Splatting is evaluated primarily in 3D generation settings. With RichDreamer, it reduces geometry optimization time to about $H_j$07 minutes versus $H_j$08 minutes for the DMTet-based baseline, and it reports Geometry CLIP $H_j$09 and Appearance CLIP $H_j$10 (Gu et al., 2024). The paper positions these gains alongside more detailed and compact meshes and negligible degradation after mesh extraction.

Several limitations recur across the literature. 3D Convex Splatting is slower than 3DGS and can trail on PSNR or SSIM in highly unstructured outdoor scenes (Held et al., 2024). Triangle Splatting may suffer from floaters in large outdoor scenes, and sorting by triangle center can induce popping and blending artifacts; the paper notes that per-pixel sorting could improve consistency (Held et al., 25 May 2025). Tetrahedron Splatting is real-time but slower than 3DGS and is limited on high-frequency textures until a later mesh-texturing stage (Gu et al., 2024). Weighted Sum Rendering approximates physical transmittance rather than reproducing it exactly, so complex occlusion regimes may expose residual bleeding depending on the learned depth weighting (Hou et al., 2024).

A common misconception is that convex splatting names a single primitive class. The literature instead shows at least four interpretations: smooth convex polytopes, triangle footprints, tetrahedral cells, and convex weighted compositing. Another misconception is that convex support alone removes all visibility artifacts. The triangle literature explicitly notes that center-based depth sorting can still cause popping, while the weighted-sum literature removes sorting but no longer performs exact front-to-back alpha blending (Held et al., 25 May 2025, Hou et al., 2024).

7. Relation to adjacent representations and research directions

Convex splatting occupies an intermediate position between Gaussian splats, SDF-based rendering, and mesh-based rasterization. Compared with Gaussian splatting, convex primitives offer bounded support, better hard-edge behavior, and in several cases lower total memory because fewer primitives are needed for structured scenes (Held et al., 2024). Compared with SDF-based methods, 3D Convex Splatting preserves direct splatting and controllable softness without meshing or baking, while Tetrahedron Splatting inherits explicit SDF structure and direct mesh extraction (Held et al., 2024, Gu et al., 2024). Compared with classical meshes, Triangle Splatting brings optimization back to triangles while avoiding the requirement of a connected mesh during training (Held et al., 25 May 2025).

The literature also indicates two partially orthogonal axes of progress. One axis concerns better primitive geometry: from Gaussians to convex polytopes, triangles, or tetrahedra. The other concerns better compositing and visibility models: from sorted alpha blending to order-independent weighted sums (Hou et al., 2024). This suggests that future convex-splatting systems may combine bounded convex support with sort-free accumulation, though that combination is not claimed in the cited works.

Empirically, the strongest performance claims currently appear task-dependent. 3D Convex Splatting emphasizes hard edges, flat surfaces, physically meaningful decompositions, and reduced primitive counts on structured scenes (Held et al., 2024). Triangle Splatting emphasizes faster rendering, higher perceptual fidelity on Mip-NeRF360, and direct compatibility with standard graphics pipelines (Held et al., 25 May 2025). Tetrahedron Splatting emphasizes topology handling, watertight mesh extraction, and integration with diffusion-driven 3D generation (Gu et al., 2024). Weighted Sum Rendering emphasizes mobile deployment, order-independence, and removal of sorting-induced popping (Hou et al., 2024).

Taken together, these results identify convex splatting as a broader research program for replacing unbounded or weakly structured splats with geometrically bounded or algebraically convex alternatives. The program is unified less by a single renderer than by a shared principle: constrain support or aggregation so that reconstruction remains fast and differentiable while more closely respecting scene structure, visibility, or hardware constraints.