2D Triangle Splatting (2DTS)

• 2DTS is a differentiable rendering technique that replaces volumetric primitives with projected triangle facelets to produce explicit, simulation-ready meshes.
• It employs image-space soft coverage functions and front-to-back alpha compositing, enabling both feed-forward scene decoding and optimization-based mesh training.
• The method bridges radiance-field optimization with standard mesh pipelines, though challenges remain in achieving watertightness and handling fine details.

2D Triangle Splatting (2DTS) denotes a family of differentiable rendering and reconstruction methods that replace volumetric or Gaussian primitives with projected triangles or triangle facelets, while retaining front-to-back alpha compositing and end-to-end gradient flow. Across the recent literature, 2DTS serves both as a rendering formulation for optimizing triangle soups or semi-connected meshes and as a supervision mechanism inside feed-forward scene decoders that output explicit surfaces in a single pass. The central premise is that triangles are native graphics primitives, admit explicit surface geometry, and can be made differentiable in image space by replacing hard visibility with smooth coverage functions or annealed opacity schedules, thereby connecting radiance-field-style optimization to mesh-ready outputs (Sheng et al., 23 Jun 2025, Held et al., 25 May 2025, Held et al., 29 Sep 2025, Jinlin et al., 6 Mar 2026, Kupyn et al., 23 Jun 2026).

1. Definition and research context

2DTS emerged from the observation that NeRF-style volumetric rendering and 3D Gaussian Splatting (3DGS) provide strong novel-view-synthesis quality but do not directly yield the kind of explicit, simulation-ready, or engine-native geometry required for relighting, shadowing, collision, and standard mesh pipelines. The original formulation titled "2D Triangle Splatting for Direct Differentiable Mesh Training" defines 2DTS as a replacement of volumetric 3D Gaussian primitives with planar 2D triangle facelets carrying opacity and color, composited front-to-back so that the primitive set naturally forms a discrete, mesh-like structure and can be exported directly as a triangle mesh (Sheng et al., 23 Jun 2025).

Subsequent work broadened the design space. "Triangle Splatting for Real-Time Radiance Field Rendering" treats each primitive as a 3D triangle rendered as a differentiable image-space splat with a bounded screen-space kernel, arguing for a "triangle comeback" relative to Gaussians and emphasizing hardware compatibility, adaptive density, and sharp surface alignment (Held et al., 25 May 2025). "Triangle Splatting+" modifies this line by introducing shared vertices, semi-connectivity, and a training strategy that anneals to opaque triangles so that the final output is immediately usable in standard graphics engines without post-processing (Held et al., 29 Sep 2025).

Feed-forward variants extend the same rendering principle beyond per-scene optimization. FTSplat predicts connected triangle surfaces from calibrated multi-view images and then rasterizes them with a differentiable triangle rasterizer and front-to-back opacity blending, which the paper explicitly characterizes as best described by 2D Triangle Splatting rather than volumetric rendering (Jinlin et al., 6 Mar 2026). FLAT decodes triangle splats directly from video diffusion latents and uses differentiable triangle rendering to supervise geometry and appearance, positioning 2DTS as an explicit, surface-aligned primitive for feed-forward scene generation (Kupyn et al., 23 Jun 2026).

A recurring conceptual distinction in the literature is therefore not whether triangles are present, but how they are rendered and optimized. Classical rasterization uses hard inside-outside tests and a z-buffer; 2DTS instead introduces image-space coverage functions and alpha compositing so that visibility remains differentiable during training, while many methods anneal toward sharp or opaque triangles at convergence (Held et al., 29 Sep 2025, Sheng et al., 23 Jun 2025).

2. Primitive representations and geometric parameterizations

Published 2DTS systems span several geometric parameterizations. In "Triangle Splatting for Real-Time Radiance Field Rendering," a primitive is a 3D triangle defined by three vertices $\mathbf{v}_i \in \mathbb{R}^3$, a color $\mathbf{c}$, a smoothness parameter $\sigma$, and an opacity $o$. Appearance is represented by learned spherical harmonics, and the implementation sets the spherical harmonics degree to 3, giving 59 parameters per triangle, matching the parameter count of a single 3D Gaussian primitive in 3DGS (Held et al., 25 May 2025).

"Triangle Splatting+" shifts the parameterization to a global shared-vertex representation,

$$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$

with per-vertex positions, spherical-harmonic color parameters, and opacities, while triangles are index triplets $\mathbf{T}_m=(i,j,k)$. In implementation, degree-3 spherical harmonics produce 48 coefficients per vertex for RGB, yielding 51 parameters per vertex when combined with 3D position. Per-triangle opacity is induced by the minimum of the incident vertex opacities, and color is barycentrically interpolated. Because triangles share vertices, gradients from adjacent faces accumulate at common vertices, producing semi-connectivity during optimization (Held et al., 29 Sep 2025).

The original 2DTS paper uses per-face triangle facelets with geometry $X_i=\{v_{i1},v_{i2},v_{i3}\}$, color $c_i$, base opacity $O_i$, compactness $\gamma$, normal $\mathbf{c}$0, and optional BRDF parameters. Here the key geometry-control variable is the compactness parameter $\mathbf{c}$1, which sharpens edges and drives the representation toward solid triangles as $\mathbf{c}$2 and $\mathbf{c}$3. Initialization comes from point clouds derived from SfM or GS, with each triangle starting as an equilateral facelet centered at a point and scaled by nearest-neighbor distance; vertices are not merged, so the final mesh is generally non-watertight (Sheng et al., 23 Jun 2025).

FTSplat adopts a markedly different, feed-forward geometry construction. From calibrated multi-view images, a shared lightweight ResNet backbone and a 6-layer multi-view Swin Transformer estimate features, multi-view depth is obtained through a cost volume over $\mathbf{c}$4 depth hypotheses, and each pixel is back-projected to 3D to form an initial point cloud. A 2D U-Net and lightweight MLP triangle head then predict per-vertex opacity $\mathbf{c}$5 and spherical-harmonics color $\mathbf{c}$6, while faces are formed by pixel-aligned connectivity. The representation uses vertex positions $\mathbf{c}$7, opacities $\mathbf{c}$8, spherical-harmonics colors $\mathbf{c}$9, and a face tensor $\sigma$0. With one vertex per input pixel across $\sigma$1 views, the method creates

$\sigma$2

vertices and two triangles per pixel via immediate-neighbor connectivity, yielding a globally connected surface over visible regions (Jinlin et al., 6 Mar 2026).

FLAT introduces a ray-centered triangle parameterization tailored to feed-forward decoding from video diffusion latents. Each decoder token predicts depth $\sigma$3, a lower-triangular shape transform

$\sigma$4

with positive diagonal entries, residual orientation angles $\sigma$5, RGB color, opacity, and sharpness. A canonical centered equilateral triangle is transformed by $\sigma$6, lifted into a ray-tangent plane, and then rotated by a residual exponential-map rotation around the local ray-centered frame:

$\sigma$7

Final vertices are

$\sigma$8

The paper argues that this ray-centered rotation parameterization stabilizes feed-forward triangle regression relative to direct global orientation prediction (Kupyn et al., 23 Jun 2026).

These parameterizations imply three broad geometric regimes in current 2DTS research: disconnected triangle soups, semi-connected shared-vertex meshes, and explicitly connected feed-forward surfaces. The literature does not treat these regimes as equivalent with respect to topology, watertightness, or downstream simulation readiness (Held et al., 25 May 2025, Held et al., 29 Sep 2025, Jinlin et al., 6 Mar 2026).

3. Image-space rendering, coverage functions, and differentiability

A defining feature of 2DTS is that triangles are projected to the image plane with a standard pinhole camera model and rendered by image-space splatting or differentiable rasterization rather than volumetric ray marching. Across the cited works, projection follows the usual form

$\sigma$9

with perspective division implicit, and gradients propagate through projected vertices to underlying 3D geometry (Held et al., 25 May 2025, Held et al., 29 Sep 2025, Kupyn et al., 23 Jun 2026).

One major formulation family uses signed edge functions and a screen-space signed distance field. In Triangle Splatting and Triangle Splatting+, each projected triangle defines half-space lines

$o$0

and

$o$1

Let $o$2 denote the incenter. The coverage window is

$o$3

This kernel is bounded by the triangle footprint, equals $o$4 at the incenter, and vanishes on the boundary and outside the triangle. Triangle Splatting explicitly rejects a sigmoid window because it violates bounded support and becomes globally supported for large $o$5, which the paper terms unsuitable for rasterization workloads (Held et al., 25 May 2025, Held et al., 29 Sep 2025).

A second formulation family, introduced in the original 2DTS paper, derives opacity from barycentric eccentricity. For barycentric coordinates $o$6 at a pixel,

$o$7

and the soft mask is

$o$8

This yields soft visibility during training, while the compactness limit

$o$9

recovers solid triangles. The same work derives an exact opacity-integral expression

$$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$0

and uses it for scale compensation as $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$1 increases (Sheng et al., 23 Jun 2025).

FLAT proposes a third coverage variant designed to improve gradient flow in feed-forward training. For normalized edge responses $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$2 and shifted clamped values

$$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$3

the product window is

$$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$4

The paper states that this routes gradients to all three edges rather than only the most active one and that the $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$5 shift preserves non-zero derivatives slightly outside triangle boundaries, improving early-training stability (Kupyn et al., 23 Jun 2026).

FTSplat uses a differentiable triangle rasterizer rather than a soft coverage kernel and sets the smoothing parameter $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$6 to zero, yielding hard triangle boundaries. The paper nonetheless retains differentiability through barycentric interpolation and the referenced differentiable rasterization framework, while photometric losses back-propagate through rasterized colors and opacities to update predicted vertices and per-vertex attributes (Jinlin et al., 6 Mar 2026).

All of these methods use front-to-back alpha compositing. A common form, stated explicitly in several papers, is

$$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$7

with $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$8 determined either by opacity times window function or by interpolated per-pixel opacity. This is the standard transmittance-based blending rule adopted from prior splatting work, and it is central to the claim that 2DTS preserves differentiable visibility while operating with explicit surface primitives (Held et al., 25 May 2025, Held et al., 29 Sep 2025, Sheng et al., 23 Jun 2025, Jinlin et al., 6 Mar 2026, Kupyn et al., 23 Jun 2026).

4. Optimization-based and feed-forward learning paradigms

Optimization-based 2DTS methods typically begin from geometric priors such as SfM points or Delaunay triangulations and then optimize triangle parameters end-to-end under photometric and auxiliary losses. Triangle Splatting initializes one triangle per sparse 3D point, with vertices sampled on a scaled sphere around the point, and combines an $$\mathcal{V}=\{\mathbf{v}_i\in\mathbb{R}^3\mid i=1,\dots,N\},$$9 plus D-SSIM objective with opacity, distortion, normals, and size regularization. Density adaptivity is handled by pruning low-importance triangles and by MCMC-inspired densification: selected triangles are split into four by connecting edge midpoints, while small triangles may be cloned with plane-aligned noise. Densification is performed every 500 steps from iteration 500 to 25,000, adding 30% new triangles each step; the last training phase anneals opacity upward and $\mathbf{T}_m=(i,j,k)$0 downward so that triangles become solid for export (Held et al., 25 May 2025).

Triangle Splatting+ also begins from SfM, but enforces connectivity through a global vertex set and a 3D Delaunay-derived initial mesh. Its training schedule centers on opacity and smoothness annealing. The window exponent $\mathbf{T}_m=(i,j,k)$1 is annealed from $\mathbf{T}_m=(i,j,k)$2 to $\mathbf{T}_m=(i,j,k)$3, while opacity is parameterized with an increasing floor

$\mathbf{T}_m=(i,j,k)$4

A hard pruning step after 5k iterations removes triangles with opacity below approximately $\mathbf{T}_m=(i,j,k)$5, typically eliminating about 70% of triangles and vertices, after which pruning uses maximum blending weight across views. Midpoint subdivision adds detail while preserving connectivity, and the final regime approaches hard visibility with effectively one top-most triangle per pixel (Held et al., 29 Sep 2025).

The original 2DTS direct-mesh-training paper adopts a three-stage schedule. In Stage 1, $\mathbf{T}_m=(i,j,k)$6 and an opacity regularizer polarizes base opacities. In Stage 2, $\mathbf{T}_m=(i,j,k)$7 increases exponentially to $\mathbf{T}_m=(i,j,k)$8, accompanied either by opacity tuning schedules or a straight-through estimator. In Stage 3, $\mathbf{T}_m=(i,j,k)$9 and $X_i=\{v_{i1},v_{i2},v_{i3}\}$0 are fixed while the system is fine-tuned to recover PSNR lost during solidification. Its total objective is

$X_i=\{v_{i1},v_{i2},v_{i3}\}$1

with normal consistency derived from depth gradients. The implementation uses custom CUDA kernels built atop the 3DGS codebase, Adam with exponential learning-rate decay, and no adaptive densification during training (Sheng et al., 23 Jun 2025).

Feed-forward 2DTS replaces per-scene optimization with direct prediction. FTSplat takes calibrated multi-view images and camera poses, extracts per-view features with a shared lightweight ResNet and a 6-layer multi-view Swin Transformer, incorporates monocular depth-aware features from Depth Anything V2, builds a cost volume from warped features over $X_i=\{v_{i1},v_{i2},v_{i3}\}$2 depth hypotheses, predicts depth maps, back-projects pixels to 3D vertices, predicts per-vertex opacity and spherical-harmonic color via a 2D U-Net plus triangle head, and assembles faces by pixel-aligned connectivity. Supervision combines photometric losses with a relative 3D point cloud loss against Depth Anything 3 or VGGT points, and training is end-to-end. At test time, inference is a single forward pass, with reported per-scene reconstruction time of $X_i=\{v_{i1},v_{i2},v_{i3}\}$3 on RealEstate10K at $X_i=\{v_{i1},v_{i2},v_{i3}\}$4 resolution (Jinlin et al., 6 Mar 2026).

FLAT uses a feed-forward generative setting rather than calibrated multi-view reconstruction alone. It reuses the RGB decoder of Wan-2.1, removes the last upsampling stage so that one triangle is predicted per $X_i=\{v_{i1},v_{i2},v_{i3}\}$5 pixels, injects camera conditioning through zero-initialized RPPC ray embeddings, and trains with RGB $X_i=\{v_{i1},v_{i2},v_{i3}\}$6, LPIPS, scale-invariant disparity, normal, and opacity losses. Training follows a progressive four-stage schedule from 320p to 768p, with 17 to 49 views, 200k iterations on $X_i=\{v_{i1},v_{i2},v_{i3}\}$7H100, and pruning of triangles with opacity below $X_i=\{v_{i1},v_{i2},v_{i3}\}$8 at the high-resolution stage. The paper further introduces a lightweight test-time refinement that performs short photometric optimization, aggressive opacity binarization, local densification near boundaries, boundary-vertex merging, and post-repair of vertex positions and colors to obtain game-engine-ready opaque geometry (Kupyn et al., 23 Jun 2026).

These two feed-forward systems illustrate an important expansion of 2DTS: the triangle splatting renderer is not only a final image formation model, but also a differentiable supervisory layer through which explicit triangle geometry can be learned from multi-view inputs or generative latents (Jinlin et al., 6 Mar 2026, Kupyn et al., 23 Jun 2026).

5. Empirical results, trade-offs, and applications

Quantitatively, the literature does not present a single universally dominant 2DTS regime; instead, different formulations prioritize perceptual sharpness, explicit geometry, connectivity, or direct engine deployment. In the original 2DTS paper, the vanilla radiance-field setting with $X_i=\{v_{i1},v_{i2},v_{i3}\}$9 and without compactness tuning, opacity regularization, or normal consistency reported on Mip-NeRF360 Outdoor a PSNR of 25.58 versus 24.71 for 3DGS and 24.65 for 2DGS, with SSIM 0.758 versus 0.732 and 0.727; on NeRF-Synthetic, it reported 33.46 PSNR, 0.969 SSIM, and 0.031 LPIPS; and in the mesh-training setting it reported on NeRF-Synthetic that 2DTS-Mesh achieved PSNR 28.79, SSIM 0.945, LPIPS 0.077, Chamfer Distance 51.9, triangle count 68k, GPU memory 2.85 GB, and time 28 min, while on MatrixCity it reported PSNR 24.15, SSIM 0.716, LPIPS 0.457, Chamfer Distance 0.83, count 5,772k, GPU memory 33.24 GB, and time 131 min (Sheng et al., 23 Jun 2025).

Triangle Splatting emphasizes perceptual quality and throughput among non-volumetric methods. On Mip-NeRF360 it reports average LPIPS 0.191 with FPS 97, improving LPIPS over 2DGS and BBSplat by 25% and 19%, and outperforming 3DGS and 3DCS by 10% and 7% in LPIPS averages. On Tanks and Temples it reports LPIPS 0.143, PSNR 23.14, SSIM 0.857, and FPS 165. The same paper reports training and rendering statistics on A100 of "Ours 39m train, 97 FPS, 795MB," and for the Garden scene it reports over 2,400 FPS at 1280×720 using an off-the-shelf mesh renderer, with supplementary RTX4090 numbers of 2,400, 1,900, and 1,050 FPS at HD, FullHD, and 4K respectively (Held et al., 25 May 2025).

Triangle Splatting+ prioritizes opaque, semi-connected meshes that can be used directly in graphics engines. It reports on Mip-NeRF360 PSNR 25.21, LPIPS 0.294, and SSIM 0.742 with about 2M vertices, and on Tanks & Temples PSNR 20.91, LPIPS 0.249, and SSIM 0.773 with about 2M vertices. Training times are reported as approximately 39 minutes on Mip-NeRF360 and 25 minutes on Tanks & Temples, while the final representation renders at approximately 400 FPS on a consumer laptop. The paper frames these results as state-of-the-art mesh-based novel view synthesis among methods that produce meshes directly usable in engines (Held et al., 29 Sep 2025).

In feed-forward reconstruction, FTSplat reports on RealEstate10K that against optimization-based triangle methods, TriangleSplatting obtained PSNR 18.53, SSIM 0.683, LPIPS 0.279, 30k iterations, and was not connected; MeshSplatting obtained PSNR 19.78, SSIM 0.685, LPIPS 0.340, 30k iterations, and was connected; and FTSplat obtained PSNR 20.39, SSIM 0.707, LPIPS 0.257 in a single forward pass while remaining connected. Against feed-forward Gaussian splatting baselines, Mvsplat and Depthsplat scored 27.03/0.891/0.106 and 27.61/0.903/0.099, whereas FTSplat scored 20.39/0.707/0.257; the paper interprets this as a trade-off in which GS achieves higher photometric metrics while FTSplat yields superior 3D geometric cleanliness and simulation-compatible surfaces. Relative point cloud supervision is reported as crucial, with an ablation without it dropping to 13.06/0.401/0.509, VGGT supervision giving 20.07/0.692/0.275, and Depth Anything 3 giving 20.39/0.707/0.257 (Jinlin et al., 6 Mar 2026).

In feed-forward generative decoding, FLAT presents perhaps the clearest geometry-quality comparison under identical training. On RealEstate10K, FLAT with 3DGS reports 22.39/0.762/0.203, FLAT with 2DGS reports 22.03/0.734/0.219, and FLAT with Triangles/2DTS reports 21.45/0.710/0.245. On DL3DV the respective numbers are 20.71/0.663/0.275, 20.44/0.647/0.284, and 20.04/0.627/0.314. However, the normals metrics favor triangles strongly: normals $c_i$0 and cosine $c_i$1 for 2DTS, versus $c_i$2 and $c_i$3 for 2DGS, while 3DGS normals are described as not meaningful surfaces and are computed via finite differences on depth. Its opaque-mesh conversion reports about 0.5M vertices and RE10K 21.23/0.749/0.388, compared with 2DGS→TSDF at 15.89/0.633/0.468 and 3DGS→GS2Mesh at 14.18/0.619/0.452 (Kupyn et al., 23 Jun 2026).

These results explain why 2DTS has been adopted in domains where explicit geometry matters. The cited works repeatedly mention direct compatibility with Blender and standard engines, Unity- or Unreal-style pipelines, VR headsets, physics colliders, walkable environments, ray tracing, relighting, shadow rendering, robotics, simulation, AR/VR, and interactive walkthroughs (Sheng et al., 23 Jun 2025, Held et al., 29 Sep 2025, Jinlin et al., 6 Mar 2026, Kupyn et al., 23 Jun 2026).

6. Limitations, misconceptions, and open research directions

A common misconception is that triangle splatting automatically produces watertight or manifold meshes. The literature is explicit that this is generally not the case. Triangle Splatting optimizes a triangle soup of unstructured, disconnected triangles and states that producing a connected, watertight mesh still needs post-processing. The original 2DTS paper states that vertices are not merged and the final mesh is generally non-watertight. FTSplat emphasizes connected visible surfaces and simulation compatibility but does not claim strict watertightness, especially across occlusions or disocclusions. FLAT reports that its mesh conversion is not fully watertight and that non-manifold areas can arise in locally dense regions (Held et al., 25 May 2025, Sheng et al., 23 Jun 2025, Jinlin et al., 6 Mar 2026, Kupyn et al., 23 Jun 2026).

Another misconception is that 2DTS is a form of volumetric rendering. In the cited formulations, triangles are projected and rasterized directly in 2D, then blended by front-to-back alpha compositing. FTSplat explicitly states that there is no volumetric accumulation or ray marching, and its rendering is therefore best described as 2D Triangle Splatting rather than volumetric rendering (Jinlin et al., 6 Mar 2026).

Failure modes are likewise consistent across papers. Optimization-based triangle soups can produce outdoor floaters and visibility popping when triangles are sorted by center depth rather than by per-pixel depth, prompting recommendations for per-pixel sorting in the style of StopThePop (Held et al., 25 May 2025). Opaque-triangle methods report degradation in sparsely observed backgrounds, artifacts when moving far outside the training-view orbit, and difficulty with transparent objects such as glass and bottles (Held et al., 29 Sep 2025). The original 2DTS notes sensitivity from piecewise depth ordering and from the piecewise differentiability of the $c_i$4 in the eccentricity term, even though soft opacity and $c_i$5 annealing mitigate this (Sheng et al., 23 Jun 2025). FTSplat reports degraded performance in occluded or weakly observed regions and acknowledges that pixel-level connectivity, while topologically stable, may struggle with fine structures and does not guarantee watertightness (Jinlin et al., 6 Mar 2026). FLAT identifies sensitivity to thin and elongated structures, reflections, semi-transparent content, very fine details, and misorientation early in training; its ray-centered residual rotation and $c_i$6-shifted product window are introduced precisely to mitigate those instabilities (Kupyn et al., 23 Jun 2026).

Open directions stated in the literature are closely tied to these limitations. They include adaptive densification based on gradient magnitudes, encouraging triangles to share vertices or edges to improve topology, triangle-tailored meshing methods, improved visibility consistency through per-pixel sorting, full PBR material modeling, differentiable shadowing, hybrid Gaussian-triangle representations, better background completion, multi-resolution tessellation, and MIP-aware coverage kernels (Sheng et al., 23 Jun 2025, Held et al., 25 May 2025, Held et al., 29 Sep 2025). A plausible implication is that future 2DTS systems will continue to occupy the interface between radiance-field optimization, differentiable rasterization, and conventional mesh pipelines, with the primary unresolved tension being the trade-off between photometric smoothness and explicit geometric fidelity.