Differentiable Triangle Soup Parameterization

• The paper introduces a differentiable framework that treats collections of triangles as learnable primitives with optimized geometry, color, and opacity.
• It leverages soft splatting and binary masking for gradient-stable differentiable rendering, enabling efficient view synthesis and scene reconstruction.
• The approach supports adaptive optimization, real-time editing, and mesh extraction while balancing connectivity, topology, and rendering performance.

A differentiable triangle soup parameterization is a representation and optimization framework in which a collection of triangles, generally unconnected and typically semi-transparent or opaque, is treated as a set of learnable primitives with all associated geometry, color, and appearance parameters directly optimized in a differentiable fashion. This paradigm stands at the intersection of mesh-based modeling, neural rendering, and radiance field simplification, allowing end-to-end optimization for high-fidelity, real-time scene reconstruction, view synthesis, mesh extraction, and interactive editing. Differentiability is achieved not only with respect to per-vertex locations and attributes, but also the rendering/rasterization process, and, in some approaches, the topology or the soft notion of connectivity.

1. Core Parameterization: The Triangle Soup Model

A triangle soup consists of a set of $N$ (typically $10^3$–$10^4$) triangle primitives, each defined by a minimal set of parameters. Two dominant parameterization families exist:

• Vertex-based indexing with shared vertices: Each vertex $v_i$ encodes a 3D position $(x_i, y_i, z_i)$, an RGB/color feature $c_i$, and opacity $o_i\in[0,1]$. Triangles are index triplets into this vertex set, supporting inherent mesh-like connectivity through shared indices. Within a triangle, color is barycentrically interpolated; triangle-level opacity may be taken as $\min(o_i,o_j,o_k)$ (Held et al., 29 Sep 2025, Fry et al., 29 May 2026).
• Disjoint per-triangle (object-centric) parameters: Each triangle $t_n$ is parameterized by a local frame: reference point $\mu_n\in\mathbb{R}^3$, per-vertex scales $10^3$0, a rotation $10^3$1, per-vertex color (typically as spherical harmonics coefficients) $10^3$2, scalar opacity $10^3$3, and a diffuse sharpness parameter $10^3$4 controlling edge smoothness. This enables each triangle to evolve independently, with no enforced global topology (Burgdorfer et al., 29 May 2025).

A hybrid approach is seen in dynamic modeling, where "core" triangles are linked to secondary ("sub") Gaussians or faces through trainable barycentric offsets, with mutable per-frame deformations for motion or editing (Waczyńska et al., 2024). Every parameter is exposed to gradient-based optimization.

2. Differentiable Rendering and Splatting Formulations

Differentiability is maintained through the rendering pipeline by employing smooth or unbiased rasterization schemes capable of providing exact or gradient-stable derivatives through triangle-based occlusion, blending, and shading.

Soft Splatting (Continuous Window Functions)

A projected triangle is described by its image-plane signed distance field (SDF):

$10^3$5

where $10^3$6 and $10^3$7 are the outward normal and offset for edge $10^3$8. A window function (splat) is defined as

$10^3$9

where $10^4$0 is the incenter, $10^4$1 a sharpness parameter. $10^4$2 supports a soft transition from triangle interior ($10^4$3) to exterior ($10^4$4), controlled by annealing $10^4$5 during training for convergence to sharp opaque boundaries (Held et al., 25 May 2025, Held et al., 29 Sep 2025, Fry et al., 29 May 2026).

Compositing across triangles follows a depth-sorted alpha-blending chain:

$10^4$6

with per-triangle color $10^4$7, opacity $10^4$8 (Held et al., 29 Sep 2025).

Binary/Discrete Masking

Alternative schemes use a non-smooth (binary) per-pixel opacity with stochastic masking to retain unbiased gradient estimates:

• Each triangle $10^4$9 exposes per-barycentric opacity $v_i$0.
• For each "fragment" (triangle-pixel intersection), sample $v_i$1.
• The first fragment with $v_i$2 and minimal depth $v_i$3 is selected as visible.
• Gradients for $v_i$4 parameters are computed via likelihood-ratio ("score function") estimators (Tojo et al., 28 Mar 2026).

This approach obviates the need for mollifiers or smooth rasterization and matches classic hardware rasterization behavior.

Neural Texturing and Per-Vertex/Face Attributes

Surface color features (with dimension $v_i$5) are either interpolated over the barycentric grid or decoded by small MLPs, and per-triangle or per-vertex alpha is used for soft blending and pruning (Tojo et al., 28 Mar 2026).

3. Optimization Pipelines and Loss Design

The differentiable triangle soup allows direct end-to-end training from multi-view image data:

• Initialization: Typical pipelines begin with sparse SfM points/camera poses, tetrahedralization or other initial triangulation, and default vertex attributes (Held et al., 29 Sep 2025, Held et al., 25 May 2025, Burgdorfer et al., 29 May 2025, Fry et al., 29 May 2026).
• Loop: Each batch samples rays/pixels, projects vertices into the image, computes SDF-based or hard-masked rasterized colors, and evaluates a compound loss with gradients assembled for all triangle/vertex parameters.
• Losses:
  • Photometric loss ($v_i$6, D-SSIM) between rendered and observed images.
  • Normal and geometric consistency losses, using rendered normals vs. sensor or depth-derived normals.
  • Opacity regularization, penalizing unnecessary or redundant triangles (opacity $v_i$7).
  • Equilateral/regularization, discouraging degenerate triangles.
  • Connectivity forces, soft penalties aligning endpoints and normals of adjacent triangle edges, encouraging watertight surfaces ('soft zipping' (Burgdorfer et al., 29 May 2025)).
  • Depth smoothness leveraging unsupervised MVS.
• Adaptive structure: Pruning of low-contribution (low-weight) triangles and densification via iterative subdivision or MCMC schemes, preserving coverage and facilitating convergence to mesh-like representations (Held et al., 29 Sep 2025, Held et al., 25 May 2025, Burgdorfer et al., 29 May 2025, Fry et al., 29 May 2026).
• Finalization: The parameter set at convergence corresponds to a set of nearly opaque, contiguous, and watertight triangles, exportable to mesh formats for rendering or downstream simulation (Held et al., 29 Sep 2025, Held et al., 25 May 2025).

4. Connectivity, Topology, and Differentiable Triangulation

The triangle soup model is, by default, unstructured — triangles may be fully disconnected, interact only by spatial adjacency or soft forces, and lack explicit mesh topology (Burgdorfer et al., 29 May 2025, Waczyńska et al., 2024, Tojo et al., 28 Mar 2026). However, several mechanisms enable structure:

• Shared vertex pools: Triangles sharing indices are coupled via parameter gradients, linking their geometry and facilitating global surface continuity (Held et al., 29 Sep 2025, Fry et al., 29 May 2026).
• Soft connectivity losses: The "connectivity force" loss penalizes pairwise endpoints or normal mismatches between proximate triangle edges, but does not enforce hard mesh manifolds (Burgdorfer et al., 29 May 2025).
• Deferred mesh extraction: At inference or for downstream tasks, the triangle soup can be triangulated into a mesh via restricted Delaunay or weighted Delaunay tesselation, using vertex positions and pruning faces with low opacity. This "postprocessing" produces manifold meshes suitable for simulation or editing (Fry et al., 29 May 2026, Rakotosaona et al., 2021).
• Differentiable surface triangulation: Recent approaches formulate the mesh connectivity itself in a differentiable manner by "soft relaxing" the (weighted) Delaunay test using sigmoid-based inclusion scores for each potential triangle, allowing optimization over both vertex locations and mesh topology (Rakotosaona et al., 2021). This methodology generalizes to 3D surfaces via patch-wise parameterization and blending along seams, and supports arbitrary geometry-aware or application-specific losses.

Parameterization	Connectivity	Differentiability	Applications
Shared-vertex index triplet	Semi-connected	Forward & backward	High-quality mesh export
Per-triangle local frame	Unstructured	Full	Editable, dynamic soups
Soft Delaunay triangulation	Topology relaxed	Full (geometry+top)	Remeshing, adaptation

5. Applications, Performance, and Integration

Differentiable triangle soup parameterizations have broad application across graphics and vision tasks:

• Novel view synthesis and radiance field simplification: Triangle-based models provide competitive or superior photometric and geometric performance compared to Gaussian splatting and volumetric approaches, with improved rendering throughput (e.g., >2,400 FPS at $v_i$8 on commodity GPUs for real scenes) and compact model size ($v_i$96–7M parameters for scenes fitting in consumer memory) (Held et al., 25 May 2025, Tojo et al., 28 Mar 2026).
• 3D dynamic scene modeling and editing: The soup formalism supports explicit per-triangle or per-group deformation and easy control of temporal trajectories at inference, facilitating interactive editing with direct gradient access (Waczyńska et al., 2024).
• SLAM and online photogrammetry: Differentiable triangle soups are integrated into real-time SLAM systems, supporting joint refinement of geometry, appearance, and camera poses online, and producing immediately usable mesh maps for collision, simulation, or robotics (Fry et al., 29 May 2026).
• High-order finite elements and mesh lifting: Parameterized edges and projection mappings allow the construction of high-order curved elements conforming to analytic or sampled boundaries, with controllable bounds on homeomorphic mapping and Jacobians (Rangarajan et al., 2011).
• Remeshing and geometric optimization: Differentiable surface triangulation extends the triangle soup paradigm to topology optimization, mesh generation with prescribed element sizing, and curvature-alignment for geometric processing (Rakotosaona et al., 2021).

6. Challenges, Advantages, and Future Directions

The differentiable triangle soup approach offers several advantages over traditional mesh or voxel/volumetric models:

• Direct compatibility with standard graphics hardware and mesh-based pipelines.
• Full end-to-end optimization of all geometry and appearance, including topology in some cases.
• Adaptive model complexity (via pruning/splitting).
• Soft enforcement of geometric regularity and connectivity without topological constraints.

However, challenges and future research directions include:

• Handling global mesh manifoldness and watertightness beyond local soft forces, especially for highly non-convex or topology-changing surfaces.
• Managing gradient stability near silhouette boundaries and under extremely sparse triangle counts.
• Extension to high-genus and multi-object scenes with interacting dynamics.
• Further improvement in efficiency and memory scaling for on-device inference and high-resource applications.

Continued development in differentiable connectivity, topology, and hierarchical parameterizations is anticipated to improve the robustness, adaptability, and accessibility of triangle soup methodologies throughout geometry processing, computer graphics, and computer vision (Fry et al., 29 May 2026, Held et al., 29 Sep 2025, Tojo et al., 28 Mar 2026, Rakotosaona et al., 2021, Burgdorfer et al., 29 May 2025).