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Honeycomb-Surfels: Compact Hexagonal Primitives for Efficient Scene Rendering and Reconstruction

Prakhar Kaushik, Soumava Paul, Alan Yuille

arXiv:2610.37668v1cs.CVcs.AIcs.GRcs.LG

TL;DR

Resource-intensive radiance-field and splatting methods motivate a compact representation that balances appearance and geometric fidelity. Hexels use planar hexagonal elements with local color variation; across 3D benchmarks, they improve quality-efficiency trade-offs and also apply to surface reconstruction and 2D images.

  • Problem

    Existing scene representations face a trade-off between how much a primitive can represent and its cost, while compact budgets must preserve both geometric fidelity and appearance expressivity.

    Source: §1 · p. 2

  • Method

    Hexels are planar surface elements that combine a soft hexagonal support with a first-order spatial color field to represent coherent regions and smooth appearance variation.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    Across three 3D benchmarks, Hexels improve PSNR by up to 7.38 dB at matched model memory, with lower peak training VRAM and rendering up to 2.31× faster than 2DGS.

    Source: §5 · p. 11

  • Takeaways & Limitations

    The same primitive performs strongly for memory-constrained surface reconstruction and directly generalizes to efficient 2D image representation.

    Source: §1 · p. 2, §5 · p. 11

    A first-order color model cannot represent arbitrary texture within one primitive; high-frequency or uncorrelated appearance requires smaller primitives or a richer color model, reducing the expected benefit of local smoothness.

    Source: §5 · p. 11

Abstract

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Novel view synthesis and 3D reconstruction have advanced rapidly with radiance fields and splatting, yet their memory and computational demands limit their use in resource-constrained settings. We introduce Hexels: novel planar hexagonal surface elements with a first-order spatial color field for efficient 3D scene and 2D image representation. Hexels exploit locally planar regions with smoothly varying appearance using fewer, larger primitives. Their hexagonal support provides an area- and boundary-efficient domain, while the local color field captures appearance variation within each primitive. Across 34 scenes, Hexels improve PSNR over memory-matched 2DGS by 4.53-7.38 dB across datasets while requiring lower training VRAM. We further ablate Hexels by showing that per primitive, Hexels outperform 2DGS by upto 4.74 dB while using lower training VRAM and having 2.31 times faster FPS. Even in unconstrained settings, Hexels offers complementary quality-memory trade-offs. For efficient 3D surface reconstruction, Hexels reduce error by 28.9% compared with memory-matched 2DGS across 15 DTU scans. On 6 Tanks and Temples surface scenes, Hexels achieves the highest mean F1 among compared methods (34.38%, versus 31.72% for 2DGS). We further also show that Hexels, as a general primitive, improves 2D image representation over Image-GS. Hexels push the Pareto frontier on the quality-efficiency trade-off of surface-aligned primitives.

1 Introduction

The paper seeks surface primitives that balance geometric fidelity, appearance expressivity, and resource use. Hexels combine a tileable hexagonal support with a local color field, and show strong quality–efficiency results across rendering, reconstruction, and image representation.

1 Introduction

  • Prior splats often use one view-conditioned color per primitive, requiring overlapping elements for spatial variation, while texture or neural-field approaches add representational complexity.The paper frames efficiency to include rendering speed, model size, and training memory.Source: §1 · p. 2
  • At constrained representation budgets, the paper targets both surface reconstruction accuracy and novel-view rendering quality because training-image agreement alone does not guarantee accurate geometry [36].Source: §1 · p. 2
  • Hexels combine a soft hexagonal support with a first-order spatial color field, letting one larger primitive represent coherent regions with smooth color variation.Smaller Hexels can capture boundaries, curvature, and fine detail; the hexagon tiles without gaps and covers more area per boundary than triangles or squares.Source: §1 · p. 2
  • At matched model sizes, Hexels improve PSNR over 2DGS by 4.53–7.38 dB with up to 3.6× less VRAM, and over Triangle Splatting by 1.70–2.90 dB with up to 1.87× higher rendering FPS.At matched primitive budgets, they improve PSNR over 2DGS by up to 4.74 dB and render up to 2.31× faster; Hexels also improve surface reconstruction and 2D image representation.Source: §1 · p. 2
  • Hexels are presented as a compact, self-contained primitive family for novel-view synthesis, surface reconstruction, and 2D image representation without textures, neural appearance fields, or decoding networks.The same underlying representation is evaluated across these tasks.Source: §1 · p. 2

2 Related Work

Related work progresses from neural radiance fields to explicit splats and planar surfels, improving rendering speed and geometry. The paper frames the remaining challenge as balancing quality with storage, training memory, primitive count, and rendering speed.

2 Related Work

  • NeRF [25], explicit Gaussian primitives [18], and planar surfels such as 2DGS [14] mark a shift toward faster rendering and more accurate geometry.Source: §2 · pp. 2–3
  • A Hexel is a planar patch with a soft hexagonal boundary, local position and orientation parameters, and a linear color field whose boundary sharpness is controllable.Source: Figure 2 · p. 3
  • Efficient representation requires quality alongside low model storage, low training VRAM, few primitives, and high FPS.Source: §2 · p. 3
  • Prior methods improve parts of this trade-off but may add memory, VRAM, or computation, while Hexels redesign the primitive to target several efficiency axes together.Hybrid methods shift capacity into textures or neural fields; Hexels also apply the same underlying design to oriented 3D surfaces and 2D images.Source: §2 · p. 3

3 Hexels: Hexagonal Surface Elements

Hexels combine hexagonal planar support with a first-order spatial color field, letting one primitive cover a larger coherent region while modeling smooth appearance variation. The section applies this representation to 3D splatting and 2D images, where Hexels outperform Image-GS on reported quality metrics and converge faster.

3 Hexels: Hexagonal Surface Elements

  • The support is a soft hexagon whose sharpness parameter controls its edge transition while its tangent scales set its spatial extent.A sigmoid weight is applied to the hexagon contour, and negligible tails are truncated by the rasterizer.Source: §3 · p. 3
  • The first-order color field models smooth within-primitive variation; view-independent V1 uses 20 parameters per Hexel versus 58 for the evaluated 2DGS primitive.V2 adds a linear view-dependent residual for 29 total parameters; V3 replaces that residual with degree-three spherical harmonics.Source: §3 · p. 4
  • Hexels use perspective-correct ray-plane intersections and depth-ordered compositing, replacing the Gaussian weight with hexagonal support and evaluating spatial color within each primitive.Training uses calibrated images and Adam; unconstrained training additionally clones or splits primitives with large position gradients and prunes mature primitives with negligible contributions.Source: §3 · p. 4

3.1 Representing 2D Images as Hexels

  • For 2D images, Hexels evaluate the support directly at each pixel and blend the up to 10 highest-weight Hexels assigned to its tile.This setting omits projection, ray-plane intersection, opacity, and depth ordering.Source: §3.1 · p. 5
  • 35.82 dB PSNR versus 33.16 dB and .0924 LPIPS versus .1372 for ImageGS; Hexels used 5.1K iterations and 157.07 seconds.In two 2048×2048 examples with 10K primitives per method, Hexels preserves finer boundaries and structure with lower RGB error.Source: Figure 3 · p. 4, §3.1 · p. 5, Table 1 · p. 5
  • Hexels converge faster than Image-GS on the Image-GS benchmark while outperforming it in PSNR and SSIM.The comparison is shown in Figure 4.Source: Figure 4 · p. 5

4 Experiments

Across novel-view synthesis and surface reconstruction, Hexels offer strong quality–efficiency trade-offs under constrained budgets and remain competitive in unconstrained comparisons. Their direct surface-cell export and image-fitting ablations further show how the primitive supports downstream use and combines shape and appearance design choices.

4.1 Novel View Synthesis

  • Across three benchmarks, matched-memory Hexels improve PSNR over 2DGS by 4.53–7.38 dB, with up to 3.6× less VRAM and 1.54–1.87× its rendering FPS.This is the paper’s primary efficiency setting; Hexels establish the strongest overall quality–efficiency trade-off among evaluated methods.Source: §4.1 · p. 6, Table 2 · p. 6
  • At equal primitive budgets, Hexels V2 improve PSNR over 2DGS by up to 4.74 dB, use up to 7× less peak training VRAM, and render up to 2.31× faster.Hexels V2 also use 1.76–1.95× less deployment memory; V3 has the highest mean PSNR among the compared methods across all three datasets.Source: §4.1 · pp. 6, 8, Table 3 · p. 6
  • In native settings, densified V3 is within 0.30 dB of 2DGS on Mip-NeRF360 while rendering 2.06× faster and using 2.00× less deployment memory.On DTU and Tanks and Temples, it improves PSNR over 2DGS by 0.90 dB on each dataset, with rendering speedups of 2.26× and 1.37×, respectively.Source: §4.1 · p. 8, Table 4 · p. 8

4.2 Efficient Surface Reconstruction

  • On 15 DTU scans, Hexels reach 0.978 mm mean Chamfer distance, versus 1.375 mm for memory-matched and 1.449 mm for primitive-matched 2DGS.On six Tanks and Temples scenes, Hexels achieve 34.38% mean F1 versus 31.72% for unconstrained 2DGS; the compared settings differ, so these values are not a matched-budget comparison.Source: §4.2 · pp. 8–9, Table 5 · p. 9, Table 6 · p. 9

4.3 Applications

  • Hexels export explicit surface cells in constant time per primitive without depth rendering, voxel grids, or Marching Cubes.The cells remain associated with parent Hexels, preserving access to geometry and appearance for downstream optimization.Source: §4.3 · pp. 9–10
  • Fixed-population Hexels retain reconstruction quality with little model memory and training VRAM, without relying on texture atlases, neural decoders, pruning, or quantization.Pruning and quantization remain optional extensions for reducing deployment storage.Source: §4 · p. 5, §4.3 · p. 11

4.4 Additional Ablation

  • In planar image fitting, the linear color-gradient Hexel reaches 37.07 dB on the shaded target versus 35.13 dB for the same-appearance Gaussian, with 2.48 versus 10.47 mean blended contributions per pixel.Hexels achieve higher PSNR than Gaussians with either appearance model; adding spatial gradients improves both shapes.Source: §4.4 · p. 11
  • On detailed artworks, Hexels reduce mean blends per pixel by 71–80% relative to Gaussian counterparts while achieving similar perceptual quality.The ablation compares the support shapes and appearance models under the same primitive count, initialization, loss, and optimization.Source: §4.4 · p. 11

5 Conclusion

Hexels establish a quality–efficiency frontier for surface-aligned representations, while the appendix details their parameterization, appearance field, support derivatives, and rasterization. Their design and rendering also have stated limits, including planar primitives and appearance that may require finer primitives or a richer model.

5 Conclusion

  • Hexels improve PSNR by up to 7.38 dB at matched model memory, use lower peak training VRAM and up to 2× less learned state at matched primitive count, and render up to 2.31× faster than 2DGS.The conclusion also reports strong performance for memory-constrained surface reconstruction and 2D image representation.Source: §5 · p. 11
  • The limitations include first-order color’s inability to represent arbitrary within-primitive texture, sparse initialization’s potential to leave coverage gaps, and independent planar cells that do not form a watertight mesh.High-frequency or uncorrelated appearance may require smaller primitives or a richer color model, reducing the expected benefit of local smoothness.Source: §5 · p. 11

A Hexel Parameterization

  • The stored-parameter variants use 20, 29, or 65 learned scalars per Hexel, depending on whether the appearance model is V1, V2, or V3.Geometry and support account for 11 scalars; V1 adds nine, while V2 and V3 add nine and 45 coefficients, respectively.Source: A Hexel Parameterization · p. 15

A.1 Appearance

  • V1 represents spatial appearance with a base RGB color and two local-coordinate RGB gradients, while V2 and V3 add view-dependent residuals that are constant across a primitive for a fixed camera.V2 uses a linear residual and V3 uses spherical-harmonic coefficients.Source: §A.1 · p. 15

A.2 Derivatives

  • The canonical hexagon is defined by six outward side normals, with its zero level set as the boundary, negative values inside, and positive values outside.At ties among maximizing projections, the support function has no ordinary gradient, but any tied maximizing normal is a valid subgradient for optimization.Source: §A.2 · p. 15

B Rasterization

  • Rasterization recovers perspective-correct local coordinates by ray–plane intersection, then uses those coordinates for both hexagonal support and the spatial color field.Tile-level depth sorting is shared across pixels; accumulated color uses transmittance-weighted contributions and the remaining transmittance for the background.Source: B Rasterization · pp. 17–18

C Densification and Pruning

The densified variants use mixed growth and contribution-based pruning, while the primary efficiency experiments keep Hexels within memory or primitive limits. Unconstrained densified comparisons are also reported but are not the focus.

C Densification and Pruning

  • Primary Hexel experiments avoid densification and pruning as a way to exceed memory or primitive limits, while unconstrained densified counterparts are reported as secondary comparisons.Baseline methods are allowed to temporarily exceed those constraints because they can perform much worse when forced to remain below them.Source: C Densification and Pruning · p. 18
  • Mixed growth clones small parent Hexels and splits large ones, moving new primitives within the parent tangent plane and copying their appearance coefficients.A split replaces the parent with two children whose scales are divided by 1.6.Source: C Densification and Pruning · p. 18
  • Contribution-based pruning scores each Hexel by its mean blending weight across training views, with scheduled pruning thresholds and a limit on removals per trigger.The schedule uses a 10^-7 threshold on q_i, a 500-iteration grace period, and removes at most 2% per pruning trigger.Source: C Densification and Pruning · p. 18

D 2D Image Representation

The 2D representation places Hexels directly in the image plane, retaining hexagonal support and first-order spatial color while replacing 3D projection and depth-ordered compositing. The evaluation also measures support overlap alongside reconstruction quality.

D 2D Image Representation

  • Hexels represent images in the image plane using hexagonal support and first-order spatial color, without 3D projection or depth-ordered compositing.Rendering retains the ten largest support weights per pixel and blends their colors without transmittance, depth ordering, or learned opacity.Source: D 2D Image Representation · pp. 18–19
  • The 45-image Image-GS benchmark is evaluated at 2048×2048 resolution, with reconstruction metrics and primitive-overdraw measures used to compare representation quality and overlap.Overdraw summarizes mean support coverage and fractions of pixels with zero or multiple supports; matched-quality comparisons test whether Hexels need fewer primitives.Source: D 2D Image Representation · p. 19

E.1 Datasets and Metrics

  • The broader evaluation covers 34 scenes across Mip-NeRF360, DTU, and Tanks and Temples, holding out every eighth camera and averaging metrics over views and then scenes.PSNR and SSIM use float RGB images, while LPIPS-VGG inputs are normalized.Source: §E.1 · p. 19

E.2 Initialization and Training

The appendix compares Hexel variants, reports implementation and evaluation settings, and describes analytical surface-cell extraction. In the appearance ablation, densified V3 achieves the highest PSNR across the three datasets.

E.2 Initialization and Training

  • Hexels start from sparse COLMAP points and colors, jointly optimize geometry and appearance for 30K iterations, and use Adam with a photometric loss weighted by SSIM.The listed learning-rate schedule is for Hexel V1–V3 NVS models.Source: §E.2 · p. 20, Table 7 · p. 20

E.3 Baseline Settings

  • Baseline memory comparisons include primitive and appearance parameters, textures, feature grids, and decoders; matched-primitive experiments cap final populations at the scene’s SfM point count.MeshSplatting is matched at initialization, though native remeshing can change its final memory.Source: §E.3 · p. 20

E.4 Timing and Memory

  • Peak training VRAM is sampled process GPU memory during training and averaged per scene for dataset values; model memory instead measures uncompressed FP32 inference parameters.FPS measurements exclude loading, image transfer, and metric computation.Source: §E.4 · p. 20

E.5 Hexel Surface-Cell Representation

  • Hexel boundaries can be mapped analytically into world-space vertices in constant time per primitive, unlike Gaussian surfels’ smooth representations without an explicitly finite boundary.The passage identifies this as a differentiable surface-cell representation.Source: §E.5 · p. 21

E.6 Spatial Appearance Ablations

  • Densified V3 achieves the highest PSNR across Mip-NeRF360, DTU, and Tanks and Temples; fixed variants require less memory and render faster.On DTU, fixed V3 has slightly higher PSNR than densified V3 (29.95 versus 29.76 dB), while densified V3 improves LPIPS (0.268 versus 0.293).Source: §E.6 · p. 21

F Surface Reconstruction Ablations

The DTU ablations identify surface losses, MVS initialization, and appearance/population choices that affect reconstruction accuracy and memory. The accompanying analysis motivates Hexels’ larger local patches through affine color fields and compares their approximation capacity.

F Surface Reconstruction Ablations

  • 0.978 mm CD with both surface losses is lower than 1.575 mm with neither, 1.246 mm with depth distortion alone, and 0.986 mm with normal consistency alone.Normal consistency provides the larger individual improvement, with a smaller complementary benefit from depth distortion; experiments use 15 DTU scans and mean Chamfer distance in millimeters.Source: F Surface Reconstruction Ablations · p. 21, Table 8 · p. 21
  • 14.6% lower mean CD results from MVS rather than SfM initialization for fixed V3 with both losses, from 1.146 mm to 0.978 mm at unchanged primitive count and model memory.The last row of Table 8(a) is the model used in Table 5.Source: F Surface Reconstruction Ablations · p. 21, Table 8 · p. 21
  • 0.922 mm is the lowest CD among appearance and population variants, achieved by densified V3 initialized from MVS, versus 8.83 MB for fixed V3 and 33.17 MB for the densified model.V3 has the lowest CD with either initialization or population; MVS beats SfM in every variant, while densification improves CD in five of six appearance-mode and initialization pairs.Source: F Surface Reconstruction Ablations · p. 21
  • 34.38% versus 31.72% mean F1 on Tanks and Temples favors Hexels over 2DGS, while unconstrained DTU CD is 0.978 mm for Hexels versus 0.794 mm for 2DGS.Hexels uses 0.61 GiB VRAM versus 1.93 GiB for 2DGS; displayed close-ups show fewer specified artifacts than Triangle Splatting and memory-matched 2DGS.Source: F Surface Reconstruction Ablations · pp. 21–22, Figure 13 · p. 24

G Why Hexagons and Spatial Color?

  • Twice the area of an affine triangle and about 30% more than an affine square can be covered by the largest affine hexagon within the same local color-error region.The comparison applies to one planar patch at a fixed viewing direction; the affine field represents local color slope, with remaining error controlled by how that slope changes across the patch.Source: G Why Hexagons and Spatial Color? · p. 22

G.1 Linear Color

  • An affine color field captures a constant and linear color change, while its remaining approximation error depends on the target’s second derivatives.The stated bound is for twice continuously differentiable functions; the analysis establishes available approximation capacity, not that optimization reaches the Taylor coefficients.Source: §G.1 · pp. 22–23
  • A nonconstant affine color ramp is represented exactly by the affine field, unlike one constant color, although blending constant-color primitives can also create spatial variation.The affine field accounts for local slope, leaving error controlled by changes in that slope.Source: §G.1 · p. 22

G.2 Patch Shape

The patch-shape analysis compares tileable affine polygons under a shared color-error bound, while the compositing analysis shows that local patch accuracy alone does not determine final pixel accuracy.

G.2 Patch Shape

  • The maximizing patch is centered, and equality holds for an orthogonal affine transform; mapping back to original coordinates scales area by 2ε/√det Q.The analysis concerns the largest affine image contained in the error ellipse.Source: §G.2 · p. 25
  • Among affine images of a regular triangle, square, and hexagon inside the same sufficient color-error ellipse, the hexagon covers the largest area.The matrix bounding color-slope changes is not a surface-curvature matrix and is not estimated during training.Source: §G.2 · pp. 24–25

G.3 3D Alpha Compositing

  • If every contributing color is within e of the pixel target, image error is at most (1 − T)e + T‖b − f‖, including remaining background transmittance.The bound applies to contributions actually accumulated by the renderer and is independent of primitive shape.Source: §G.3 · p. 25
  • A local Taylor bound applies only when the intersection’s target is the desired pixel color; unclamped Hexel colors mean D = 1 cannot be assumed for other contributions.The soft support extends beyond the nominal hexagon, where the local Taylor bound may not apply.Source: §G.3 · p. 25

H Extended Related Work

The related-work discussion frames Hexels as a native surface primitive for jointly addressing representation quality and efficiency, rather than moving capacity into separate appearance representations or post-hoc compression. The full results report memory-matched comparisons and the effects of densifying Hexels.

H Extended Related Work

  • Hexels use an affine hexagonal patch with a first-order color field to represent larger, locally smooth surface regions with fewer elements.The analysis reports twice the area of the corresponding affine triangle and about 1.3× that of the square under the same local approximation-error ellipse.Source: H Extended Related Work · pp. 26–27
  • Hybrid approaches move appearance capacity into textures, feature fields, or neural networks, adding memory, training state, or computation beyond geometric primitives.The comparison counts each method’s full learned representation; these approaches are not the objective of this work.Source: H Extended Related Work · p. 26
  • Post-hoc pruning and compression begin with a chosen primitive, whereas Hexels target efficiency in the native surface primitive before those additions.The paper notes that pruning, quantization, texture atlases, and neural decoders could still be applied on top of Hexels.Source: H Extended Related Work · pp. 26–27
  • Hexels also serve as 2D image primitives, outperforming Image-GS in reconstruction quality and training convergence under the evaluated settings.The same hexagonal support and appearance model is placed in the image plane for 2D or oriented on a surface patch for 3D.Source: H Extended Related Work · p. 27

I.1 Matched Model Memory

  • At V2, Hexels exceeds Triangle Splatting by 1.40 dB outdoors and 2.19 dB indoors on Mip-NeRF360; at V3, PSNR gains are 0.54/0.55 dB there and 2.73/0.83 dB on DTU/T&T.Across V2/V3 budgets, Hexels has higher PSNR than MeshSplatting across all three datasets, with higher SSIM and lower LPIPS.Source: §I.1 · p. 27

I.2 Fixed and Densified Hexels

  • Densified V3 gains 1.37 dB outdoors and 0.78 dB indoors on Mip-NeRF360, while deployment sizes rise from 17.0 to 337.6 MB and 48.2 to 133.2 MB, respectively.On DTU, densified V3 has 0.20 dB lower PSNR than fixed V3 but 0.026 lower LPIPS; on Tanks and Temples it gains 0.76 dB.Source: §I.2 · p. 27

I.3 Deployment Compression analysis

The compression analysis covers lossless compression and appearance quantization for 3D models, plus mixed-bit storage for 2D models. The main NVS comparisons use uncompressed FP32 models.

I.3 Deployment Compression analysis

  • The main NVS experiments use uncompressed FP32 models; in separate PQ1024 experiments, RGB appearance vectors are replaced by 1,024-entry codebooks, while geometry and base color remain lossless.Parameters are decoded to FP32 before rendering, without retraining.Source: §I.3 · p. 28
  • 0.01674 dB mean PSNR loss accompanies V3-G PQ1024, whose mean deployment size across 34 scenes is 43.87 MB, 60.09% below lossless compression.The reported mean PSNR/SSIM/LPIPS changes are −0.01674 dB/−0.000259/+0.000849; the largest dataset-mean PSNR loss is 0.0303 dB on Mip-NeRF360.Source: §I.3 · p. 28
  • 0.03307 dB for constant-color Hexels and 0.01091 dB for linear-color Hexels are the mean PSNR losses from mixed-bit storage relative to FP32 parameters.At 10K primitives, the constant- and linear-color files occupy 128524 and 253524 bytes, respectively, versus 161024 bytes for 16-bit Image-GS.Source: §I.3 · pp. 28–29

J Per-Scene NVS Results

The appendix reports V3 densified results across Mip-NeRF360, DTU, and Tanks and Temples, alongside additional qualitative comparisons with other surface methods.

J Per-Scene NVS Results

K Additional Qualitative Comparisons

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