Deformable Beta Splatting

• 3DGS’s Gaussian kernels blur fine geometry and require hard-coded cutoffs, while low-order SH limits view-dependent color representation.Source: §1 · p. 2
• Regularized opacity enables distribution-preserved densification independent of kernel type, while a redefined Beta noise function maintains consistency with opacity constraints.Source: §1 · p. 2
• DBS uses 45% of the parameters required by 3DGS and renders 1.5x faster than 3DGS-MCMC, while outperforming 3DGS- and NeRF-based methods across all benchmarks in visual quality.Source: §1 · p. 2
• DBS uses bounded, shape-adaptive Beta Kernels to represent flat surfaces, sharp edges, and smooth regions with higher geometric fidelity.Source: §1 · p. 2
• Spherical Beta decouples diffuse and specular color, models sharp highlights, and uses 31% of the parameters of degree-3 SH.Source: §1 · p. 2

2.1 Neural Radiance Fields

• NeRF research improves convergence, efficiency, memory use, and scalability, but dense ray sampling and network evaluations hinder real-time use, while implicit geometry complicates integration with explicit-geometry pipelines.Source: §2.1 · pp. 2–3

2 RELATED WORK

• 3DGS offers an explicit, computationally efficient alternative to NeRF, but can lag in photorealism, fine detail, aliasing, and complex view-dependent effects.Source: §2.2 · p. 3

2.2 3D Gaussian Splatting

• 3DGS extensions use anti-aliasing and multiscale methods to reduce jagged edges and shimmering, and deblurring methods to sharpen reconstructions from blurred inputs.Source: §2.2 · p. 3
• Alternative splatting kernels target sharper edges, better occlusion modeling, or faster computation, but deformable kernels lack aligned view-dependent color models and dedicated densification strategies.Source: §2.2 · p. 3
• Appearance methods trade speed against fidelity, while Gaussian densification methods remain kernel-specific, motivating explicit diffuse-specular modeling and kernel-agnostic densification.Source: §2.2 · p. 3

3.1 The Beta Kernel

• The Beta Kernel has bounded support and adaptive shape control, and its parameterization starts Gaussian-like before adapting to different geometry forms.Negative b produces flatter surfaces with sharper cutoffs; positive b produces sharper peaks for high-frequency details.Source: §3.1 · p. 4, Figure 2 · p. 4

3.2 3D Ellipsoidal Beta Primitive

• Each 3D Ellipsoidal Beta Primitive adds a shape parameter b that deforms the kernel from a Gaussian-like form toward sharper edges and flat surfaces with minimal overhead.The primitive’s center, opacity, rotation, and scale mirror 3DGS parameters.Source: §3.2 · p. 4

3.3 Spherical Beta

• Spherical Beta models view-dependent color with bounded, learnable specular lobes and a feature dimension that grows linearly with their number.Its 3 + 6M feature dimension is more efficient than SH, and bounded support avoids truncation artifacts while allowing varied specular sharpness.Source: §3.3 · p. 5

3.4 Kernel-Agnostic Markov Chain Monte Carlo

• Kernel-Agnostic MCMC preserves the distribution during densification by regularizing and adjusting opacity alone, regardless of clone count or kernel choice.The method also replaces logit noise with a Beta Kernel component that scales positional noise by primitive opacity.Source: §3.4 · pp. 5, 7

4.1 Experimental Setup

• Evaluation covers four datasets with PSNR, SSIM, and LPIPS; implementation initializes b=0 and uses sb=2 unless otherwise specified.Source: §4.2 · p. 7, §4.1 · p. 7

4.2 Results and Comparisons

• DBS-30K and DBS-Full outperform prior implicit and explicit state-of-the-art methods across all metrics on almost all datasets.Source: §4.2 · p. 7, Table 1 · p. 9
• DBS renders 1.5x faster than 3DGS-MCMC at the same primitive count and uses less than half the storage when sb=2.Source: §4.2 · p. 9, Table 2 · p. 9
• With sb=2, DBS uses 200.3 MB for 24.79/30.10 dB; sb=4 or 6 reaches 24.81/30.12 dB at 289–378 MB.Source: §4.2 · p. 9, Table 3 · p. 9
• Flexible stopping lifts the sb=2 setup to 24.85 dB on Tanks & Temples and 30.12 dB on Deep Blending.Source: §4.2 · p. 9

4.3 Beta Decomposition

• Beta parameter b separates low-frequency solid geometry from high-frequency texture, illustrated by texture removal in treehill and garden.Source: §4.3 · p. 9
• Spherical Beta separates diffuse base color from view-dependent specular effects; it captures bicycle and bonsai highlights that Gaussian Splatting misses.Source: Figure 6 · p. 8, §4.3 · p. 9

• DBS combines adaptive geometry representation, efficient view-dependent color encoding, and MCMC optimization relying solely on regularized opacity.Source: §5 · p. 10
• Rasterization sorting can occasionally produce popping; Spherical Betas struggle with mirror-like reflections and anisotropic highlights, while the frustum model affects distant-background beta geometry.Source: §5 · p. 10

A.1 Splatting Condition

• Splatting projects a continuous 3D kernel onto a 2D plane, typically by integrating along one axis such as z.Source: §A.1 · p. 12
• For a separable kernel, deriving the corresponding 2D projection is straightforward.Source: §A.1 · p. 12
• Adjustable-shape deformable kernels can make the projection integral difficult; the radially symmetric Beta example is non-separable and depends on β.Source: §A.1 · p. 12

A.2 Inverse Splatting

• Defining a 2D Beta kernel for splatted billboards yields the multi-view-consistent geometry observed in 3D.The construction assumes radial symmetry and finite support.Source: §A.2 · p. 12

A.3 General Conditions for 2D Kernel Functions

• For radial 2D functions meeting the stated conditions, inverse Abel transformation guarantees a unique radial 3D kernel without requiring a closed-form derivation.The result applies to any qualifying 2D splatting kernel.Source: §A.3 · p. 12

B PER-SCENE RESULTS

• Over 1 dB in PSNR on “counter”: DBS surpasses 3DGS-MCMC and Zip-NeRF, while matching prior-work performance on diffuse scenes.The diffuse examples are “bicycle,” “stump,” and “flowers”; “counter” has strong specular reflections.Source: B PER-SCENE RESULTS · p. 13

C KERNEL COMPARISON

• The ablation disables view-dependent color terms (sh=0, sb=0), trains both kernels on Mip-NeRF 360, and evaluates test views under MCMC densification schedules.Source: C KERNEL COMPARISON · p. 13
• Across 0.1 M–5 M primitives, DBS achieves higher PSNR and lower MSE than 3DGS.Figure 7 plots PSNR and MSE against the total number of primitives.Source: Figure 7 · p. 13, C KERNEL COMPARISON · p. 14
• Around 1.5 M primitives, the Gaussian baseline plateaus while the Beta kernel continues to improve.Both kernels improve as primitive count increases.Source: C KERNEL COMPARISON · p. 14

D.1 Detailed Proof

• The proof compares the original primitive’s contribution with the combined contribution of N cloned copies, each assigned a new opacity.Source: §D.1 · p. 14
• Gaussian-specific scale adjustments used for boundary preservation in 3DGS-MCMC are complex and do not directly apply to deformable kernels.Source: §D.1 · p. 14
• With small opacity, adjusting opacity alone preserves the distribution to first order, regardless of copy count or splatting kernel.The difference diminishes as opacity decreases because the error term is O(o^2).Source: §D.1 · p. 14

D.2 Validity of the Small Opacity Assumption

• In volumetric rendering, many low-opacity primitives accumulate to form the image, so a moderate opacity prior aligns with that premise and supports distribution-preserving densification.The passage contrasts this with surface-based representations, where higher opacity is typically desirable for clean geometry.Source: §D.2 · p. 14
• Near opacity 0.1, the densified and original distributions nearly align across clone counts and kernel functions.The discrepancy diminishes as opacity decreases, supporting distribution preservation in the small-opacity regime.Source: §D.2 · p. 14, Figure 8 · p. 15
• λ_o=0.01 yields peak PSNR with average opacity around 0.1; omitting the opacity term or setting it excessively high damages reconstruction quality.The ablation reports consistent PSNR gains with the opacity prior, while excessive regularization drives primitives toward transparency.Source: §D.2 · p. 14, Table 6 · p. 14

E COMPRESSION

• Post-training compression reorders parameters for lossless PNG compression, quantizes most parameters from 32-bit floats to 8-bit, and retains 3D positions at 32-bit precision.The pipeline is adapted from [Morgenstern et al. 2025]; the authors report severe quality degradation when 3D positions are reduced to 8 bits.Source: E COMPRESSION · p. 15
• Pruning from 3.11 M to 1 M to 0.49 M primitives lowers mean PSNR from 28.75 dB to 28.42 dB to 28.03 dB, while raw footprint falls from 356 MB to 114 MB to 56 MB.The corresponding SSIM values are 0.845, 0.831, and 0.816; LPIPS values are 0.179, 0.205, and 0.232.Source: E COMPRESSION · p. 15
• 63 MB, 22 MB, and 11 MB final sizes correspond to overall compression ratios of 11.9x, 33.7x, and 66.2x relative to 3DGS, with PSNR drops of 0.15 dB, 0.10 dB, and 0.23 dB.The respective DBS-full, DBS-1M, and DBS-490K variants require 41.7 s, 14.3 s, and 9.2 s per scene for compression; naively compressed DBS-490K matches vanilla 3DGS visual quality at approximately 10 MB.Source: E COMPRESSION · p. 15

F HYPERPARAMETERS

• Training uses position learning rates of 0.00016 initially and 0.0000016 finally, color and Spherical Beta rates of 0.0025, and opacity and beta rates of 0.05 and 0.001.The DSSIM weight is 0.2; densification runs every 100 iterations from 500 through 25,000, with scale and opacity regularization weights of 0.01.Source: F HYPERPARAMETERS · p. 15