Laplacian Frequency Hierarchies for Efficient 3D Gaussian Splatting Training

1. Introduction

• As scene complexity and resolution grow, 3DGS typically requires more Gaussian primitives and Gaussian–tile interactions, increasing computation and peak memory.Existing acceleration approaches often reduce average Gaussian workload without separating coarse and fine representation into distinct fields.Source: §1 · p. 1
• Laplacian decomposition separates a low-frequency base from high-frequency residuals, which are represented by frequency-specific Gaussian fields and composed in the image domain.Training proceeds coarse to fine, archiving each completed lower-frequency field so later stages optimize only the active band.Source: §1 · pp. 2–3, Figure 1 · p. 2
• The method improves efficiency across 1K, 2K, and 4K settings while maintaining comparable reconstruction quality, with gains more pronounced at higher resolutions.It is plug-and-play across multiple 3DGS backbones.Source: §1 · pp. 2–3
• Archiving earlier levels reduces training workload and peak memory usage by keeping only the current frequency band’s Gaussians actively optimized.This addresses the overhead of carrying coarse components that have converged while later training focuses on high-frequency details.Source: §1 · pp. 1–3

Novel view synthesis

• 3DGS uses explicit Gaussian primitives and differentiable splatting for GPU-friendly rasterization, supporting applications including reconstruction, avatars, SLAM, and digital twinning.It follows earlier NeRF-based implicit representations for novel view synthesis.Source: §2 · p. 3

Acceleration for 3DGS training

• Existing training accelerations span backpropagation and optimizer changes, scheduling, densification control and pruning, and system-level parallelization.Examples include Taming 3DGS, DashGaussian, Mini-Splatting, and FastGS.Source: §2 · p. 3

Frequency-based methods in 3DGS

• Unlike frequency scheduling or reweighting, the proposed approach decomposes the 3DGS scene into frequency-specific fields and composes renderings through Laplacian-style image reconstruction.Completed fields are archived so later stages optimize only the active frequency band.Source: §2 · p. 3
• Frequency-aware approaches include Fourier-domain regularization, coarse-to-fine supervision with a single field, frequency-aware densification, and Laplacian subbands for LOD and editing.The proposed method differs by factorizing the scene representation and archiving completed fields.Source: §2 · p. 3

3.2. Laplacian Frequency Hierarchy

• At inference, the fields are rendered separately and composed through Laplacian reconstruction to produce the full-resolution image.The framework renders each field using its corresponding splatting renderer.Source: §3.1 · p. 4, §3.2 · p. 5

3.1. Preliminaries.

• 3DGS represents scenes with Gaussian primitives, and growing primitive counts and resolution raise rendering, backpropagation, and memory costs.These costs become a key bottleneck for efficient training.Source: §3.1 · pp. 3–4

3.2. Laplacian Frequency Hierarchy

• The hierarchy pairs a coarsest Gaussian field with low-frequency image content and finer fields with Laplacian residual bands.Residual fields support signed outputs by removing the standard color shift and clamp.Source: §3.2 · pp. 4–5

3.3. Frequency-staged training with archiving

• After each frequency stage, archiving removes its field from further optimization and memory residency, leaving fewer active GPU Gaussians for later stages.The paper states this improves training throughput and lowers peak VRAM usage.Source: §3.3 · p. 5
• Because direct per-level target fitting can accumulate errors, each finer stage is supervised using the reconstructed image at its corresponding resolution.The prior-level prediction is supplied to the dataloader, so its archived field need not remain in memory.Source: §3.3 · p. 5
• Training finer fields from inherited Gaussian positions reinitializes their other attributes, with an inheritance ratio controlling how many positions carry over.At a ratio of 1.0 all Gaussians are inherited; below 1.0 a random subset is inherited.Source: §3.3 · p. 5

4.1. Experimental Setup

• Experiments cover Mip-NeRF 360, Deep Blending, and Tanks & Temples, measuring PSNR, SSIM, LPIPS, training time, and Gaussian counts on Taming-3DGS and FastGS.The Mip-NeRF 360 tests include 1K, 2K, and full-resolution 4K settings; the default uses two Laplacian levels and a 10K/20K iteration split.Source: §4.1.1 · pp. 5–6, §4.1.2 · p. 6, §4.2.1 · p. 6

4.2. Comparison with Fast Optimization Methods

• 1.73× speedup for Ours(Taming) on Mip-NeRF 360 at 1K, with a 0.34 dB PSNR drop; Ours(FastGS) reduces time from 4.82 to 3.96 minutes with a 0.29 dB PSNR drop.Across Deep Blending and Tanks & Temples, the method is also faster, and it achieves the fastest training time across the reported benchmarks.Source: Table 1 · p. 6, §4.2.2 · pp. 6–7

4.2.5. COMPARISONS ON HIGH-RESOLUTION SETTINGS (2K AND 4K)

• At 2K and 4K, the efficiency advantage becomes more pronounced while quality gaps become negligible; at 4K, comparable quality is reached earlier on both backbones.The 4K PSNR-versus-time plot highlights early convergence, and the 2K and 4K tables report Mip-NeRF 360 results.Source: §4.2.5 · p. 8, Table 2 · p. 8, Table 3 · p. 8

4.3. Ablation study

• Iteration splits are relatively insensitive: the default (10000, 20000) achieves the fastest training time with comparable metrics, as residual error is delegated to the next stage.Moderate reallocations mainly shift where the error is corrected rather than fundamentally changing final quality.Source: §4.3.1 · p. 8, Table 4 · p. 8
• At 1K, two levels best balance efficiency and quality; at 4K, four levels reduce peak Gaussians from 1.74M to 0.49M, with less pronounced quality degradation.Deeper hierarchies tend to speed training but degrade quality, with structural misalignment accumulating across frequency bands; the quality penalty is less pronounced at higher resolution.Source: §4.3.2 · p. 8, Table 5 · p. 9

• The framework factorizes scenes into a low-frequency base field and high-frequency residual fields, reducing active Gaussians during optimization.The conclusion reports improved training throughput and lower peak GPU memory, with more pronounced gains at higher resolutions.Source: §6 · p. 9
• GT Laplacian supervision directly compares the rendered residual with the fixed ground-truth Laplacian residual.The target residual is defined in Eq. 4.Source: §6 · p. 9
• Reconstructed-image supervision yields the best overall performance; both residual-supervision variants underperform this design.The authors therefore supervise the reconstructed image at each level rather than fitting the residual alone.Source: §6 · p. 9

• Coarse structures may converge early yet remain active and consume optimization resources in later stages.The method targets this persistent workload by reducing the active burden of coarse structures.Source: §5 · p. 9
• Cross-level bridging and high-frequency residual modeling add costs that must be balanced against the reduced active workload.Source: §5 · p. 9
• Deeper hierarchies can make finer residuals harder to represent efficiently, reducing gains and potentially degrading quality.The authors find relatively shallow hierarchies provide a better speed-quality balance.Source: §5 · p. 9

A. Implementation Details and Reproducibility

• The default setup uses a two-level hierarchy with a 10K/20K iteration split, inherits Gaussian positions with ρ = 1.0, and reinitializes other attributes.Experiments use 30K iterations; 1K runs use an NVIDIA L40 GPU, while 2K and 4K runs use an NVIDIA Pro6000 GPU.Source: A. Implementation Details and… · p. 12
• Trajectory comparisons report stable image-domain Laplacian composition without visible flickering under continuous viewpoint changes.The video also includes the training pipeline, representative scenes, and low-frequency, residual, and reconstructed components.Source: A. Implementation Details and… · p. 12

B.1. Inheritance strategy: xyz-only vs. full inheritance.

• Final reconstruction quality is relatively insensitive to xyz-only versus full inheritance, but xyz-only consistently performs slightly better and is the default.Full inheritance retains parameters such as scale, rotation, and optimizer states; later optimization can largely correct initialization differences.Source: §B.1 · p. 12, B. Ablation Studies on Inheritance… · p. 12

B.2. Analysis of the inheritance ratio ρ.

• As inheritance ratio ρ decreases, training becomes faster but reconstruction quality drops because fewer primitives remain to represent fine details.The default ρ = 1.0 provides the best reported trade-off between efficiency and rendering performance.Source: §B.2 · p. 12, Table 7 · p. 12

C. Edge Preservation at Object Boundaries

• The high-frequency residual restores edge details smoothed in the low-frequency prediction, preserving boundary sharpness and contrast comparable to FastGS and ground truth.This comparison is reported across three test views.Source: C. Edge Preservation at Object… · p. 13
• Although the image-space Laplacian is not strictly view-invariant, errors from earlier levels, including view-dependent errors, can be corrected by the finer field.The reconstructed image is directly supervised at each finer stage.Source: C. Edge Preservation at Object… · p. 13

D. Scene-wise Quantitative Results

• At 1K, the method tends to perform better on challenging scenes requiring more Gaussians, such as garden.The scene-wise comparisons cover Mip-NeRF 360, Deep Blending, and Tanks & Temples.Source: D. Scene-wise Quantitative Results · p. 13
• At 2K and 4K, reconstruction metrics remain comparable or improve in many scenes while acceleration gains become more pronounced.The authors attribute this pattern to smaller Gaussian fields reducing per-stage burden and helping stabilize high-resolution training.Source: D. Scene-wise Quantitative Results · p. 13

E. Scene-wise Qualitative Results

• Across 1K, 2K, and 4K, qualitative results maintain high visual quality without noticeable artifacts from accumulating rendered components.The comparisons are shown in Figs. 8–10.Source: E. Scene-wise Qualitative Results · p. 13

F. Analysis of peak Gaussian count and peak GPU memory

• Ours(FastGS) achieves the lowest average peak Gaussian count and peak GPU memory usage among the methods.The memory measure covers the 3DGS training pipeline and excludes GPU memory used to store the dataset.Source: F. Analysis of peak Gaussian count and… · p. 13, Table 8 · p. 13

G. Runtime Breakdown and Timing Boundary

• 56.5% of iterative optimization time is spent on backward propagation, versus 18.2% on forward rendering, 14.3% on the optimizer step, and 3.5% on composition and loss computation.The profile is for a 30K-iteration Ours(FastGS) run on garden.Source: G. Runtime Breakdown and Timing Boundary · p. 14, Table 9 · p. 14
• One-time pyramid construction, stage-cache construction, and inheritance total 0.559 s, or 0.15% of iterative optimization time.Reported training times exclude pyramid preprocessing and archive I/O; two archive writes take 16.72 s and are excluded consistently with model-saving and checkpoint I/O for compared methods.Source: G. Runtime Breakdown and Timing Boundary · p. 14

1K ablation

• At 1K, a two-level hierarchy gives the best overall trade-off, while deeper hierarchies reduce training time but gradually degrade reconstruction quality and rendering speed.The detailed level ablation is reported in Table 15.Source: H. Additional analysis on Laplacian… · p. 14

4K ablation

• At 4K, deeper hierarchies consistently reduce peak Gaussian count and training time, with quality degradation becoming less noticeable at higher resolutions.The 30K- and 45K-iteration settings use identical split ratios across levels; increasing the iteration budget slightly improves quality across level choices.Source: H. Additional analysis on Laplacian… · p. 14
• Across the 4K hierarchy choices, increasing the iteration budget slightly improves reconstruction quality while preserving the relative trends among hierarchies.The comparison uses 30k and 45k iterations.Source: H. Additional analysis on Laplacian… · p. 14

Qualitative results

• At object boundaries, the smoother low-frequency prediction is supplemented by a high-frequency residual that restores missing edge details.The final reconstructions are described as comparable to FastGS and ground truth.Source: Figure 7 · p. 16

PSNR versus training time

• The 1K and 2K PSNR-versus-training-time curves show comparable reconstruction quality reached earlier than with the original Taming-3DGS and FastGS backbones.The reported advantage is more evident at higher resolutions, including the 4K results in the main paper.Source: §I · pp. 14, 21, Figure 11 · p. 21
• The 1K and 2K curves are presented for Mip-NeRF 360 scenes, complementing the 4K PSNR-versus-time curves in the main paper.Figure 11 gives the 1K and 2K plot scope; the supplied 4K discussion reports comparable quality reached earlier than the original backbones.Source: §I · p. 14, Figure 11 · p. 21