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CubicSplat: Differentiable Vector Graphics via Error-Bounded Forward Relaxation

Chenglong Liu, Xin Zhang, Yimeng Zhu, Liyang He, Yixiao Ma, Yu Su, Zhenya Huang, Qi Liu

arXiv:2608.20803v1cs.GRcs.CVcs.LG

TL;DR

Differentiable vector rasterization must balance forward geometric fidelity against gradient conditioning, because exact or overly coarse renderers can each hinder optimization. CubicSplat uses uniform polyline surrogates with bounded O(S^-2) error and compositing-derived visibility pruning, achieving strong reconstruction quality with faster training. Its scope includes open strokes and closed fills, while the paper identifies scale-equivariance failure as a limitation of point-sampled alternatives rather than CubicSplat itself.

  • Problem

    Differentiable vector graphics optimization is hindered by discontinuous rasterization and a gradient seesaw between forward exactness and gradient quality.

  • Method

    CubicSplat replaces Bézier closest-point solvers with uniform polyline surrogates, bounded geometric bias, a static computation graph, and compositing-derived visibility pruning.

  • Results

    25.78 dB closed / 25.93 dB open PSNR versus Bézier Splatting’s 23.45 / 25.45 dB, with 4× and 1.8× lower training time, respectively.

  • Takeaways & Limitations

    CubicSplat provides a controlled relaxation that maintains reconstruction quality while improving optimization outcomes at substantially lower computational cost.

  • Takeaways & Limitations

    Point-sampled curve splatting can violate scale equivariance, coupling optimization outcomes to sampling hyperparameters.

Abstract

from arXiv · show

Vector graphics are prized for their resolution independence, compact storage, and direct editability, making differentiable optimization of their parametric primitives an attractive goal. Yet classical rasterization is discontinuous with respect to geometry, and existing remedies that smooth the forward pass demand increasingly elaborate heuristics as scene complexity grows. We trace this fragility to a gradient seesaw: design choices that improve forward geometric exactness can systematically degrade the induced gradient signal, and vice versa. To navigate this tension we introduce CubicSplat, a differentiable vector rasterizer that replaces Bézier closest-point solvers with uniform polyline surrogates whose geometric error is bounded at $O(S^{-2})$. The resulting static computation graph yields well-conditioned gradients by construction, while a compositing-derived visibility mechanism prunes degenerate primitives without auxiliary regularization. On DIV2K and Kodak benchmarks CubicSplat achieves state-of-the-art reconstruction quality with over 2 dB PSNR gain in the closed-fill setting, while training up to 4x faster than prior methods. The code is available at https://github.com/CubicSplat/repo

1 Introduction

Differentiable vector rendering faces a gradient seesaw: improving forward geometric fidelity can worsen gradient quality, while excessive relaxation biases the optimized objective. CubicSplat addresses this tension with bounded polyline relaxation, visibility-driven adaptation, and strong quality–efficiency results.

  • Classical rasterization produces gradients that are zero almost everywhere or explosive near boundaries.
  • Increasing forward fidelity reduces geometric error but can plateau representation quality, increase runtime, and destabilize optimization.
  • The gradient seesaw captures the trade-off between forward exactness and gradient quality, while overly coarse approximations optimize the wrong objective.
  • CubicSplat replaces iterative Bézier closest-point solving with uniform polyline distance queries, bounding geometric bias at O(S^-2) and providing a static gradient graph.
  • CubicSplat uses compositing-derived visibility statistics for pruning and densification, avoiding auxiliary geometric regularization.
  • 25.78 dB closed / 25.93 dB open PSNR versus Bézier Splatting’s 23.45 / 25.45 dB, with 4× and 1.8× lower training time, respectively.At N=1024 on a 200-image DIV2K subset, CubicSplat also matches DiffVG’s 1024-curve PSNR using 256 curves and runs 1.4× faster than Bézier Splatting and 120× faster than DiffVG per step.

2 Related Work

Prior differentiable renderers variously soften images, defer antialiasing, sample boundaries, or use point-based splats to obtain gradients. CubicSplat targets the resulting sampling, solver, and heuristic limitations with an error-bounded polyline surrogate.

  • SoftRas smooths gradients by softening the forward image, whereas nvdiffrast preserves crisp rendering through deferred antialiasing.
  • DiffVG offers analytical and Monte Carlo antialiasing, but its closest-point solvers can be ill-conditioned and sampling can be costly at high resolution.
  • LIVE, LIVSS, and SGLIVE reduce intercurve gradient interference by incrementally adding paths.
  • Bézier Splatting uses Gaussian point samples along curves but inherits phase- and density-dependent aliasing and sampling-sensitive optimization.
  • Related 3D Gaussian methods expose pathologies involving compositing, gradient collision, densification bias, and depth-order mismatch.

3 Method

CubicSplat treats differentiable vector rendering as gradient-oracle design: it relaxes training rasterization with bounded bias while retaining stable optimization and renderer-derived adaptation. Bézier primitives become uniform polylines, coverage becomes smooth opacity, and compositing supplies visibility signals.

  • Framework: CubicSplat designs relaxed forward passes for stable, well-conditioned gradients rather than maximal geometric exactness, while keeping forward bias bounded.
  • Compositing and Adaptation: Front-to-back alpha compositing produces transmittance values that provide renderer-derived visibility statistics for pruning and densification without explicit geometric regularization.
  • Error-Bounded Forward Relaxation: The relaxed renderer maps parameters through a pixel loss, with sampling density S controlling the surrogate and geometric error decreasing at O(S^-2).
  • Error-Bounded Forward Relaxation: Each Bézier primitive is converted into a uniform S-segment polyline, reducing distance queries to closed-form point-to-segment computations and removing iterative solvers from backpropagation.
  • Differentiable Coverage: A smooth monotone coverage kernel maps signed residuals to soft opacity, while open strokes use a width threshold and closed fills use winding-number inside/outside labeling.
  • Sampling-Density Behavior: Beyond a threshold S0, learned representations and reconstruction quality remain nearly unchanged despite increasing surrogate fidelity, making S a relaxation knob rather than an approximation budget.

4 Experiments

CubicSplat is evaluated on high-resolution DIV2K and Kodak benchmarks for reconstruction quality, speed, scalability, and ablations. Across settings, it combines strong image quality with efficient training, while analyses examine sampling, visibility pruning, regularization, subdivision, magnification, and primitive-count scaling.

  • Datasets and evaluation: CubicSplat is evaluated on a 200-image DIV2K subset and the full 24-image Kodak dataset at original resolution, using PSNR, SSIM, and LPIPS.Comparisons include DiffVG, LIVE, LIVSS, and Bézier Splatting.
  • Single training-step speed: 6.77 ms per forward–backward pass makes CubicSplat 1.4× faster than Bézier Splatting and over 120× faster than DiffVG on open curves.The measurement uses a 2040×1344 image with 2048 curves on an RTX 4090.
  • Parameter efficiency: 256 curves match the PSNR of DiffVG with 1024 curves in both topology modes because visibility-driven pruning concentrates capacity on contributing primitives.Pruning removes redundant and occluded curves.
  • Quantitative reconstruction results: Over 2 dB PSNR improvement and roughly 4× lower per-image training cost occur over Bézier Splatting in closed mode at N=1024.CubicSplat also improves PSNR and SSIM at every open-mode budget while reducing training time markedly.
  • Ablations and analysis: Sampling gains saturate beyond S=24 in open mode, while closed mode reaches essentially the same quality at S=12 as S=24 with higher throughput.The result supports treating S as a relaxation knob rather than a direct optimization-quality budget.

5 Conclusion

CubicSplat addresses the gradient seesaw with error-bounded forward relaxation, combining stable optimization with scalable rendering. Its efficiency supports potential real-time design and video vectorization applications, while integration with perceptual and semantic priors remains a future direction.

  • 5 Conclusion: CubicSplat replaces Bézier closest-point root solving with closed-form polyline distance queries, producing a stable static computation graph with O(S^-2) geometric error.The controlled relaxation is reported to preserve reconstruction quality while reducing computational cost.
  • 5 Conclusion: High efficiency may bring real-time differentiable vector rendering and video vectorization within reach, where fast rendering and stable gradients are required.These are presented as natural extensions suggested by the tile-parallel architecture and low per-frame cost.
  • 5 Conclusion: Scaling experiments show improving PSNR with sublinear training time and manageable memory overhead as closed-fill primitive counts increase.Figure 6 reports scale-consistent fine structures under extreme primitive counts.
  • 5 Conclusion: Combining CubicSplat with perceptual losses, semantic constraints, and learned priors is identified as a direction toward controllable, real-time vector-content creation.The paper frames this combination as a promising extension rather than a demonstrated result.

S1 Formal Analysis of Differentiability and Gradient Structure

This supplement analyzes differentiability failures in exact and point-sampled vector rasterizers, then contrasts them with CubicSplat’s bounded, fixed-polyline formulation.

  • S1.2 Ill-Conditioning of the DiffVG Analytical Prefilter: Approximate root solving introduces gradient error proportional to the stationarity defect when numerical solvers are treated with stop-gradient.The error vanishes at a true stationary point but not generally in floating-point arithmetic.
  • S1.2 Ill-Conditioning of the DiffVG Analytical Prefilter: Near multiple roots or flat distance landscapes, residual terms dominate and produce gradient noise that drives optimization instability.The analysis connects this behavior to regularization or more expensive Monte Carlo fallbacks in DiffVG.
  • S1.3 Discretization Artifacts in Point-Sampled Curve Splatting: Point-sampled curve splatting introduces phase-sensitive aliasing whose structured artifacts depend on sampling spacing and sub-pixel phase.The resulting deviations differ qualitatively from CubicSplat’s geometric O(S^-2) polyline error.
  • S1.3 Discretization Artifacts in Point-Sampled Curve Splatting: Discrete sampling violates scale equivariance because image scaling changes effective spacing while the sample comb remains tied to the discrete lattice.Resolution changes, sampling-rate changes, and sample redistribution can therefore alter both rendered images and loss surfaces.
  • S1.3 Discretization Artifacts in Point-Sampled Curve Splatting: Moving discrete samples generate oscillatory, schedule-dependent gradients that can require careful density tuning and lose fine structure under extreme scaling.The mechanism arises from high-frequency kernel-gradient contributions whose phase shifts relative to pixel centers.
  • S1.3 Discretization Artifacts in Point-Sampled Curve Splatting: Structured sampling micro-textures can artificially improve perceptual scores: injected uncorrelated noise lowered PSNR and SSIM but improved LPIPS from 0.5016 to 0.4636.This controlled result shows that perceptual metrics may reward texture-like artifacts.
  • S1.3 Discretization Artifacts in Point-Sampled Curve Splatting: CubicSplat avoids a sampling comb by using a deterministic polyline Hausdorff bound and uniformly bounded coverage error.Its reconstruction quality is reported as nearly invariant across sampling rates and resolutions.
  • S1.4 Error Bounds of CubicSplat: The fixed-segment surrogate is differentiable wherever the minimizing segment is unique, with nondifferentiability confined to measure-zero transition sets under genericity assumptions.Transitions occur when minimizing segments tie or closest-point status switches between an interior projection and an endpoint.

S2 Adaptive Subdivision Introduces Gradient Discontinuities

Adaptive subdivision makes the surrogate topology change discontinuously as control points cross a flatness threshold, creating gradient discontinuities and implementation overhead.

  • S2 Adaptive Subdivision Introduces Gradient Discontinuities: Adaptive de Casteljau subdivision recursively splits Bézier segments until a flatness criterion is met, producing a parameter-dependent vertex count S(θ).The implementation uses a flatness tolerance of 0.5 px.
  • S2 Adaptive Subdivision Introduces Gradient Discontinuities: When the flatness threshold is crossed, S jumps and the minimizing surrogate segment can change combinatorially.The resulting Clarke subdifferential may exclude the true Bézier-distance gradient.
  • S2 Adaptive Subdivision Introduces Gradient Discontinuities: Uniform fixed-S sampling avoids topological jumps, while adaptive subdivision reduces throughput and trails uniform S=24 by −0.06 dB open and −0.07 dB closed at N=1024.Adaptive subdivision also has lower throughput in both modes.

S3 Low-Budget Color-Gradient Fitting

The low-budget fitting experiment indicates that stable oracle behavior, rather than per-primitive color capacity alone, determines whether scarce primitives are allocated to useful regions.

  • S3 Low-Budget Color-Gradient Fitting: Optimization traces show that primitive allocation to useful regions depends on oracle stability, not only on per-primitive color capacity.Overlapping coverage can efficiently express smooth gradients when optimization remains stable.
  • S3 Low-Budget Color-Gradient Fitting: Near-identical outcomes across sampling rates indicate that, once the oracle is well-conditioned, S mainly controls forward fidelity and rendering cost.The supplied figure reports LPIPS↓ and PSNR↑ for low-budget fitting with 32 closed primitives.
  • S3 Low-Budget Color-Gradient Fitting: CubicSplat’s resolution-scalability comparison reports consistent PSNR across rendering resolutions from 2K to 24K, including at K = 8.The comparison covers both unclosed and closed curves.

S4 Original Experiment Data

The original experiment data reports per-image PSNR and rendering time for 2K-trained models evaluated through 24K rendering, enabling raw resolution-scaling analysis.

  • S4 Original Experiment Data: Tables 5 and 6 report per-image PSNR and rendering time for models trained at 2K with 1024 curves and rendered up to 24K on one RTX 4090.Rendered outputs are bicubically downsampled to training resolution before PSNR computation.
  • S4 Original Experiment Data: The supplementary rendering-time table compares CubicSplat latency across resolutions and sampling rates K, reporting real-time standard-resolution rendering and graceful ultrahigh-resolution scaling.The table covers both unclosed and closed curves.
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