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arxiv: 2606.31764 · v1 · pith:ISBKBLVRnew · submitted 2026-06-30 · 💻 cs.GR · cs.CV

NURBS Splatting: A Unified Differentiable Rendering Framework for Vector Graphics

Jingye Qiu , Shizhe Zhou This is my paper

Pith reviewed 2026-07-01 02:24 UTC · model grok-4.3

classification 💻 cs.GR cs.CV
keywords NURBSdifferentiable renderingvector graphicsGaussian splattingspline renderingcurve vectorizationplanar curves
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The pith

Representing NURBS curves as sampled Gaussian fields reformulates vector graphics rendering as a stable differentiable accumulation process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to make rendering of planar rational splines differentiable by converting them into continuous fields of Gaussians. Sampling occurs both along the curve's parameter domain and inside closed regions, turning the process into smooth accumulation rather than direct analytic evaluation. This approach supports features like rational weights, non-uniform knots, long splines, and region filling that prior analytic methods struggle with. A sympathetic reader would care because vector graphics appear in design, reconstruction, and abstraction tasks where gradient stability directly affects optimization quality and flexibility.

Core claim

NURBS Splatting represents planar rational curves as continuous Gaussian fields. By sampling Gaussians along the curve parameter domain and inside closed regions, rendering is reformulated as a smooth accumulation process with stable gradients. The method naturally supports long splines, rational weights, non-uniform knots, and closed-region filling, and is demonstrated on calligraphy reconstruction, vectorization frameworks, and long-spline image abstraction.

What carries the argument

NURBS Splatting as continuous Gaussian fields sampled along the parameter domain and inside closed regions, which converts analytic curve rendering into differentiable accumulation.

If this is right

  • Calligraphy reconstruction becomes feasible with improved stability over analytic baselines.
  • Vectorization frameworks gain the ability to handle rational weights and non-uniform knots without custom gradient fixes.
  • Long-spline image abstraction produces higher-quality results because the accumulation process avoids gradient instability.
  • Closed regions can be filled differentiably by sampling inside boundaries rather than boundary-only evaluation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sampling idea might extend to other parametric curve families if their parameter domains admit similar Gaussian placement.
  • Optimization loops that currently avoid long splines due to instability could now incorporate them directly.
  • The framework suggests a general pattern where any curve representation could be turned into a splattable field for differentiability.

Load-bearing premise

That discrete sampling of Gaussians along the NURBS parameter domain yields gradients stable enough and faithful enough to the analytic curve for the shown reconstruction tasks.

What would settle it

A side-by-side optimization run on long splines where the Gaussian method produces visibly worse reconstruction error or diverges while an analytic baseline converges.

Figures

Figures reproduced from arXiv: 2606.31764 by Jingye Qiu, Shizhe Zhou.

Figure 1
Figure 1. Figure 1: We optimize NURBS parameters (control points Pi, weights wi, knot intervals τi) by sampling isotropic Gaussians along the curve and interior. These primitives are rendered via a differentiable tile-based rasterizer, enabling end-to-end optimization through image and geometric losses. support B-splines but do not introduce rational weights or non-uniform knots. Consequently, differentiable renderers cannot … view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of grid-step annealing. The grid resolution progressively increases, allowing coarse-to-fine optimization [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Single-stroke area-filling results. Letters are stylized with different style images; digits use coverage loss only (no stylization). where Limg is the distance-weighted MSE loss, L 3 deriv is the third-order derivative smoothing loss, and Lxing is a self-crossing penalty adapted from LIVE that pe￾nalizes control-point configurations whose consecutive segments reverse winding direction. Results are present… view at source ↗
Figure 4
Figure 4. Figure 4: Image abstraction with semantic stylization. Style images are inset in the bot￾tom corners of the inputs. Our method matches stylization quality while running 1.27× faster (single NVIDIA RTX 4090, 512×512, 300 iterations). where ˆϵ(xt, t, y) is the predicted denoising direction of the frozen diffusion model for a latent xt at time step t, ϵ is the sampled noise, and w(t) is a weighting function dependent o… view at source ↗
Figure 5
Figure 5. Figure 5: Rendering comparison on three diagnostic cases. Bézier Splatting (BS) shows blurred edges on thick strokes (left pair), spike artifacts near high-curvature regions (middle pair, local crop), and spikes at filled-region junctions (right pair). Bézier splines are equivalently converted to NURBS for fair geometric comparison [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative comparison of calligraphy reconstruction. Top row: Japanese char￾acters. Bottom row: Chinese characters. Our method better preserves stroke trajectories and width variations compared to the baseline [2] [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative comparison of vector reconstruction. Our method faithfully recon￾structs complex facial expressions and fine details where the DiffVG-based baseline tends to fail. 6 Conclusion We presented NURBS Splatting, the first differentiable rendering framework capable of optimizing rational B-splines in 2D image space. By representing curves and filled regions as continuous Gaussian fields, our method u… view at source ↗
read the original abstract

Differentiable rendering of planar rational splines remains largely underexplored, despite their widespread use in vector graphics and design. Existing differentiable vector renderers primarily focus on B\'ezier curves and rely on analytic rasterization, which can suffer from gradient instability and limited flexibility. We propose NURBS Splatting, a unified framework that represents planar rational curves as continuous Gaussian fields. By sampling Gaussians along the curve parameter domain and inside closed regions, rendering is reformulated as a smooth accumulation process with stable gradients. Our method naturally supports long splines, rational weights, non-uniform knots, and closed-region filling. We demonstrate its effectiveness in calligraphy reconstruction, vectorization frameworks, and long-spline image abstraction, showing improved stability and reconstruction quality over existing approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper proposes NURBS Splatting, a unified differentiable rendering framework for planar rational splines. It represents NURBS curves as continuous Gaussian fields by sampling Gaussians along the curve parameter domain and inside closed regions, reformulating rendering as a smooth accumulation process. The method is claimed to yield stable gradients while naturally supporting long splines, rational weights, non-uniform knots, and closed-region filling. Effectiveness is demonstrated on calligraphy reconstruction, vectorization frameworks, and long-spline image abstraction, with reported improvements in stability and reconstruction quality over existing approaches.

Significance. If the stability and faithfulness claims hold under quantitative validation, the approach could offer a practical alternative to analytic rasterization for differentiable vector graphics, enabling more robust optimization in applications involving complex NURBS geometry.

major comments (1)
  1. [Abstract] Abstract: The central claim that discrete Gaussian sampling along the NURBS parameter domain produces gradients that remain stable and faithful to the analytic curve (including for rational weights, non-uniform knots, and long splines) is load-bearing but unsupported by any derivation details, error bounds on approximation error (as a function of sample count, Gaussian variance, or curvature), or side-by-side comparisons against analytic rasterization. Without these, the stability advantage cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the single major comment below, acknowledging the need for additional support on the stability claims while clarifying the empirical evidence already present.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that discrete Gaussian sampling along the NURBS parameter domain produces gradients that remain stable and faithful to the analytic curve (including for rational weights, non-uniform knots, and long splines) is load-bearing but unsupported by any derivation details, error bounds on approximation error (as a function of sample count, Gaussian variance, or curvature), or side-by-side comparisons against analytic rasterization. Without these, the stability advantage cannot be assessed.

    Authors: We agree that the manuscript does not provide a formal derivation of error bounds on the approximation error (as a function of sample count, Gaussian variance, or curvature) or a theoretical analysis of gradient faithfulness to the analytic curve. The stability claim rests on the formulation of rendering as a smooth accumulation of Gaussians (detailed in Section 3), which ensures differentiability by construction, together with empirical demonstrations across the experiments. These include successful optimization on long splines, rational weights, and non-uniform knots (Sections 4.1–4.3), with quantitative improvements in reconstruction quality and stability over analytic baselines. Visual and numerical side-by-side comparisons of rendered results and optimization trajectories are included in the main figures and supplementary material. We will add a dedicated subsection on approximation properties and expanded gradient-stability comparisons in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a modeling reformulation independent of its outputs

full rationale

The paper's core step is to represent NURBS curves as sampled Gaussian fields and reformulate rendering as accumulation; this is an explicit modeling choice presented in the abstract as a direct reformulation, not a quantity fitted or defined in terms of its own predictions. No equations, self-citations, or uniqueness claims appear in the provided text that would reduce the claimed stable gradients or support for rational weights/non-uniform knots back to the method's own inputs by construction. The approach is self-contained as a proposed framework with external demonstrations, meeting the criteria for an honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The method rests on the modeling choice that Gaussian sampling along the parameter domain is a sufficient continuous approximation; no free parameters, axioms, or invented entities are explicitly listed in the abstract.

pith-pipeline@v0.9.1-grok · 5657 in / 1007 out tokens · 38199 ms · 2026-07-01T02:24:10.275755+00:00 · methodology

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Reference graph

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