NURBS Splatting: A Unified Differentiable Rendering Framework for Vector Graphics

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 Bézier 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.

• The paper introduces a unified differentiable rendering framework for NURBS that enables exact conic representations and local geometric control.
• It employs adaptive Gaussian splatting for contour sampling and SDF-based region filling to improve reconstruction fidelity and optimization speed.
• Extensive experiments reveal significant reductions in MSE and Hausdorff distance, with enhanced SSIM, benefiting CAD and vectorization applications.

NURBS Splatting: A Unified Differentiable Rendering Framework for Vector Graphics

Motivation and Context

The paper introduces "NURBS Splatting" (2606.31764), an approach to differentiable rendering of planar rational splines—specifically, Non-Uniform Rational B-Splines (NURBS). NURBS are the canonical representation for curves and surfaces in CAD, enabling exact conics, local control, and flexible knot placement. Despite their ubiquity, existing differentiable renderers have focused on Bézier and polynomial B-spline curves, neglecting the rational parameterization and non-uniform knots essential for compatibility with industry-standard CAD systems and for high-fidelity geometric modeling. Consequently, shapes native to CAD, like true circular arcs and analytic conics, are only approximated or unsupported in current differentiable rendering systems. This paper directly addresses the gap by formulating a differentiable rasterization mechanism for NURBS, bridging learning-based image reconstruction with CAD geometry.

Technical Architecture

NURBS Parameterization and Gaussian Splatting

Each curve is parameterized as a NURBS of degree $p$, with learnable control points, rational weights, and knot intervals. The basis function evaluation leverages the Cox-de Boor recursion, which ensures regularity and local support. Key points are mapped to control points and knot vectors, while for open and closed curves, knot construction guarantees geometric continuity and periodicity.

Rendering is approached by adaptively sampling isotropic Gaussian kernels along curve contours and within closed regions. For contour sampling, density is linked to arc length, ensuring coverage proportional to geometric complexity. For filled closed regions, a signed distance field (SDF) guides interior Gaussian placement on grid points, modulating opacity via a boundary sigmoid. All Gaussians inherit color and opacity from the parent curve and are composited in a tile-based rasterizer via alpha blending. The entire pipeline is differentiable: losses propagate gradients to curve parameters, enabling end-to-end optimization.

Differentiable Loss Functions

The optimization objective combines pixel-level reconstruction (typically MSE or semantic losses) with geometric regularization. Specifically:

• Derivative Loss: Penalizes squared magnitude of higher-order derivatives (typically third-order) to encourage smoothness without sacrificing shape fidelity.
• Bounding Box Loss: Ensures curve remains within image bounds, applying soft penalties for violations.
• Self-Crossing Penalty: In vectorization applications, prevents pathological crossings.
• Grid-Step Annealing: For region fills, spatial resolution is increased progressively to accelerate early-stage optimization without losing boundary detail.

Experimental Results

Extensive experiments demonstrate practical and numerical superiority over prior approaches utilizing polynomial splines and Bézier curves. Key findings include:

• Calligraphy Reconstruction: On a benchmark of 192 Chinese and Japanese characters, the NURBS Splatting method achieves superior quantitative metrics (MSE, PSNR, Hausdorff, F1 score) compared to the baseline [2]. At contour density $d_c=18$, MSE is reduced by 33%, PSNR is increased by +1.6 dB, and Hausdorff distance is lowered by 30%, with a 1.2x speedup in optimization runtime. Rational weights provide substantial improvement in modulating local curvature and edge fidelity. High density sampling ($d_c=30$) further enhances SSIM and F1, with only modest computation overhead.
• Layer-wise Image Vectorization: On a set of 100 Noto Emoji images, NURBS Splatting outperforms the LIVE baseline [28] in every metric: MSE is reduced by 32%, SSIM improved by +0.017, and LPIPS decreased by 41%, with 1.4x faster runtime. Step annealing and splatting strategies yield robustness to complex topologies, mitigating artifacts seen in the DiffVG-based baseline.
• Neural Image Abstraction: When integrated with diffusion-guided Score Distillation Sampling pipelines, NURBS Splatting achieves comparable stylization fidelity to prior work while offering 1.27x faster optimization—even though NURBS evaluation is inherently more complex than polynomial B-splines.

Ablation studies confirm the individual contributions of rational weights and non-uniform knots, with weights being critical for local control and knots for concentrating basis support at regions of high curvature.

Implications and Limitations

Practical Impact

The proposed framework unifies vector graphics rendering and differentiable optimization, enabling direct editing and refinement of NURBS parameters for image-space tasks. This directly aligns learning-based vectorization with CAD workflows, facilitating downstream applications such as robotic pen plotting, semi-automated font design, and geometric abstraction. The isotropic Gaussian splatting mechanism addresses several pathology artifacts found in prior anisotropic splatting methods—such as blurred edges and tangent discontinuities—yielding sharp boundaries and stable gradients.

Theoretical Impact

By supporting rational weights and non-uniform knots, NURBS Splatting expands the family of differentiable primitives to exact conics and analytic curves, enhancing expressivity and fidelity. This brings differentiable rendering closer to the standards of isogeometric analysis and CAD-integrated simulation pipelines.

Limitations and Future Directions

The region-filling strategy relies on SDF-based grid annealing, which may be less efficient than analytic scanline methods for regular shapes. As curve complexity rises, the number of generated Gaussians may lead to increased memory consumption, especially for highly detailed scenes. Current work is limited to 2D planar curves; extending to NURBS surfaces would further enable native support for CAD geometry, mesh refinement, and generative CAD models.

Future avenues include:

• Efficient analytic region-filling algorithms
• Synthesis of editable vector designs from text/image prompts via generative models
• Extending splatting to nonplanar NURBS surfaces and volumetric domains
• Integration with CAD kernels for downstream manufacturing and analysis

Conclusion

NURBS Splatting establishes a differentiable rendering backbone compatible with rational parametric curves, closing the gap between learning-based image abstraction and CAD-standard geometry. The method consistently outperforms polynomial baselines in reconstruction fidelity and optimization speed, enabling practical workflows for vector graphics, calligraphy, and abstraction. Its architectural extensibility and geometric flexibility position it as a foundational tool for future research at the intersection of differentiable graphics, geometric modeling, and generative design (2606.31764).