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

cs.GRAug 21, 2026

CubicSplat: Differentiable Vector Graphics via Error-Bounded Forward Relaxation

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

Organizations: State Key Laboratory of Cognitive Intelligence, University of Science and Technology of China, Hefei, 230088, China · Hefei Normal University, Hefei, 230061, China · Zhejiang Key Laboratory of Intelligent Education Technology and Application, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

Abstract

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

Figures & tables

Figure 1: Overview of the CubicSplat pipeline. 1) Each primitive is encoded as a set of learnable Bézier parameters. 2) The forward pass applies the error-bounded relaxation: a polyline surrogate $\Pi_{S}$ converts Bézier geometry into closed-form distance queries, and a smooth coverage kernel $\phi$ maps the signed residual to per-pixel opacity, which is assembled into the final image via alpha compositing. 3) The raster loss drives gradient updates back to the vector parameters, while compositing-derived visibility statistics feed into Gini pruning and densification.

	Open curves			Closed curves
	DiffVG	Bézier Splatting	Ours	DiffVG	Bézier Splatting	Ours
Forward	141.3ms	4.5ms	2.70ms	85.2ms	14.1ms	10.68ms
Backward	701.3ms	4.7ms	4.07ms	448.3ms	24.58ms	8.96ms
Total step	842.6ms	9.2ms	6.77ms	533.5ms	38.68ms	19.64ms

Table 1: Single training-step time averaged over a

2040{\times}1344

image with 2048 curves on an RTX 4090.

Figure 2: Qualitative results across varying budgets, topology modes, and datasets [ 1 , 9 , 30 ] . At low budgets (closed 128, open/closed 256), CubicSplat preserves clean boundaries and smooth color gradients, producing coherent stylization rather than noisy artifacts. At higher budgets (open 1024, closed 2048), fine details such as the half-timbered façade emerge faithfully, confirming that the gradient oracle scales from compact stylization to high-fidelity reconstruction.

	Method	256 curves				512 curves				1024 curves
	Method	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	Time $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	Time $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	Time $\downarrow$
Open	DiffVG [ 23 ]	0.552	19.83	0.563	18.9min	0.587	21.47	0.537	22.0min	0.616	22.62	0.517	30.6min
	Bézier Splatting [ 26 ]	0.600	22.17	0.540	3.4min	0.646	23.79	0.498	3.3min	0.699	25.45	0.448	3.2min
	CubicSplat (Ours, $S=24$ )	0.624	23.07	0.589	1.8min	0.662	24.43	0.548	1.8min	0.708	25.93	0.502	1.8min
Closed	DiffVG [ 23 ]	0.578	20.69	0.548	16.4min	0.601	21.82	0.531	18.5min	0.631	22.95	0.509	25.1min
	LIVE [ 29 ]	0.576	20.09	0.543	2.6h	0.611	21.70	0.521	4.2h	0.648	23.11	0.495	5.1h

Table 2: Quantitative evaluation on the DIV2K dataset [ 1 ] (200 images). Best results in bold , second-best underlined . Training time is the average per image on an RTX 4090. CubicSplat achieves the highest PSNR and SSIM across all budgets and both modes while training in under 2.1 min per image; in closed mode at

N{=}1024

it surpasses Bézier Splatting by over 2 dB PSNR at roughly

4{\times}

lower cost.

	Method	256			512			1024
	Method	$SSIM^{\uparrow}$	$PSNR^{\uparrow}$	$LPIPS^{\downarrow}$	$SSIM^{\uparrow}$	$PSNR^{\uparrow}$	$LPIPS^{\downarrow}$	$SSIM^{\uparrow}$	$PSNR^{\uparrow}$	$LPIPS^{\downarrow}$
Open	DiffVG	0.601	23.43	0.535	0.645	24.70	0.495	0.699	26.04	0.439
	Bézier Splatting	0.679	26.18	0.457	0.743	27.90	0.383	0.797	29.24	0.310
	CubicSplat(Ours, S=24)	0.681	26.75	0.498	0.740	28.28	0.432	0.803	29.94	0.355
Closed	DiffVG	0.622	24.11	0.513	0.666	25.34	0.475	0.719	26.66	0.420
	Bézier Splatting	0.621	24.19	0.519	0.664	25.61	0.485	0.708	26.91	0.448

Table 3: A quantitative evaluation on the Kodak dataset [ 9 ] with 256 to 1024 curves on RTX 4090. CubicSplat leads in PSNR and SSIM across all settings and achieves the best LPIPS in closed mode.

Figure 3: Qualitative comparison with state-of-the-art differentiable VG rasterization methods, including DiffVG [ 23 ] , LIVE [ 29 ] , LIVSS [ 38 ] , and Bézier Splatting [ 26 ] . The lake scene illustrates gradient and reflection reconstruction, while the dog example highlights preservation of fine details such as eye color and facial structure.

Figure 4: Close-up detail comparison. Raster input (left) and CubicSplat reconstruction with 1024 closed fills (right). Insets highlight preserved sharp architectural edges, coherent layer structure, and clean color boundaries using only parametric vector primitives, yielding stable fine details under magnification.

	Setting	256 curves				512 curves				1024 curves
	Setting	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	TPM $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	TPM $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	LPIPS $\downarrow$	TPM $\uparrow$
Open	$S=24$	0.624	23.07	0.589	0.861	0.662	24.43	0.548	0.883	0.708	25.93	0.502	0.873
	w. Regularization	0.614	22.60	0.597	0.894	0.650	23.89	0.561	0.938	0.693	25.28	0.522	0.917
	w.o Gini-Pruning	0.611	22.38	0.596	0.971	0.645	23.56	0.554	0.982	0.686	24.83	0.508	0.952
	$S=32$	0.624	23.06	0.589	0.859	0.662	24.42	0.548	0.868	0.708	25.94	0.502	0.830
	$S=12$	0.622	22.93	0.591	0.814	0.658	24.27	0.550	0.794	0.704	25.75	0.504	0.832

Table 4: Ablation on the uniformly sampled DIV2K subset [ 1 ] (200 images). We report SSIM/PSNR/LPIPS and training throughput in Tasks Per Minute of a single GPU(TPM, higher is better), measured with 2 workers per GPU. Best results in bold , second-best underlined .

Figure 5: Resolution scalability of CubicSplat. (a) Render time versus resolution scale at different sampling densities

S

. (b) PSNR versus render time for all tested (resolution,

S

) configurations. At native 2K,

S{=}8

achieves near-optimal quality in 2.9 ms; increasing

S

to 48 yields

<

0.03 dB gain at

4{\times}

the cost. Original data refers to Appendix D.

Figure 6: Scaling with closed-fill count $N$ on a detailed illustration from the DanbooRegion dataset [ 44 ] . Insets rendered at 8 $\times$ magnification show scale-consistent fine structures under extreme primitive counts. PSNR improves steadily with sublinear training time and manageable memory overhead.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : Single-primitive artifacts of Bézier Splatting. We isolate failure modes that persist even without multi-primitive occlusion or scheduling effects. Left: A typical open-stroke reconstruction (256 curves) exhibits texture-like blotches arising from sample-based image formation rather than geometric structure. Right (top): A single closed fill reveals coherent diagonal banding inside the primitive (zoomed inset), showing that the discretized Gaussian sum induces structured micro-texture even in the simplest setting. Right (bottom): A single open stroke shows a pronounced spike near the endpoint curvature transition (zoomed inset), where local sampling density and compositing amplify spurious intensity. These artifacts are intrinsic to discrete splatting and vary with rescaling or sampling schedules.

Figure S2 : Low-budget gradient/text fitting with 32 closed primitives. Gray labels report LPIPS $\downarrow$ / PSNR $\uparrow$ .

Image	S	Unclosed					Closed
Image	S	2K	4K	8K	16K	24K	2K	4K	8K	16K	24K
0004	8	28.3370	28.5303	28.5274	28.5274	28.5273	29.2560	29.4434	29.4404	29.4404	29.4404
	12	29.0966	29.3242	29.3207	29.3207	29.3207	29.2969	29.4829	29.4799	29.4800	29.4800
	24	29.2937	29.5277	29.5241	29.5241	29.5241	29.3063	29.4909	29.4879	29.4880	29.4880
	32	29.2755	29.5098	29.5062	29.5062	29.5062	29.3046	29.4899	29.4869	29.4870	29.4870
	48	29.2717	29.5056	29.5020	29.5020	29.5020	29.3045	29.4899	29.4869	29.4869	29.4869

Appendix

Table S1 : Resolution scalability comparison of CubicSplat. PSNR (dB) at different rendering resolutions ranging from 2K (

2040\times 1344

) to 24K (

24480\times 16128

) and sampling rates

K

. Left: unclosed curves; Right: closed curves. CubicSplat maintains consistent quality across all resolutions even at the lowest sampling rate (

K=8

). Bold : best in column; underline : second best.

Image	S	Unclosed					Closed
Image	S	2K	4K	8K	16K	24K	2K	4K	8K	16K	24K
0004	8	1.3366	3.6481	12.6969	47.5781	144.7014	3.4261	10.3347	94.7455	144.1402	323.6840
	12	1.3722	3.8530	13.7628	53.6563	118.3065	4.2291	13.4328	49.9176	196.6026	684.1346
	24	1.7198	5.0681	18.3116	72.4286	161.6611	7.2113	24.9134	95.0149	372.1476	833.3861
	32	1.9629	5.9953	21.5207	85.8643	191.2358	9.3311	33.1704	125.7391	492.9564	1103.9075
	48	2.5748	7.5838	28.4253	113.0126	252.4241	13.7372	48.9079	186.9312	733.0095	1639.3223

Appendix

Table S2 : Rendering time (ms) of CubicSplat at different resolutions and sampling rates

K

. Left: unclosed curves; Right: closed curves. CubicSplat achieves real-time rendering at standard resolutions and scales gracefully to ultra-high resolutions with moderate sampling rates.

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