Bézier Splatting for Fast and Differentiable Vector Graphics Rendering
Authors: Xi Liu
Chaoyi Zhou
Nanxuan Zhao
Siyu Huang
Venue: NeurIPS 2025 poster ()
Keywords: differentiable vector graphics
Gaussian splatting
Primary area: applications
TL;DR: We propose Bézier Splatting, a new differentiable vector graphics representation that achieves an order-of-magnitude computational speedup compared to the state-of-the-art methods.

Abstract: Differentiable vector graphics (VGs) are widely used in image vectorization and vector synthesis, while existing representations are costly to optimize and struggle to achieve high-quality rendering results for high-resolution images. This work introduces a new differentiable VG representation, dubbed Bézier Splatting, that enables fast yet high-fidelity VG rasterization. Bézier Splatting samples 2D Gaussians along Bézier curves, which naturally provide positional gradients at object boundaries. Thanks to the efficient splatting-based differentiable rasterizer, Bézier Splatting achieves 30× and 150× faster per forward and backward rasterization step for open curves compared to DiffVG. Additionally, we introduce an adaptive pruning and densification strategy that dynamically adjusts the spatial distribution of curves to escape local minima, further improving VG quality. Furthermore, our new VG representation supports conversion to standard XML-based SVG format, enhancing interoperability with existing VG tools and pipelines. Experimental results show that Bézier Splatting significantly outperforms existing methods with better visual fidelity and significant optimization speedup.

Bézier Splatting for Fast and Differentiable Vector Graphics Rendering
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Figure 1: This work proposes Bézier Splatting, new differentiable vector graphics (VGs) renderer that achieves magnitude computational speedup in comparison with the state-of- ofte-of-the-art method DiffVG [15] (tested on a NVIDIA RTX 4090 GPU).
Abstract
Differentiable vector graphics (VGs) are widely used in image vectorization and vector synthesis, while existing representations costly to optimize and struggle to achieve high-quality rendering results for high-resolution images. This work introduces
differentiable VG representation, dubbed Bézier Splatting, that enables fast yet high-fidelity VG rasterization. Bézier Splatting samples 2D Gaus- sians along Bézier curves, which naturally provide positional gradients at object boundaries. Thanks to the efficient splatting-base differentiable rasterizer, Bézier Splatting achieves 30x and 150x faster per forward and backward rasterization step for open curves compared to DifVVG. Additionally, we introduce an adaptive pruning and densification strategy that dynamically adjusts the spatial distribution of curves to escape local minima, further improving VG quality. Furthermore, our new VG representation supports conversion standard XML-based SVG format, enhancing interoperability with existing VG tools and pipelines. Experimental results show that Bézier Splatting significantly outperforms existing methods with better visual fidelity and significant optimization speedup. The project page is xiliu8006 gitthu..io/Beiiieeeee..o.ee wwtpsssttttppooppei...
Introduction
Vector graphics (VGs) represent images through parametric primitives such as points, curves, and shapes. Unlike raster images, they enable structured representations, lossless resizing, compact storage, and precise content editing, making them crucial for various applications such as user interfaces and animation.
Recently, differentiable VG rasterization gains significant attention, as it allows raster-based al- gorithms edit synthesize VGs through gradient-based optimization. DiffVG [15] is the first differentiable VG framework that leverages the anti-aliasing algorithm to differentiate the vector curves that are inherently discontinuous pixel space. However, suffers from slow training and low-fidelity rendering, particularly for high-resolution images. LIVE [18] further introduces
Corresponding author: Siyu Huang
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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layer-wise coarse-to-fing fine strategy to improve quality and topology. However, it is highly computation- ally expensive, requiring hours to vectorize a 2K-resolution image. Additionally, learning- based methods such as Im2 [21] train neural networks to map pixels to VGs, but they are restricted to simple graphies and struggle with out-of-domain generalization. Towards scalable applications of VGs, these challenges highlight the need for a differentiable VG method with better efficiency, fidelity, and generalization capability.
This work presents Bézier Splatting, new differentiable VG representation that optimizes Bézier curves through Gaussian splatting- based rasterization. We sample 2D Gaussian points along Bézier curves and their interior regions, then leverage the Gaussian Splatting framework [12] for efficient curve rasterization. Unlike DiffVG [15] which requires computationally intensive boundary sampling and gradient computation, 2D Gaussians inherently provide direct positional gradients at object boundaries through its differentiable Gaussian formulation, enabling over 150x faster backward computation over DifVVG for open curves. We further introduce an adaptive pruning and densification strategy that adaptively removes redundant curves while adding new ones to necessary regions during optimization. It helps the optimization process escape the local minima of current spatial distributions of curves, formulating a "global receptive field" for further improving the VG optimization process. As shown in Fig. 1, Bézier Splatting outperforms the state-of-the-an differentiable VG rasterizer DifTVG [15] for both open and closed curves in terms of efficiency and rendering quality. We summarize the contribution of this work as follows.
We propose a novel differentiable vector graphic representation, Bézier Splatting, which achieves order-of-magnitude computational speedup while producing high-quality ren- dering results.
We present an adaptive pruning and densification strategy to improve the optimization process of Bézier curves by escaping the local minima of the spatial distributions of curves.
Extensive experiments demonstrate that Bézier Splatting outperforms existing diffcrentiable VG rendering methods in efficiency and visual quality.
2
Related works
2.1 Vectorization and Rasterization
Image vectorization and vector graphic (VG) rasterization are important research topics in computer graphics and computer vision. DiffVG [15] introduces the first differentiable VG rasterization framework, laying a foundation for optimizing and generating vectorized representations through gradient-based methods. It expands the applicability of VGs to broader range of tasks including image vectorization, text-to- generation, and painterly rendering. Based on DiffVG [15], LIVE [18) and O&R [7] further proposes layer-wise path initialization strategy, vectorizing raster images into compact and semantically consistent VG representations while preserving image topology. Du er al. [4] use linear gradients fill the regional colors, and Chen al. [3] propose a specific implicit neural representation to model regional color distributions, enhancing the color representation within closed Bézier curves. However, these DiffVG-based VG rasterization approaches [15, 18, 4, 3] suffer from slow optimization, often requiring several hours to process 2K-resolution image with 1,024 curves.
With the advancement of deep learning, another line of approaches directly learns deep neural networks for vector synthesis. Lopes al. [16] combine image-based encoder and VG decoder to generate fonts. Img2 [21] integrates the encoder with recurrent neural networks (RNNs) by leveraging sequential modeling for structured vector synthesis. SVGFormer [2] further adopts Transformer-based architecture [25] to improve the capacity for representing complex geometric structures. More recently, diffusion models [8] have been applied text-to-VG synthesis [33, 11, 30, 29]. However, existing sarming-basee VG synthesis approaches are limited to generating simple graphics, and struggle with oft-of-domain data. In this work, we propose novel VG representation dubbed Bézier Splatting that achieves high-fidelity VG rendering within minutes of optimization for high-resolution images.
2.2 Gaussian Splatting
3D Gaussian Splatting [8] emerges as a promising approach for novel view synthesis (NVS) and attracts significant attention from the community. Its explicit 3D Gaussian-bans volumetric

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representation enables high- fidelity 3D reconstruction, while the differentiable tile- blse rasterization pipeline ensures real-time and high-quality rendering. It has been applied to various domains and tasks, such as 4D modeling [17, 28] and 3D scene generation [31, 23]. Several works further improve the Gaussian primitives for better representation quality. SuGaR [5] proposes to approximate 31) Gaussians with 2D Gaussians for enhanced surface reconstruction. 2D Gaussian Splatting [9] directly adopts 2D Gaussians for 3D reconstruction for simplified optimization and improved geometric fidelity. TetSphere splatting[6] further employs tetrahedral meshes as the geometric primitives to achieve high-quality geometry. Particularly for 2D image representation, Gaussianlmage [35] adopts 2D Gaussians [9] for efficient image representation, achieving a compact and expressive alternative to rasters or implicit representations [22, 19]. Image-GS [36] further enhances it through content- adaptive compression approach. This work proposes to integrate Gaussian splatting with vector representations for differentiable VG rasterization. By sampling 2D Gaussians on Bézier curves and rasterizing through the efficient Gaussian splatting pipeline, the proposed Bézier Splatting representation enables fast yet high-quality VG rendering, even for high-resolution images with complex structures.
Method
3.1 Overall
Given a raster image, our goal is to efficiently vectorize it into a VG representation that closely resembles the input while preserving the details. Existing methods, including DifVV [15] and its following work [18, 31, incur substantial computational costs due to the pixel color accumulation computation. Specifically, DiffVG first constructs bounding volume hierarchy (BVH) tree to determine the curves that intersect with each individual pixel, then solves equations to precisely determine whether pixel lies within a region and to compute the inward or outward gradients at boundary points.
To overcome this inefficiency, this work proposes a novel VG representation, Bézier Splatting, which inspired by the high computational efficiency and expressive fitting capacity of Gaussian splatting [35] for rasterization. Bézier Splatting samples 2D Gaussian points along Bézier curves and their interior regions, then leverages the Gaussian splatting method for efficient rasterization, enjoying the following advantages:
This design significantly accelerates the forward and backward pass of Bézier curve rasteriza- tion, achieving an order-of- mnder-of-magniude speedup without specialized optmizattion techniques;
The 2D Gaussian representation inherently provides direct position gradients for object boundaries, eliminating the need for additional computations such as boundary sampling and gradient derivation via the Reynolds transport theorem in DiffVG [15];
It supports richer texture representation by allowing properties such as spatially varying opacity and width, facilitating complex effects such as linear-gradient color transitions within single curve.
To further improve the fidelity and expressiveness of Bézier Splatting, we introduce a pruning and densification approach (Sec. 3.5), which dynamically removes redundant Bézier curves while adaptively adding necessary eurves in regions that have high reconstruction error. Fig. 2 illustrates the algorithm flow of Bézier Splatting. More details are discussed in the following sections.
3.2 Primitives of Bézier Splatting
Bézier curves. We adopt Bézier curves the parametric primitives of VGs. The representation includes Bézier curves with a degree of M, as:
B((1) Σ ( LE [0,1], i (I....,N),
where is normalized position the curve, curve, and B344 (t) is the Bernstein polynomial of degree M, given by:
of represents the j-th control point of the i-th Bézier
(M (M (1 M
(1)
(2)
3

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Differentiable rasterization splatting
Initialication
Désier
Sampling 2D Goussians
Ground truth
Pruning and deneification
An illustration of the algorithm flow Bézier Splatting. begins by randomly initializing Bézier curves and uniformly sampling Gaussians points along them. These Gaussians then rasterized into the image, enabling gradient-basee based computation to optimize parameters of both Bézier curves and Gaussians. Curves with negligible opacity extremely small shapes removed, while new curves are adaptively added into areas with high reconstruction error, ensuring curves are placed map.
Figure 2:
in areas requiring finer details. forward, backpropagation,
Each Bézier curve (4) is associated with an RGB color parameter E IRS, and an opacity parameter Og E [0, 1] that defines the transparency of the curve.
To formulate an open curve, we follow DifVVG [15] to adopt three sequentially connected Bézier requires additional width parameter to define the stroke thickness. For a closed curve, we adopt two connected Bézier curves, enabling a more efficient sampling in enclosed regions. For closed curves, color filling is applied. The connected Bézier curves in either an open or closed curve share the color parameters, but they have separate opacity parameters better model the opacity changes along the curves within closed areas for enriching the texture representation capacity.
curves with two control points each. An open
2D Gaussians curves. Our Bézier Splatting novelly associates 2D Gaussians with each Bézier curve. The standard formulation of 2DGS [9] parameterizes each 2D Gaussian by position, color, rotation, scale, opacity, and depth. However, ensure compact and differentiable representation, these parameters of Gaussians in our Bézier Splatting are inherited from the corresponding control points. We discuss more details of the sampling of Gaussians, rasterization process, and backaward computation the following sections.
3.3 Sampling Gaussians on Bézier Curves
This work introduces fast differentiable VG rasterizer based Gaussian splatting, allowing gradients from raster images to be backpropagated to the 2D Gaussians, then further backpropagated to the Bézier eurves through differentiable sampling strategy, resulting in highly efficient optimization of the Bézier curves.
Specifically, for each Bézier curve (t),), uniformly sample K points along it based Eq. 1. The sampled point set b, is:
b;
where t is uniformly sampled from [0, 1].
Sampling 2D Gaussians open curves. We represent open curve by using 3 sequential Bézier curves (t) with degree of 4 to form a single continuous stroke, by following the same setting as DiffVG [15]. The stroke consists of 10 control points, the end point of each Bézier curve serves as the start point of the next. To ensure that the final rendering result generates a continuous stroke with consistent width and color, calculate the x-direction scale of Gaussian point by the distance y
between neighboring points by following Fq. 7, while the y-direction scale is learnable parameter that represents the stroke width.
For both closed and open curves, the depth of Gaussian is assigned based on the area curves, ensuring smaller curves not occluded by larger ones. It prevents them from being ignored during optimization, as the gradient would not count Gaussians when the accumulated opacity exceeds
Sampling 2D Gaussians on clased curves. Achieving accurate color filling is non-trivial for closed curves. straightforward approach involves uniformly sampling a large number of points, identifying intersecting curves, solving equations to determine which curves cover them, and then computing scaling factors based on the sampled points. However, this process is computationally expensive, making it inefficient for curve rasterization and optimization.

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To address this limitation, we propose new structure named paired Bézier curve structure, which supports two groups of Bézier curves with equal cardinality and arbitrary degrees, forming a closed region. This structure enables efficient and flexible sampling within the enclosed region between the two curve groups. Specifically, the two Bézier curves B1 (t) and BR+1(t) share the same start and end points, forming the inside curves and boundaries of a closed region. A total of R intermediate Bézier curves are generated by linearly interpolating the corresponding control points p,0) and pf p(R+1) as:
PSA (1 (1- _ +
+taP)R+ k- R,
where is E [0 are sampled from a normalized cumulative distribution function (CDF). The interpolated control points (Pf()) define intermediate curves that form dense strip between the two boundaries. Note that this paired structure can naturally extend to cases where each curve is composed of multiple connected Bézier segments, provided that all segments have the same number of control points, similar to the representation used in DiffVG [15].
This non-uniform sampling ensures that points near the boundary curves have small scales, mitigating the influence of interior Gaussians to the exterior of closed area. The interpolated Bézier curves are:
M Σ By (A)Ps))
B
= R.
(5)
Then, 2D Gaussians are sampled on these interpolated curves, by following the same procedure as open curves (Eq. 3), resulting in a structured and efficient point sampling within the enclosed region. Furthermore, to mitigate artifacts caused by non-convex curve shapes during interpolation, we incorporate the Xing loss from LIVE [18], which enforces convexity constraints on curve shapes, to improve the stability of the interpolation process. The full set of sampled points on the i-th Bézier curve is:
X [bo bI......................... bR+1] R [(R+2)xXx2,
(6)
Let X,, denote the 2D position of the k-th sampled point on the r-th interpolated curve. To construct anisotropic 2D Gaussian primitives aligned with the local geometry, we define the spatial scales as follows. The x-direction of each 2D Gaussian is defined to align with the local tangent direction of the curve (i.e., along the curve), while the y-direction is perpendicular to it (i.e., across adjacent curves).
The scale T (r, k) is computed from the Euclidean distance between consecutive points along the curve, and o, (r, k)) is computed from the distance between corresponding points on adjacent curves:
on (r, k) = X X,,k 0g k:) = X, Xx,,///pp.
Here, p a global constant that controls the overall density and overlap of the Gaussians. The rotation 0r of each Gaussian is defined by the angle of the local tangent vector, estimated from neighboring points as
0,,6 = alaan2 (yr,4+1 1r,k- Tr.411 xr)k-1).
For boundary points, the rotation is set to align with the nearest available neighbor.
3.4 Splatting-based Differentiable Rasterization
Once all Gaussians on Bézier curves are sampled, the rasterization process follows the Gaussian splatting pipeline [12], which is very fast in both forward and backward computation. Different from GaussianImage [35], we use a-blending for pixel rendering instead of the accumulation-base blending. Accumulation-based blending calculates the pixel value based all overlapping Gaussians, such that it conflicts with the rendering principles of VGs, where occlusion plays a crucial role in defining vector structures. a-blending ensures proper occlusion handling, it allows foreground elements to contribute more to the final pixel value. The rendering of a pixel is:
On = Σ c. (1 (1-a,).
(9)
iEM
j=1
5

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On is the n-th pixel color, c₁ is the color of corresponding Gaussians, a₁ is computed by the projected
2D covariance Σj:
1 oi = 2 d =, d,,
0g3 = 04 exp
2
(10)
where du is the distance between the pixel and Gaussian center, and Σg can be modeled by 0, og and y as:
Σ; = (R,S)).
(11)
cos( 0.)
sin( cos(0.)
o 0 0
R.
S
(12)
sin (.))
Discussion on method efficiency. The rasterization process of Bézier Splatting is very efficient, since we directly sample 2D Gaussians from Bézier curves in differentiable manner, then splat them to the 2D plane. The rasterization pipeline remains end-to-end differentiable while being highly optimized for parallel computation and large-scale matrix operations. As result, Bézier Splatting is a highly efficient and fully differentiable VG representation, which ensures both fast rendering and gradient-based optimization. It not only preserves the flexibility of VGs but also allows seamless integration into deep learning frameworks, making it well-suited for tasks requiring high-quality and editable vector representations.
3.5 Optimization
Training objective. Given raster image I E Hx the goal of image vectorization is to vectorize the image into Bézier curves while ensuring high-fidelity reconstruction. We first randomly generate set of Bézier curves to lay on the canvas, then employ the differentiable rasterizer discussed in Sec. 3.4 to render a raster image Î E " The Bézier curves can then be optimized through any gradient-based based loss functions. In this work, we formulate the optimization objective as minimizing the loss between I and Ý while enforcing the curves to be convex. Therefore, we only adopt an L2 loss and a Xing loss Lxing [18]. as:
L A A1 I A2
(13)
where a1 and 22 are hyperparameters that trade off the two loss functions.
Adaptive curve pruning and densification. The gradients of 2D Gaussians are influenced by local pixels only. Therefore, it is hard for 2D Gaussians to dynamically reallocate to regions requiring finer- grained details. The Gaussians would be trapped in local minima, leading to redundant Gaussians that are optimized as either low opacity or excessively large size, resulting in artifacts in rendering results.
In standard 3D Gaussian Splatting pipeline [12), Gaussians with low opacity or excessive size pruned, whereas those with high gradient responses are split into two. However, this strategy is not directly applicable to Bézier curves. Note that in volumetric representations. large Gaussians are usually unnecessary for modeling any particular structure. In contrast, VG representations often encompass large uniform regions such as backgrounds or areas with homogeneous colors (e.g., walls). Consequently, the size-based pruning strategy would remove critical structures in VG representations. Similarly, splitting Bézier curve into two can introduce significant randomness, as regions do not always indicate poor reconstruction in VGs. Since VGs assume a uniform color within each enclosed region, complex textures naturally produce high gradients. This does not imply the region should be split, as it may disrupt the semantic consistency of the VG representation.
To address this issue, this work introduces new pruning and densification strategy dynamically adjust the density of Bézier curves throughout the optimization process. For pruning redundant Bézier curves, we apply three criteria to ensure a stable and precise optimization process. First, we remove curves with opacity below a dynamic threshold that gradually decreases as optimization progresses, ensuring that weakly contributing curves are eliminated while preserving essential structures. Second, we remove curves with an area below a predefined threshold, as they contribute minimally to the final representation. To remove visually insignificant or noisy Bézier curves, we apply an opacity- based filtering strategy: for per-segment opacities, we discard curves whose middle segment is significantly fainter than both ends-often indicating that single curve improperly spans a region better represented by multiple curves; otherwise, we discard curves with overall opacity below threshold (e-g., 0.2) to eliminate globally low-visibility curves. Third, we remove curves that exhibit high color similarity with surrounding curves and have significant overlap, they provide little

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Table 1: Computational speed for rendering 2,040 1,344 image with 2,048 curves.
Open curve
Closed curve
DIIVG [15]
Bezier Splatting
Speedup
DIIVG (15)
Bézier Splatting
Speedup
Forward pass
141.3ms
4.5ms 4.7ms
31.4%
85.2ms
14.1ms 24.58ms
0.0%
Backward pass
701.3ms
149.2x
448.3ms
18.2x
1024 closed curves
DIIYG, PSNR: 26.25
LIVE, PSNR: 26.61
Raster Input
LIVSS, PSNR: 27.30
Ours, PSNR: 27.93
2,048 open curves
Input
Ours,
Figure 3: A qualitative comparison of our method and the state-of-thh-an art differentiable VG rasteriza- tion methods, including DifVV [15], LIVE [18], and LIVSS [27].
additional information and can be pruned without affecting the final representation. By removing such curves, we allow the optimization to introduce more appropriate replacements, leading to more accurate and visually coherent vectorized representation. For curve densification, we adopt an error-driven curve allocation strategy inspired by LIVE [18]. Specifically, we compute connected error regions, rank them by area, and add new Bézier eurves into the highest-error regions. This adaptive redistribution mechanism ensures that curves are allocated where they would contribute the most to the reconstruction fidelity. The pruning and densification strategy enables a "global receptive field" for redistributing curve density, effectively preserving visual details for high-fidelity rendering while maintaining compact vector representation.
4 Experiments
4.1 Experimental Setups
Implementation details. We implement Bézier Splatting in PyTorch [20] and optimize it by using the Adam optimizer [13] with a StepLR learning rate scheduler. The learning rate is initialized at 0.01 for color, 2e-4 for Bézier curve control points, and 0.1 for opacity. For the pruning and densification strategy, the opacity threshold is set to 0.02, and the overlap threshold based on Axis- Aligned Bounding Boxes (AABB) is set to 0.9. Following the approach in LIVE [18], new curves are initialized in a circular pattern, and the number added curves matches the number of removed ones maintain constant total curve count. Open curves are optimized for 15,000 iterations, and closed curves are optimized for 10,000 iterations. Pruning and densification are applied every 400 steps until the last but 1,000 steps, after which they are halted to stabilize the representation.
Datasets. We comprehensively evaluate our method across different image domains. We use publicly available DIV2K [24] dataset for evaluating natural images. Due to the high computational cost of baseline method LIVE [18], we uniformly subsample the DIV2K dataset by selecting one out of every four images, resulting in final evaluation set of 200 images from the original 800-image

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Table 2: Quantitative evaluation and optimization efficiency of differentiable VG methods the
DIV2K dataset [24].
Method
curves
curves
curves
SSIM
PSNR
LPIPS-
SSTM!
PSNR
L/IPS
Opt.
SSIM
PSNR!
LPIPS-
Opt.
Open
Diavo [15]
Ours
0.600
0.552
22.17
19.83
0.563
0.540
18.9min
3.4min
0.646
0.587
23.79
21.47
0.498
0.537
22.0min
3.3min
0.699
0.616
25.45
22.62
0.448
0.517
3.2min
6mii
DirVG
20.69
0.548
16.4min
0.601
21.82
18:5min
0.631
22.95
0.509
25.1
Closed
LIVE [18]
0.576
20.09
0.543
2.6h
0.611
21.70
0.521
4:2h
0.648
23.11
0.495
5.1h
LIVSS [27]
0.586
17.71
0.542
59 2min
0.633
18.71
0.530
54.3min
0.678
19.83
0.517
1.4h
7.8min
0.507
8.6min
Open 200
Cliok 2122
Figure 4: Our Bézier Splatting achieves high-quality image vectorization results various types of images including artworks, cartoons, and natural images. Curve type and count are indicated at the bottom right of each sample.
dataset. Additionally, we test our method on non-photorealistic images, including the artwork images from C dataset [10] and cartoon images from Danbooregions dataset [32]. to demonstrate its effectiveness for diverse types of images, as shown in Fig. 4.
4.2 Method Comparison
Table reports the forward and backward runtime for processing 2,048 curves on an image with a resolution of 2,040 1,344. Compared to DiffVG [15], our method significantly accelerates VG rasterization by 31.4x faster per forward step and 149.2 faster per backward step for open curves. For closed curves, our color filling strategy requires sampling 20 additional Bézier curves per closed curve, while our method remains highly efficient, achieving 6x faster forward and 18.2x faster backward computation.
Table 2 quantitatively evaluates the quality of differentiable VG representations by three commonly used metries, MS-SSIM [26), PSNR, and LPIPS [34]. Our method demonstrates higher optimization efficiency and rendering fidelity for both open and closed curves with different curve numbers.
Fig. visually compares DiffVG [15), LIVE [4), [27], and Bézier Splatting under 512 closed curves and 2048 open curves. Compared to DifVVG [15], our method captures significantly more fine-grained textures. Compared to LIVE [18], method effectively enhances rendering quality, resulting in higher-fidelity results. This improvement stems from our method's ability to globally optimize all curves, rather than LIVE's layer-by-layer path-adding strategy. Compared to LIVSS [27], our method demonstrates superior ability in preserving fine structural details, especially in text regions. Due its heavy reliance on layer-wise semantic simplification, LIVSS struggles to optimize regions where semantic information is ambiguous or difficult to extract. Moreover, by optimizing the entire VG simultaneously, our approach ensures a cohesive and natural appearance, avoiding the accumulation of errors and inconsistencies introduced by layer-wise updates.
As shown in Fig. 4, Bézier Splatting consistently renders high-quality VGs across various image domains from photorealistic natural images, watercolor paintings, to cartoon images. Unlike learning- based image vectorization methods [16, 21, 2], which can not easily generalize out-of-domain data, our method can flexibly handle different types of images.
4.3 Layer-wise Image Vectorization
Existing works such as LIVE [18), SGLIVE (37] and LIVSS [27] have demonstrated that layer-wise vectorization strategies significantly improve the topology and compositionality of DifIVG-

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Raster Input
Bézier Splacting
Bézier Splatting Layer-wiie Strategy
Figure 5: As a differentiable VGs renderer, Bézier Splatting can integrate with topology-awarg strategies such layer- wise vectorization [18] to further improve compositionality and detail.
Bézier Splatting Rendering
Converted SVG
Remove
Move
Bézier Splatting Rendering
Converted
Deform Replicate
Recolor
Figure 6: Our Bézier Splatting representation is compatible with the standard SVG XML format and supports flexible vector editing operations.
generated vector graphies [15]. Since the proposed Bézier Splatting a fast and differentiable VGs renderer alternative to DiffVG, is naturally compatible with these topology-aware strategies and can leverage them further enhance the structural quality of the optimized vector outputs. As shown in Fig. 5, we incorporate the layer-wise vectorization approach from LIVE [18] into the Bézier Splatting optimization process This integration yields improved compositional integrity and finer details for regions with complex patterns such as the butterfly's wings.
4.4 Editability and Compatibility with SVG XML Format
Our Bézier Splatting representation is fully compatible with the standard SVG XML format, ensuring the same editability as standard SVGs and seamless integration with existing SVG tools. As shown in Fig. 6, a high-fidelity SVG be obtained via a simple conversion algorithm (see Appendix for details). Once converted to SVG, the vectorized output is fully editable at the primitive level, allowing users to freely deform, remove, move, replicate, and recolor individual elements.
Conclusion
This work has presented Bézier Splatting. novel differentiable vector graphics (VGs) representation that leverages Gaussian splatting for efficient Bézier curve optimization. Our method achieves x0x faster forward computation and 150x faster backward computation rasterization compared to existing methods, while also delivering high rendering fidelity. Additionally, our adaptive pruning and densification strategy improves optimization by dynamically adjusting curve placement during optimization. Extensive experiments have demonstrated that Bézier Splatting outperforms existing differentiable VG methods both training efficiency and visual quality, making a promising solution for scalable applications of VGs.
Limitation and future work. The closed curves in Bézier Splatting are enforced to have convex shapes using Xing loss [18] to prevent false interpolation results, which may slightly reduce the capacity of VG representations. Additionally, the closed curves require sampling more Gaussian points to represent area boundaries precisely, leading to slower computation compared to open curves. interesting future direction to leverage the efficiency and differentiability of our approach for high-quality VG synthesis applications such text-to- VG generation text-guided VG editing.

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6
Acknowledgement
This work supported in part by the National Science Foundation (NSF) and SC EPSCoR Program under award number #O1A-2242812. This research used in part resources on the Palmetto Cluster Clemson University under NSF awards MRI 1228312. II NEW 1405767, MRI 1725573, and MRI 201 8069. The views expressed in this article do not necessarily represent the views of NSF or the United States government. The authors also thank Dr. Yiqi Zhong and Dr. Luming Liang for beneficial discussions.
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A
Ablation Study
The effectiveness of adaptive pruning and densification and layer-wise vectorization strategy. We conduct an ablation study to evaluate the effectiveness of our adaptive pruning and densification strategy. Quantitative results are shown in Table 3. To further investigate the flexibility our framework, we incorporate layer-wise curve addition strategy inspired by LIVE [18], where curves are progressively added during training. Although this strategy achieves slightly better reconstruction metrics, it doubles the training time compared to our adaptive pruning and densification approach, making less favorable for simessensitiee applications. Fig. shows our result with layer-wise image vectorization strategy.
Table 3: An ablation study on adaptive pruning and densification and layer-wise training strategy. evaluated on 512 closed curves with DIV2K dataset [24].
Method
SSIM
PSNR
LPIPS-
Opt
No Strategy
0.590
21.10
0,530
8.3 min
w/ Prune&Densify
0.607
22.11
0.528
8.3 min
Layer-wise
0.613
22.21
0.521
16.2 min
GT
Closed, 32
Closed, 64
Closed, 128
Closed, 256
Closed, 512
Figure 7: Our Bézier Splatting is fully compatible with the layer-wise vectorization strategies [18].
Comparing different numbers curves. We progressively increase the number of curves for vectorizing watercolor image (Fig. 8) with numerous small spots. As the number of curves grows, our approach first reconstructs the foreground object, the bird, then gradually refines the smaller spots. This demonstrates that our method prioritizes the primary structure before optimizing finer details, effectively distributing curves to balance global structure and local texture representation, thanks to our adaptive pruning and densification strategy.
256
512
1024
2048
Raster Input
Figure 8: A comparison of different curve numbers by Bézier Splatting.
systematic evaluation of computation time. systematically evaluate the computational efficiency of our framework. report detailed timing statistics across varying configurations of
13

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Table 4: A systematic evaluation of computation time. We test the forward and backward time (ms) under varying numbers of sampled curves within one closed Bézier eurve (Inter. #), and the total number of Bézier curves (Curve #).
Inter: #
Curve #
Forward w/ gradient
Sample Gaussians
Splatting
Backward
Render (FPS)
20
2048
10.66
4.12
6.11
21.64
103.45
40
2048
14.79
5.11
9.32
25.06
68.30
80
2048
27.01
8.22
17.74
33.40
37,90
256
7.20
3,81
3.09
16.72
150.30
512
7.70
3.88
3,67
17.42
144.30
1024
10.39
4.06
5.96
19.96
114.30
2048
14.79
5.11
9.32
25.06
68.30
interpolation curve numbers and curve resolutions, as summarized in Table 4. The total forward time includes two major parts: the Gaussian sampling and the Gaussian splatting. Increasing the number of interpolated curves within the closed Bézier curves leads to near-linear growth in both forward and backward runtimes, demonstrating good computational scalability. When fixing the interpolation number (e.g., at 40) and varying the number of Bézier curves, the forward time increases moderately, reflecting the additional cost of handling finer spatial detail. The splatting stage dominates the forward time at higher number of curves due the increased number of sampled points per region, while the Gaussian sampling time is the primary bottleneck at lower number of curves. This attributed to overheads from sequential memory allocation and data transfer operations, which are not effectively parallelized on the GPU, suggesting potential for further optimization.
Table quantitative comparison between Adobe Image Trace [1] and our method across different images.
Image ID
0004
0008
0012
0016
0020
Method
Adobe [1]
Ours
Adobe [1]
Ours
Adobe [1]
Ours
Adobe [1]
Ours
Adobe [1]
Ours
SSIM
0.835
0.849
0.645
0.637
0.658
0.658
0.627
0.628
0.814
0.822
PSNR
26.43
28.62
23.26
24.33
22.29
22.87
23.87
24.54
26.67
28.31
LPIPST
0.380
0.387
0.489
0.489
0.523
0.518
0.534
0.543
0.505
0.503
Comparison with conventional image vectorization methods. We conduct a comparison against the Image Trace from Adobe Illustrator [1] on five images from the DIV2K dataset. Both methods are evaluated using the same number of parameters (around 20K). shown Table 5, our method achieves higher fidelity, with average PSNR of 734 db compared to 24.904 db of Image Trace.
Arbitrary resolution rendering. Our Bézier Splatting representation naturally supports rendering arbitrary resolutions because all Gaussian parameters are analytically derived from the underlying Bézier curves. When higher-resolution images required, proportionally increasing the sampling rate is sufficient to maintain reconstruction fidelity. For example, doubling the resolution simply doubles the sampling density. As shown in Table 6, rendering higher resolutions (4K or 8K) with increased sampling preserves comparable level of visual quality to the 2K baseline, without introducing noticeable artifacts. We report results on the first four images from the [24] to illustrate this property.
Table 6: Results of Bézier Splatting rendered at different resolutions and sampling settings. Notation: "orig." denotes the baseline sampling rate used during optimization, and "xS" indicates S-times higher sampling density.
4K
8K
Image ID
2K (orig.)
orig.
(x2S)
orig.
(x4S)
0004
26,8976
27. 27.1553
27.2761
25,9800
26.6594
0008
21.2063
21.0614
21.4340
19.8318
21.1214
0012
19.8816
19,9079
20.0900
18.9389
19.7760
0016
22.3456
22.4327
22.5408
21.7088
22.3140
14

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Algorithm Adaptive Pruning and Densification for Closed Bézier Curves
Notation:
opacity(b) - mean opacity of curve b.
area(() area enclosed by curve b.
(b, bj) Euclidean color difference between curves b; and by-
[b, intersection-over-uiioon between the bounding box of by and neighboring curves with colordiff < 0.03.
ConnectedComponnntss connected enror regions extracted from the quantized error map.
Input: Closed Bézier curve set B (bI..... ..t....,ba. by J; opacity map oz; error map E; iteration index t. Output: Updated curve set B".
Initialize B' < B.
Pruning phase for i + to N do if Roppcityyb (b:) < Topacity (L) then
Remove b, from B". (Low-opacity pruning] end if
if (b;) Tarea then Remove b; from B². (Small-area pruning) end if
if [b; {b colordif bj) < 0.03 > 0.9 then
Remove b, from B'. (Redundant-overllap pruning)
end if end for
Densification phase
R + ConnoctedComppoeents/SP EE).
Rxorted <- SortBy for each region E Rorttd do
if curve budget allows then
Insert a new closed Bézier curve into ry.
end if end for return B
B Details of Adaptive Pruning and Densification
To provide a clearer understanding of our optimization behavior, present here the detailed algorithm used in our Bézier-splatting framework. This algorithm is designed to mitigate the local-minima issue observed in 3D Gaussian Splatting (3DGS) and is adapted in our Bézier-splatting framework, where suboptimal primitive initialization leaves certain regions insufficiently covered. Inspired by the pruning- pruning-splitting splitting mechanism of 3DGS [1], method dynamically reallocates Bézier curves according to the reconstruction eтror map: curves with low opacity, small area, high overlap with nearby curves of similar color pruned, while new curves are inserted into regions with high reconstruction emror to improve coverage. This reallocation keeps the total curve budget fixed, enhances reconstruction quality, and helps the optimization escape poor local minima. For open curves, we additionally introduce a splitting rule: if the middle opacity of curve is more than 0.5 lower than that of both endpoints, the curve split into two segments to better adapt to local structure variations.
procedure of the
C Convert Bézier Splatting to Standard SVG Format
As described in Algorithm 2, we convert our optimized Bézier Splatting representation into a standard SVG file for compatibility with downstream vector editing tools. Each curve is represented by set of 34 control points C, E (3++11xx corresponding to & continuous cubic Bézier segments. These control points are normalized the range [-1, and are first transformed into pixel coordinates according to the canvas size (W, H).
15

---

For each curve i, we initialize the SVG path string d with [M P. command. We then iteratively con- struct each cubic segment using three internal control points: (p1, P2. pa) (P: p p()++), and append the segment as a path command [C Pr, P2. ps] to d. After all segments are processed, [Z] is appended to close the path.
Each path d then associated with fill color (r, g. b), computed via sigmoid activation on the optimized feature vector Fi, and opacity 1.0. The result is stored in global path set S, and all paths are assembled into the final SVG file S. This output directly compatible with standard vector graphics tools such as Adobe Illustrator.
Algorithm 2 Convert Bézier Splatting to Standard SVG
Input: Optimized control points C [=3xx33-11x2 feature colors F R canvas size (W, H)
Output: SVG file S containing cubic Bézier paths
for each curve = 1 to N do
P. Ca+ll (W, .I) (r.g.b) - sigmoid(Fi) x 255 Initialize path d [M P
for each segment = 1 to k do
(PI,P2 P2.P..P Ps) (P, p33422 p,)++3
de [C pı, P2, ps]
end for d
S + S (path(d, fill (r,g,b), opacity = 1.0)
end
Return: SVG file assembled from all paths
D Bézier Splatting Supports Flexible Curve Attributes
All 2D Gaussians in our method are generated via a differentiable sampling algorithm, which makes it straightforward to incorporate user-defined shape attributes, such as linear-gradient fills in color or opacity. To demonstrate the extensibility our model, we evaluate the multi-opacity scheme for open curves: each Bézier segment within open curve is assigned an independent opacity value. This design enables users to either maintain consistent sampling point appearance across segments or apply customized interpolation strategies. This strategy significantly improves vectorization quality for open curves, as shown in Table Since closed curves are the primary representation format SVG, adopt uniform setting for opacity and color in those regions to ensure full compatibility with existing vector graphic tools. Nonetheless, users can still follow the above multi-opacity scheme to implement iinar-gradient fills for both opacity and color, enhancing the flexibility and expressiveness of vector graphics.
Table 7: An ablation study on the number of opacity parameters per curve, evaluated on 512 open curves with DIV2K dataset [24].
Method
SSIM
PSN R
LPIPS
3-opacity
0.65
23.79
0.50
1-opacity
0.63
22.97
0.51
E
More Comparisons
We present more qualitative comparisons on the DIV2K [24] dataset. As shown in Fig. Fig. 10, Fig. 11, Fig. 12, and Fig. 13, our method shows consistent improvements over the baselines, including better preservation fine details, higher rendering fidelity, fewer visual artifacts, and more accurate geometric structures. Furthermore, as the number of curves increases, our method demonstrates a significantly improved representational ability, enabling more precise reconstruction of complex shapes, and fine-grain -nn-igri structures.
16

---

We further evaluate our method on another natural image dataset, Kodak [14]. Due to the slow vectorization speed LIVE [18], we compare with DifVV on open curves only. As shown in Table 8, our method consistently outperforms DiftVG across quantitative metrics including PSNR, SSIM, and LPIPS.
Table 8: A quantitative evaluation the Kodak dataset [14] with 256 1024 curves. We report
results on open curves.
256
512 PSNR
1024
Method
SSIM?
PSNR
LPIPS*
SSIM
LPIPS
SSIM
PSNR
LPIP51
DiffVG
0.601
23.43
0.535
0.645
24.70
0.495
0.699
26.04
Open
Ours
0.679
26.18
0.457
0.743
27.90
0,383
0.797
29.24
0.310
DifVVG
0.622
24.11
0.513
0,666
25.34
0.475
0.719
26.66
0.420
Closed
Ours
0.621
24.19
0.519
0.664
25.61
0.485
0.708
26.91
0,448
F More Image Vectorization Results
Fig. 14, Fig. 15, and Fig. 16 show more results on natural images from DIV2K [24] and Kodak [14], well animation images from DenbooRegion [32], respectively. The results demonstrate that both the global structure and local texture the images are well reconstructed, highlighting the effectiveness of our approach in capturing fine details and complex shapes.
17

---

Raster input
DIlLG
Open
Ours
= 256
N 512
N= 1024
DilvvG
LIVE
Closed curve
LIVSS
Ours
- 256
N= 512
= 1024
Figure 9: qualitative comparison of our method and the existing differentiable VG rasterization method on DIV2K dataset [24].
18

---

Raster input
DiliGG
Open curve
Ours
N 256
N 512
N 1024
DiliVG
LIVE
Closed curve
LIVSS
Ours
- 256
= 512
= 1024
Figure 10: More qualitative comparisons of our method and the existing differentiable VG rasteriza- tion method on DIV2K dataset [24].
19

---

Raster input
Open curve
Ours
= 256
N 512
1024
LIVE
Closed curve
LIVSS
Ours
N 256
== 512
= 1024
Figure 11: More qualitative comparisons of our method and the existing differentiable VG rasteriza- tion method on DIV2K dataset [24].
20

---

Raster input
Open curve
Ours
N = 256
= 512
= 1024
DiLva
LIVE
Closed curve
LIVSS
Ours
= 256
N= 512
= 1024
Figure 12: More qualitative comparisons of our method and the existing differentiable VG rasteriza- tion method on DIV2K dataset [24].
21

---

Raster input
DiliGG
Open
Ours
= 256
512
= 1024
DirvVG
LIVE
Closed curve
LIVSS
Ours
= 256
512
= 1024
Figure 13: More qualitative comparisons of our method and the existing differentiable VG rasteriza- tion method on DIV2K dataset [24].
22

---

GT
Closed, 256
Closed, 512
Closed, 1024
Closed, 2048
Figure 14: More image vectorization results by Bézier Splatting on DIV2K dataset [24].
23

---

GT
Closed, 256
Closed, 512
Closed, 1024
Closed. 2048
Figure 15: More image vectorization results by Bézier Splatting on Kodak [14] dataset.
24

---

Pe verg
Dmc ma
I' sorg
I'm werg
GT
Closed, 256
Closed. 512
Closed, 1024
Closed, 2048
Figure 16: More image vectorization results by Bézier Splatting DanbooRegion dataset [32].
25

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Reviews:
[{"review_id": "I6koWEuWUI", "reviewer": "Reviewer_BJtc", "summary": "This work has presented Bézier Splatting, a novel differentiable vector graphics (VGs) representation that leverages Gaussian splatting for efficient Bézier curve optimization. The proposed method achieves faster forward computation and faster backward computation in rasterization compared to the baselines, while also delivering high rendering fidelity. Overall, the idea of this paper is interesting.", "questions": "Please see the weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n1. The authors propose a novel differentiable vector graphic representation, Bézier Splatting, which achieves 30× faster forward and 150× faster backward computation while producing high-quality rendering results.\n2. The authors introduce an adaptive pruning and densification strategy to improve the optimization process of Bézier curves by escaping the local minima of the spatial distributions of curves.\n3. Extensive experimental results demonstrate the effectiveness of the proposed method.\n\n# Weaknesses\n1. The reviewer wants more explanation about the statement \"It helps the optimization process escape the local minima of current spatial distributions of curves\". Is there any theoretical analysis that can support this argument?\n2. In the experiments, how to determine the hyperparameters \\lambda_1 and \\lambda_2 in (13)? Is it necessary to change the hyperparameters to fit different datasets?", "was_revised": true, "final_justification": "I appreciate the authors’ rebuttal, which addresses most of my concerns. I have raised the score from 4 to 5.", "rebuttal": "We thank the reviewer BJtc‘s valuable comments and suggestions, which have helped us improve the clarity and quality of the paper. We have carefully addressed the identified weaknesses, and our responses are summarized as follows:\n\n**Weakness 1**: The reviewer wants more explanation about the statement \"It helps the optimization process escape the local minima of current spatial distributions of curves\". Is there any theoretical analysis that can support this argument?\n\nIn this work, we provide an intuitive analysis of the optimization landscape: both 3D Gaussian Splatting (3DGS) and Bézier‑splatting suffer from local minima because suboptimal primitive (Gaussian/curve) initialization leaves some regions insufficiently covered. In 3DGS [1], this issue is alleviated by pruning large‑volume and low‑opacity Gaussians and splitting those with high gradients to densify underrepresented regions.\n\nIn image vectorization, smooth or homogeneous regions only need a few curves to be well reconstructed, whereas complex textures typically require many more. However, random initialization can again leave some regions underrepresented and cause the optimization to fall into local minima. Inspired by 3DGS [1], we propose a pruning and densification method for image vectorization that uses the error map as guidance to reallocate curves: removing redundant or low‑impact ones and relocating them to regions where additional curves are needed. \n\nSince our method is based on Gaussian-based alpha blending, the contribution of each Gaussian to the final color is defined as Equation (9):\n$$\nC_n =  \\sum_{i \\in M} c_i \\alpha_i  \\prod_{j=1}^{i-1}(1 - \\alpha_j)\n$$\nwhere $c_i$ and $\\alpha_i$ represent the color and opacity of the $i$-th Gaussian, respectively. $M$ is the set of Gaussians along the compositing order. This formulation directly implies that Gaussians with very low opacity, or those heavily occluded by earlier Gaussians, have almost no impact on the final rendering.\n\nGuided by this observation, during optimization we prune curves with extremely low opacity, very small area, or those that largely overlap with nearby curves of similar color, as they are redundant and contribute little to the result. At the same time, we introduce new curves into regions with large reconstruction errors. This reallocation keeps the total number of curves fixed, improves overall reconstruction quality, and helps the optimization escape local minima caused by imperfect initial positions. Table 3 in supplemental material demonstrates the effectiveness of our pruning and densification method, which improves the PSNR from 21.10 to 22.11.\n\n**Weakness 2:** In the experiments, how to determine the hyperparameters \\lambda_1 and \\lambda_2 in (13)? Is it necessary to change the hyperparameters to fit different datasets?\n\nThe choice of hyperparameters​ $\\lambda_1$ and $\\lambda_2$  in Equation (13) depends primarily on user preference and the specific characteristics of the target vector graphics. If users expect little or no self-intersection, or prefer smoother vector graphics, increasing the weight of the crossing loss $\\lambda_2$ can be beneficial. However, this often comes at the cost of a slight reduction in PSNR. Setting the weights too high may lead to optimization failure, which aligns with observations reported in LIVE [2].\n\nIn our experiments, we consistently adopt the same set of hyperparameters ($\\lambda_1 = 1.0$, $\\lambda_2 = 0.01$). Empirically, this set performs well across different datasets, so it is generally not necessary to adjust $\\lambda_1$ and$\\lambda_2$ for different datasets.\n\n**References**\n\n[1] Kerbl, B., Kopanas, G., Leimkühler, T., & Drettakis, G. (2023). 3D Gaussian splatting for real-time radiance field rendering. ACM Trans. Graph., 42(4), 139-1.\n\n[2]Ma, X., Zhou, Y., Xu, X., Sun, B., Filev, V., Orlov, N., ... & Shi, H. (2022). Towards layer-wise image vectorization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 16314-16323)."}, {"review_id": "KEVnwgR4wy", "reviewer": "Reviewer_ZXL5", "summary": "This paper proposes a curve splatting method with 2D Gaussian rendering. Both open and closed curve representations are converted to a set of Gaussians, and those Gaussians are rendered via differentiable rasterization, yielding back-propagation from the image loss to the Bezier curve parameters. Three pruning conditions discard redundant or inefficient curves and replace them with the regions that need more curves.", "questions": "I have the following questions and suggestions:\n\n- Does 2D grid Gaussian sampling for closed curves suffer from over-sampling near the starting and ending points? How does it affect the rendering performance, and how could we mitigate this problem? \n\n- Can the current method handle images of various scales without changing sampling parameters? If not, the relation between sampling resolution and image size should be discussed.\n\n- What is the reconstruction policy for curve splatting about depth? \n\n- Please clarify the pruning and densification process.\n\n- For the offline method, the gain in quality is more important than the speed gain. Is there any quality improvement against the state-of-the-art method?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper’s strengths are as follows;\n\n-The curve-to-Gaussians conversion achieves fast rendering speed compared to the state-of-the-art method.\n- The proposed splatting method achieves the state-of-the-art results for open curves.\n\nThere are the following weak points:\n\n-It would be great to add some figures about open and closed curves with notations.\n\n-A 2D Gaussian splat has a round shape, which poses a problem in representing rectangular shapes. To this end, Gaussian splat representation requires dense sampling as it is a discrete representation.\n\n-The sampling resolution for Gaussians has not been mentioned in this paper. This is crucial as the proposed method aims to reconstruct vector graphics at any image scale. This paper should discuss and examine how dense Gaussian sampling should be with respect to the image resolution.\n\n-Lines 170 and 175 seem duplicated.\n\n-Which probability density function has been integrated into the cumulative distribution function mentioned at Line 170?\n\n-2D grid sampling for the closed curve suffers from non-uniform sampling near the starting and ending points. This increases the rendering time of Gaussians without quality gain. \n\n-A proposed curve pruning method should be explained in more detail. The current explanation looks vague; the overlap threshold at Line 266 and AABB are not mentioned in the “pruning and densification” paragraph. \n\n-Although the rendering time becomes faster, there is no significant improvement in quality.", "was_revised": true, "final_justification": "After the rebuttal, the authors resolved my concerns with solid answers. I will change my stance to accept this paper.", "rebuttal": "We sincerely thank reviewer ZXL5 for the detailed comments and suggestions. \n\n**Weakness 1:** It would be great to add some figures about open and closed curves with notations.\n \nFigures 1, 3, and 4 of the paper include both open and closed curves with corresponding open/closed notations. We will add additional notations to Figures 5 and 6 in the revised version for clarity. Furthermore, Figures 7–16 in the supplementary material provide more examples with notations. \n\n**Weakness 2:** A 2D Gaussian splat is round in shape and requires dense sampling.\n\nWe agree that Bézier splatting, similar to 2D Gaussian splatting, requires dense sampling to accurately capture complex geometry. In our implementation, each individual Bézier curve is sampled with 64 points. However, despite this relatively dense sampling, our method achieves highly efficient rasterization as shown in Table 1 with 4.5 ms for open curves and 14.1 ms for closed curves, significantly faster than DiffVG (141.0 ms and 85.4 ms, respectively). Furthermore, our approach yields much faster backpropagation times (4.7 ms and 24.6 ms) compared to DiffVG (701.3 ms and 225.8 ms). These results demonstrate that our method strikes an effective balance between geometric expressiveness and computational efficiency. \n\n**Weakness 3 and Question 2:** The paper should discuss Gaussian sampling density vs. image resolution. Can the method works across any scales?\n\nThe density of Gaussian sampling is not directly related to image resolution. It depends on the geometric complexity of the current curves and the size of the region they represent. In general, more complex and larger curves require denser sampling. To ensure that our method can adapt to the vast majority of datasets, we adopted relatively dense sampling parameters (64). Varying the per-segment sampling resolution (32, 64, 128) improves PSNR by less than 0.1. Regarding image resolution, the number of curves has a greater impact on reconstruction quality, as images with more complex textures generally require more Bézier curves to represent.\n\nOur method can handle images of arbitrary resolution without any modification. One of the key advantages of vector graphics is their resolution-free representation, meaning they can be rendered at any target resolution without quality loss. In addition, because our method is fully differentiable, we render the vector graphics at the same resolution as the target image to compute the loss and backpropagate accurate gradients, making our method inherently support arbitrary resolutions. We conducted experiments on datasets covering a variety of resolutions, including high-resolution photographs (e.g., DIV2K_HR, KIDAK) and low-resolution animation images (e.g., Danbooregions), using the same sampling number (64). \n\n**Weakness 4:** Lines 170 and 175 seem duplicated.\n\nThank you for pointing this out. We will correct it in the camera-ready version and carefully review the entire paper to revise any remaining typos.\n\n**Weakness 5:** Which probability density function for CDF?\n\nWe use a normal distribution with mean=0 and standard deviation=0.85 for the CDF-based strategy to sample interpolated Bézier curve positions. This results in smaller Gaussian scales near the boundaries, which helps prevent unwanted influence beyond the true boundaries.\n\n**Weakness 6:** Non-uniform sampling near the starting and ending points.\n\nAlthough 2D grid sampling may introduce some non-uniformity near the start and end of closed curves, we adopt this strategy to ensure that all curves have the same number of sampling points and a consistent structure. This design enables efficient matrix operations for computing Gaussian scales, rotations, and region-based sampling, all of which benefit greatly from GPU parallelism. In contrast, adaptive sampling strategies, while potentially reducing redundancy near curve endpoints, lead to varying numbers of samples per curve and make it harder to utilize GPU acceleration effectively. \n\n**Weakness 7 and Question 4:** The pruning and densification process should be explained in more detail.\n\nThank you for the helpful suggestion. We have outlined the pruning and densification process in three steps in the paper. We will add an algorithm diagram (as shown below) in the supplement. The relevant hyperparameters (e.g., overlap threshold and AABB pruning) are currently listed in the “Implementation Details” section, as we follow the standard practice in deep learning papers to place all hyperparameters in this section. We will update the “Pruning and Densification” paragraph to explicitly mention these hyperparameters for better clarity.\n\n---\n### Algorithm of Pruning and Densification for Closed Curves\n**Definitions:**\n- **opacity($b$):** Opacity of curve $b$.  \n- **area($b$):** Area of closed curve $b$.  \n- **colordiff($b_i, b_j$):** Color difference between curves $b_i$ and $b_j$.  \n- **IoU($b_i$):** Intersection-over-union between $b_i$'s bounding box and all other curves.\n- **ConnectedComponents($E$):** Extracts connected error regions by quantizing the error map.\n---\n**Input:**  \n- Closed Bézier curve set $\\mathcal{B} = \\{b_1, b_2, \\dots, b_N\\}$  \n- Error map $E$, optimization iteration $t$\n\n**Output:**  \n- Updated curve set $\\mathcal{B}'$\n---\n\nInitialize $\\mathcal{B}' \\leftarrow \\mathcal{B}$\n\n**// Pruning**  \nfor each curve $b_i$, $i = 1$ to $N$ do  \n if $\\text{opacity}(b_i) < \\tau_{\\text{opacity}}(t)$ then  \n  remove $b_i$ from $\\mathcal{B}'$  // low-opacity pruning  \n\n if $\\text{area}(b_i) < \\tau_{\\text{area}}$ then  \n  remove $b_i$ from $\\mathcal{B}'$  // small-area pruning  \n\n if   $\\text{IoU}(b_i, \\{b_j \\in \\mathcal{B} \\mid \\text{colordiff}(b_i, b_j) < 0.03\\}) > 0.9$ then  \n  remove $b_i$ from $\\mathcal{B}'$  // redundant overlap pruning  \nend for\n\n**// Densification**  \n$\\mathcal{R} \\leftarrow \\text{ConnectedComponents}(E)$  \n$\\mathcal{R}_{\\text{sorted}} \\leftarrow \\text{SortByArea}(\\mathcal{R})$  \n\nfor each region $r_j \\in \\mathcal{R}_{\\text{sorted}}$ do  \n if curve budget allows then  \n  insert a new closed Bézier curve initialized by $r_j$  into $\\mathcal{B}'$\nend for\n\n**Return**: $\\mathcal{B}'$\n\n---\n\nFor open curves, we add a splitting rule: if the middle opacity is over 0.5 lower than both ends, the curve is split to better fit the region.\n\n**Weakness 8 and Question 5:** Although rendering is faster, quality improvement is not significant. \n\n**For differentiable rendering methods, efficiency is critically important,** as it directly affects the scalability of deep learning–based applications such as feedforward image-to-SVG and text-to-SVG generation. These neural networks typically require fast forward and backward passes, but existing differentiable VG rasterizers (DiffVG) are very slow, often taking over 700 ms per iteration (compared to network inference time <100ms per iteration), which makes training on large datasets impractical.\nFor optimization-based methods like LIVSS that initialize compact vectors from SegmentAnything, replacing their rasterization backend (DiffVG) with ours reduces the time to reach a semantically meaningful result from ~45s to 2–3s, which greatly improves the user experience.\n\nA similar case is 3D Gaussian Splatting, which is proposed for offline tasks like novel view synthesis. However, the differentiable rendering of 3DGS has enabled a wide range of downstream applications, including feedforward 3D reconstruction (e.g., LSM, AnySplat), text-to-3D generation (e.g., DreamFusion), and the transfer of 3D knowledge into 2D foundation models (e.g., Improving 2D Feature Representations by 3D-Aware Fine-Tuning). Likewise, we hope our fast and differentiable vector graphics representation can open up similar opportunities in the vector graphics domain.\n\nRather than only focusing on higher PSNR, we want readers to appreciate the high efficiency of our approach. There are several ways to further improve the PSNR of our method. For example, increasing the number of interpolated lines within closed curves can improve PSNR (see the table below), and using the layer-wise strategy can also further improve PSNR (Table 3). Using 80 lines improves PSNR to 23.93, nearly 1 point higher than DiffVG (22.95), at the cost of doubling the rendering time. However, for the best trade-off between efficiency and quality, we recommend using 40 lines as the default.\n\n| Inter. | SSIM↑  | PSNR↑  | LPIPS↓ | FPS↑|\n| ------ | ------ | ------ | ------ | ---------------- |\n| 20     | 0.636  | 23.15  | 0.504  | 103.45           |\n| 40     | 0.639  | 23.45  | 0.507  | 68.30            |\n| 80     | 0.654  | 23.93  | 0.495  | 37.90            |\n\n**Question 1:** Does 2D grid Gaussian sampling for closed curves suffer from over-sampling near the starting and ending points?\n\nNo, the 2D grid sampling strategy does not negatively affect rendering performance. Although the sampling points are denser near the starting and ending positions of closed curves, this does not lead to over-sampling artifacts. This is because the scale of each 2D Gaussian is computed based on the distance to its neighboring samples, which naturally balances the density during rendering. As a result, denser sampling in certain areas does not increase visual artifacts. Extensive qualitative results are provided in the supplementary material, with no noticeable artifacts near the start or end points of closed curves.\n\nFurthermore, as shown in Section 4.4, our representation can be fully converted into standard XML-based SVG files and rendered using traditional vector graphics engines, which also confirms the absence of sampling-related visual issues.\n\n**Question 3:** What is the reconstruction policy for curve splatting about depth?\n\nAs mentioned in Line 156, the depth is determined by the spatial area of the curve. Larger area indicates larger depth. For open curves, it is computed as length$\\times$width; for closed curves, we use the area of the rotated AABB."}, {"review_id": "P3dT5TNCXm", "reviewer": "Reviewer_qWw9", "summary": "This work presents Bezier Splatting, a new differentiable Vector Graph representation that optimizes Bezier curves through Gaussian splatting-based rasterization. This method provide direct positional gradients at object boundaries so that over 150x faster than DiffVG. Overall, given a raster image, the author efficiently vectorize the input into a VG representation that closely resembles the input while preserving the details.", "questions": "I wonder if number of curve and other parameters are changeable: \n1. In Line 150, is 10 control points for each curve is fixed or flexible?\n2. In table 1, is it necessary to use so many curves? How to determine the best number of curve?\n3. Is it possible to make the result more compact? Using fewer curves?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strength: \n1. Vector graph differentiable rendering that make it available for machine learning.\n2. Using 2D Gaussian representation, it can provide direct position gradients and it is 150x faster than former State of Art result (DiffVG).\n3. Better vectorization results. In both qualitative and quantitative.\n4. Vectorization results support flexible editing operations.\n\nWeakness: \n1. Lacking ablation study in Section 4.\n2. Lacking experiments on binary images, and comparison with traditional vectorization algorithms.", "was_revised": true, "final_justification": "The author's answer in weakness 2 seems solid. I decide to keep my score.", "rebuttal": "We sincerely thank the reviewer qWw9's valuable comments and suggestions, which have helped us improve the clarity and quality of the paper. We also thank the reviewers for pointing out that our work can benefit downstream machine learning applications, which is indeed one of the key motivations behind this paper. Our responses to the weaknesses and questions are as follows.\n\n**Weakness 1:** Lacking ablation study in Section 4.  \n\nThank you for pointing this out. Due to the page limitation, we provided the ablation studies in the supplemental material. Specifically, they include:  \n\n- **Tab. 3** studies the **effectiveness of adaptive pruning and densification**, demonstrating how they improve reconstruction quality and avoid local minima.  \n- **Figs. 8-16** show the comparison across **different numbers of Bézier curves**, visually illustrating how increasing the number of curves progressively recovers more image details. \n- **Tab. 4** conducts a systematic analysis of **method efficiency**, including the impact of varying the number of Gaussian samples per curve in closed regions.  \n- **Tab. 5** shows an ablation study of the **multi-opacity scheme** for open curves, comparing the use of a single opacity vs. three opacities for each Bézier segment of an open curve.  \n- **Fig. 7** visualizes the results of our **layer-wise vectorization strategy**, demonstrating its adaptability with improvement techniques developed for DiffVG [1].  \n\n**Weakness 2:** Lacking experiments on binary images, and comparison with traditional vectorization algorithms.  \n\nDifferentiable VG rendering methods starting from DiffVG generally struggle with complex binary images. We conjecture this is because the pixel gradients in the binary space are inherently sparse. Differentiable methods often rely on anti-aliasing (DiffVG-based) or Gaussian splatting (ours) to compute stable positional gradients in the color image domain, but these approaches do not well suit with the extreme case of binary images. In addition, the assumptions about opacity and overlaps between curves made by differentiable methods are not applicable to binary images. Most recent works do not evaluate on binary images and focus primarily on performance in the color image domain, we also did not include binary image experiments in our comparisons.  To better handle binary image representations, we could leverage clustering-based method to generate guidances to help determine the positions and shapes of curves. In addition, we have verified that our method can be combined with LIVSS [2], which uses SegmentAnything [3] masks to initialize Bézier curves and then optimizes their shapes using the masks as supervision signals. This further demonstrates that our method can operate on binary images. Our method combined with LIVSS [2] will also be released together with our current codebase.  \n\nRegarding the comparison with traditional vectorization algorithms, we conduct a comparison using the Image Trace from Adobe Illustrator [4] on several images from the DIV2K dataset. Both methods were evaluated using the same number of parameters (~20K). As shown in the table below, our method achieves higher fidelity, with an average PSNR of **25.734** compared to **24.904** of Image Trace.\n\n| Image ID       | 0004       |           | 0008       |           | 0012       |           | 0016       |           | 0020       |           |\n|----------|------------|-----------|------------|-----------|------------|-----------|------------|-----------|------------|-----------|\n| **Method** | Adobe      | Ours      | Adobe      | Ours      | Adobe      | Ours      | Adobe      | Ours      | Adobe      | Ours      |\n| **SSIM↑**   | 0.835      | 0.849     | 0.645      | 0.637     | 0.658      | 0.658     | 0.627      | 0.628     | 0.814      | 0.822     |\n| **PSNR↑**   | 26.43      | 28.62     | 23.26      | 24.33     | 22.29      | 22.87     | 23.87      | 24.54     | 26.67      | 28.31     |\n| **LPIPS↓**  | 0.380      | 0.387     | 0.489      | 0.489     | 0.523      | 0.518     | 0.534      | 0.543     | 0.505      | 0.503     |\n\n**Question 1:** In Line 150, is 10 control points for each curve is fixed or flexible?\n\nThe design is flexible and supports arbitrary Bézier degrees and any number of connected segments. The number of control points is determined by the Bézier curve degree and the number of connected segments. In the implementation of this paper, each open curve is composed of three connected cubic Bézier segments, where each segment is defined by four control points. Due to continuity between segments, adjacent curves share one control point, resulting in a total of 10 control points per open curve (i.e., 3×3+1). For closed curves, our framework only requires the number of segments to be even, which facilitates efficient and consistent interior sampling during rasterization.\n\n**Question 2:** In table 1, is it necessary to use so many curves? How to determine the best number of curve?\n\nThank you for the question. In general, high-resolution images (e.g., DIV2K_HR at 2K resolution) require a relatively large number of vector primitives (typically 1K–2K Bézier curves) to preserve fine-grained visual details. The best number of curves depends on the user's goal: fewer curves yield more abstract or stylistic representations, while more curves are needed for faithful reconstruction of complex textures. Moreover, the optimal number of curves can vary significantly depending on the image content. For example, highly textured images naturally require more curves than flat or homogeneous ones to achieve similar PSNR or perceptual quality.\nIn practice, determining the best number of curves could also benefit from external guidance mechanisms. For instance, LIVSS [2] uses SAM [3] to provide semantically meaningful curve initialization with fewer but more informative paths. Our method is also compatible with this mechanism.\n\n**Question 3:** Is it possible to make the result more compact? Using fewer curves?\n\nWe have shown more compact results (32 to 512 curves) in Figure 7 of the supplement. To achieve more compact results, we can combine our method with the layerwise training strategy, as shown in Section 4.3 and supplement Table 3. To further ensure better semantic consistency when using fewer curves, we could combine our method with LVISS [2], which uses SAM [3] to provide semantically meaningful curve initialization, with a higher computational efficiency compared to the original LVISS+DiffVG.\n\n\n**References**\n\n[1] Tzu-Mao Li, Michal Lukáč, Gharbi Michaël, and Jonathan Ragan-Kelley. Differentiable vector graphics rasterization for editing and learning. ACM Trans. Graph. (Proc. SIGGRAPH Asia), 39(6):193:1–193:15, 2020.\n\n[2] Wang, Z., Huang, J., Sun, Z., Gong, Y., Cohen-Or, D., & Lu, M. (2025). Layered image vectorization via semantic simplification. In Proceedings of the Computer Vision and Pattern Recognition Conference (pp. 7728-7738).\n\n[3] Kirillov, A., Mintun, E., Ravi, N., Mao, H., Rolland, C., Gustafson, L., ... & Girshick, R. (2023). Segment anything. In Proceedings of the IEEE/CVF international conference on computer vision (pp. 4015-4026).\n \n[4] Adobe Illustrator."}]

Author rebuttal:

Meta-review: The work presents a method for representing a bitmap image with a set of Bezier 2d surfaces. It is significantly faster, and results in better quality than the state of the art.  Overall, all 3 reviewers were happy with the submission.

(3.1 Overall -> Overview)
Decision: Accept (poster)