CVTSplat: Multi-view Reconstruction Using an Adaptive Centroidal Voronoi Tessellation

Briac Toussaint; Diego Thomas; Jean-Sébastien Franco; Edmond Boyer; Christian Theobalt

cs.CVOct 26, 2026

CVTSplat: Multi-view Reconstruction Using an Adaptive Centroidal Voronoi Tessellation

Authors: Briac Toussaint, Diego Thomas, Jean-Sébastien Franco, Edmond Boyer, Christian Theobalt

Organizations: Max Planck Institute for Informatics, Saarbrücken, Germany · Kyushu University, Fukuoka, Japan · Inria, Grenoble, France

Abstract

Voronoi tessellations have recently emerged as an adaptive, space-partitioning discretization for shape and appearance reconstruction from multi-view images. However, the irregular cells they produce make the repeated reconstruction of the tessellation during optimization costly. We therefore explore adaptive centroidal Voronoi tessellations (CVTs), which extend standard CVTs with non-uniform cell sizes while preserving the regular, well-shaped cells that make them efficient to compute. In CVTSplat, cell size is modulated by the evolving signed distance field, concentrating sites near the zero level set while keeping cells nearly centroidal. This regularity lets us specialize the whole pipeline, with GPU algorithms for nearest-neighbor queries and approximate Voronoi construction, and a rasterizer tailored to small, nearly isotropic cells. Our Voronoi construction is 5 to 10 times faster than general-purpose alternatives in this regime. On DTU and BlendedMVS, CVTSplat reaches geometric accuracy competitive with the best Gaussian-splatting surface methods while training in about 2.5 minutes on a single GPU.

Figures & tables

Figure 1. We represent the scene with Voronoi cells carrying trainable signed-distance and view-dependent appearance attributes, and directly rasterize them for differentiable volume rendering (middle). Our adaptive CVT combines regular cells with increased spatial resolution near the zero level set, concentrating representation capacity where it is most needed (bottom right). Top left: signed-distance values mapped to hue; bottom right: randomized cell colors illustrate the adaptive refinement near the surface.

Figure 2. Overview of our approach during optimization. Top row: Construction of the tessellation. (a) The sites are processed to obtain a list of their 32 nearest neighbors, (b) before building the Voronoi diagram as an adjacency graph. (c) The center of mass of each cell is then computed to regularize the distribution of the sites via the CVT loss. (d) The Voronoi diagram is optionally converted to a Delaunay triangulation to extract the zero-level set with marching tetrahedra. Bottom row: Differentiable volume rendering from posed images. (e) The sites’ attributes (learnable SDF and features) are smoothed with a Gaussian kernel. (f) The gradient of the SDF is also computed to produce normal vectors, which define a piecewise linear SDF over the cells. Then, for all viewpoints in the mini-batch, (g) a per-site color is computed based on the camera’s position, (h) before rasterizing the scene and comparing against the ground-truth image.

Figure 3. Left: The volume and centroid of the cell are computed by casting rays from the site’s location. Right: Our biased CVT loss maintains the sites slightly closer to the implicit surface than the centroids

\bar{\mathbf{x}}

, resulting in stretched cells along the SDF gradient.

Figure 4. Voronoi cell in 2D. The bisector plane

\mathcal{P}_{k}

is inactive since it does not participate in defining the cell

\mathcal{C}_{i}

. Any neighboring point beyond the security radius, defined as twice the cell radius, is guaranteed to not participate.

Figure 5. The 2D inequalities enclose the feasibility region, which corresponds to a facet of the polyhedron wrapping the cell. Each line is the intersection of the reference plane

\mathcal{P}_{r}

with another bisector plane

\mathcal{P}_{j}

Figure 6. The adjacency matrix of the site

i

encodes which of its neighbors are adjacent to each other.

Figure 7. The light probes, parametrized with spherical harmonics, are interpolated to produce view-dependent features

\mathbf{F}^{a,\mathbf{v}}

and

\mathbf{F}^{a,\mathbf{w}}

, based on the local view direction

\mathbf{v}

and reflected vector

\mathbf{w}

Figure 8. The color is constant but the density varies inside the cells. They are sorted by distance to the camera for pop-free rasterization.

Figure 9. The pixel tiles are split into chunks of 32 consecutive cells. All chunks are rasterized in parallel but the ones marked in red have been skipped by detecting occlusions dynamically. The final result (top left) is the $\alpha$ -blending of all the chunks in front-to-back order, evaluated in a subsequent pass.

Method	Mean	Dog	Bear	Clock	Durian	Man	Sculpture	Stone	Jade
NeuS2 ( Wang et al., 2023 )	2.93	2.78	2.71	2.63	4.23	2.25	2.50	1.91	4.43
Voxurf ( Wu et al., 2023 )	2.64	2.28	2.20	1.88	2.98	2.11	1.75	3.96	3.99
ProbeSDF ( Toussaint et al., 2025 )	2.21	2.24	2.03	1.69	3.26	1.81	1.61	1.36	3.71
PGSR ( Chen et al., 2024 )	1.56	1.21	1.20	N/A	2.15	1.39	1.81	1.58	N/A
GGGS ( Zhang et al., 2026b )	0.88	0.55	0.37	0.71	0.93	0.85	0.99	0.40	2.28
CVTSplat	1.13	0.91	0.49	0.97	1.44	0.91	1.21	0.60	2.53

Table 1. Chamfer distance on BlendedMVS with 4M sites.

	Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	Mean	Runtime	GPU
Implicit	VolSDF ( Yariv et al., 2021 )	1.14	1.26	0.81	0.49	1.25	0.70	0.72	1.29	1.18	0.70	0.66	1.08	0.42	0.61	0.55	0.86	>12 h	RTX 2080 Ti
	NeuS ( Wang et al., 2021 )	1.00	1.37	0.93	0.43	1.10	0.65	0.57	1.48	1.09	0.83	0.52	1.20	0.35	0.49	0.54	0.84	8 h	RTX 2080 Ti
	Neuralangelo ( Li et al., 2023 )	0.37	0.72	0.35	0.35	0.87	0.54	0.53	1.29	0.97	0.73	0.47	0.74	0.32	0.41	0.43	0.61	>128h	V100
	NeuS2 ( Wang et al., 2023 )	0.56	0.76	0.49	0.37	0.92	0.71	0.76	1.22	1.08	0.63	0.59	0.89	0.40	0.48	0.55	0.70	5 min	RTX 3090
	Voxurf ( Wu et al., 2023 )	0.91	0.73	0.45	0.34	0.99	0.66	0.83	1.36	1.31	0.78	0.53	1.12	0.37	0.53	0.51	0.76	15 min	A100
	ProbeSDF ( Toussaint et al., 2025 )	0.65	0.74	0.34	0.34	1.02	0.71	0.62	1.34	0.94	0.70	0.53	0.96	0.36	0.45	0.47	0.68	2.5 min	RTX 3090

Table 2. Chamfer distance on the DTU benchmark in millimeters. Lower is better. Best, second-best, and third-best results are highlighted.

Figure 10. Qualitative comparison on DTU against GGGS ( Zhang et al., 2026b ) and PGSR ( Chen et al., 2024 ) . The ground truth meshes were obtained by applying Poisson reconstruction ( Kazhdan and Hoppe, 2013 ) on the ground truth point clouds. Our approach recovers finer details, especially visible on scans 24, 106 and 122.

Figure 11. Qualitative comparison on BlendedMVS against GGGS ( Zhang et al., 2026b ) and PGSR ( Chen et al., 2024 ) . Errors are mesured relative to the unit bounding box enclosing the objects. Our approach produces smoother geometry than GGGS (see close-ups on man and clock, heatmaps on jade), although some concavities are better reconstructed with GGGS (clock, jade, dog). PGSR fails on two examples and is less accurate overall.

Method (#Sites, #Optim. steps)	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	Mean	Time (s)
1M Sites, 10K	0.38	0.54	0.30	0.30	0.82	0.50	0.47	1.02	0.75	0.53	0.33	0.66	0.30	0.34	0.34	0.50	87
2M Sites, 10K	0.36	0.51	0.30	0.30	0.82	0.48	0.47	1.00	0.74	0.51	0.33	0.65	0.30	0.33	0.33	0.50	110
4M Sites, 5K	0.36	0.58	0.30	0.30	0.75	0.46	0.46	0.97	0.74	0.50	0.39	0.75	0.29	0.32	0.31	0.50	79
4M Sites, 7.5K	0.35	0.50	0.30	0.30	0.75	0.46	0.46	0.97	0.74	0.49	0.39	0.65	0.29	0.32	0.31	0.48	115
Basic Rasterizer (4M Sites, 10K)	0.35	0.49	0.30	0.29	0.76	0.45	0.46	0.97	0.76	0.49	0.41	0.58	0.29	0.32	0.31	0.48	280
Chunked Rasterizer (4M Sites, 10K)	0.35	0.49	0.30	0.29	0.76	0.45	0.46	0.97	0.75	0.49	0.38	0.58	0.29	0.32	0.31	0.48	147

Table 3. Training speed analysis on DTU.

Figure 12. Test with fewer sites or fewer optimization steps. Top: Using fewer sites produces a predictable loss of geometric details with faster optimization. Bottom: Our model converges after 10K training steps, with marginal improvement beyond 7.5K steps.

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	Mean
No Linear Radiance	0.35	0.49	0.30	0.29	0.79	1.78	0.44	0.99	0.84	0.50	0.32	0.59	0.29	0.32	0.31	0.57
No ZNCC	0.40	0.57	0.33	0.31	0.80	0.52	0.52	1.05	0.83	0.54	0.38	1.07	0.29	0.35	0.32	0.55
No Depth Smoothing	0.44	0.50	0.46	0.29	0.75	0.45	0.46	0.97	0.75	0.50	0.50	0.70	0.29	0.36	0.31	0.52
Biased CVT ( $a=0.05$ )	0.36	0.51	0.34	0.30	0.81	0.50	0.48	1.02	0.77	0.52	0.34	0.80	0.30	0.34	0.34	0.52
No Ref. Probes	0.35	0.56	0.30	0.29	0.76	0.45	0.50	0.97	0.82	0.49	0.38	0.81	0.30	0.32	0.31	0.51
No Envmap	0.35	0.52	0.29	0.29	0.80	0.44	0.44	1.12	0.76	0.53	0.39	0.64	0.28	0.31	0.30	0.50

Table 4. Ablation study on DTU.

Figure 13. We remove one loss on each example to ablate our model. Bottom: The multi-view (ZNCC), depth and normal losses as well as the reflection probes are important for geometric accuracy. Top: Training with a linear radiance avoids a failure case on scan 65. Not replacing the planar ZNCC with a nonplanar variant in the last phase of training suppresses fine details (scan 106). The envmap alleviates silhouette artifacts from inaccurate masks (scan 83).

	CVTSplat with/without warmup	RestrictedVoronoi ( Basselin et al., 2021 )	CuPy-KNN ( Igelbrink, )	Torch-kdtree ( Grandits, )	Sklearn ( Pedregosa et al., 2011 )	TrueKNN ( Nagarajan et al., 2023 )
Build Tree	1.3	<1	17.3	6742*	1925*	-
Query Tree	13.7 / 46.8	97.9	720	1026	25630*	-
Total	15.0 / 48.1	97.9	737	7768	27556	28543

Table 5. KNN timings (ms) on our DTU point clouds with 4M points. The * symbol denotes CPU operations.

	CVTSplat (Ours)	MeshlessVoronoi ( Ray et al., 2018 )	RestrictedVoronoi ( Basselin et al., 2021 )	Paragram ( Taveira et al., 2026a )	gDel3D ( Cao et al., 2014 )	RadFoam ( Govindarajan et al., 2025 )
KNN	15.0	126.6	97.9	-	-	-
Adjacency	11.9	44.5	40.5	275.5	617.4	18995
Centroids	3.0	44.5	40.5	-	-	-
Total	29.9	171.0	138.4	275.5	617.4	18995

Table 6. Voronoi diagram construction timings (ms) on our trained DTU point clouds (4M points).

Figure 14. Timings for Voronoi diagram construction, evaluated on DTU checkpoints saved every 500 optimization steps. Reported numbers include KNN and centroids computations if applicable.

	Adjacency $\rightarrow$ Del.	Marching Tet.	Total
All cells	11.5 ms	4.0 ms	15.5 ms
Zero-crossing only	4.5 ms	0.6 ms	5.1 ms

Table 7. Mesh extraction timings on DTU.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Training phases. 10,000 iterations: 400 initialization, 9200 main training, and 400 cleanup. No sites are added during initialization or cleanup.

Point population. 200k initial sites up to 4M sites. Densification every 10 iterations.

Data and optimization. 4 image levels of detail, batch size: 4 images per batch using gradient accumulation, learning rate [0.025, 0.001] .

Structure updates. KNN every 10 iterations, Voronoi diagram construction frequency [1, 2, 5] .

Geometric regularization.

\mathcal{L}_{\text{SDF}}=

[0.4, 0.1] ,

\mathcal{L}_{\mathbf{n}}=

[0.05, 0.1] ,

\mathcal{L}_{\text{Eik}}=

[0.005] , depth smoothing

\mathcal{L}_{\text{D}}=

[0.5] .

CVT Densification.

\mathcal{L}_{\text{CVT}}=1

a=

[0.05, 0.10] ,

b=

[0.01, 0.001, 0.0001] .

Image losses.

\mathcal{L}_{\text{RGB}}=

[5, 10] ,

\mathcal{L}_{\text{DSSIM}}=

[1, 2] ,

\mathcal{L}_{\text{MV}}=

[0.1, 10] ,

\mathcal{L}_{\text{Mask}}=

[5, 0.1] .

Appendix

Table 8. Optimization schedule. Array values are linearly interpolated throughout training.

Dataset	Non-manifold edges	Boundary edges	Triangles
DTU	2.7	2.3	6.4M
BlendedMVS	2.4	1.4	6.7M

Appendix

Table 9. Average mesh extraction defects by dataset.

Figure 15. Relative error curves when computing cell volume, centroid and bounding sphere radius using a varying number of ray samples. The volume and centroid error curves are nearly overlapped in the plot.

Figure 16. Histograms pooled over all 23 DTU and BlendedMVS scenes (92 million sites). The blue one queries KNN at each site while the red one queries at the cell centroid computed at the previous update. Left : Number of Voronoi neighbors per site. The mean is

\approx 14.2

, corresponding to

14

bisector planes contributing to each cell on average. Middle and Right : zero-based index of the last detected Voronoi neighbor in the KNN list, with linear and logarithmic frequency scales respectively. For

\approx 60\%

of sites, all Voronoi neighbors lie among the first 16 nearest neighbors. The frequency decreases roughly exponentially beyond index 15, with

\approx 0.1\%

of cells not finding their last Voronoi neighbor within the 32-KNN list with the base strategy. Using shifted ball queries reduces the frequency of incomplete cells by a factor of 170 in comparison (

\approx 600

cells in total out of 92M).

Figure 17. Cell-radius statistics averaged over all scenes. Left : the distribution of cell radius $r$ against SDF $d$ . The cyan curve fits the median radius in each distance bin, showing approximately linear growth with $|d|$ . Its similar branches on either side of zero indicate comparable cell size inside and outside the surface at equal distance. Middle : the distribution of $\log_{2}(r)$ . The red curve fits a power-law tail over $2^{-8}\leq r\leq 2^{-4}$ , containing 26.2% of sites. Within this range, each doubling of $r$ corresponds to approximately $3.9\times$ fewer sites per histogram bin ( $h(r)/h(2r)\approx 3.9$ ). Right : The number of cells decreases very quickly away from the implicit surface.

Figure 18. Adaptive discretization of a known SDF. Top : 2D slice with random colors and superimposed extracted mesh. Bottom : 2D slice with SDF mapped to hue. We fit the SDF using $N=10^{5}$ sites and vary the parameter $a$ in the CVT loss. The mesh approximates the zero level set more accurately for $a>0$ . The proportion of cells $e$ with missing bisector planes increases sharply for $a>0.1$ when using 32 Euclidean nearest neighbors. Missing bisector planes can cause cells to clearly overlap for larger values of $a$ (see $a=0.2$ , center of the horseshoe in the top panel).

Dataset	Image size	Cell radius (px)	Render time (ms)	FPS
DTU	$1600\times 1200$	$1.55\pm 0.31$	$3.63\pm 0.29$	$277\pm 21$
BlendedMVS	$768\times 576$	$1.21\pm 0.15$	$2.40\pm 0.21$	$419\pm 36$

Appendix

Table 10. Average projected cell radius and rendering speed (mean

\pm

standard deviation across scenes).

Figure 19. Qualitative comparison of renderings on DTU. Our approach can represent high-frequency texture patterns (scan 24, house roof) but misses some specularities compared to GGGS and PGSR.

Figure 20. Qualitative comparison of renderings on BlendedMVS. PGSR failed to converge on the clock scene.

Dataset	Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
	CVTSplat	30.32	0.888	0.147
DTU	GGGS	30.76	0.872	0.253
	PGSR	30.33	0.866	0.265
	CVTSplat	29.52	0.894	0.073
BMVS	GGGS	32.43	0.953	0.046
	PGSR	30.50	0.902	0.108

Appendix

Table 11. Masked image metrics on training views.

Dataset	Method	Primitives	Parameters
	CVTSplat	4M sites	39.35M
BMVS	GGGS	1.1M Gaussians	64.21M
	PGSR	0.5M Gaussians	29.95M
	CVTSplat	4M sites	39.35M
DTU	GGGS	0.6M Gaussians	36.16M
	PGSR	0.3M Gaussians	16.11M

Appendix

Table 12. Average model size.

Component	Total
Site positions ( $3\times 4$ M)	12M
Site distances ( $1\times 4$ M)	4M
Site features ( $4\times 4$ M)	16M
Two SH probes grids	4.2M
Environment map	3.1M
MLP	595

Appendix

Table 13. Parameter count breakdown.

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