EntON: Eigenentropy-Optimized Neighborhood Densification in 3D Gaussian Splatting

Miriam Jäger; Boris Jutzi

cs.CVMar 6, 2026

EntON: Eigenentropy-Optimized Neighborhood Densification in 3D Gaussian Splatting

Authors: Miriam Jäger, Boris Jutzi

Organizations: Institute of Photogrammetry and Remote Sensing (IPF), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Abstract

We present a novel Eigenentropy-optimized neighboorhood densification strategy EntON in 3D Gaussian Splatting (3DGS) for geometrically accurate and high-quality rendered 3D reconstruction. While standard 3DGS produces Gaussians whose centers and surfaces are poorly aligned with the underlying object geometry, surface-focused reconstruction methods frequently sacrifice photometric accuracy. In contrast to the conventional densification strategy, which relies on the magnitude of the view-space position gradient, our approach introduces a geometry-aware strategy to guide adaptive splitting and pruning. Specifically, we compute the 3D shape feature Eigenentropy from the eigenvalues of the covariance matrix in the k-nearest neighborhood of each Gaussian center, which quantifies the local structural order. These Eigenentropy values are integrated into an alternating optimization framework: During the optimization process, the algorithm alternates between (i) standard gradient-based densification, which refines regions via view-space gradients, and (ii) Eigenentropy-aware densification, which preferentially densifies Gaussians in low-Eigenentropy (ordered, flat) neighborhoods to better capture fine geometric details on the object surface, and prunes those in high-Eigenentropy (disordered, spherical) regions. We provide quantitative and qualitative evaluations on two benchmark datasets: small-scale DTU dataset and large-scale TUM2TWIN dataset, covering man-made objects and urban scenes. Experiments demonstrate that our Eigenentropy-aware alternating densification strategy improves geometric accuracy by up to 33% and rendering quality by up to 7%, while reducing the number of Gaussians by up to 50% and training time by up to 23%. Overall, EnTON achieves a favorable balance between geometric accuracy, rendering quality and efficiency by avoiding unnecessary scene expansion.

Figures & tables

Figure 1: Methodology EntON. Gaussians are adapted based on the Eigenentropy of their local neighborhood: low Eigenentropy leads to splitting, medium Eigenentropy results in unchanged Gaussians, and high Eigenentropy triggers pruning. In constrast, 3DGS triggers densification based on the view-space position gradient: small Gaussians are cloned, large Gaussians are splitted. EntON uses the level of Eigenentropy to focus densification on object surfaces, avoiding unnecessary scene expansion and thus efficiently compressing the information content of the scene representation.

Figure 2: Representation of an ellipsoid from the neighborhood points represented by the Gaussian centers with the three eigenvectors ( $\mathbf{\epsilon_{1}},\mathbf{\epsilon_{2}},\mathbf{\epsilon_{3}}$ ) and the corresponding eigenvalues ( $\lambda_{1},\lambda_{2},\lambda_{3}$ ) in the three-dimensional coordinate system.

Figure 3: Eigenentropy

E(\lambda_{1},\lambda_{2},\lambda_{3})

as a function of the largest normalized eigenvalue

\lambda_{1}

(with

\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 0

\sum_{i=1}^{3}\lambda_{i}=1

). Curves represent different fixed values of

\lambda_{3}

(ideal planar:

\lambda_{3}=0

; near-planar:

\lambda_{3}=0.1

; transitional between planar and spherical:

\lambda_{3}=1/6\approx 0.1667

; and higher spherical distribution). Markers indicate minimum and maximum Eigenentropy for each case. The dashed line at

\ln 2\approx 0.693

corresponds to the ideal planar case (

\lambda_{1}=\lambda_{2}=0.5

\lambda_{3}=0

Figure 4: Low Eigenentropy leads to splitting, medium Eigenentropy results in unchanged Gaussians, and high Eigenentropy triggers pruning. Representation of the ellipsoids based on neighboring Gaussian centers with the three eigenvectors ( $\mathbf{\epsilon_{1}},\mathbf{\epsilon_{2}},\mathbf{\epsilon_{3}}$ ) and the corresponding eigenvalues ( $\lambda_{1},\lambda_{2},\lambda_{3}$ ) in the three-dimensional coordinate system.

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	mean
3DGS	0.95	0.96	0.99	0.97	0.86	0.90	0.92	0.95	0.91	0.96	0.96	0.95	0.96	0.95	0.95	0.96
EntON	0.82	0.82	0.82	0.82	0.81	0.81	0.82	0.82	0.82	0.81	0.83	0.83	0.82	0.82	0.82	0.82

Table 1: Eigenentropy comparison on the DTU dataset: Mean Eigenentropy of all Gaussian centers within a local neighborhoods after the training process using 15 000 iterations.

Figure 5: Trend of the mean Eigenentropy

\downarrow

and mean cloud to cloud (C2C) distance

\downarrow

of all DTU scenes during the training process. Comparison of 3DGS and EntON. As EntON starts at iteration 3000, C2C is first reported at iteration 2500.

Panel of Figure 6 — Figure 6: Comparison of Eigenentropy and geometric accuracy for DTU scene scan55. Point cloud of 3DGS and our method for mean Eigenentropy with different neighborhood sizes and cloud-to-cloud distance (C2C).

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	mean
3DGS	1.82	1.84	1.72	1.55	2.17	1.70	1.66	2.37	1.90	1.71	1.50	1.51	1.47	1.30	1.34	1.61
2DGS	1.25	1.51	0.97	0.73	2.03	1.46	1.26	1.97	1.76	1.51	0.78	1.37	0.94	0.69	0.75	1.33
PGSR	0.92	1.15	0.77	0.67	1.43	0.99	0.97	1.28	1.79	1.13	0.60	0.96	0.66	0.68	0.79	1.00
EntON( $knn_{adaptive}$ )	1.13	1.22	0.79	0.57	1.73	1.13	1.06	1.82	1.79	1.27	0.72	1.09	0.73	0.66	0.71	1.14
EntON( $knn_{100}$ )	1.17	1.32	0.88	0.62	1.77	1.23	1.12	1.85	1.80	1.32	0.82	1.11	0.80	0.73	0.77	1.19
EntON( $knn_{75}$ )	1.13	1.29	0.82	0.58	1.68	1.15	1.05	1.71	1.79	1.24	0.76	1.14	0.75	0.68	0.72	1.13

Table 2: Surface accuracy. Geometric accuracy comparison on the DTU dataset with Chamfer cloud-to-cloud distances

\downarrow

in mm for points

\leq

10 mm from the reference, according to the DTU evaluation script. Best results are highlighted as 1st , 2nd , and 3rd . Mean scores are listed at the bottom. The training incorporates 15 000 iterations.

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	mean
3DGS	35.86	30.28	34.59	34.06	37.22	34.02	33.28	34.51	33.53	37.59	37.97	31.55	35.39	36.47	36.33	34.84
2DGS	32.69	28.51	31.74	32.13	35.10	31.92	31.01	31.51	31.27	34.15	35.92	30.17	32.60	35.06	34.38	32.54
PGSR	31.77	27.67	30.92	31.99	34.20	31.82	30.15	30.71	30.83	34.63	36.00	30.90	32.55	35.84	34.80	32.32
EntON( $knn_{adaptive}$ )	35.38	28.70	33.51	35.32	37.01	35.66	32.05	35.05	33.02	37.49	37.74	30.46	34.46	36.24	35.99	34.60
EntON( $knn_{100}$ )	34.36	29.44	34.51	34.50	36.87	35.93	31.47	35.36	32.95	37.24	37.86	30.52	34.06	36.32	36.24	34.71
EntON( $knn_{75}$ )	34.68	28.72	34.90	35.13	36.45	34.94	31.60	34.94	32.93	37.42	37.87	30.25	34.47	35.94	36.47	34.75

Table 3: Rendering quality comparison on the DTU dataset. We report the PSNR

\uparrow

in dB. Mean scores are listed. The training incorporates 15 000 iterations. Best results are highlighted as 1st , 2nd , and 3rd

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	Mean
3DGS	611 749	695 233	778 308	622 019	223 472	233 292	249 407	200 975	556 486	238 905	215 615	200 024	276 819	291 762	266 299	392 129
2DGS	317 153	391 404	354 076	341 317	151 758	159 384	169 785	123 622	240 807	127 348	105 397	102 635	132 201	160 033	144 754	208 572
PGSR	338 583	368 363	394 813	376 698	127 514	221 565	253 433	122 908	211 997	205 436	134 614	175 274	166 209	211 044	195 750	232 004
EntON( $knn_{adaptive}$ )	328401	370351	414967	373220	156504	224221	179262	140561	260293	186702	125623	116045	155316	171382	164187	228578
EntON( $knn_{100}$ )	329222	403205	409569	365176	156361	207048	173000	151514	285136	185796	136097	114874	159414	175780	169528	233116
EntON( $knn_{75}$ )	316775	346756	409775	366452	150623	184157	153795	148258	287247	173638	130814	104196	163856	165875	158206	211925

Table 4: Number of Gaussians on the DTU dataset. We report the total number of Gaussians

\downarrow

resulting from the EntON, compared to 3DGS, 2DGS, PGSR, and the number of SfM points used for initialization. The training incorporates 15 000 iterations. Mean scores are listed. Best results (low total number is better) are highlighted as 1st , 2nd , and 3rd .

Method	24	37	40	55	63	65	69	83	97	105	106	110	114	118	122	Mean
3DGS	18.48	15.61	15.95	13.05	11.86	11.97	11.85	11.22	14.20	11.88	11.03	10.38	11.84	11.02	10.93	12.88
2DGS	21.87	20.24	19.53	18.22	17.22	18.27	17.44	18.00	19.37	18.12	17.32	17.16	18.17	17.19	17.51	18.11
PGSR	33.45	30.04	32.58	28.86	27.55	27.04	31.04	28.08	28.38	40.17	38.07	38.00	40.03	22.20	37.34	32.89
EntON( $knn_{adaptive}$ )	12.35	10.97	11.91	10.76	10.06	9.60	9.57	9.07	9.77	9.46	9.28	9.09	9.36	9.23	9.28	10.52
EntON( $knn_{100}$ )	13.02	11.77	12.74	11.18	10.33	9.75	9.80	9.33	10.26	9.71	9.80	8.80	9.66	9.67	9.95	10.52
EntON( $knn_{75}$ )	12.49	11.27	12.34	10.68	10.17	9.54	9.65	9.39	9.93	9.41	9.78	8.73	9.80	9.18	9.34	10.21

Table 5: Training time comparison on the DTU dataset. We report the minutes for 15 000 iterations in min. Mean scores are listed. Best results (low training time is better) are highlighted as 1st , 2nd , and 3rd . The training incorporates 15 000 iterations.

Figure 7: Trend of the mean PSNR (plotted as its inverse to illustrate the trade-off) and mean cloud to cloud (C2C) distance in EntON of all DTU scenes over different neighborhood sizes

k_{\text{nn}}

, in comparison to 3DGS, 2DGS and PGSR.

Panel of Figure 8 — Figure 8: Geometric accuracy comparison of 3DGS, 2DGS, PGSR and EntON on the DTU dataset with Chamfer cloud-to-cloud distances $\downarrow$ for the same PSNR. Color values are cropped at 10mm distance.

Panel of Figure 9 — Figure 9: Rendering quality comparison of 3DGS, 2DGS, PGSR and EntON on the small-scale DTU dataset, as well as ground truth (GT) images in original resolution.

scene	3DGS	2DGS	PGSR	EntON( $knn_{100}$ )	EntON( $knn_{75}$ )	EntON( $knn_{50}$ )	EntON( $knn_{25}$ )
building1	0.184	0.193	0.197	0.184	0.179	0.182	0.187
building2	0.185	0.160	0.177	0.168	0.169	0.167	0.163
mean	0.185	0.177	0.187	0.176	0.174	0.175	0.175

Table 6: Surface accuracy. Geometric accuracy comparison on the TUM2TWIN dataset with Chamfer cloud-to-cloud distances

\downarrow

in m for points

\leq

0.5 m from the reference, according to the DTU evaluation script. Best results are highlighted as 1st , 2nd , and 3rd . Mean scores are listed at the bottom. The training incorporates 15 000 iterations.

scene	3DGS	2DGS	PGSR	EntON( $knn_{100}$ )	EntON( $knn_{75}$ )	EntON( $knn_{50}$ )	EntON( $knn_{25}$ )
building1	32.36	29.90	30.05	31.55	32.57	32.09	31.94
building2	31.00	28.00	27.69	30.84	30.50	30.56	30.78
mean	31.68	28.95	28.87	31.20	31.54	31.26	31.36

Table 7: Rendering quality comparison on the TUM2TWIN dataset. We report the PSNR

\uparrow

in dB. Best results are highlighted as 1st , 2nd , and 3rd . Mean scores are listed at the bottom. The training incorporates 15 000 iterations.

scene	3DGS	2DGS	PGSR	EntON( $knn_{100}$ )	EntON( $knn_{75}$ )	EntON( $knn_{50}$ )	EntON( $knn_{25}$ )
building1	3 622 623	3 457 686	3 495 252	2 716 037	2 693 175	2 500 641	2 475 797
building2	3 037 856	2 087 933	1 625 555	2 306 556	2 247 377	2 225 013	2 278 891
mean	3 330 240	2 772 810	2 560 404	2 511 297	2 470 276	2 362 827	2 377 344

Table 8: Number of Gaussians on the TUM2TWIN dataset. We report the total number of Gaussians

\downarrow

. Best results are highlighted as 1st , 2nd , and 3rd . Mean scores are listed at the bottom. The training incorporates 15 000 iterations.

scene	3DGS	2DGS	PGSR	EntON( $knn_{100}$ )	EntON( $knn_{75}$ )	EntON( $knn_{50}$ )	EntON( $knn_{25}$ )
building1	18.35	25.32	40.48	27.75	26.24	22.82	20.56
building2	17.42	22.77	37.97	22.21	20.93	19.19	18.04
mean	17.89	24.05	39.23	24.98	23.59	21.00	19.30

Table 9: Training time comparison on the TUM2TWIN dataset. We report the minutes for 15 000 iterations in min. Best results are highlighted as 1st , 2nd , and 3rd . Mean scores are listed at the bottom.

Panel of Figure 10 — Figure 10: Geometric accuracy comparison of 3DGS, 2DGS, PGSR and EntON TUM2TWIN dataset with Chamfer cloud-to-cloud distances $\downarrow$ . Color values are cropped at 2 m distance.

Panel of Figure 11 — Figure 11: Rendering quality comparison of 3DGS, 2DGS, PGSR and EntON on the large-scale TUM2TWIN dataset, as well as ground truth (GT) images in original resolution.

Scene	scan40	scan55	scan106	scan122
Mean Eigenentropy (Gaussians, >1 mm distance)	0.9863	0.9617	0.9594	0.9531

Table 10: Mean Eigenentropy of Gaussians in 3DGS corresponding to points with cloud-to-cloud distance over 1 mm distance to the reference. Values guide the selection of the upper Eigenentropy threshold for pruning.

Panel of Figure 12 — Figure 12: Comparison of Eigenentropy of points with a cloud-to-cloud distance (C2C) over 1mm to the reference point cloud. Blue points are below Eigenentropy $E=0.95$ , red points above Eigenentropy $E=0.95$ and therefore candidates for pruning.

3D Gaussian Splatting

3D Gaussian Splatting represents scenes as optimizable 3D Gaussian primitives for fast, photorealistic rendering from new viewpoints.

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3D Gaussian Splatting 3DGS Compression 3D Gaussian Pruning Adaptive Gaussian Densification

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3D Gaussian Splatting (3DGS) has demonstrated breakthrough performance in novel view synthesis and real-time rendering. Nevertheless, its practicality is constrained by the high memory cost due to a huge number of Gaussian points. Many pruning-based 3DGS variants have been proposed for memory saving, but often compromise spatial consistency and may lead to rendering artifacts. To address this issue, we propose graph-based spatial distribution optimization for compact 3D Gaussian Splatting (GS\textasciicircum2), which enhances reconstruction quality by optimizing the spatial distribution of Gaussian points. Specifically, we introduce an evidence lower bound (ELBO)-based adaptive densification strategy that automatically controls the densification process. In addition, an opacity-aware progressive pruning strategy is proposed to further reduce memory consumption by dynamically removing low-opacity Gaussian points. Furthermore, we propose a graph-based feature encoding module to adjust the spatial distribution via feature-guided point shifting. Extensive experiments validate that GS\textasciicircum2 achieves a compact Gaussian representation while delivering superior rendering quality. Compared with 3DGS, it achieves higher PSNR with only about 12.5% Gaussian points. Furthermore, it outperforms all compared baselines in both rendering quality and memory efficiency.

Abstract

3D Gaussian Splatting

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