NDSplat — N-Dimensional Splatting

The big picture

One framework, many dimensions

Static 3D Gaussian Splatting cannot express specular reflections, temporal dynamics, or view-dependent geometry. N-dimensional splatting solves this by lifting each primitive into a joint distribution over 3D position and conditioning variables, then conditionally slicing it at render time to recover a 3D primitive whose mean, opacity, and covariance vary continuously with view or time.

ICLR 2025 6DGS + view direction → ICCV 2025 7DGS + time → ICLR 2026 UBS Beta kernels → latest dGS / dBS direct conditioning

application ECCV 2026 Render-FM feedforward medical
volumetric rendering

Brings the same 6D representation to clinical CT — predicting Gaussian parameters in a single forward pass, a 500× speedup over per-scan optimization.

◳

Dimensionality

Lift primitives from 3D to 6D (view) and 7D (view + time) so a single primitive can carry view-dependent and dynamic behavior.

◐

Kernel shape

Swap the Gaussian opacity decay for an anisotropic Beta kernel with per-dimension bandwidth control for richer light transport.

⚡

Conditioning

Replace covariance-derived slicing (inversion + regression) with a directly parameterized Cholesky precision — kernel-agnostic and far faster.

Latest work · under review

Direct Conditional Parameterization
for N-Dimensional Splatting

Direct Gaussian Splatting (dGS) & Direct Beta Splatting (dBS)

N-DGS (6DGS/7DGS) and UBS are the leading N-dimensional formulations across Gaussian and Beta kernels — but they share the same expensive conditional-slicing step: every forward pass inverts a covariance block and multiplies a regression matrix, once per primitive, scaling linearly with primitive count.

Our insight: the slicing step needs far less than the full joint covariance. Opacity only needs a precision matrix in the query space; position only needs a small displacement matrix. We parameterize both directly: a Cholesky precision factor for opacity and a spatially-scaled displacement matrix with learnable per-dimension coupling for position.

Because the change touches only the conditioning step, it is kernel-agnostic: it instantiates as dGS (Gaussian, the direct counterpart of N-DGS) and dBS (Beta, the direct counterpart of UBS). Matched-pair evaluation isolates the gain of direct conditioning from the choice of kernel.

            At a glance
            6.9–7.7× faster conditional slicing
up to 2.66× faster end-to-end rendering
+1.26 dB top PSNR gain (7DGS-PBR, dBS)
no inversion — direct Cholesky precision
drop-in — covariance-derived replacement

          

Direct conditioning, given query deviation δ = q − μ_q:

Opacity α_cond = α · exp( −λ_o · ‖ L^⊤δ ‖² )

Position μ_cond = μ_p + V_pq · diag(Λ) · V_qq · δ

L is a lower-triangular Cholesky factor, so V_qq = LL^⊤ is the precision matrix with no inversion. Λ ∈ [0,1]^C is a per-dimension coupling: 0 decouples position from opacity, 1 recovers full N-DGS-style coupling. For the Beta kernel, the opacity decay becomes ∏_i (1 − tanh(z_i²))^β_i with z = L^⊤δ.

MCMC densification results — matched Gaussian budgets

Each direct method paired against its covariance-derived counterpart, so any difference reflects the conditioning parameterization. FPS uses a PyTorch reference slicer; FPS_CUDA uses the fused CUDA slicing kernel. Best of each pair in bold.

Dataset	Method	#Gauss (K)	Time (m)↓	FPS↑	FPS_CUDA↑	PSNR↑	SSIM↑	LPIPS↓
NeRF Synthetic	N-DGS	300	10.43	183.58	348.95	34.04	0.973	0.027
	dGS	300	10.26	245.18	593.94	34.34	0.973	0.026
	UBS	300	6.80	155.96	306.33	34.48	0.975	0.026
	dBS	300	5.48	239.09	415.66	34.53	0.975	0.025
Mip-NeRF 360	N-DGS	2,852	51.23	35.69	75.31	26.88	0.775	0.263
	dGS	2,574	49.85	42.49	200.56	27.67	0.794	0.227
	UBS	3,057	28.59	40.94	76.18	28.50	0.841	0.183
	dBS	3,057	24.42	62.05	157.45	28.70	0.842	0.185
6DGS-PBR	N-DGS	163	54.80	328.75	476.79	40.06	0.976	0.039
	dGS	163	47.54	356.15	551.02	40.34	0.976	0.038
	UBS	163	8.70	239.87	348.63	39.53	0.975	0.042
	dBS	163	8.47	305.33	401.93	39.72	0.976	0.042
D-NeRFdynamic	N-DGS	150	10.31	306.51	501.52	34.63	0.976	0.033
	dGS	150	10.61	407.40	746.65	34.64	0.976	0.030
	UBS	150	8.37	229.83	371.81	32.91	0.967	0.040
	dBS	150	9.34	323.75	463.09	33.55	0.972	0.033
7DGS-PBRdynamic	N-DGS	175	28.44	294.48	511.12	29.20	0.944	0.063
	dGS	175	31.12	366.06	577.38	29.90	0.948	0.061
	UBS	175	9.90	232.17	396.33	30.61	0.951	0.057
	dBS	175	8.79	347.93	539.57	31.87	0.955	0.053

All methods: NVIDIA A100, 30K iterations, TC-GS rasterizer, identical learning rates and densification schedules per dataset.

Comparison with state-of-the-art methods

dBS situated against implicit and explicit static-scene methods on Mip-NeRF 360 and NeRF Synthetic. It achieves the best PSNR on Mip-NeRF 360 (28.70 dB) and ties for best PSNR on NeRF Synthetic (34.53 dB). UBS and dBS are re-evaluated under the 3DGS protocol, so UBS numbers differ from the original paper.

	Method	Mip-NeRF 360			NeRF Synthetic
	Method	PSNR↑	SSIM↑	LPIPS↓	PSNR↑	SSIM↑	LPIPS↓
Implicit	Instant-NGP	25.51	0.684	0.398	33.18	0.959	0.055
	Mip-NeRF 360	27.69	0.792	0.237	33.25	0.962	0.039
	Zip-NeRF	28.54	0.828	0.189	33.10	0.971	0.031
Explicit	3DGS	27.20	0.815	0.214	33.31	0.969	0.037
	GES	26.91	0.794	0.250	–	–	–
	2DGS	27.04	0.805	0.297	33.07	–	–
	Mip-Splatting	27.79	0.827	0.203	33.33	0.969	0.039
	3DGS-MCMC	28.29	0.840	0.210	33.80	0.970	0.040
	DRKS	26.76	0.787	0.236	33.82	–	–
	Textured GS	27.35	0.827	0.186	33.24	0.967	0.043
	Quadratic GS	27.39	0.813	0.213	–	–	–
	Disc-GS	28.01	0.833	0.189	–	–	–
	Triangle Splatting	27.00	0.808	0.231	–	–	–
	Radiance Meshes	27.15	0.810	0.274	–	–	–
	Spherical Voronoi	28.57	0.835	0.230	34.53	0.973	0.032
	UBS	28.50	0.841	0.183	34.48	0.975	0.026
	dBS	28.70	0.842	0.185	34.53	0.975	0.025

Best Second best

ICLR 2026

Universal Beta Splatting

Rong Liu, Zhongpai Gao, Benjamin Planche, Meida Chen, Van Nguyen Nguyen, Meng Zheng, Anwesa Choudhuri, Terrence Chen, Yue Wang, Andrew Feng, Ziyan Wu

UBS generalizes 3D Gaussian Splatting to N-dimensional anisotropic Beta kernels. Unlike fixed Gaussian primitives, Beta kernels enable controllable dependency modeling across spatial, angular, and temporal dimensions. The learned Beta parameters automatically decompose scene properties — without explicit supervision — into spatial, angular, and temporal components, achieving real-time rendering while remaining backward-compatible with Gaussian Splatting and outperforming prior methods across static, view-dependent, and dynamic benchmarks.

Project page arXiv 2510.03312 Code

N-DGS · the Gaussian-kernel lineage

6DGS & 7DGS

Where N-dimensional splatting began: lifting Gaussian primitives along view direction (6D) and then time (7D), with an efficient conditional-slicing mechanism.

ICLR 2025

6DGS: Enhanced Direction-Aware Gaussian Splatting for Volumetric Rendering

Zhongpai Gao, Benjamin Planche, Meng Zheng, Anwesa Choudhuri, Terrence Chen, Ziyan Wu

6DGS extends 3D Gaussian Splatting into 6D space to capture view-dependent effects, jointly modeling spatial position and view direction. By enhancing color and opacity representations and leveraging additional directional information, it achieves up to a 15.73 dB PSNR improvement with 66.5% fewer Gaussian points than 3DGS — while staying compatible with the established 3DGS framework.

Project page arXiv 2410.04974

ICCV 2025

7DGS: Unified Spatial-Temporal-Angular Gaussian Splatting

Zhongpai Gao, Benjamin Planche, Meng Zheng, Anwesa Choudhuri, Terrence Chen, Ziyan Wu

7DGS represents scenes as seven-dimensional Gaussians unifying spatial position (3D), time (1D), and viewing direction (3D). An efficient conditional slicing mechanism converts these 7D primitives into view- and time-conditioned 3D Gaussians, enabling real-time rendering of dynamic scenes with view-dependent effects — surpassing prior methods by up to 7.36 dB PSNR at 401 FPS.

Project page arXiv 2503.07946

ECCV 2026

Render-FM: Feedforward Model for Real-time Photorealistic Volumetric Rendering

Zhongpai Gao, Benjamin Planche, Meng Zheng, Anwesa Choudhuri, Van Nguyen Nguyen, Terrence Chen, Ziyan Wu

Render-FM brings the 6D representation to medicine. It directly predicts 6D Gaussian Splatting parameters from CT volumes in 2.8 seconds — a 500× speedup over per-scan optimization — while maintaining quality. Anatomy-Guided Priming incorporates segmentation masks and transfer functions as structural priors, so the feedforward model generalizes to unseen anatomies and supports real-time interactive rendering on standard GPUs without per-scan optimization.

Project page arXiv 2505.17338

Cite this line of work

BibTeX

@inproceedings{gao20246dgs,
  title     = {6DGS: Enhanced Direction-Aware Gaussian Splatting for Volumetric Rendering},
  author    = {Gao, Zhongpai and Planche, Benjamin and Zheng, Meng and
               Choudhuri, Anwesa and Chen, Terrence and Wu, Ziyan},
  booktitle = {International Conference on Learning Representations (ICLR)},
  year      = {2025}
}

@inproceedings{gao20257dgs,
  title     = {7DGS: Unified Spatial-Temporal-Angular Gaussian Splatting},
  author    = {Gao, Zhongpai and Planche, Benjamin and Zheng, Meng and
               Choudhuri, Anwesa and Chen, Terrence and Wu, Ziyan},
  booktitle = {Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)},
  pages     = {26316--26325},
  year      = {2025}
}

@inproceedings{liu2025universal,
  title     = {Universal Beta Splatting},
  author    = {Liu, Rong and Gao, Zhongpai and Planche, Benjamin and Chen, Meida and
               Nguyen, Van Nguyen and Zheng, Meng and Choudhuri, Anwesa and
               Chen, Terrence and Wang, Yue and Feng, Andrew and Wu, Ziyan},
  booktitle = {International Conference on Learning Representations (ICLR)},
  year      = {2026}
}

@inproceedings{gao2026renderfm,
  title     = {Render-FM: Feedforward Model for Real-time Photorealistic Volumetric Rendering},
  author    = {Gao, Zhongpai and Planche, Benjamin and Zheng, Meng and Choudhuri, Anwesa and
               Nguyen, Van Nguyen and Chen, Terrence and Wu, Ziyan},
  booktitle = {European Conference on Computer Vision (ECCV)},
  year      = {2026}
}

N-Dimensional Splatting