Hierarchical Incremental Beta Splatting for Online Reconstruction

Linus Franke

cs.CVNov 5, 2026

Hierarchical Incremental Beta Splatting for Online Reconstruction

Authors: Linus Franke

Organizations: University of Würzburg

Abstract

Online 3D Gaussian Splatting reconstructs a scene during capture: each keyframe spawns primitives once and refines them for only a few dozen iterations, leaving little chance to fix poor initialization or missing detail. Gaussians are also inherently soft, needing many overlaps for solid surfaces. We present Hierarchical Incremental Beta Splatting (HIBS), which brings beta kernels to incremental reconstruction. Their shape parameter interpolates between a soft, Gaussian-like profile and a flat-topped, near-opaque surfel, acting as a confidence dial for geometry. Our confidence-aware spawning uses geometric observability and multi-view depth agreement to initialize both opacity and shape, producing solid primitives where geometry is verifiable and soft, correctable primitives where it is not. As online densification, error-driven relocation recycles stale primitives into photometric-error hotspots. Bounded beta kernel support enables exact tile culling and faster rasterization, and a closed-form, shape-preserving merging formulation provides a level-of-detail (LOD) hierarchy for fast optimization and real-time rendering. On Mip-NeRF 360 in the on-the-fly protocol, HIBS improves PSNR by up to 0.8 dB while reducing optimization time by up to 40%. Project page: https://linus-franke.com/hibs/

Figures & tables

Figure 2: Overview of our pipeline. An RGB stream is registered with camera parameters estimated online [ 40 ] . Each keyframe spawns beta primitives using geometric observability and depth agreement, keeping uncertain primitives soft. Near-dead primitives are relocated to high-error regions, followed by a short 30-iteration optimization burst with our fused rasterizer. Periodically, small-footprint primitives are merged hierarchically using closed-form moment matching that preserves their solidity.

Figure 3: Spawning/parameter initialization. Based on depth observability

c_{\mathrm{z}}

and agreement of depths (orange window)

c_{\mathrm{mvs}}

, the beta primitives are initialized as hard patches or soft probes.

Figure 4: Online confidence field. After the first orbit (left), many background primitives remain uncertain, and the table reconstruction is fuzzy. After the second orbit (middle), accumulated multi-view evidence solidifies most primitives. In the final reconstruction (right, novel view rendering), these areas are largely solidified, while uncertainty remains around the reflective table center.

Figure 5: Merging beta primitives . KL-style merging of beta primitives (middle) similar to merging of Gaussians (left) leads to oversized and solid primitives. Instead, we use moment merging (right), keeping the footprint.

	TUM				MipNeRF360 (otf)				StaticHikes
	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓
G+Beta-Splatting (7k)	22.73	0.842	0.214	0:01:50	27.75	0.863	0.232	0:04:34	20.36	0.538	0.470	0:59:22
G+Faster-GS (7k)	23.08	0.845	0.207	0:01:42	27.81	0.876	0.210	0:04:22	20.57	0.547	0.459	0:59:38
MonoGS †	16.60	0.682	0.381	$\sim$ 15min	14.46	0.436	0.663	$\sim$ 4min	15.46	0.301	0.659	$\sim$ 9min
S3PO-GS with refinement †	21.88	0.777	0.309	$\sim$ 1h	19.82	0.553	0.576	$\sim$ 20min	18.09	0.368	0.611	$\sim$ 50min
LongSplat †	26.51	0.875	0.176	$\sim$ 1.5h	23.63	0.657	0.415	$\sim$ 2h	20.05	0.461	0.481	$\sim$ 20h

Table 1: Results for novel view synthesis on pose-free methods. The best and second best are color coded for pose-free methods. Times for G+Beta-Splatting and G+Faster-GS (offline baselines at 7k as a reference) include GLOMAP pose estimation (about 50s for TUM, 174s for MipNeRF360 ( otf ) and 58min for StaticHikes). † numbers taken from i3DGS [ 40 ] , with times roughly scaled to our setup.

Panel of Figure 6 — Figure 6: Qualitative comparison against baselines on the 3DGS benchmark: Truck , and Flowers scenes.

		MipNeRF360				Tanks and Temples				Deep Blending
		PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓
G ( seq.)	BetaSplatting (30K)	25.90	0.703	0.333	0:09:07	23.31	0.831	0.190	0:08:27	$\times$	$\times$	$\times$	$\times$
	FasterGS (30K)	25.53	0.702	0.356	0:05:29	22.17	0.802	0.250	0:04:21	$\times$	$\times$	$\times$	$\times$
	BetaSplatting (7K)	24.05	0.656	0.396	0:03:17	20.81	0.733	0.321	0:02:55	$\times$	$\times$	$\times$	$\times$
	FasterGS (7K)	24.57	0.678	0.411	0:02:29	20.79	0.742	0.343	0:02:10	$\times$	$\times$	$\times$	$\times$
	i3DGS	24.75	0.721	0.280	0:01:59	22.08	0.783	0.226	0:01:25	27.16	0.856	0.285	0:02:03

Table 2: Results on the full 3DGS benchmark . We compare with online method i3DGS as well as FasterGS and BetaSplatting with GLOMAP sequential mapping (requiring 1:52 on MipNeRF360, 1:42 on Tanks&Temples and failing on Deep Blending). More expensive exhaustive mapping is in Tab. 16 . The best and second best are color coded.

	SmallCity				CityWalk
	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓
H3DGS	21.17	0.679	0.285	$\sim$ 2.5h	11.78	0.557	0.560	$\sim$ 20h
Octree-GS	24.18	0.831	0.273	$\sim$ 1h	17.33	0.638	0.520	$\sim$ 2.5h
S3PO-GS	19.71	0.652	0.462	$\sim$ 0.7h	10.72	0.439	0.579	$\sim$ 5.5h
OtF NVS	23.59	0.789	0.323	02:21	21.71	0.712	0.395	29:35
i3DGS	23.56	0.800	0.302	02:54	22.03	0.772	0.336	45:20

Table 3: Results for large scale scenes. Offline methods (H3DGS, Octree-GS) include COLMAP times. Comparing image metrics are from [ 40 ] , non-approximate times measured on our machine. S3PO-GS includes refinement. The best and second best are color coded.

Configuration		PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	t (s)
Full (ours)		25.30	0.810	0.213	101
w/o SB lobes ( $\to$ SH)	[ Sec. 4.1 ]	25.12	0.807	0.214	106
DBS lobe init prior	[ Sec. 4.1 ]	25.19	0.809	0.214	100
w/o confidence	[ Sec. 4.2 ]	25.14	0.808	0.215	99
w/o solidification	[ Sec. 4.3 ]	25.15	0.807	0.216	100
w/o error relocation	[ Sec. 4.4 ]	25.09	0.804	0.219	95

Table 4: Component ablation (leave-one-out from full method). Top table: Metrics are averaged over all nine OtF scenes ( Tab. 1 ).

Configuration	Confidence	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	t (s)
i3DGS [ 40 ]	none	24.73	0.797	0.227	129
Gaussian	$o_{\mathrm{spawn}}$	25.09	0.808	0.214	130
$\beta$ frozen at 4	$o_{\mathrm{spawn}}$	25.06	0.806	0.217	100
learnable $\beta$	$o_{\mathrm{spawn}}$	25.11	0.807	0.215	100
learnable $\beta$	$o_{\mathrm{spawn}}$ , $\beta_{\mathrm{spawn}}$	25.15	0.807	0.216	100
Full (ours)	$o_{\mathrm{spawn}}$ , $\beta_{\mathrm{spawn}}$ , solidify	25.30	0.810	0.213	101

Table 5: Kernel ablation . Metrics averaged over all OtF scenes.

Figure 7: Quality during reconstruction. Top : mean PSNR over test views registered so far versus reconstruction progress, for our method and i3DGS. Bottom : PSNR immediately after each view’s registration versus just before the next test view arrives. Points above the diagonal show improvement after registration.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Stage (ms/call)	Garden	fr1/desk	Truck	University2	SmallCity
Image load, features	13.4	11.8	13.4	11.4	13.2
Pose estimation	119.2	149.2	117.6	114.6	119.7
Loop closure	101.3	–	164.1	–	–
Primitive spawning	47.1	49.3	46.0	44.6	36.4
Optimization burst	170.6	146.5	129.2	174.9	329.6
Hierarchy update	2.7	2.7	3.0	41.4	56.5

Appendix

Table 6: Per-stage runtime , in milliseconds (CUDA-synchronized timers, so stage costs are exact but totals are overestimated). Image load and feature extraction run once per streamed frame; all other stages run per registered keyframe (loop closure rarely

1-3\times

	1st image	Frames	Boot.	Mean latency/
Scene	(s)	used	KFs	frame (ms)
Garden	1.2	8	8	336
fr1/desk	1.5	17	8	106
Truck	1.0	10	8	276
University2	1.2	9	8	176
SmallCity	1.4	9	8	237

Appendix

Table 7: Time to first image. Time from the first streamed frame to the first renderable reconstruction, including bootstrap, mini-BA, and the first optimization burst. “Frames used” denotes the number of streamed frames consumed by bootstrap.

Stage	ms	share
blend backward	1.67	37%
preprocess backward (fused Adam)	1.31	29%
blend forward	0.85	19%
preprocess forward	0.36	8%
sort / instancing	0.37	8%
preprocess backward (no fused Adam)	0.66

Appendix

Table 8: Fused rasterization step profile on a converged Garden reconstruction (2.2M primitives,

1297{\times}840

). The fused backward integrates the sparse Adam update in-kernel, so the per-primitive stages carry the optimizer’s memory traffic: running the preprocess backward with plain gradient stores instead of the fused Adam update halves it (

0.66

vs.

1.31

ms) — i.e. half that kernel is the optimizer. End-to-end, the fused path reduces the 30-iteration burst from

287

170

ms per keyframe ( Tab. 6 ).

Table 15

Scene	Colour	PSNR ↑	SSIM ↑	LPIPS ↓
drjohnson	SB lobes	25.67	0.828	0.315
	SH	25.88	0.832	0.309
playroom	SB lobes	27.28	0.865	0.276
	SH	27.58	0.868	0.268

Appendix

Table 11: Colour basis on DeepBlending. Comparison of the default five zero-initialized spherical-beta lobes with degree-3 SH using the identical reconstruction pipeline.

Dataset	Scene	$\Delta$ PSNR	$\Delta$ SSIM	$\Delta$ LPIPS
db	drjohnson	-0.21	-0.004	+0.007
db	playroom	-0.29	-0.003	+0.008
MipNeRF360	garden	+0.18	+0.005	-0.000
MipNeRF360	bonsai	+0.23	+0.003	-0.002
MipNeRF360	counter	+0.41	+0.007	-0.006
MipNeRF360	bicycle	+0.16	+0.013	-0.001

Appendix

Table 12: Spherical-beta lobes vs. SH colour, per scene.

\Delta

denotes lobes minus SH for PSNR and LPIPS. Both variants use the same reconstruction configuration except color representation.

Figure 8: Half-maximum scale compensation. Without compensation, lowering

\beta

broadens the kernel; with the proposed compensation, all profiles share the same half-maximum radius, isolating shape changes from footprint changes.

Scene	Method	ATE(%) ↓	R.APE( ∘ ) ↓	T.RPE(%) ↓	R.RPE( ∘ ) ↓
fr1/desk	Ours	0.55	1.90	0.344	0.644
	i3DGS	0.47	1.42	0.314	0.548
fr2/xyz	Ours	0.33	0.41	0.178	0.128
	i3DGS	0.37	0.42	0.185	0.133
fr3/office	Ours	0.32	0.94	0.127	0.326
	i3DGS	0.38	1.02	0.150	0.347

Appendix

Table 13: TUM pose accuracy. Our method and i3DGS evaluated against the same COLMAP pseudo-ground truth using aligned trajectories. Translation errors are reported as a percentage of trajectory extent and rotation errors in degrees; results are averaged over multiple runs.

Bootstrap	ATE (%) ↓	R. APE ( ∘ ) ↓	PSNR ↑	f.i. (s)	$f$
ordered first-8	0.55	1.90	23.93	1.5	519
ordered + parallax	1.78	3.17	23.67	1.5	527
spread, pool 30	0.51	1.21	24.20	2.6	511

Appendix

Table 14: Conditioning the fr1/desk bootstrap. Three bootstrap variants under the same scoring protocol. The spread selector improves pose accuracy and PSNR at the cost of

1.1

s additional latency; the parallax floor degrades accuracy.

Figure 9: Online confidence field on a walking capture ( University2 ), analogous to Fig. 4 : renders over their per-primitive shape field ( $\beta$ , warm = soft). As the walk advances, accumulated multi-view evidence solidifies the traversed corridor of the scene behind the camera, while freshly entered and peripheral regions remain soft probes — the field marks the reconstruction frontier of an unbounded capture live, at no quality cost.

Variant	PSNR ↑	SSIM ↑	LPIPS ↓
Full (error-weighted + consistency gate)	28.45	0.893	0.133
w/o error weighting (opacity-sampled)	27.89	0.885	0.148
w/o relocation	28.21	0.890	0.136

Appendix

Table 15: Relocation design choices on fr2/xyz . Each row removes one component of the error-driven relocation.

Figure 10: Spawn-confidence signals on University2 . Primitive locations are colored by geometric observability $c_{z}$ , MVS–monocular agreement $c_{\mathrm{mvs}}$ , and their product $c$ . The two signals fail in different regions: distant trees have low observability, while dense foliage has poor depth agreement despite sufficient baseline. Their product therefore assigns solid, opaque initialization to well-constrained surfaces and soft initialization to ambiguous regions.

		MipNeRF360				Tanks and Temples				Deep Blending
		PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓	PSNR ↑	SSIM ↑	LPIPS ↓	Time ↓
G ( exh.)	BetaSplatting (30K)	28.09	0.822	0.222	0:12:34	22.18	0.818	0.209	0:11:14	30.49	0.915	0.250	0:10:11
	FasterGS (30K)	27.66	0.818	0.251	0:09:01	20.45	0.781	0.274	0:07:03	29.78	0.908	0.306	0:07:14
	BetaSplatting (7K)	25.92	0.750	0.308	0:06:21	19.72	0.710	0.347	0:05:36	26.85	0.861	0.350	0:04:56
	FasterGS (7K)	25.36	0.740	0.347	0:05:34	19.07	0.712	0.373	0:04:48	28.91	0.891	0.358	0:04:17
G ( seq.)	BetaSplatting (30K)	25.34	0.679	0.341	0:09:07	23.31	0.831	0.190	0:08:27	$\times$	$\times$	$\times$	$\times$

Appendix

Table 16: Results on the full 3DGS benchmark. Nine MipNeRF360, two Tanks&Temples, and two Deep Blending scenes. We evaluate BetaSplatting and FasterGS with GLOMAP poses using sequential and exhaustive matching, and compare against i3DGS and our method. GLOMAP sequential requires 0:01:52 on MipNeRF360, 0:01:42 on Tanks&Temples and fails on Deep Blending. GLOMAP exhaustive takes respectively 0:04:53, 0:04:22 and 0:03:42. The best and second best are color coded for non-exhaustive matching methods.

Scene	Order	PSNR ↑	SSIM ↑	LPIPS ↓
Garden	ordered	25.82	0.786	0.187
	shuffled	25.34	0.774	0.191
Counter	ordered	26.98	0.860	0.222
	shuffled	27.54	0.877	0.204
Bonsai	ordered	29.61	0.909	0.204
	shuffled	27.11	0.850	0.261

Appendix

Table 17: Unordered input. Identical configuration.

Spawning ( Sec. 4.2 )
Constant	Value	Role	Provenance
$\theta_{0}$	$2^{\circ}$	full-observability angle	probed ( $\times/\div\,2$ )
$o_{0}$	$0.1$	spawn opacity, unverified	fixed heuristic
$o_{1}$	$0.3$	spawn opacity, fully verified	probed ( $0.2/0.5$ )
$o_{\mathrm{floor}}$	$0.02$	opacity without MVS support	fixed heuristic
$\beta_{\mathrm{solid},\mathrm{soft}}$	$1,5$	shape band around init $\beta{=}4$	structural

Appendix

Table 18: Method constants and their provenance. Values are grouped by spawning, solidity control, relocation, optimization/rasterization, and the hierarchy. Derived quantities follow from other parameters, structural values are fixed by the kernel construction, and inherited values are taken unchanged from prior work.

$\Delta b$	PSNR ↑	LPIPS ↓	Solid.	Soft.
0.1	25.77	0.188	229k	81k
0.2	25.65	0.190	226k	79k
0.4 (ours)	25.82	0.187	224k	81k
0.8	25.80	0.188	220k	82k

Appendix

Table 19: Solidity-controller step sweep on Garden .

\Delta b

denotes the per-keyframe shape update. “Solid.” and “Soft.” report the corresponding update counts.

Figure 11: Error-driven relocation on Garden . Left: incoming keyframe and per-pixel error at $22.6$ dB. Middle: dead primitives relocated to high-error contributors, with render-preserving opacity and scale correction. Right: the subsequent 30-iteration burst increases quality to $24.5$ dB.

Scene	Constant	shipped $\to$ probe	PSNR ↑	$\Delta$
Garden	$o_{\mathrm{solid}}$ (solidify opacity gate)	0.7 $\to$ 0.5	25.78 $\pm$ 0.05	-0.04 $\pm$ 0.06
Garden	$o_{\mathrm{solid}}$ (solidify opacity gate)	0.7 $\to$ 0.9	25.75 $\pm$ 0.08	-0.07 $\pm$ 0.09
Garden	$\theta_{0}$ (observability angle)	$2^{\circ}$ $\to$ $1^{\circ}$	25.78 $\pm$ 0.02	-0.04 $\pm$ 0.04
Garden	$\theta_{0}$ (observability angle)	$2^{\circ}$ $\to$ $4^{\circ}$	25.77 $\pm$ 0.09	-0.05 $\pm$ 0.10
Garden	$o_{\mathrm{1}}$ (verified spawn opacity)	0.3 $\to$ 0.2	25.83 $\pm$ 0.03	+0.01 $\pm$ 0.05
Garden	$o_{\mathrm{1}}$ (verified spawn opacity)	0.3 $\to$ 0.5	25.81 $\pm$ 0.08	-0.01 $\pm$ 0.09

Appendix

Table 20: One-at-a-time sensitivity study. Each row varies one constant around its default while keeping all other settings fixed. Results are averaged over three runs;

\Delta

reports the PSNR change relative to the default configuration.

Figure 12: Memory and capacity over reconstruction. GPU memory remains bounded while active capacity saturates on Garden . Relocation redistributes this capacity over time, whereas hierarchy offloading allows the model to grow on SmallCity without increasing the GPU footprint.

Panel of Figure 13 — Figure 13: Qualitative comparison against online reconstruction methods on Garden , Forest1 , University2 , fr1/desk , and SmallCity (top to bottom).

Panel of Figure 14 — Figure 14: Additional qualitative comparisons against online reconstruction methods: Bonsai , Counter , Forest2 , fr2/xyz , and fr3/office (top to bottom).

Panel of Figure 15 — Figure 15: Additional qualitative comparisons against offline baselines: Bicycle , Kitchen , Room , Treehill , and Garden (top to bottom).

Panel of Figure 16 — Figure 16: Additional qualitative comparisons against offline baselines: Bonsai , Counter , Dr. Johnson , Train , Stump , and Playroom (top to bottom).

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We introduce a probabilistic splat-based radiance field framework that retains the fast rasterization and test-time efficiency of 3D Gaussian Splatting (3DGS) while replacing heuristic primitive manipulation with gradient-based optimization of a volumetric probability density. Rather than relocating, splitting, or culling Gaussians via hand-tuned densification (e.g., ADC), we treat primitive locations as samples drawn from a persistent, learnable density. We instantiate this density using a novel, memory-efficient multi-scale hierarchical grid that enables end-to-end gradient-based optimization. To stabilize the optimization, we derive an unbiased gradient estimator with control variates that markedly reduces variance. By allowing probability mass to flow to where the loss demands, our framework eliminates brittle priors and naturally explores the volume, achieving state-of-the-art reconstruction quality on mip-NeRF 360 while preserving 3DGS-level rendering speed.

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