Ground Sample Distance Constrained k-Coverage Viewpoint Planning via Set Multi-Cover Formulation

Waleed Bin Khalid; Chul Min Yeum

cs.ROOct 27, 2026

Ground Sample Distance Constrained k-Coverage Viewpoint Planning via Set Multi-Cover Formulation

Authors: Waleed Bin Khalid, Chul Min Yeum

Organizations: Luddy School of Informatics, Computing, and Engineering, Indiana University Bloomington, IN, USA. · Civil and Environmental Engineering, University of Waterloo, ON, Canada.

Abstract

Drone-based 3D reconstruction and inspection require viewpoints that satisfy image-quality constraints while observing each surface region multiple times, which overlap-based planning heuristics do not guarantee. We study ground sample distance (GSD)-constrained $k$ -coverage viewpoint selection for structures represented by surface voxels. Candidate viewpoints are generated along outward surface directions, screened with distance, viewing-angle, field-of-view, and occlusion tests, and augmented for voxels with insufficient visibility. Selection is then posed as set multi-cover with per-voxel coverage demands. We propose a dynamically weighted greedy heuristic that prioritizes remaining coverage deficits and scarce viewing alternatives, and compare it with sequential overlap-based pruning and an integer linear programming (ILP) formulation on the same candidate pools. Across two structures, four voxel resolutions, and three coverage requirements, the proposed approach achieves full or near-full $k$ -coverage with compact viewpoint sets, using only 3.00--16.39% more viewpoints than ILP while offering an interpretable selection process based on coverage deficits and viewpoint availability.

Figures & tables

Fig. 1: Overview of the proposed GSD-constrained $k$ -coverage pipeline, from voxelization and sensing-compliant candidate generation to visibility assignment, augmentation, and final viewpoint selection.

Panel of Figure 2 — Fig. 2: Evaluation structures with voxelized representations

Parameter	Values
Voxel size ( $s$ )	$\{0.04,\,0.06,\,0.08,\,0.10\}\ \mathrm{m}$
GSD max	$0.5\mathrm{mm/pixel}$
GSD setpoint	$0.25\mathrm{mm/pixel}$
$\alpha$	$45^{\circ}$
$R_{s}$	$0.3394\ \mathrm{m}$
$h$	$0.2400\ \mathrm{m}$

TABLE I: Experimental parameters.

Fig. 3: Multi-normal voxels and their retained outward-facing normals.

	Angel Statue				Rooster
VS	0.04	0.06	0.08	0.10	0.04	0.06	0.08	0.10
TV	4178	1905	1064	695	2131	952	536	332
IV	1397	653	350	222	599	217	114	49
SV	2781	1252	714	473	1532	735	422	283
IW	2781	1252	714	473	1540	741	435	294
BW	67	20	7	4	47	27	9	4

TABLE II: Voxelization statistics across resolutions for both structures

		Angel Statue					Rooster
$k$	VS	NC	PC	AW	NC ∗	PC ∗	NC	PC	AW	NC ∗	PC ∗
1	0.04	5	0	291	1	0	2	0	635	1	0
	0.06	2	0	755	0	0	8	0	3490	0	0
	0.08	0	0	0	0	0	5	0	2004	0	0
	0.10	2	0	500	0	0	3	0	994	0	0
3	0.04	5	9	1797	1	1	2	18	8992	1	0

TABLE III: Visibility Statistics Before and After Viewpoint Augmentation

$k$	VS	Angel Statue				Rooster
$k$	VS	Min	Max	Mean	Std	Min	Max	Mean	Std
1	0.04	0.102	0.500	0.330	0.069	0.083	0.500	0.331	0.071
	0.06	0.118	0.500	0.335	0.064	0.105	0.500	0.325	0.069
	0.08	0.167	0.500	0.335	0.072	0.082	0.500	0.326	0.063
	0.10	0.150	0.495	0.328	0.080	0.174	0.492	0.330	0.067
3	0.04	0.102	0.500	0.325	0.069	0.082	0.500	0.321	0.069

TABLE IV: GSD statistics over viewpoint–voxel pairs (all values are reported in mm/px, Std = standard deviation).

O	S	Angel Statue				Rooster
		TS	FC (%)	PC (%)	NC (%)	TS	FC (%)	PC (%)	NC (%)
50	Vertical	171	32.14	63.00	4.86	135	32.99	64.29	2.72
50	Visibility	71	21.99	75.90	2.11	56	27.89	69.73	2.38
70	Vertical	253	32.98	66.60	0.42	194	39.12	60.54	0.34
70	Visibility	80	13.32	86.26	0.42	70	19.05	80.27	0.68

TABLE V: Sequential Overlap-Based Viewpoint Pruning results at voxel size 0.10 m

		Globally Informed Greedy $k$ -Coverage								ILP-Based Viewpoint Selection										Difference of WS (%)
		Angel Statue				Rooster				Angel Statue					Rooster					Angel Statue	Rooster
$k$	VS	WA	WS	CR	FC(%)	WA	WS	CR	FC(%)	WA	WS	CR	FC(%)	Opt.	WA	WS	CR	FC(%)	Opt.
1	0.04	3005	142	5	99.96	2128	110	3	99.93	3005	122	0	99.96	False	2128	95	0	99.93	True	16.39	15.79
	0.06	1987	111	0	100.00	4204	85	1	100.00	1987	96	0	100.00	True	4204	76	0	100.00	True	15.62	11.84
	0.08	707	92	1	100.00	2430	78	1	100.00	707	87	0	100.00	True	2430	73	0	100.00	True	5.75	6.85
	0.10	969	81	0	100.00	1284	69	0	100.00	969	76	0	100.00	True	1284	64	0	100.00	True	6.58	7.81

TABLE VI: Comparison between Globally Informed Greedy

k

-Coverage and ILP-Based Minimum-Cardinality Viewpoint Selection across voxel sizes and

k

-coverage levels. Difference (%) is computed relative to the reported ILP viewpoint count; Opt. indicates whether ILP optimality was certified.

Next-Best-View Planning

Next-best-view planning selects camera or sensor viewpoints that gather useful evidence while balancing coverage, reconstruction quality, and motion cost.

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What makes a good viewpoint? The quality of the data used to learn 3D reconstructions is crucial for enabling efficient and accurate scene modeling. We study the active view selection problem and develop a principled analysis that yields a simple and interpretable criterion for selecting informative camera poses. Our key insight is that informative views can be obtained by minimizing a tractable approximation of the Fisher Information Gain, which reduces to favoring viewpoints that cover geometry that has been insufficiently observed by past cameras. This leads to a lightweight coverage-based view selection metric that avoids expensive transmittance estimation and is robust to noise and training dynamics. We call this metric COVER (Camera Optimization for View Exploration and Reconstruction). We integrate our method into the Nerfstudio framework and evaluate it on real datasets within fixed and embodied data acquisition scenarios. Across multiple datasets and radiance-field baselines, our method consistently improves reconstruction quality compared to state-of-the-art active view selection methods. Additional visualizations and our Nerfstudio package can be found at https://chengine.github.io/nbv_gym/.

Abstract

Next-Best-View Planning

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