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.
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.
Fig. 2: Evaluation structures with voxelized representations
Parameter
Values
Voxel size ( s )
{0.04,0.06,0.08,0.10}m
GSD max
0.5mm/pixel
GSD setpoint
0.25mm/pixel
α
45∘
Rs
0.3394m
h
0.2400m
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
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
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
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.
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