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On The Computational Complexity for Minimizing Aerial Photographs for Full Coverage of a Planar Region

Si Wei Feng

TL;DR

The paper investigates the computational complexity of covering a simple polygon with k circular or square footprints, modeling aerial photography constraints. It establishes strong inapproximability results for both circle and square coverage, including restricted-location variants, via gadget-based reductions from planar vertex cover. It also proposes practical polynomial-time algorithms using sampling and k-center heuristics, achieving about $2$- or $2\sqrt{2}$-approximation factors, with IP-based refinements for small instances. The work highlights fundamental limits and practical heuristics for real-world coverage problems in aerial imaging and sensor placement, with implications beyond drones to areas like pesticide spraying and security surveillance.

Abstract

With the popularity of drone technologies, aerial photography have become prevalent in many daily scenarios such as environment monitoring, structure inspection, law enforcement etc. A central challenge in this domain is the efficient coverage of a target area with photographs that can entirely capture the region, while respecting constraints such as the image resolution, and limited number of pictures that can be taken. This work investigates the computational complexity of several fundamental problems arised from this challenge. By abstracting the aerial photography problem into the coverage problems in computational geometry, we demonstrate that most of these problems are in fact computationally intractable, with the implication that traditional algorithms cannot solve them efficiently. The intuitions of this work can extend beyond aerial photography to broader applications such as pesticide spraying, and strategic sensor placement.

On The Computational Complexity for Minimizing Aerial Photographs for Full Coverage of a Planar Region

TL;DR

The paper investigates the computational complexity of covering a simple polygon with k circular or square footprints, modeling aerial photography constraints. It establishes strong inapproximability results for both circle and square coverage, including restricted-location variants, via gadget-based reductions from planar vertex cover. It also proposes practical polynomial-time algorithms using sampling and k-center heuristics, achieving about - or -approximation factors, with IP-based refinements for small instances. The work highlights fundamental limits and practical heuristics for real-world coverage problems in aerial imaging and sensor placement, with implications beyond drones to areas like pesticide spraying and security surveillance.

Abstract

With the popularity of drone technologies, aerial photography have become prevalent in many daily scenarios such as environment monitoring, structure inspection, law enforcement etc. A central challenge in this domain is the efficient coverage of a target area with photographs that can entirely capture the region, while respecting constraints such as the image resolution, and limited number of pictures that can be taken. This work investigates the computational complexity of several fundamental problems arised from this challenge. By abstracting the aerial photography problem into the coverage problems in computational geometry, we demonstrate that most of these problems are in fact computationally intractable, with the implication that traditional algorithms cannot solve them efficiently. The intuitions of this work can extend beyond aerial photography to broader applications such as pesticide spraying, and strategic sensor placement.

Paper Structure

This paper contains 10 sections, 4 theorems, 3 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulations
  3. Preliminaries
  4. Intermediate structure construction
  5. Hardness of approximate circle coverage
  6. Hardness of approximate square coverage
  7. Hardness of approximate coverage with restricted locations inside the region
  8. Constant factor approximation algorithm
  9. Limitations and future work
  10. Acknowledgements

Key Result

Theorem 1

Feng2020Optimally It is NP-hard to approximate problem p:cir_cov within a factor of $1.152$.

Figures (9)

  • Figure 1: Coverage footprint of the region boundary or interior with circle or square coverage footprints.
  • Figure 2: Structure within the odd length path and attached perpendicular "bars" with length $\zeta = \sqrt{3}$. Regarding the representation of such non-integral coordinates in the problem input, we may scale the coordinates to some certain extent and round them to integers so that the relative distance between each other is precise enough for the proof.
  • Figure 3: Circle coverage pattern at the junction crossing with unit radius.
  • Figure 4: Circle coverage pattern at the extreme circle radius at the junction crossing.
  • Figure 5: Structure within the odd length path, where each unit length segment is now shrinked to $\sqrt{2}/2$, and the vertical bar length is $\zeta = \sqrt{2}/2$.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4