A nearly optimal sensor placement algorithm for boundary coverage

Abstract

Locating visual sensors in 2D can be often modeled as an Art Gallery problem. Tasks such as surveillance require observing or “covering” the interior of a polygon with a minimum number of sensors or “guards”. For other tasks, such as inspection and image based rendering, observing the boundaries of the environment is sufficient. As Interior Covering (IC), also Edge Covering (EC) is NP-hard, and no finite algorithm is known for its exact solution. Approximate EC solutions are provided by many heuristic algorithms, but their performances with respect to optimality (minimum number of sensors) is unknown. In this paper, we propose a new EC sensors location technique. The algorithm is incremental, and converges toward the optimal solution. It refines an initial approximation provided by an integer covering algorithm (IEC) where each edge is observed entirely by at least one sensor. A lower bound for the number of sensors, specific of the polygon considered, is used at each step for evaluating the quality of the current solution, and a set of rules are provided for performing a local refinement to reduce the computational burden. The algorithm has been implemented, and tests over hundreds of random polygons show that it supplies solutions very close to and often coincident with the lower bound, and then suboptimal or optimal. In addition, the approximate starting solutions provided by the IEC algorithms are, on the average, close to optimum. The tight lower bound can also be used for testing other EC sensor location algorithms. Running times allow dealing with polygons with up to a few hundreds of edges, which appears adequate for many practical cases. An enhanced version of the algorithm, also taking into account range and incidence constraints, has also been implemented and tested.

Introduction

Several computer vision and robotics tasks require multiple visual sensors, or the displacement of one visual sensor in multiple positions. Sensor placement is an active area of research. The survey [1] on sensor positioning refers in particular to tasks as reconstruction and inspection. The surveys [2], [3], [4] are mainly devoted to object and environment inspection. Tasks such as surveillance, inspection and image based rendering, usually refer to known objects or environments, and are categorized as model-based view planning or robot vision [1], [5], [6].

Practical sensor planning problems require considering several constraints, such as minimum and maximum distance, incidence, feature visibility and lighting. Visibility is clearly the fundamental constraint for any kind of task where optical sensors are involved. The sensor is usually modeled as a point. A feature of an object is said to be visible from the sensor, or not occluded, if any segment joining a point of the feature and the sensor does not intersect the environment or the object itself (usually excluding boundary points).

Although in general the sensor location problem is 3D, in several cases it can be restricted to 2D. This is, for instance, the case of buildings, which can be modeled as objects obtained by extrusion.

Assuming that the sensors are omni-directional or rotating, for tasks such as surveillance, the 2D visibility constraint is modeled by the classic Art Gallery problem, which requires observing or “covering” the interior of a polygonal environment with a minimum set of sensors. We call this the interior covering (IC) problem.

Other tasks, such as inspection or image based rendering when the geometry, but not the texture, of the environment is known, require observing only the boundary, in 2D the edges of a polygonal environment. Observing the edges with a minimum set of sensors is the problem tackled in this paper. We call this the edge covering (EC) problem. As it will be discussed in detail in Section 2, EC and IC are different problems, which is not always recognized in the literature.

Joining the locations of a set of IC or EC sensor is a technique used for robot motion planning in static environments (see, for instance, Refs. [7], [8]).

Unfortunately, the EC problem, as well as the original IC problem and several of their variations, is NP-hard, and no exact finite algorithm, not even exponential, is known for locating a minimum set of sensors. However, this is an important practical problem and “good” approximation algorithms for average practical cases are sorely needed. In our view, a “good” practical algorithm should not only be computationally feasible, but also provide a set of sensors whose cardinality, on the average, is not far from optimum.

Several heuristic algorithms for this problem have been presented, but they do not match these requirements, since it is unknown how close to the optimum is the cardinality of the solution provided.

In this paper, after discussing the available approximation algorithms, we present a new EC sensor positioning technique. The algorithm is incremental and converges toward the optimal solution. A key feature of the algorithm is that it computes a lower bound, specific of the polygonal environment considered, for the minimum number of guards. It allows evaluating the quality of the solution obtained at each step, and halting the algorithm if the solution is satisfactory.

The algorithm refines a starting approximate solution provided by an integer edge covering (IEC) algorithm, where each edge must be observed entirely by at least one sensor. A set of rules, aimed to reduce the computation, is provided for refining locally the current solution. The proposed technique can also easily take into account other geometrical constraints such as range and incidence.

The algorithm has been implemented and tested on several hundreds of polygonal floor maps. According to the experimental results, the algorithm is a “good” practical algorithm. In fact, the cardinality of the covers supplied by the algorithm is always very close to, and in several cases coincident with, the lower bound, and therefore optimal or nearly optimal. Concerning running times, the algorithm works well for several tens of edges. When the number of edges increases, a simplification of the algorithm reduces running times affecting only slightly the quality of the solution. This allows dealing with polygons with a few hundreds of edges, which appears sufficient for many practical cases. A preliminary version of the algorithm has been presented in Ref. [9], and some preliminary results in Ref. [10]. An enhanced version of the algorithm, also taking into account range and incidence constraints, has also been implemented and tested.

The paper is organized as follows. In Section 2, we summarize the relevant Art Gallery theory and discuss the existing sensor location approximate algorithms. Section 3 describes our sensor positioning algorithm. In Section 4, we present and discuss the experimental results. Section 5 is devoted to the enhanced algorithm taking into account range and incidence constraints. Conclusions and directions of future research are discussed in Section 6.