A Preliminary Study on Optimal Placement of Cameras

10/26/2019
by   Lin Xu, et al.
12

This paper primarily focuses on figuring out the best array of cameras, or visual sensors, so that such a placement enables the maximum utilization of these visual sensors. Maximizing the utilization of these cameras can convert to another problem that is simpler for the formulation, that is, maximizing the total coverage with these cameras. To solve the problem, the coverage problem is first defined subject to the capabilities and limits of cameras. Then, poses of cameras are analyzed for the best arrangement.

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I Introduction

Cameras are ubiquitous in our lives, as we can almost see it everywhere we go. Streets, restaurants, offices, schools, areas where people go and stay, are under these ”eyes.” Cameras, or in a broad way of definition, visual sensors, are widely utilized for video surveillance, recording, military uses, and so on. Given the area that the cameras need to watch, if all cameras are the same, the size of the area is proportional to the number of cameras. Therefore, the cost of setting cameras is positively related to the number of cameras employed. Sometimes, designers make coverage mistakes, ineffective setting visual sensors, as shown below. In this case, optimal placement of cameras is critical.

Fig. 1: Different camera set

In 1973, mathematician Victor Klee proposed the art gallery problem, a real-world problem trying to find a minimum number of guards who together can observe the whole gallery. These guards are at fixed positions, and the gallery shapes like a polygon. Similarly, there is also the Floodlight illumination problem, coping with the illumination of planar regions by light sources. For more information, please refer to [1],[2]. However, these solutions contain unrealistic factors, such as unlimited field of view, the infinite depth of field, making these algorithms unsuitable for most real-world computer vision applications.

Ii Problem Formulation

In this paper, as shown in Figure 2, the problem of optimal camera placement for a given region is formulated. We focus on the static camera placement problem, where the goal is to determine optimal position and number of cameras for a region to be observed, given a set of task-specific constraints, and a set of possible cameras to use in the layout.

Fig. 2: Camera placement problem

For simplicity, we first assume that the detection range of each visual sensor is unbounded. However, there are several constraints needed to consider: (1) the number of cameras that given, (2) the each camera’s limited field-of-view, (3) fixed position, (4) the room without obstacles. The optimal placement is defined as the cameras are placed so that they cover as much space as possible.

The field-of-view of a camera can be described by a triangle, as shown in Figure 3. There are three parameters, with representing the length of sight, determining the pose of a camera, and defining the field-of-view angle.

Fig. 3: Different camera set

Iii Main Result

For a cuboid 3D space, consider the camera working in 2D space such as 2D space from the side view and 2D space from the top view as shown in Figure 4. As for the coverage problem, we can claim that if a place is covered by both the side view and the top view of a camera, then it can be covered by the field-of-view angle of the camera in 3D space. With this result, we divide the coverage problem in 3D space into two subproblems, namely from the top view and side view.

Fig. 4: 2D space coverage scenario

Iii-a From the top view

From the top view, the best arrangement should be that all spaces are covered ideally, and the overlapped areas should be minimal. Due to such an analysis, the cameras should be placed in a staggered way such that these visual sensors would not monitor a space multiple times, shown in Figure 5.

Fig. 5: Staggered arrangement of cameras

Proposition 1. As shown in Figure 6, given that cameras with the same field-of-view angle 2 and unbounded the depth of view, the staggered arrangement is optimal for a rectangular space.

Proof. In order to get the maximum coverage, the pose of a camera should be changed such that all field-of-view angle is included in the given space, as shown in Figure 6. Six cameras are labelled by A to F, and their field-of-view angles also correspond to1 to 6 in Figure 6, respectively.

Fig. 6: Maximum coverage

At that time, since all field-of-view angles are in the space, so

Therefore, AJ // CB // FG // HD// IE. We can see that no fields of view overlap until the fifth camera is in position, so they maximize the coverage while no space is counted twice and no space is out of sight. The dashed area is the fields of view overlap due to the insufficient length of the space, but it is indeed the best one because applying five cameras will not coverage all spaces. The dashed area can be more even if we slightly adjust the pose of the first camera at the corner, and the rest sensors still form sets of a rectangle. For such an arrangement, the length of a given space, , and the vertex angle of the sensor, , do matter. The position of the cameras is determined by them if follows the optimal placements, for which the first camera should be at one corner of a space, and the others follow, as shown in Figure 7. Let us name them from top to bottom as number 1, number 2, number 3, etc.

Fig. 7: Optimal arrangement

Since Camera #1 is at the corner, where we can see it as the origin (0, 0), camera #2’s position could be calculated by camera #1. Let us assume that refers to the horizontal position, and represents the vertical position of a camera. Then

So the position of camera #2 is , with position vertically shifts downward at the point where camera #1’s sight edge intersects that of camera #2. Similarly, we can calculate other cameras’ positions by each moving downward a distance of and horizontal positions switch between and .

Iii-B From the side view

For analyzing from the side view, it is relatively more straightforward, since all visual sensors are located on the top of the space. In order to achieve the maximum coverage, cameras have to be equally spaced so that they can spread area as much as they can, shown in Figure 8.

Fig. 8: Side view

In this case, each camera is separated by a distance of one-fifth of the length horizontally. Combined the analysis of side view and top view, we can assume that both are the optimal placements that depend on real situations.

Iv Conclusion and future work

We have presented an initial study for optimal visual sensor arrangement in a given space. We have simplified the three-dimension problem to be a two-dimension problem. However, the camera models are still ideal, and the space is somewhat simple, such as some variables like the pitch angles of cameras are not considered. Future works will include these variables and make it more useful to the practice.


References

  • [1] E. Horster, and R. Lienhart. On the optimal placement of multiple visual sensors. Proceeding VSSN ’06 Proceedings of the 4th ACM international workshop on Video surveillance and sensor networks, Santa Barbara, California, USA, October 27, 2006, Pages 111-120.
  • [2] T.C. Shermer. Recent Results in Art Galleries. Proceedings of the IEEE, vol. 80, no. 9, 1992, Pages 1384-1399.