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As the use of camera networks has expanded, camera placement to satisfy some quality assurance parameters (such as a good coverage ratio, an acceptable resolution constraints, an acceptable cost as low as possible,

Camera networks are used in many novel applications, such as video surveillance [

The camera network deployment problem can be defined as how to place the cameras in the appropriate places to maximize the coverage of the camera network under some constraints. The constraints can be categorized into three main types: task constraints, camera constraints and scene constraints. The task constraints include continuous tracking (enough overlap between cameras), people identification (image resolution and focus), complete coverage of the surveillance area (field of view of each camera) and so on. The camera constraints include the camera network type (a homogeneous or heterogeneous camera network), camera type (PTZ or static camera), camera intrinsic parameters (focus length, CCD size,

Since the camera network placement problem is a NP-hard combinatorial optimization problem [

There are some weaknesses in these results, mainly due to the overly simple assumptions used. From the perspective of the three constraints mentioned above, we give a brief explanation of the limitations of the earlier works. From the perspective of task and camera constraints, most of the works only consider the coverage of the area while the video resolution and focus are seldom considered; From the perspective of scene constraints, most of the scenes are modeled as a 2D case which is too simple to conduct the real camera network placement, or modeled as a 3D case which is too restrictive because in most of the cases we are only concerned with the surveillance plane area.

We give several examples. The surveillance area of [

In this paper, we consider the deployment of homogeneous camera network in the 3D space to surveil a 2D ground plane. For simplicity considerations, the surveillance plane is modeled as a rectangle area which is not essential to our work. We separate the surveillance plane area into

We take a more synthetic constraints set, including the surveillance video resolution, video focus, the camera field of view

The main contributions of this paper can be summarized as follows:

We consider a more realistic problem in that we deploy the cameras in a 3D space to surveil a plane area. Some of the previous works consider the problem in a 2D plane and the FOV of the camera is modeled as a sector which is too simple an assumption, while some works consider the problem in the 3D space and model the FOV of the camera as a cone which is too restrictive an assumption. We can get a more accurate result to solve the camera deployment problem in the 3D space to surveillance of a 2D plane and instruct the real life camera network placement;

We take more constraints into consideration than others, including resolution, focus, FOV. Most of the previous works only consider the FOV of the camera to get a good coverage while the other constraints are important to get a good surveillance video;

We propose a probability-inspired PSO algorithm to solve the camera network placement problem heuristically. In the algorithm, we introduce a regulated item in the fitness function to optimize the coverage ratio and the number of the cameras simultaneously. The experimental results show the effectiveness of the algorithm.

The rest of the paper is organized as follows: we review recent progress in camera network deployment in Section 2. In Section 3, we give the camera network placement problem from three perspectives. In Section 4, we propose a PI-BPSO algorithm and discuss the representation and fitness computation of particles in the algorithm. Simulation results are given in Section 5. We give the conclusions and discuss some future work in the last section.

In Computational Geometry, there is a well-known problem called the Art Gallery problem (AGP) [

The other interesting research problem related to the camera network placement problem is the wireless sensor network (WSN) placement problem [

Researches on the camera network placement problem can be divided into two disciplines, one is the coverage problem and the other is the optimization problem. The two problems can be related through the optimization framework issued by Zhao

The visual coverage of the camera network describes what can be seen and what can't in the surveillance area. It is so fundamental to many computer vision tasks that different works have suggested different visual coverage models according to the different surveillance tasks.

The standard coverage model is defined as the area of the surveillance area. It can be classified as two cases, from the perspective of the area and the camera. From the perspective of the area [

As we have stated that the camera network placement problem is NP-hard, researchers have put forward various approximate optimization algorithms to solve the problem [

We put forward an optimal camera network placement problem to satisfy the need of different surveillance tasks on a specific surveillance area. The problem can be modeled as a multi objective optimization problem that must satisfy multiple constraints. In this work, we are interested in the static camera network placement problem, where the objective is to determine the number of cameras, their positions and poses for an rectangle surveillance area, given the intrinsic parameters of the cameras (such as focal length, the diameter of the lens's aperture

In the real world circumstances, we often layout some linked cameras to surveillance a square which is the main subject of this article. We establish a world frame as that: we model the surveillance square as the

We assume that the various cameras used in the layout share the same intrinsic parameters, such as the resolution of the camera is _{h}_{v}

We use a pyramid to represent the camera's FOV, which is shown in _{C}_{C}_{C}

For the surveillance application, we desire that the angle between the object and the direction of the camera is less than a constant angle _{0} (which is critical for feature point extraction and the other applications), such as 60°. We assume that the object in the surveillance video is vertical to the ground (which is almost true because Pisa tower is seldom), then we get a constraints on the (_{0}. As _{i}_{i}

If we know that a camera's configuration (

In this section we describe the relationship between the surveillance video resolution constraints and the camera's position. For a specific surveillance camera, the required resolution provides an upper bound on the distance between the camera and the surveillance area.

The surveillance video resolution of object

The distance between the position of the camera len's center and the surveillance area

In this section, we determine the constraints on the camera's viewpoints when we request that all the points of the surveillance area must be sharp (in focus) in some surveillance camera's FOV. For any camera, there is only one plane on which the camera can precisely focus, and the point object in any other plane is imaged as a disk (known as the blur spot) rather than a point. When the diameter of blur spot is sufficiently small, the image disk is indistinguishable from a point. The diameter of the blur spot is known as the acceptable circle of confusion, or simply as the circle of confusion (

Here we assume the _{Max}_{Min}

If we want the video to be sharp, then:

Then we have that the height of the camera must satisfy (illustrated in

In theory, cameras can be located anywhere in the space since the camera position variables _{C}_{C}_{C}_{x}_{y}_{z}_{φ}, f_{θ}_{ψ}

In the real situation, there are some other different constraints on the allowed deployable area such as the constraints that the cameras must be located on the walls of the indoor surveillance environment or the constraints that the cameras must be located at some restricted height for the consideration of their management, but in this article we only restrict the position of the cameras according to the constraints of surveillance video resolution, FOV (field of view) and DOF (depth of field), which are described in the previous section.

We can model the camera network placement as an optimization problem which is defined as maximization of some utility function given some cameras and task constraints. From the definition of the model, we can analyze the problem from four perspectives: the utility function (or cost function), the task constraints, the space constraints and the camera's parameters. Let _{T}

Let the vector ^{I}^{E}^{T}_{i}

In this work we are mainly interested in the 2D plane surveillance by a camera network located in a 3D environment. In this special surveillance scene, we want to place the minimum number of cameras in the optimal locations (position and pose) to maximize the visual coverage under the surveillance task constraints. Based on the assumptions we make above, we give the following specific model to solve this visual sensor placement.

Suppose that we have _{i}_{C}_{i}_{S}_{i}_{T}_{i}_{T}

We define a binary variable {_{i}_{,}_{j}_{S},j_{C}

_{i}_{,}_{j}_{j}_{i}

_{i}_{,}_{j}

We define a binary variable {_{i}_{,}_{j}_{T},j_{C}

_{i}_{,}_{j}_{j}_{i}

_{i}_{,}_{j}

Based on the notations we define above, we define the utility function

When the number of grids in the surveillance area is too large, then we can apply a sample method to determine the coverage. When we choose

_{C}

Based on the augment above, we turn the camera network placement problem to an optimization problem as follows:

Satisfies:

_{0}.

Evolutionary computation technique, motivated by the evolution of Nature, is a powerful tool to approximately solve many NP-hard problems. There are many kind of evolution computing methods, such as genetic algorithm (GA) [

The PSO algorithm consists of a population of agents called particles, each of which is a potential solution to the optimization problem. The particle ^{t}_{1} and _{2} are the weights of the personal influence and the society influence,

Despite the different form of the various PSO algorithms that can be used, the main task during the solving process is to determine the representation of the particles and the fitness of each particle. That is to say, to solve the camera network placement with the PSO approach, we should build a mapping between the camera network deployment solution to the particle state and calculate the fitness of each particle's state. Because we represent the solution space in a discrete space, we must have a mechanism to transform the new position of the particle to a legal place. Like any optimization method, the initialization is very important for the convergence speed and solution quality. We will illustrate the important factors above in the next sections.

In this section, we describe the state representation of the camera network placement problem. Each of the PSO particles is represented as a _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Next we show how to represent a camera network placement problem as an example. Suppose that we deploy four cameras to surveil a rectangular area and the cameras can be placed in a
_{z}_{ψ}

^{3} + 0 × 2^{2}+ 0 × 2^{1} + 1 × 2^{0} = 1 of 1 in the particle space, and the other camera state can also be transformed to the position of 1 in the particle state.

The fitness of each particle is calculated according to the cost function

Then we can get the fitness of the camera network configuration.

The flows of most PSO algorithms are very similar, that is a four step iterations of “initialization, evaluate, update, stop”. The main difference between them is the strategy of new particles generation and the update strategy. In this work, we propose a probability induced binary particle swarm optimization algorithm (PI-BPSO) which is a extension of the algorithm propose in [_{ij}_{ij}

^{t}

First we calculate the probability

Secondly, we generate two uniformly distributed random numbers

Thirdly, we determine the velocity at time

Fourthly, we determine the new state of the particle
_{ij}

In the algorithm aspect, the main difference between our algorithm and [

There is another key difference between the two methods: PI-BPSO uses a regulation item in the cost function to determine the appropriate number of cameras while BPSO-PI [

Most of the constrained optimization problems solved by PSO [

For the problems in this article, we apply the following strategy:

During the initialization stage, all particles are started with feasible solutions;

During the updating stage, only the feasible particles are kept in their memories.

To evaluate the effectiveness and reliability of the proposed algorithm, we adopted the following experiment configuration:

The surveillance area is a 50 m × 50 m square;

The camera intrinsic parameter is

The minimum resolution requirement is

The CCD size is 1/4^{″} (3.2 mm × 2.4 mm) and the resolution of the camera is 1,024 × 768;

The Euler angle of the camera is restricted in the following range:

We do two kinds of experiments. One kind of experiment is to determine the influence of the different sampling frequencies on the solution of the camera network deployment problem; the second kind of the experiment is to determine the influence of the

We present some results obtained by the algorithms (_{x}_{y}_{φ}_{ψ}_{z}_{θ}_{x}_{y}_{z}_{φ}_{ψ}_{θ}

In the utility function, we set a regulization item which is a penalty for the camera number. We do an experiment to determine the effect of the _{x}_{y}_{z}_{φ}_{ψ}_{θ}

In

We do an experiment to compare PI-BPSO with BPSO-IP, GA and ABC from three perspectives: iteration times, fitness and computation time. We repeat the experiment 50 times and the value of the iteration times, fitness and computation time are the average values of the 50 tests. Because the codes of the other methods are all unlnown, the experimental results may not be same as the original author stated. We show the result in

We do an experiment with real cameras in a hall building with the simulation result that we have obtained from the simulation. The surveillance area is a basketball court and a man is roller skating. We determine the position and pose of the surveillence cameras using the PI-BPSO algorithm. The experimental result shows the effect of the algorithm is that the video is in focus and the people in the video can be identified easily. We show the result in

In this paper, we discuss the automatic camera network placement problem which is solved by an evolution-like method, PI-BPSO. The different simulation results show the effectiveness of the proposed algorithm. The algorithm is guaranteed to get a global optimum with high probability due to two reasons:

The initialization and the update process are both determined randomly which assures that we will now land in the local minimum with high probability;

The introduction of the regulation item eases the optimization process and allows us to optimize the number of the cameras and the configuration of the cameras at the same time.

In the future, we will continue our study on the camera network placement problem from several directions, such as a more accurate initialization or integration of the PSO and other optimization methods to speed up the convergence.

This work is supported by the National Natural Science Foundation of China under Grants 61225008 and 61020106004, and by the Ministry of Education of China under Grant 20120002110033. This work is also supported by Tsinghua University Initiative Scientific Research Program.

The authors declare no conflict of interest.

The surveillance plane area is divided into

The coordinate frame we establish to model the surveillance scene where the ground plane (the surveillance area) is modeled as the

The FOV of the camera in 3D space and the camera's yaw, pitch, roll angles which is shown as (_{camera}_{world_t(C)}

Camera and spatial resolution for the camera.

DOF for the camera.

The feasible height of the cameras. We can see that the point _{Min}Cosθ_{0} ≤ _{Max}_{Min}Cosθ_{0} ≤ _{Max}_{0} is the biggest feasible angle between the direction of the camera and the object which is discussed above.

A particle state representation for a camera network placement example where we can see that the number of 1 in the particle state is 4.

A camera network placement strategy for a particle state which is a inverse process of the particle representation.

PI-BPSO flowchart. We should point out that we should calculate both the personal best and global best of the particles

Optimal placement of cameras (six cameras). The surveillance area is 10 m × 10 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Optimal placement of cameras (24 cameras). The surveillance area is 50 m × 50 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Optimal placement of cameras (16 cameras). The surveillance area is 50 m × 50 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Optimal placement of cameras (14 cameras). The ssurveillance area is 50 m × 50 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Optimal placement of cameras (18 cameras). The surveillance area is 50 m × 50 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

Optimal placement of cameras (20 cameras). The surveillance area is 50 m × 50 m, _{x}_{y}_{z}_{φ}_{θ}_{ψ}

The comparison result of different optimization method. (

Some images from the real surveillence video to validate the effectiveness of the algorithm. (