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  • Abstract

Application of Halftoning Algorithms to Location Dependent Sensor Placement

We consider a sensor network placement problem where the sensing range of a sensor depends on its location in order to model the effect of terrain features. We study how sensors should be placed in order to maximize the coverage and illustrate how digital halftoning algorithms from the field of image processing can be useful in this respect. In particular, we reduce the sensor placement problem to a corresponding image halftoning problem and then apply two well known halftoning algorithms to the problem: dither mask halftoning and direct binary search. We illustrate our approach with experimental results and show that this approach is also applicable to the problem of preferential coverage.

SECTION I

Introduction

AD-HOC sensor networks is an important area of study in both military and civilian applications and has generated many challenging problems. One such problem is to determine how sensors are to be positioned to maximize coverage [1], [2], [3], [4], [5]. However, relatively little attention is being paid to how the terrain can affect the sensor's effectiveness and how this can in turn affect the placement of sensors. In [6] sensor placement algorithms are proposed that take into account the presence of obstacles in a scenario where the targets are on grid points. In [7] a model of terrain dependence is introduced where each sensor has a circular coverage region and the radius depends on the location of the sensor center. A coverage algorithm is studied such that in areas where the radius is small, a denser grid deployment is used. In this paper we continue studying this model of terrain dependence and utilize tools from image processing to address this problem. In particular, we use image halftoning algorithms to determine where the sensors should be placed. The link between sensor placement and image halftoning was first discussed in [5].

SECTION II

Terrain Effects in Sensor Deployment

The basic formulation of the problem is as follows. Given a 2-dimensional region R, sensors are placed within this region. In the sequel, we choose R to be a rectangular region. Each sensor centered at location x has a coverage region. Because of different aspects of the terrain, such as elevation, obstructions such as building and mountains, etc., the coverage region of a sensor will change depending on where it is placed. We will use the simple model introduced in [7] where the coverage region is a circular region whose radius depends on the location x (Fig. 1). The goal is to place sensors in the 2-dimensional region (i.e., find a covering) that maximizes the coverage. We can describe the location dependent coverage region radius with the following coverage radius map: Formula where f (x) is the radius of the coverage region of a sensor centered at position x.

Figure 1
Fig. 1. Sensors whose coverage region depends on its location.

In [7] the region is covered by placing the sensors on a square grid and the grid spacing is reduced, i.e., the sensors are deployed more densely, for regions where the corresponding coverage region is small. This method is less effective for cases where the terrain (i.e., the function f) varies significantly across the region. The purpose of this paper is to present placement algorithms derived from image halftoning that can provide a more optimal sensor deployment strategy.

SECTION III

Sensor Placement and Image Halftoning

Digital halftoning is the art of positioning ink drops (or other types of colorants) of a few colors on paper (or displays) such that the resulting image appears to consist of many more colors when viewed at an appropriate distance [8]. This is necessary in digital printers since they use only a few colors. An example of halftoning a grayscale image is shown in Fig. 2.

Figure 2
Fig. 2. (a) A continuous tone image (b) The corresponding halftone image.

As can be seen in Fig. 2, darker regions result in denser arrangement of dots such that they appear as a darker shade when viewed from a distance. One of the features of halftoning algorithms is that it can handle abrupt changes in intensity among the image pixels (caused by e.g., edges) well, i.e., the halftone image can still render details in the original image.

Let us consider the sensor placement problem. For regions that results in sensors having a small coverage region, it is intuitive that sensors should be deployed more densely. This corresponds to the same effect in halftoned images discussed above. In other words, if we relate coverage area radius with the intensity of an image pixel, then the terrain can be considered as an image and we can apply digital halftoning solutions to the sensor placement problem. One constraint here is that the digital halftoning algorithm assumes that the pixels live on a grid which in most cases is rectangular. However, by choosing the grid fine enough, this approximates the requirement that the sensors can be arbitrarily placed. Furthermore, there are scenarios where the sensors are deployed on prescribed locations on a grid.

In digital halftoning, the density of black dots is proportional to the graylevel of the input image pixels. Whereas in the sensor placement problem the density of sensors is inversely proportional to the coverage region area. Thus the first step is to transform the coverage region radius map f into a corresponding image.

The general framework for sensor deployment using digital halftoning is as follows:

  1. Generate image T, such that at each pixel, the graylevel T(p) is proportional to f (p)−2.

  2. Apply halftoning algorithm to image T.

  3. Deploy a sensor at each grid location where the halftone image has a halftone dot.

We will denote the map which maps f (p) to T(p) as Φ. In the next 2 sections we will consider two well-known halftoning algorithms: dither mask halftoning and direct binary search.

SECTION IV

Stochastic Dither Mask

One of the fastest and simplest image halftoning algorithm is the dither mask algorithm. Given an continuous-tone image I, this algorithm uses a dither mask A, tiles the image with this mask and prints a dot at pixel p if I(p) ≥ A(p). Both I and A are sets of pixel values, represented as a matrix of real numbers (or more commonly 8-bit integers). The type of dither mask which generates dispersed halftone patterns such as in Fig. 2 are called stochastic or blue noise dither masks [9]. Even though the method is extremely fast, each location makes its decision on whether to deploy a sensor there or not without regard of what neighboring locations are doing. Thus the algorithm does not take into account the relationship between neighboring pixels and is not optimal in resolving edges and abrupt spatial changes in intensity. However, for relatively gradual spatial changes, the performance is quite adequate.

Let us assume that the grayscale image in Fig. 3 is a representation of a terrain as generated by the map Φ in Section III. The graylevel of each pixel is proportional to the coverage region radius with a lighter area corresponding to sensors having a larger sensing radius. Applying the above framework using a stochastic dither mask results in the deployment shown in Fig. 4. We allow sensors to be placed on a grid that is 4 times finer in both the horizontal and vertical direction than the pixels in Fig. 3. A total of 339 sensors is used. We see that it places more sensors in areas where the sensing radius is small. However, because of the lack of communication between sensors in the dither mask method, there are areas with significant gaps and areas with significant overlaps, especially in areas where there are sharp transitions between light and dark pixels. This corresponds to terrain features where the sensing region of the sensors changes dramatically. Another issue of this approach is that the number of sensors used is not predetermined.

Figure 3
Fig. 3. Terrain expressed as a grayscale image. Sensors positioned in lighter areas have a larger sensing radius.
Figure 4
Fig. 4. Sensor deployment based on the stochastic dither mask halftoning algorithm.
SECTION V

Direct Binary Search

Direct binary search (DBS) [10] and related methods [11] are iterative algorithms that produce some of the best halftones. It is however time-consuming and requires several passes through the entire image. DBS can be considered as a heuristic to solve the integer programming problem of finding the best halftone that approximates the original continuous-tone image. It operates by comparing pairs of pixels and determine if swapping produces a better halftone. It also toggles pixels to check if that generates a better halftone. It measures the goodness of a halftone by comparing low-pass filtered versions of the halftone image and the original image. Because of the interaction between neighboring pixels, it produces better halftones than the dither mask method, especially if the image has sharp edges and variations. Applying this algorithm to the terrain in Fig. 3 results in Fig. 5. Again the grid is 4 times finer in each direction and the total number of sensors is the same as in Fig. 4. We see that the sensor placement is better than in the dither mask case, especially in the transition areas.

Figure 5
Fig. 5. Sensor deployment based on the DBS halftoning algorithm.

A. Fixing the Number of Sensors

One feature of DBS is that you can control the number of sensors. By starting with an initial halftone with a specified number of black pixels and utilizing only the swapping operation, the number of black pixels will remain fixed in the final halftone, i.e., the number of sensors deployed can be fixed. Applying this algorithm to the terrain in Fig. 3, but limiting the number of sensors to be 300 and 400 respectively, we obtain the deployments shown in Fig. 6.

Figure 6
Fig. 6. Sensor deployment based on the DBS halftoning algorithm and restricting the number of sensors to be (a) 300 and (b) 400.
SECTION VI

Preferential Coverage

Ref. [6] also considers preferential coverage where some target points should be given more attention. This can also be taken into account in our approach by modifying the image that is to be halftoned, i.e., by modifying the map Φ which maps the radius f(p) to the graylevel T(p) of the image to be halftoned. For instance, instead of setting T(p) = α f(p)−2 as before, we can add a (multiplicative) term to take the preferential coverage into account. For instance, we can set T(p) = α f(p)−2 c(p) where c(p) describes the importance of sensing pixel p, i.e., c(p) is larger the more important sensing p is. For instance, consider the case where the left half of the terrain c(p) = 1 and the right half of the terrain c(p) = c2 < 1 and we apply the DBS algorithm with 300 sensors. The result is shown in Fig. 7. One can see that the left half of the terrain is covered more densely than the right half. Other forms of Φ are possible; we can add additive terms or change the exponent −2 to deal with different kinds of preferential coverage.

Figure 7
Fig. 7. Sensor deployment based on the DBS halftoning algorithm and restricting the number of sensors to be 300, with a modified Φ to take into account preferential coverage to the left half of the terrain.
SECTION VII

Conclusion

We continue the study of problem of deploying sensors where the performance of the sensor is location dependent. We approach this problem from an image processing perspective and apply digital image halftoning algorithms to the problem. In particular, we show how stochastic dither mask methods and direct binary search halftoning methods can be useful in this regard.

Acknowledgment

This research was sponsored by US Army Research laboratory and the UK Ministry of Defence and was accomplished under Agreement Number W911NF-06-3-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the US Army Research Laboratory, the US Government, the UK Ministry of Defence, or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.

Footnotes

Dinesh Chandra Verma and Chai Wah Wu are with the IBM T. J. Watson Research Center, 19 Skyline Drive, Hawthorne, NY 10532, USA, email: chaiwahwu@ieee.org.

Theodore Brown, Amotz Bar-Noy, and Simon Shamoun are with the City University of New York Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA.

Mark Nixon is with the School of Electronics and Computer Science, University of Southampton, Southampton, SO171BJ, UK.

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Authors

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Dinesh Chandra Verma

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Chai Wah Wu

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Theodore Brown

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Amotz Bar-Noy

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Simon Shamoun

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Mark Nixon

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iterative methods, image processing, image sensors

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