An Optimal Edge Detection using Universal Law of

Gravity and Ant Colony Algorithm

Om Prakash Verma

Department of Information Technology

Delhi Technological University

Delhi, India

opverma.dce@gmail.com

Rishabh Sharma

Ericsson India Global Services Pvt. Ltd, Software

Engineer, Noida

UP, India

rishabhs.32@gmail.com

Abstract— An approach for edge detection using ant colony

optimization (ACO) and universal law of gravity is presented

in this paper. The direct application of the edge detector

operator in an image requires a huge search space, therefore

the task of edge detection is time consuming and memory

exhausting without optimization. Ant colony optimization is an

optimization algorithm inspired by the natural behavior of ant

species that ants deposit pheromone on the ground for

foraging. Ant colonies and more generally social insects act as

a distributed system presenting a highly structured social

organization. In the proposed approach the heuristic function

is calculated using law of universal gravity which acts as the

way to a food source for the artificial ants to detect the edge

pixels.

Keywords- Ant colony optimization, Edge, Universal law of

gravity, Pheromone and Probabilistic transition matrix.

I. INTRODUCTION

Edge detection is a fundamental tool used in most image processing applications to obtain information from the image as a precursor step to feature extraction and object segmentation. Physical edges are produced by variation in the reflectance, illumination, orientation, and depth of scene surfaces. Since image intensity is often proportional to scene radiance, physical edges are represented by changes in the intensity function of an image [1].

Several approaches to the edge detection problem have been proposed. The earliest works in this category include the algorithms developed by Sobel [2], Prewitt [3], Kirsch [4], Frei-Chen [5], Canny [6] etc. Wu et al. [7] introduced a fast multilevel fuzzy edge detection algorithm that realizes the fast and accurate detection of the edges from the blurry images. Wenlong Fu et al. [8] proposed a global approach to edge detection using genetic programming (GP) which directly uses an entire image as input and classifies pixels directly as edges or non-edges without preprocessing or post processing. Verma et al. [9] introduced a new approch to edge and corner detection based on fuzzy logic. Recently Verma et al.[10] have developed an algorithm for edge detection using bacterial foraging (BF) in which direction of

movement of bacteria is found using a directional probability matrix derived from ant colony optimization (ACO). Setayesh et al. [11] introduced a new homogeneity-based approach to edge detection using PSO.

In 2007 [12], another edge detection algorithm based on the law of universal gravity was introduced. In this approach, every image point is assumed as a celestial body, which has relationships with other neighboring image points. In this method magnitude and the direction of the vector sum of all gravitational forces of a pixel exerted on by neighboring pixels is used to detect the edges in the image. Direct application of this approach on an image of size 1024 by 1024 pixels requires solution space of the order of 2

1024*1024

which leads to problems such as memory exhaustion and time consuming.

Ant Colony Optimization (ACO), proposed by Dorigo et al. [13] is inspired by foraging behavior of an ant colony. ACO exploits the sensing capabilities of the group and avoids the premature convergence by way of distributed computing but due to slow convergence and inability to tackle large problems, modified ACO approaches have come into picture like that of Ho et al. [14], Verma et al. [15, 16] etc.

In this paper, ACO and law of universal gravity is used to tackle the edge detection problem. Theory of universal gravity is used to calculate the heuristic function which guides the ant towards the most promising solution.

The rest of the paper is organized as the follows. The Ant colony optimization is briefly reviewed in Section II. Edge detection based on the law of universal gravity is briefly reviewed in Section III. Then, the algorithm of the proposed edge detector is presented in Section IV. The experimental results are given in Section V and conclusions are presented in Section VI.

II. ANT COLONY OPTIMIZATION

In ACO, several artificial ants find solutions to an optimization problem and they exchange information on

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their quality via a communication scheme that is reminiscent of the one adopted by the real ants. Individual ants with limited memory are capable of performing simple actions. However, the collective behavior of ants provides intelligent solutions to problems such as finding the shortest path from the nest to the food source. Ants foraging for the food deposit volatile chemical substance, named pheromone, marking their path of search. The ants thus accomplish an indirect communication through the pheromone which is accumulative but also evaporative to achieve the final goal cooperatively. The ants take the path where the pheromone concertration is large. The first ACO algorithm, called the ant system, was proposed by Dorigo et al. [13]. Since then, a number of ACO algorithms have been developed, such as the ant colony system [17] and the Max-Min ant system [18]. ACO aims to iteratively find the optimal solution of the target problem through the movements of a number of ants over the solution space, by constructing the pheromone information.

The algorithm for ACO Let the total of K ants be en-thrusted to find the optimal

solution in the solution space that consists of M1× M2nodes.

1) Initialize the positions of K ants, as well as the pheromone matrix (0).

2) For the construction-step index n = 1: N, for the ant index k = 1: K, move the kth ant for L steps, according to a

probabilistic transition matrix ( )

,

n

i jP given by [17]

, ,( )

,

, ,

( ) ( )

( ) ( )j i

n

i j i jn

i j n

i j i j

P

α β

α β

τ η

τ η∈Ω

Σ

= (1)

where ,

n

i jτ is the pheromone value of the arc linking the

node i to the node j, iΩ is the set of neighborhood nodes of

ant k given that it is on the node I,,i jη represents the

heuristic information in going from node i to node j (which is fixed for each construction-step), α represent the influence

of the pheromone and β represent the influence of heuristic

information,

3) Update the pheromone matrix ,

n

i jτ after the movement

of each ant in every construction step using [17]

( 1), ,

( 1),

,

(1 ). . ( , ){n k

i j i j

ni j

n

i j

if i j belongstothebesttour

otherwise

ρ τ ρ

ττ

−

−

− + Δ= (2)

where ρ is the evaporation rate, and ,

k

i jΔ is the amount

of pheromone deposited by the kth ant. The determination of best tour is subject to the user-defined criterion; it could be either the best tour found in the current construction-step, or the best solution found since the start of the algorithm, or a combination of both.

4) Make the decision according to the final pheromone

matrix nτ after the movement of all K ants within each

construction-step and update the pheromone matrix by [17]

( 1)(1 ). (0)n nτ ψ τ ψτ−= − + (3)

where ψ is the pheromone decay coefficient.

III. EDGE DETECTION BASED ON LAW OF

UNIVERSAL GRAVITY

In the proposed method the edge detection technique proposed in [12] using the concept of universal gravity is used as heuristic information for the movement of ants. The algorithm assumes that each image pixel is a celestial body with a mass represented by its grayscale intensity. Accordingly, each celestial body exerts forces onto its neighboring pixels and in return receives forces from the neighboring pixels. These forces can be calculated by the law of universal gravity. The vector sums of all gravitational forces along, respectively, the horizontal and the vertical directions are used to compute the magnitude and the direction of signal variations. Edges are characterized by high magnitude of gravitational forces along a particular direction and can therefore be detected. The edge detection technique of [12] is briefly described below.

1). For each image point g(i, j), consider an m×nneighborhood Ω with pixels ( , ) & ( , ) ( , )k l k l i j∈Ω ≠ . For

each point, the gravitational force of the point exerts on its neighboring pixels is computed using

, ,

, ; , 3|| ||

i j k l

i j k l

Gm m rf

r= (4)

where , ; ,i j k l

f is the force of gravity exerted by neighboring

pixels g(k, l) on g(i, j), ,i jm , ,k lm are the gray values of the

pixel g(i, j) and g(k, l), r is the distance vector from g(k, l)to g(i, j) and its magnitude is

2 2|| || ( ) ( )r k i l j= − + − (5)

Fig. 1. Basic edge structure oriented in 90 degree directions

[12].

508 2011 World Congress on Information and Communication Technologies

2) Vector , ; ,i j k l

f can be calculated by calculating the

forces in x and y direction using the following formula

, ; , , ; ,, ; ,

x y

i j k l i j k li j k lf f x f y= + (6)

The vector sum of all gravitational forces which the neighborhoods exert on g(i, j) is expressed as

, , ; ,

( , ) & ( , ) ( , )

x yi j i j k l

F f F x F y

k l k l i j

= = +

∈Ω ≠

(7)

3) The edge strength response of point g(i, j) is produced

by the magnitude of the vector ,i jF ,its edge direction is

given by the direction of ,i jF

2 2|| || ( ) ( ) ,

arctan( / ),

x y

x y

F F F

F Fθ

= +

=

(8)

where || ||F is the magnitude of the vector sum ,i jF

and θ is the direction of the vector sum ,i jF .

4) By applying an appropriate threshold an edge map is produced.

IV. THE PROPOSED IMAGE EDGE DETECTION

APPROACH

The proposed ACO-based image edge detection approach aims to utilize a number of ants to move on a 2-D image for constructing a pheromone matrix, each entry of which represents the edge information at each pixel location of the image.

Steps of the proposed approach 1) Initialize the positions of K ants by randomly

distributing them on an image I with a size of M1 ×M2.Pheromone matrix (0) is initialized by a constant initial value.

2) At the n-th construction-step, one ant is randomly selected from the above-mentioned total K ants, and this ant will consecutively move on the image for L movement-steps. This ant moves from the node (i0,j0) to its neighboring node (i, j) according to a transition probability that is defined as

0

( , )0 0

( 1 )

, ,( )

( , ),( , ) ( 1 )

( , ) , ,

( ) ( )

( ) ( )o

i j

n

i j i jn

i j i j n

i j i j i j

P

α β

α β

τ η

τ η

−

−∈Ω

=Σ

(9)

Where ( 1)

,

n

i jτ − is the pheromone value of the node (i, j),

( , )0 0i jΩ is the neighborhood nodes of the node (i0 ,j0),

,i jη represents the heuristic information at the node (i, j).

The constants α and β represent the influence of the

pheromone matrix and the heuristic matrix, respectively.

The value of a heuristic function at a given node in the search process gives a good estimate of that node being on the desired path to solution. The purpose of a heuristic

function is to guide the search process in the most profitable directions, by suggesting which path to follow first when more than one path is available. In this approach we calculate this function using theory of Universal Gravity

Determination of heuristic function using theory of Universal Gravity

In the proposed approach the heuristic function is calculated using the theory of Universal Gravity. Total force exerted by neighboring pixels on the central pixel is calculated by the vector sum of the forces exerted individually on the central pixel by the neighboring pixels. The magnitude of the total force is calculated using Eq. 8 and is used as heuristic function .

, || ||i j Fη = (10)

3) After the movement of each ant in every construction

step, the pheromone matrix is updated using Eq. 2 4) After the movement of all ants within each

construction-step pheromone matrix ,

n

i jτ is updated using

Eq. 3

V. EXPERIMENTAL RESULTS

The computer simulations are performed using MATLAB 7.10.The Experiments are conducted to evaluate the

performance of the proposed approach using two test

images. The Fig. 2 shows the original Cameraman and Lena

images. results of the proposed method. The results of the

proposed approach are compared Sobel edge detector and

edge detector based on the universal law of gravity [12]

(Fig. 3-4). The threshold value for the standard Sobel

operator is selected as default value which gives the ideal

edge map. Although the number of edges found using the

Sobel operator and method present in [12] is more than that

of the proposed approach, the proposed approach is much

faster, less time consuming and requires less memory. The

proposed approach is more focused on the optimization

concerns of the edge detection problem. Furthermore,

various parameters of the proposed approach are set as

follows [15]:

• K = : the total number of ants. • init = 0.0001: the initial value of each component of the pheromone matrix. • α =2: the weighting factor of the pheromone information.

• β = 0.1: the weighting factor of the heuristic information.

• Ω =8-connectivity neighborhood. • = 1: the adjusting factor of the functions. • ρ = 0.05: the evaporation rate.

• L = 40: total number of ant’s movement-steps within each construction-step. • ψ = 1.5: the pheromone decay coefficient.

• e =0.1: the user-defined tolerance value used in the decision process of the proposed method.

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• N = 4: total number of construction-steps.

VI. CONCLUSIONS

In this paper, a new ACO and universal law of gravitational

based image edge detection approach has been successfully

developed. The proposed approach uses the universal

gravitational law to find the heuristic information for the

movement of ant. The proposed approach leads to less time

consumption and less memory usage as compared to edge

detection algorithm based on theory of law universal gravity

[12].

REFERENCES

[1] D.Ziou, S.Tabbone, “Edge Detection Techniques – An Overview”, International Journal on Pattern Recognition and Image Analysis, Vol. 8, pp. 537-559, 1998.

[2] SOBEL, I., “Camera Models and Perception”, Ph.D. thesis, Stanford University, Stanford, CA, 1970.

[3] J.M.S. Prewitt, “Object enhancement and extraction”, in: B.S. Lipkin, A. Rosenfeld (Eds.), Picture Analysis and Psychopictorics, Academic Press, New York, NY, pp. 75-149, 1970.

[4] Kirsch, R., “Computer determination of the constituent structure of biological images”, Computers and Biomedical Research, Vol. 4, pp. 315-328, 1971.

[5] Frei and Chen, “Fast boundary detection: A generalization and a new algorithm”, IEEE Trans. Computers, Vol. 26, pp. 988-998, 1977.

[6] J. Canny, “Computational Approach to Edge Detection”, IEEE Transaction on Pattern Analysis and Machine intelligence, No. 6, pp. 679-698, 1986.

[7] J. Wu, Z. Yin, and Y. Xiong, “The Fast Multilevel Fuzzy Edge Detection of Blurry Images”, IEEE Signal Processing Letters, Vol. 14, No. 5, pp. 344-347,2007.

[8] Wenlong Fu, Mark Johnston and Mengjie Zhang,"Genetic Programming for Edge Detection: A Global Approach", Proceeding

of the 2011 IEEE Congress on Evolutionary Computation. IEEE Press. New Orleans, USA, pp. 254-261, 2011.

[9] Hanmandlu M., Verma O.P., Gangwar P., and Vasikarla S., "Fuzzy Edge and Corner Detector for Color Images", in Proc. ITNG, pp.1301-1306, 2009.

[10] Verma O.P., Hanmandlu M., Kumar P., and Jindal A, “A Novel Bacterial Foraging Technique for Edge Detection”, Pattern Recognition Letters, No.8, pp.1187-1196, 2011.

[11] Setayesh M., Zang M. and Jonhnton M., “A new homogeneity-based approach to edge detection using PSO”, 24th International Conference Image and Vision Computing, New Zealand (IVCNZ), pp. 231-236, 2009.

[12] Genyun Suna, Qinhuo Liua, Qiang Liua, Changyuan Jib, Xiaowen Lia ,“A novel approach for edge detection based on the theory of universal gravity” Pattern Recognition, Vol. 40(10), pp . 2766-2775, 2007.

[13] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: Optimization by a colony of cooperating agents”, IEEE Transactions on System, Man, and Cybernetics, Part B, Vol.26, pp. 29-41, 1996.

[14] Ho, S.L., Shiyou Yang, Guangzheng Ni, Wong, H.C., “ A Modified Ant Colony Optimization Algorithm Modeled on Tabu Search Methods”, IEEE Transactions on Magnetics, Vol. 42, pp.1195-1198, 2006.

[15] Verma O.P., Hanmandlu M., Kumar P., Srivashtav S., “A novel approach for edge detection using ant colony optimization and fuzzy derivative technique” ,Advance Computing Conference, 2009.IEEE international, pp. 1206 – 1212 , 2009.

[16] Verma, O.P., Hanmandlu. M, Sultania A.K.,Dhruv, “A Novel Fuzzy Ant System for Edge Detection” IEEE/ACIS 9th International Conference on Computer and Information Science, pp. 228 – 233, 2010.

[17] M. Dorigo and L. M. Gambardella, “Ant colony system: A cooperative learning approach to the traveling salesman problem,” IEEE Trans. On Evolutionary Computation, Vol. 1, pp. 53–66, 1997.

[18] T. Stutzle and H. Holger H, “Max-Min ant system,” Future Generation Computer Systems, Vol. 16, pp. 889–914 ,2000.

(a) (b)

Fig 2. Original Image: (a) Cameraman (b) Lena

510 2011 World Congress on Information and Communication Technologies

(a) (b) (c) Fig. 3. (a) Sobel detector (threshold: default) (b) edge detection based on the theory of universal gravity and (c) Resultant Image of Proposed method (threshold: 0.9)

(a) (b) (c) Fig.4. (a) Sobel detector (threshold: default) (b) edge detection based on the theory of universal gravity and (c) Resultant Image of Proposed method (threshold: 0.4)

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