[intelligent systems reference library] handbook of optimization volume 38 || circle detection...

I. Zelinka et al. (Eds.): Handbook of Optimization, ISRL 38, pp. 907–934. springerlink.com © Springer-Verlag Berlin Heidelberg 2013

Circle Detection Algorithm Based on Electromagnetism-Like Optimization

Erik Cuevas, Diego Oliva, Daniel Zaldivar, Marco Pérez, and Raúl Rojas*

Abstract. Optimization approaches, inspired by different metaphors, have recent-ly attracted the interest of the scientist community. On the other hand, circle detec-tion over digital images has received considerable attention from the computer vision community over the last few years as tremendous efforts have been directed towards seeking for an optimal detector. This chapter presents an algorithm for the automatic detection of circular shapes embedded into cluttered and noisy images with no consideration of conventional Hough transform techniques. The approach is based on a physics-inspired technique known as the Electromagnetism-like Op-timization (EMO). It follows the Electromagnetism principle regarding a attrac-tion-repulsion mechanism which manages particles towards an optimal solution. Each particle represents a solution by holding a charge which is related to the ob-jective function to be optimized. The algorithm uses the encoding of three non-collinear points embedded into the edge map as candidate circles. Guided by the values of the objective function, the set of encoded candidate circles (charged par-ticles) are evolved using the EMO algorithm so that they can fit into actual circu-lar shapes over the edge map. Experimental evidence from several tests on synthetic and natural images which provide a varying range of complexity validates the efficiency of our approach regarding accuracy, speed and robustness.

Erik Cuevas · Diego Oliva · Daniel Zaldivar · Marco Pérez Departamento de Ciencias Computacionales Universidad de Guadalajara, CUCEI Av. Revolución 1500, Guadalajara, Jal, México e-mail: {erik.cuevas,diego.oliva,daniel.zaldivar, marco.perez}@cucei.udg.mx Raúl Rojas Freie Universität Berlin, Institut für Informatik Arnimallee 7, 14195, Berlin, Germany e-mail: [email protected]

908 E. Cuevas et al.

1 Introduction

The application of some biology concepts to solve demanding problems by assum-ing that natural world experiences may hold some answers to real-life technical challenges [16], this field then is called Nature-inspired computing [23]. Engi-neering is being challenged everyday by more complex, large, ill-structured and distributed systems yielding a renovated interest on the subject. However, nature is providing simple structures and organizations which are capable of dealing with most complex systems and tasks. Collective intelligence is a concept commonly used by many nature-inspired approaches [14,26,39]; by the way this concept has lately been developed and applied to problems holding a complex behavioral pat-tern [13,41]. The approach follows the idea that a system is composed of decentra-lized individuals that may effectively interact with other elements according to their localized knowledge. Therefore, the overall image of the system emerges from aggregating individual interactions. Special kinds of artificial collective-individuals are the elements created by analogy with bees [20], ants [7] or charged particles [30]. Nature-inspired algorithms with collective intelligence characteris-tics have recently gained considerable interest from the computer vision communi-ty as they effectively add interesting features from other subjects such as optimiza-tion, pattern recognition, shape detection and machine learning.

An important problem for image analysis is the detection of circular shapes, in particular for industrial applications such as automatic inspection of manufactured products and components, aided vectorization of drawings, target detection, etc.[5,38]. Two sorts of techniques are commonly applied to solve the object loca-tion challenge: first hand deterministic techniques including the application of Hough transform based methods [50], geometric hashing and template or model matching techniques [1,32]. On the other hand, stochastic techniques including random sample consensus techniques [12], simulated annealing [8] and Genetic Algorithms (GA) [37], have been also used.

Template and model matching techniques are the first approaches to be applied to shape detection yielding a considerable amount of publications [22,27,48]. Shape coding techniques and combination of shape properties have been common-ly used to represent objects. Their main drawback is related to the contour extrac-tion step from a real image which hardly deals with pose invariance, except for very simple objects.

Typically, the problem of circles detection in digital images is performed by the Circular Hough Transform [29]. A typical Hough-based approach employs edge information obtained by means of an edge detector to infer locations and radius values. Peak detection is then performed by averaging, filtering and histo-gramming the transformed space. However, such an approach requires a large storage space given the required 3-D cells to cover all parameters (x, y, r) and a high computational complexity which yields a low processing speed. The accu-racy of the extracted parameters for the detected circle is poor, particularly under the presence of noise [3]. For a digital image holding a significant width and height and a densely populated edge pixel map, the required processing time for Circular Hough Transform makes it prohibitive to be deployed in real time

Circle Detection Algorithm Based on Electromagnetism-Like Optimization 909

applications. In order to overcome this problem, some other researchers have pro-posed new approaches based on the Hough transform, for instance the probabilis-tic Hough transform [40], the randomized Hough transform (RHT) [46] and the fuzzy Hough transform [15]. Although those new approaches demonstrated faster processing speeds in comparison to the original Hough Transform, they are still highly sensitive to noise. Stochastic search methods such as Genetic Algorithms (GA) are other alternatives to shape recognition in computer vision. Ayala–Ramirez et al., presented a GA based circle detector [4] which is capable of de-tecting multiple circles on real images albeit failing on detecting imperfect circular shapes.

On the other hand, nature-inspired methods with collective behaviour have been successfully applied for solving constrained global optimization problems [13, 41, and 53]. The Electromagnetism-like optimization (EMO) algorithm has been recently proposed [17]. It is a flexible and effective nature-based approach for single objective optimization problems. EMO originates from the electro-magnetism theory of physics by assuming potential solutions as electrically charged particles which spread around the solution space. The charge of each par-ticle depends on its objective function value. This algorithm employs a collective attraction-repulsion mechanism to move the particles towards optimality. Such re-lationship among particles corresponds to the interaction among individuals pro-duced by reproduction, crossover and mutation in Genetic Algorithms (GA) [10]. Likewise, EMO moves particles around the search space following similar paths from other methods such as the Particle Swarm Optimization (PSO) [11 and 49] and the Ant Colony Optimization (ACO) [42].

(a)

(b)

Fig. 1 Circle detection applied to two images, (a) the detected circle in a real-life images and (b) the detected circle in a synthetic complex image.

Although EMO algorithm shares some characteristics to other nature-inspired

approaches, recent works (see [33, 34, 43 and 45]) have exhibited EMO’s great savings on computation time and memory allocation as to surpass other methods such as GA, PSO and ACO. EMO’s low computational cost and its reassured con-vergence [18] have both been successfully applied to the solution of different sorts of engineering problems such as flow-shop scheduling [31], vehicle routing [52], array pattern optimization in circuits [19], neural network training [45] and control


systems [21]. However, to the best of our knowledge, EMO has not yet been ap-plied to any computer-vision related task.

An EMO algorithm based circle detector method which assumes the overall de-tection process as an optimization problem, is presented in this chapter. The algo-rithm uses the encoding of three non-collinear edge points lying on the edge-only image of the scene as candidate circles (x,y,r). Guided by the values of the objec-tive function, the set of encoded candidate circles are evolved using an EMO algo-rithm so that they can fit into the actual circles on the edge map (see examples in Figure 1). The approach generates a sub-pixel circle detector which can effectively identify circles despite circular objects exhibiting significant occluded portions. Experimental results show performance evidence for detecting circles under dif-ferent conditions.

This chapter has been organized as follows: Section 2 provides a brief outline of the EMO method. Section 3 explains the application of the EMO algorithm for circle detection. Section 4 discusses the results after designing an EMO applica-tion for a number of images exhibiting different conditions. Finally, Section 5 of-fers some insights into the algorithm’s performance and some conclusions.

2 Electromagnetism-Like Optimization (EMO)

In order to describe the EMO algorithm, some important concepts about global optimization and electromagnetism should be reviewed.

Global stochastic methods employ random sampling from a feasible region to initialize possible solutions for a problem. Therefore, new samples are generated considering locations over which it is possible to find best solutions (regarding to their objective function values). The stochastic method starts by exploiting the points in order to localize a global solution. Similarly, the EMO algorithm em-ploys the analog physical principle to exploit feasible locations within the search space.

2.1 Fundamental Concepts of Electromagnetism in Physics

By considering electromagnetism principles, it is possible to describe macroscopic physical phenomena of interacting electrical charges. Such principles are meta-phorically followed to build the EMO algorithm as follows:

Electrical charge: It is a feature of elemental particles. It depends directly on the electrostatic force between particles. Such force can be attractive or repulsive de-pending on the charge’s polarity.

Coulomb’s law: It was initially proposed in 1785. It states that the force between two charges is directly proportional to the product of both charges and inversely proportional to the square distance between the charges. The force’s direction (vector) follows the line connecting both charges. The magnitude of such vector is modeled as follows:


( )( )2

0

' '1

4 '

qq r rF

r rπε

− = −

(1)

where: 12

0 8.85419 10 Fmε −= × , and it is the vacuum permittivity.

r =

Is the position of the charge q .

'r =

Is the position of the charge 'q .

Superposition principle: It states that the force exerted by a charged system over other charge is equal to the vector sum of forces over the coming charge. Such principle can be modeled as follows:

( ) ( )2

1 0

' '1

4 '

Ni i

ii

q q r rF

r rπε=

− = −

(2)

The index i indicates the charge number.

2.2 Electromagnetism-Like Optimization Algorithm

The EMO algorithm is defined as a simple and direct search algorithm which has been inspired by the electro-magnetism phenomenon. It is based on a given popu-lation and the optimization of global multi-modal functions. In comparison to GA, it does not use crossover or mutation operators to explore feasible regions; instead it does implement a collective attraction–repulsion mechanism yielding a reduced computational cost with respect to memory allocation and execution time. More-over, no gradient information is required and it employs a decimal system which clearly contrasts to GA. Few particles are required to reach converge as has been already verified in [18].

Problems with bounded variables can be solved effectively by EMO algorithm, these problems can be considered as a special class of unconstraint optimization and they should be defined as follow:

[ ]ulx

xf

,

)(min

∈ (3)

where [ ] { }, | , 1,2...nd d dl u x l x u d n= ∈ℜ ≤ ≤ = and n being the dimension of the

variable x, [ ], nl u ⊂ ℜ , a nonempty subset and a real-value function [ ]: ,f l u → ℜ .

Hence, the following problem’s features are known:

• n : Dimensional size of the problem.

• du : The highest bound of the thk dimension.


• dl : The lowest bound of the thk dimension.

• ( )f x : The function to be minimized.

Under these conditions EMO algorithm uses two processes in order to optimize an objective function. First the algorithm explores feasible regions, taking samples randomly; these samples are called attractive regions since possible solutions may exist in them. The second process consists in exploiting the attractive regions. To make this, the algorithm uses operators inspired by the electromagnetism prin-ciples. Employing such operators, the particles are attracted to highly attractive fitness values or repelled from bad fitness values. The EMO algorithm has four main phases [17], i.e., initialization, local search, computation of the total force vector and movement. Each stage is discussed below.

Initialization, a number of m particles is gathered as their highest (u) and lowest

limit (l) are identified. Local search, gathers local information for a given point pg ,

where (1, , )p m∈ .

Calculation of the total force vector, charges and forces are calculated for every particle.

Movement, each particle is displaced accordingly, matching the corresponding force vector.

2.2.1 Initialization

First, m solutions are randomly produced at an initial state, these solutions are considered as a group called G . Each n-dimensional solution is regarded as a charged particle holding a uniform distribution between the highest (u) and the lowest (l) limits. The optimum particle (solution) is thus defined by the objective function to be optimized. The procedure ends when all the m samples are evalu-ated, choosing the sample (particle) that has gathered the best function value.

2.2.2 Local Search

Theoretically, the local search should be able to find a better solution when it is applied. However it may be unnecessary for some problems [19]. Considering this, a broad classification of the EMO algorithms can be formulated as follows: EMO with no local search, EMO with local search applied to all particles and EMO with local search applied only to the current best particle.

In its operation, the local search procedure uses some parameters, they are: a determined number of iterations, known as ITER, and a feasible neighbourhood search δ . The procedure iterates as follows: Point pg is assigned to a temporary

point t to store the initial information. Next, for a given coordinate d, a random number is selected ( 1λ ) and combined with δ as a step length, which in turns

moves the point t. along the direction d, with a randomly determined sign ( 2λ ).


If point t. observes a better performance within the iteration number ITER, point pg . is replaced by t. and the neighbourhood search for point pg . finishes, oth-

erwise pg is held. Finally the current best point is updated. The pseudo-code is

shown in Fig.2.

1: 1Counter ← 12: ( )2d d Lengthλ← −t t

2: { }( )max d dLength u lδ← − 13: end if

3: for 1p = to m do 14: if ( ) ( )pf f<t g then

4: for 1d = to n do 15: p ←g t 5: ( )1 U 0,1λ ← 16: 1Counter ITER← −

6: whileCounter ITER< do 17: end if

7: p←t g 18: 1Counter Counter← +

8: ( )2 U 0,1λ ← 19: end while

9: if 1 0.5λ > then 20: end for

10: ( )2d d Lengthλ← +t t 21: end for

11: else 22: { }argmin ( ),best pf p← ∀g g

Fig. 2 Pseudo-code list for the local search algorithm.

The local search for all particles can reduce the risk of falling onto a local solu-tion but it represent a problem because is relatively time consuming. Keeping the local search focused only on the current best particle is a rather convenient proce-dure in order to maintain the computational efficiency and precision. The step length for local search is therefore an important factor that depends on the limits for each dimension and determines the performance of the overall local search.

2.2.3 Total Force Vector Computation

The total force vector computation is based on the superposition principle (Fig. 3) from the electro-magnetism theory which states: „the force exerted on a point via other points is inversely proportional to the distance between the points and di-rectly proportional to the product of their charges‰ [28]. The resultant Coulomb’s force has been generated among particles as a charge-like value, the particle moves following this force value. In the EMO implementation, the charge for each particle is determined by its fitness value as follows:

( ) ( )( ) ( )( )

1

exp ,p best

pm

h best

h

f fq n p

f f=

− = − ∀

−

g g

g g

(4)


where n denotes the dimension of pg and m represents the population size. A

higher dimensional problem usually requires a larger population. In Eq. (4), the particle showing the best fitness function value bestg is called the „best particle‰,

getting the highest charge and attracting other particles holding high fitness val-ues. The repulsion effect is applied to all other particles exhibiting lower fitness values. Both effects, attraction-repulsion are accordingly applied depending on the actual proximity between a given particle and the best-graded element.

Fig. 3 The superposition principle.

The overall resultant force between all particles determines the actual effect of the optimization process. A final force vector is generated for each particle and it is evaluated under the EMO superposition principle, which is defined as follows.

( )

( )

2

2

( ) ( )

,

( ) ( )

p hh p h p

h pm

p

p hh p p h h p

h p

q qif f f

pq q

if f f≠

− <

− = ∀ − ≥ −

g g g g

g gF

g g g gg g

(5)

where ( ) ( )h pf f<g g represents the attraction effect and ( ) ( )h pf f≥g g repre-

sents repulsion force (see Fig. 4). The resultant force of each particle is propor-tional to the product between charges and is inversely proportional to the distance between particles. In order to keep feasibility, the vector in expression Eq. (5) should be normalized as follows:

ˆ , .p

p

pp= ∀F

FF

(6)

2.2.4 Movement

Based on the total force vector, each particle p changes its position of the d-coordinate. Such new values are computed as follows:

1q

2q

3q

2,3F

1,3F

3F


( )( )

ˆ ˆif 0, ,

ˆ ˆif 0

p p p pd d d d dp

d p p p pd d d d d

g F u g Fg p best d

g F g l F

λ

λ

+ ⋅ ⋅ − > = ∀ ≠ ∀ + ⋅ ⋅ − ≤

(7)

where λ is a random step length that is uniformly distributed between zero and one. du and dl represent the upper and lower boundary for the d-coordinate, re-

spectively. ˆ pdF represents the d element of ˆ pF . The particle moves towards the

highest boundary by a random step length if the resultant force is positive. Other-wise, it moves toward the lowest boundary. The best particle does not move at all, because it holds the absolute attraction pulling or repelling all others in the population.

Fig. 4 Coulomb law: α represents the distance between charged particles, 1 2,q q are the

charges, and F is the exerted force as has been generated by charge interaction.

The process is halted when a maximum iteration number is reached or when the value ( )bestf g is near zero or the required optimal value.

2.3 A Numerical Example Using EMO Algorithm

A common way to test the efficiency of Global Optimization Algorithms is using mathematical functions. One of the most popular benchmark functions used for testing the optimization capabilities is the Rosenbrock function [55] which is a 2-D function defined as follows:

2 2 2( , ) (1 ) 100( )f x y x y x= − + − (8)

The main characteristics of the Rosenbrock function, used in this example, are de-fined in Table 1.

α

1q 2q

1q 2q

2,1F 1,2F

α

1,2F 2,1F


Table 1 Limits and global minima of Rosenbrock function.

Higher limit (u) Lower limit(l) Global Minima 2

2

x

y

==

2

2

x

y

= −= −

1

1

xbest

ybest

==

Therefore, the intervals of each parameter are defined as follows:

[ 2, 2]

[2,2]

= − −=

l

u (9)

where ,l u contain the lower and upper limits of each 2-dimension respectively. For the local search procedure, it is used the parameters suggested in [17].

Table 2 EMO parameters selected to find the minima of Rosenbrock function.

Population size ( )m

Length step ( )δ

MAXITER

10 0.0001 50

Considering the parameters defined in Table 2 as the initial configuration,

EMO algorithm begins with a random population of m particles, a possible initial population for the Rosenbrock problem is shown in Table 3.

Table 3 Initial population ( G ), generated by the initialization procedure.

-1.6109 0.7127 -0.8361 1.7886 1.0436 -0.6682 0.4100 -1.7861 1.5771 -0.8316

-1.8580 0.8332 -1.9183 1.5444 -0.2486 -1.0087 -1.4834 0.6282 -0.8488 -0.8769

Each element of the population (particle) is evaluated with regard to the Rosen-

brock function (fitness function). As a result of these evaluations, it is obtained the values shown in Table 4.

Table 4 Fitness values ( )f G of the G population.

1989.7924 10.6587 688.4320 274.4003 178.9637 214.5376 273.0979 664.1470 1113.2490 249.3679

The initial values of the population are evolved using all operators of the EMO

algorithm, when a stop criterion is satisfied, EMO finishes its execution. For this example the stop criteria is a maximum number of iterations. The evolution results and the Rosenbrock function are shown in the Fig.5.


(a) (b)

Fig. 5 (a) Shows the Rosenbrock function, (b) the initial population (black dots), evolved population (blue cross), global minima (red dot).

3 Circle Detection Using EMO

Using the parameters of the well-known second degree equation shown in Eq. (10), it is possible to represent a circle. This equation [3] considers three points ex-tracted directly from the edge-only space of the image. In order to achieve this, a pre-processing stage marks the object’s contour by applying a single-pixel edge detection method. In this chapter, the classical Canny’s algorithm is applied to do this task; then the locations for each edge point are stored. All points are potential candidates to define circles by considering triplets. They are stored within a vector

array { }1 2, , , Np= …E e e e with Np being the total number of edge pixels in the im-

age. In turn, the algorithm stores the ( , )v vx y coordinates of each edge pixel yield-

ing the edge vector ve ( 1 2,v vv ve x e y= = ).

In order to construct each candidate circle (or charged particles within the EMO framework), the indexes 1v , 2v and 3v of three non-collinear edge points must be

combined, assuming that the circle’s circumference includes the points1v

e ; 2ve ;

3ve . A number of candidate solutions are generated randomly for the initial pool

of particles. Solutions will thus evolve through the recursive application of the EMO algorithm affecting all particles until a minimum is reached and the best par-ticle is considered as a solution. The function improves at each generation step by discriminating non-plausible circles, locating others and avoiding useless visits to other image points. The discussion below explains the required steps to formulate the circle detection task from EMO’s perspective.

3.1 Particle Representation

Each particle or circle candidate C is defined by the combination of three edge points; these points proceed from the edge map. Under such representation, points


are stored by assigning an index which refers to their relative position within the edge array E . In turn, the procedure will encode each particle as the circle passing through three points: ie , je and ke ( { , , }p

i j k=C e e e ). Each circle C is thus rep-

resented by three parameters: 0x , 0y and r, with representing the circle’s centre

coordinates and r its radius. The circle equation can thus be computed and ex-plained as follows:

2 2 20 0( ) ( )x x y y r− + − = (10)

considering:

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

( ) 2 ( ),

( ) 2 ( )

2 ( ) ( )

2 ( ) ( )

j j j i j i

k k i i k i

j i j j i i

k i k k i i

x y x y y y

x y x y y y

x x x y x y

x x x y x y

+ − + ⋅ −= + − + ⋅ − ⋅ − + − +

= ⋅ − + − +

A

B

(11)

0

0

det( ),

4(( )( ) ( )( ))

det( )

4(( )( ) ( )( ))

j i k i k i j i

j i k i k i j i

xx x y y x x y y

yx x y y x x y y

=− − − − −

=− − − − −

A

B (12)

and

2 20 0( ) ( )b br x x y y= − + − (13)

where det(.) stands for the determinant and { }, ,b i j k∈ . Figure 6 illustrates the pa-

rameters defined by equations 11 to 13.

Fig. 6 Circle candidate (charged-particle) following the combination of points

ip , jp and kp .

rie je

ke

0 0( , )x y


The process of representing the set of shape parameters of each circle [ 0x , 0y ,

r], by edge vector indexes i, j and k, can be considered as a space transformation T which is defined by Eq. (14).

[ ]0 0, , ( , , )x y r T i j k= (14)

EMO algorithm explores the space by assuming each circle as a particle, searching effective circular shape parameters and eliminating unfeasible solutions.

3.2 Objective Function

A virtual shape can be used to calculate a circumference, this shape is applied as a way to measure the matching factor between the candidate circle and the actual circle within the image, i.e. it must be validated if such circular shape actually ex-ists within the edge-only image. The test for such points is verified using the vec-tor 1 2{ , , , }Ns=S s s s , where Ns represents the number of testing points recalling

that the vector ws contains the pixel coordinates 1w

ws x= and 2w

ws y= .

The test vector S is generated by means of Midpoint Circle Algorithm (MCA) [9] which calculates the vector S representing a circle from parameters 0 0( , )x y

and r . For details refer to [44]. Although the algorithm is considered to be the fastest providing a sub-pixel precision, it is important to assure it does not con-sider points lying outside the image plane. The matching function, also known as the objective function ( )J C , represents the error resulting from pixels S of a

given circle candidate and C including only pixels that really exist in the edge im-age, yielding:

1

( )

( ) 1

Nsv

v

V

JNs

== − s

C (16)

where ( )vV s is a function that verifies the pixel existence in vs , such as:

1 if the pixel in position ( , ) exists( )

0 otherwisev vv x y

V

=

s (17)

Eq. (16) accumulates the number of identified edge points (points in S ) that are actually present in the edge image. Ns holds for the number of pixels lying on the circle’s perimeter that correspond to C , currently under testing.

In each iteration, the algorithm tries to minimize ( )J C , a smaller value of

( )J C implies a better response (minimum error) of the “circularity” operator. The

optimization process can thus be stopped either because the maximum number of epochs is reached or because the best individual is found.


3.3 EMO Implementation

The following steps summarizing the implementation of the proposed algorithm:

Step 1: Canny’s filter is applied to obtain image edges, storing them into

the { }1 2, , , Np=E e e e vector. The iteration count is set to 0.

Step 2: m initial particles are generated (each one holding ie , je and ke

elements, where ie , je and ke ∈ E ). Particles belonging to a very

small or to a quite long radius are eliminated (collinear points are discarded). The objective function ( )pJ C is evaluated to determine

the best particle bestC (where { }argmin ( ),best pJ p← ∀C C ).

Step 3: For a given coordinate ( , , )d i j k∈ , the particle pC is assigned into

a temporary point t. to store the initial information. Next, a ran-dom number is selected and combined with δ yielding the step length. Therefore, point t. is moved along the direction d with the sign being randomly determined. If ( )J t is minimized, the particle

pC . is replaced by t. and the neighbourhood searching for a parti-

cle . . finishes, otherwise pC is held. Finally the current best parti-

cle bestC is updated. Step 4: The charge between particles is calculated using Eq. (4), and its

vector force is calculated using Eq. (5). The particle featuring a better objective function value holds a bigger charge and therefore a bigger attraction force.

Step 5: The new particle’s position is calculated by Eq.(7). bestC is not moved because it has the biggest force and it attracts others parti-cles to itself.

Step 6: The Iteration index is increased. If Iteration MAXITER= or if ( )J C value is as small as the pre-defined threshold value, then the

algorithm is stopped and the flow jumps to step 7. Otherwise, it jumps to step 3.

Step 7: The best bestC particle is selected from the last iteration. Step 8: From the original edge map, the algorithm marks points corre-

sponding to bestC . In case of multi-circle detection, it jumps to step 2.

Step 9:

Finally the best particle bestNcC from each circle is used to draw

(over the original image) the detected circles, considering Nc as the number of successfully found circles.


Fig. 7 shows an analogy to the Coulomb’s law. The original circle to be de-tected is represented by a solid line while the discontinuous line represents some circles holding the most attractive force, i.e. they have the lowest error value. Other repelled circles are represented by dotted lines as they hold a larger error.

Fig. 7 A circle-based analogy for Coulomb’s law.

4 Experimental Results

The performance of the EMO-based circle detector is measured using several tests which are presented as follows: (4.1) circle localization, (4.2) shape discrimina-tion, (4.3) multiple circle detection, (4.4) circular approximation, (4.5) approxima-tion from occluded or imperfect circles or arc detection and (4.6) accuracy and computational time. Each test is performed with the following parameters: a pool of particles of size 10m = , a maximum iteration for the local search 2ITER = , a step length for the local search 3δ = . Parameters 1λ and 2λ are random values

uniformly distributed while the maximum iteration value 20MAXITER = . For the particles movement (Eq. (7)), the step length λ is also set as a uniformly distrib-uted random number. Finally, the search space (edge-only pixels) implies bounda-ries 1u = , l Np= for each variable: ie , je and ke .

4.1 Circle Localization

4.1.1 Synthetic Images

The experimental setup includes a number of synthetic images of 256 x 256 pix-els. Each image has been generated by drawing only a randomly located imperfect circle (ellipse). Some of these images are contaminated by adding noise to in-crease complexity in the detection process. The parameters to be estimated are the


centre of the circle position (x, y) and its radius (r). The algorithm is set to 20 it-erations for each test image. In all cases, the algorithm has been able to detect the circle’s parameters despite noise presence. The detection is robust to translation and scale, keeping a reasonable low elapsed time (typically under 1ms). Fig. 8 shows the circle detection over two different synthetic images.

(a)

(b)

Fig. 8 Circle detection over synthetic images: (a) an original circle image with noise and (b) its corresponding detected circle.

4.1.2 Real-Life Images

This experiment tests the circle detection algorithm upon real-life images. All im-ages have been captured using an 8-bit colour digital camera and each natural scene includes a circle shape among other objects. All images are pre-processed using an edge detection algorithm before applying the EMO’s circle detector. Fig. 9 shows two cases from 25 tested images.

Real-life images rarely contain perfect circles. Therefore the detection algo-rithm can approximates the circle that adapts better to the imperfect circle, i.e. the circle corresponding to the smallest error in the objective function ( )J C . All re-

sults have been statistically analyzed to ease comparison. The detection time for the image shown in Fig. 9(a) is 13.540807 seconds while it is 27.020633 seconds for the image in Fig. 9(b). The detection algorithm has been executed over 20 times on the same image (Fig. 9(b)), yielding the same parameter set: 0 214x = ,

0 322y = , and 948r = . The proposed EMO algorithm was able to converge to the

minimum solution as is referred by the objective function ( )J C .


(a)

(b)

Fig. 9 Circle detection applied to two real-life images, (a) the detected circle is shown nearby the ring periphery, (b) the detected circle is shown nearby the ball’s periphery.

(a)

(b)

(c)

(d)

Fig. 10 Circle detection on synthetic images: illustrations (a) and (c) show the original im-ages while (b) and (d) show their corresponding detected circles.


4.2 Shape Discrimination

In this section we focus on the ability for detecting circles despite the presence of any other shape on the image. Five synthetic images of 540x300 pixels are consid-ered for this experiment. Noise has been added to all images and a maximum of 20 iterations was used for the detection. Two examples of circle detection on syn-thetic images are shown by Fig. 10.

This experiment is repeated for some real-life images (see Fig. 11) confirming that the circle detection is completely feasible on natural images despite other shapes being present on the scene.

(a)

(b)

(c)

(d)

Fig. 11 Different shapes embedded into real-life images, (a) the test image, (b) the corre-sponding edge map, (c) the detected circle and (d) the detected circle overlaid on the origi-nal image.

4.3 Multiple Circle Detection

The presented approach was also capable of detecting several circles over real-life images; this advantage is possible making a small modification in EMO-based cir-cle detector. In the first experiment, a maximum number for circular shapes to be


detected had been set. . The algorithm works on the original edge-only image until the first circle is detected. The first circle represents the circle with the minimum objective function value ( )J C . Such shape is then masked (eliminated) and the EMO circle detector continues operating over the modified image. The procedure was repeated until the maximum number of detected circular shapes was reached. As a final step, all detected circles were validated by analyzing their circumfer-ence continuity as proposed in [4]. Such procedure becomes necessary in case more circular shapes are required in the future.

The optimized image is taken as a new EMO’s input. It is important to con-sider that the new input does not contain any complete circle, because any other has already been detected and masked at the previous step. Therefore, the algo-rithm focuses on detecting other potential circles until the maximum of 20 itera-tions is reached.

(a)

(b)

(c)

(d)

Fig. 12 Multiple circle detection on real images: pictures in (a) and (c) show edge images as they are obtained from applying Canny’s algorithm. Images (b) and (d) present original scenes with the red overlaid of detected circles.

Fig. 12(a) shows the edge image after applying the Canny’s operator and Fig. 12(b) presents the actual image including several detected circles which have been overlaid for presentation. Figures 12(c) and 12(d) show similar cases.


4.4 Circular Approximation

Since circle detection has been approached as an optimization problem, it is possi-ble to approximate a given shape as the concatenation of a number of circles. By using the multiple-circle feature of the EMO algorithm presented in the previous section, the EMO algorithm may continually find circular shapes that may be gathered to resemble a non-circular shape according to the values from the objec-tive function ( )J C .

Figure 13 shows some examples of circular approximation. Figure 13(a) pre-sents a circular-like shape which may host a circular superposition. In that case, Figure 13(b) features the approximation with 3 circles after applying the EMO multi-circle detector. Fig. 13(c) presents an ellipse that has been approximated by the concatenation of four circles as shown by Fig. 13(d).

(a)

(b)

(c)

(d)

Fig. 13 Circular approximation: (a) The original image, (b) its circular approximation con-sidering 3 circles. (c) Another testing image, and (d) its circular approximation considering 4 circles.


4.5 Circle Localization from Occluded or Imperfect Circles and Arc Detection

An interesting feature of this approach is the ability to approximate circular shapes from arc segments, occluded circular segments or imperfect circles. Such cases represent common challenges for typical computer vision problems. Again, EMO algorithm was able to find the circle shape that approaches an arc segment accord-ing to the values in the objective function ( )J C . Figure 14 shows some examples

of such functionality.

(a)

(b)

(c)

(d)

Fig. 14 Approximation of circles from occluded shapes, imperfect circles an arc detec-tion:(a) the original images with 2 arcs, (b) the circle approximation for the first image, (c) occluded natural image of the moon and (d) circle approximation.

4.6 Accuracy and Computational Time

This section discusses the algorithm’s ability to find the best circle considering synthetic images under noise presence. The experiment gets ten contaminated im-ages of 256 x 256 pixels which contain only one circle centre at 128x = , 128y = ,

with 64r = pixels. Salt & Pepper noise distribution is considered for this test.


EMO algorithm iterates 35 times per image and the particle showing the best fitness value is regarded as the best circle matching to the actual one in the image. The process was repeated over 25 times per image to test consistency. The accu-racy evaluation was achieved by the Error sum (Es) which measures the difference between the ground truth circle (actual circle) and the detected one. The error sum is defined as follows:

d true d true d trueEs x x y x r r= − + − + − (18)

where , ,true true truex y r are the coordinates of the centre and the radius value of the

ground truth circle in the image. Moreover , ,d d dx y r correspond to the values of

centre and radius respectively after being detected by the method. One experiment considers contaminated synthetic images doped with Salt &

Pepper noise. The EMO parameters are: maximum iterations 35MAXITER = ; for the local search, 4δ = and 4ITER = . The added noise is produced by MatLab©, considering noise levels between 1% and 10%. The resulting values for Es and the computing elapsed time are reported in Table 5. Fig. 15 shows three different im-ages as examples, including 25 detected circles that are overlaid on each original image. It is evident from Fig. 15 that a higher added noise yields a higher disper-sion on the detected shapes.

Table 5 Data analysis for images containing added Salt & Pepper noise.

Image Properties

Results

Error sum (Es) Computational time

(Seconds)

Size Circle center

Ra-dius

Salt and pepper

level Total Mean Standard deviation Mode Mean Standard deviation

256x256 (128,128) 64 0.01 0 0 0 0 12.5505 0.5426

256x256 (128,128) 64 0.02 0 0 0 0 13.0491 0.7455

256x256 (128,128) 64 0.03 0 0 0 0 13.3401 0.7365

256x256 (128,128) 64 0.04 0 0 0 0 14.3228 0.655

256x256 (128,128) 64 0.05 2 0.0571 0.2355 0 14.2141 0.7982

256x256 (128,128) 64 0.06 0 0 0 0 14.8669 0.9604

256x256 (128,128) 64 0.07 3 0.0857 0.5071 0 15.3725 0.5147

256x256 (128,128) 64 0.08 7 0.2 0.6774 0 14.5373 0.5147

256x256 (128,128) 64 0.09 6 0.1714 0.5137 0 14.5747 0.8907

256x256 (128,128) 64 0.10 28 0.8 2.1666 0 14.7288 0.8116


Table 6 Experimental results after applying the EMO circle detector to real-life images.

Image Properties

Results

Matching Error (%) Computational time

(Seconds)

Image name Size Total Mean Standard Deviation Mode Mean Standard Deviation

Cue Ball 430 x 473 28.1865 0.805 0.0061 0.8035 26.6641 1.4018

Street Lamp 474 x 442 19.0638 0.545 0.0408 0.5326 23.4763 1.7857

Wheel 640 x 480 22.265 0.636 0.0062 0.6336 31.356 1.4351

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 15 Test images with added Salt and Pepper noise: (a) image holding 0.01 of added noise, (b) image including 0.05 of added noise, (c) image containing 0.1 of added noise, and (d) , (e) and (f) show the final twenty five detected circles for each test image.

In a different test, the EMO approach is applied to different real images. It is

important to mention that the Error sum (Es) is not used because the coordinates for the centre and the radio for each circle are unknown. Table 6 shows the results after applying the EMO circle detector considering the algorithm for natural im-ages presented in Fig. 16. The final twenty five detected circles are overlaid over the original images.


5 Conclusions

This chapter has presented an algorithm for the automatic detection of circular shapes among cluttered and noisy images with no consideration of the conven-tional Hough transform principles. The work describes a circle detection method which is based on a nature-inspired technique known as the Electro-magnetism Optimization (EMO). It is a heuristic method for solving complex optimization problems inspired by electro-magnetism principles. The algorithm uses the encod-ing of three non-collinear edge points as candidate circles ( ), ,x y r in the edge-

only image of the scene. An objective function evaluates if such candidate circles (charged particles) are actually present. Guided by the values of this objective function, the set of encoded candidate circles are evolved using the EMO algo-rithm so that they can fit into the actual circles on the edge-only map of the image. From our results, the EMO approach has been capable of effectively detecting cir-cular shapes embedded into complex images with little visual distortion despite the presence of noisy background pixels and still shows good accuracy and consistency.

(a)

(b)

(c)

Fig. 16 Real-life images used in the experiments, (a) cue ball, (b) wheel, (c) street ball, in-cluding the overlaid detected circles.

An important feature in this method is to consider the circle detection problem as an optimization approach. Such a view enables the EMO algorithm to find cir-cle parameters according to ( )J C instead of voting all possible chances for oc-

cluded or imperfect circles as other methods do.


Although Hough Transform methods for circle detection also uses three edge points to cast one vote for the potential circular shape in the parameter space, they would require huge amounts of memory and longer computational times to obtain a sub-pixel resolution. In the HT-based methods, the parameter space is quantized and the exact parameters for a valid circle are often not equal to the quantized val-ues. This fact evidently yields a non-exact determination for a circle actually pre-sent in the image. However, the proposed EMO method does not employ the quantization of the parameter space. In our approach, the detected circles are di-rectly obtained from Eqs. (10)–(13) effectively detecting the circle with sub-pixel accuracy.

The results offer evidence to demonstrate that the EMO method can yield good results on complicated and noisy images, however the aim of this chapter is not to devise a circle-detection algorithm that could beat all currently available circle de-tectors, but to show that Electro-magnetism systems can be effectively considered as an attractive alternative for detecting circular shapes.

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