Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 6494390, 13 pages

https://doi.org/10.1155/2017/6494390

## Fast Parabola Detection Using Estimation of Distribution Algorithms

^{1}Division de Ingenierias, Campus Irapuato-Salamanca (DICIS), Universidad de Guanajuato, Carr. Salamanca-Valle Km 3.5+1.8, Palo Blanco, 36885 Salamanca, GTO, Mexico^{2}CONACYT, Centro de Investigacion en Matematicas (CIMAT), A.C., Jalisco S/N, Col. Valenciana, 36000 Guanajuato, GTO, Mexico^{3}CONACYT, Centro de Investigacion en Matematicas (CIMAT), A.C., Fray Bartolome de las Casas 314, Barrio La Estacion, 20259 Aguascalientes, AGS, Mexico

Correspondence should be addressed to Juan Gabriel Avina-Cervantes

Received 24 September 2016; Revised 4 January 2017; Accepted 15 January 2017; Published 21 February 2017

Academic Editor: Amparo Alonso-Betanzos

Copyright © 2017 Jose de Jesus Guerrero-Turrubiates et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new method based on Estimation of Distribution Algorithms (EDAs) to detect parabolic shapes in synthetic and medical images. The method computes a virtual parabola using three random boundary pixels to calculate the constant values of the generic parabola equation. The resulting parabola is evaluated by matching it with the parabolic shape in the input image by using the Hadamard product as fitness function. This proposed method is evaluated in terms of computational time and compared with two implementations of the generalized Hough transform and RANSAC method for parabola detection. Experimental results show that the proposed method outperforms the comparative methods in terms of execution time about on synthetic images and on retinal fundus and human plantar arch images. In addition, experimental results have also shown that the proposed method can be highly suitable for different medical applications.

#### 1. Introduction

In the pattern recognition field, detection of curves in natural or medical images is a significant and challenging problem since relevant information about an object is highly related to the shape of its boundary. Any curve can be detected by using the Hough transform (HT), if this curve can be represented by a parametric equation [1, 2].

Circular Hough transform (CHT) is based on the Hough transform principle and it has been adapted for the detection of circles [3]. For this particular problem, three different parameters that define a circle have to be determined. The parameters represent the coordinate of the center and represents the radius [4]. Although the CHT is able to detect suitable approximation to circles in different type of images, the computational time to detect a single curve is high. This disadvantage is due to the fact that each single pixel represents a potential center , and a range of possible radii have to be tested for each particular pixel. On the other hand, the voting accumulator [5] that indicates the parameters of a curve that exists in the image is computed on a matrix where each cell has the number of intersections of the circles formed by taking a single pixel as the center; if the matrix is plotted in a three-dimensional graph, sharp peaks will be visualized in the cells that have more intersections. The parameters of the circle are computed by taking the highest peak in the accumulator. Since the representation of an accumulator is a matrix, two of the three parameters of the circle are represented by the rows and columns; in order to cover the third parameter, a three-dimensional matrix is needed. Although it is possible that several sharp peaks will be formed in the accumulator, when the radius of the tested circle is close to the radius of the circle, these peaks do not represent a real circle. Therefore, once the accumulator has been computed, the next challenge is to find the peak that actually represents the real circle. Due to the nature of the CHT, it would be virtually impossible to find multiple circles in an image; therefore, some studies have proposed different techniques to solve this issue by using techniques such as genetic algorithms [6] or Harmony Search (HSA) [7]. Ayala-Ramirez et al. [6] optimize the computational time of the CHT, by applying a strategy consisting in selecting three random pixels from the edge image as the chromosomes for the genetic algorithm. These three points are used to calculate a center and radius of a circle, and the fitness function evaluates how many pixels of a virtual circle are actually present in the real edge image.

The problem of detecting parabolas can be accomplished by finding the basic parameters of the general equation for parabolic shapes, which are the vertex in , the angle, and its aperture. In literature, different methods have been proposed to detect parabolas using the Hough transform as a baseline. Maalmi et al. [8] proposed a method that applies genetic algorithms to perform the voting process of the Hough transform. The detected curves are horizontal and the aperture is fixed. This implementation was reported for the detection of crack defects in B-scan images. Oloumi and Rangayyan [9] applied the Hough transform for parabola detection using three parameters: that represent the vertex and to describe the latus rectum or aperture. The main application reported of this method is the detection of temporal arcade in retinal fundus images, where parabolas fit retina vessels [10, 11]. Other reported works also have been focused on the human eye, for finding the iris [12], and in retina angiography [13].

Another potential application for parabola detection can be seen in medical images for orthopedic diagnostics in the plantar arch. Some of the basic human movements are walking and running; these movements are possible thanks to a complete set of muscles working together. However, if the plantar arch of the human feet is not of the correct size, a set of problems (e.g., back problems) can lead to surgery and prosthetics [14]. In patients with diabetes, it was observed that there is a relationship between forefoot and rear foot pressure with the ulceration on the human foot [15]. Different methods have been proposed to address problems in the orthopedic area. P. S. Kulkarni and V. B. Kulkarni [16] proposed a method to classify the human footprint using a parameter that they call Footprint Index (FPI). First, they found the minimum distance from -axis on the lateral upper and lower part of the footprint; with those two points, they draw a line , where the middle point of will be the center of the arch. Considering this center, two lines and , which are perpendicular to , are computed. These lines will provide the distances from to the arch vertex and from the arch vertex to the edge foot, respectively. The FPI parameter is calculated as the ratio of to . Then, the footprint can be classified as flat foot, normal foot, or high arch foot. Chu et al. [17] proposed a similar process as in [16]; however, some other measurements were computed and were added to the ratios proposed by P. S. Kulkarni and V. B. Kulkarni [16] in an effort to compute the arch height. Zheng et al. [18] used the footprint to perform an analysis and match a subject’s gait with the corresponding footprint.

Some other approaches have been proposed for parabola detection. Salehin et al. [19] proposed conic detection by applying Pascal’s theorem (i.e., approximating the curve from two tangent lines and a point from the conic). Merazi-Meksen et al. [20] detected parabolic forms from Time-Of-Flight Diffraction images in order to analyze material defects. Detection procedure is named randomized Hough transform and is a combination of Least Squares, Randomized Sampled Consensus (RANSAC), and Hough transform. Certainly, robust fitting may be seen as a nonlinear optimization problem that could be solved iteratively by RANSAC method; in particular, this method is able to cope with outliers to estimate the parameter of a desired mathematical method. RANSAC is a simple and powerful method that could be useful in many applications; depending on the model it could have many parameters to tune, but the probability of convergence increases as more iterations are used. The convergence is not warranted because initialization is chosen randomly from a small data subset (i.e., results are not repeatable). In some special cases, RANSAC is not always capable of obtaining the optimal results for well-conditioned data [21, 22].

Most state-of-the-art algorithms use Hough transform for model fitting which is very time demanding. EDAs represent a stochastic optimization technique similar to genetic algorithms, which has begun to attract more attention for solving different problems in the area of image analysis [23, 24]. One of the main advantages of EDAs (UMDA) is that they use global statistical information of the best solutions instead of a crossover or mutation operators. On the other hand, UMDA has only two parameters to be tuned, number of individuals and selection rate, since the number of generations can be replaced by another convergence criterion such as the average or standard deviation of the population.

In this paper, a new method for the parabola detection problem based on Estimation of Distribution Algorithms (EDAs) is proposed. The method is evaluated in terms of computational time on synthetic and medical images of the retina and human plantar arch. Since EDAs represent an evolutionary computation technique, the fitness function used in this work is based on the Hadamard product. EDAs have shown remarkable advantages in order to solve optimization and model fitting problems. In our proposed approach, Univariate Marginal Distribution Algorithm (UMDA) [23] performs a single detection of a parabola faster than HT and 45% faster than RANSAC.

Finally, the results of the proposed method are compared with those obtained by using the Hough transform implementation of Sanchez found in the MATLAB® central [25], the parabolic shapes detection provided by the MIPAV® [26] software, and additionally the well-known RANSAC method proposed by Fischler and Bolles [27]. The remainder of this paper is organized as follows: in Section 2, the fundamentals of the Hough transform, the Estimation of Distribution Algorithms, and the proposed method for detecting parabolic shapes are explained in detail. Experimental results are presented and discussed in Section 3, and conclusions are given in Section 4.

#### 2. Methods

##### 2.1. Parabola Detection

The detection of curves can be achieved by exploring the duality between points on a curve and the parameters representing that curve; this method is known as Hough transform (HT) [28]. Geometric curves such as circles, ellipses, parabolas, and hyperbolas can be parameterized in a polar coordinate system , if the equation that represents the curve satisfies one of the following two equations:where represents the eccentricity and is the shortest distance between the focal points and the directrix of the curve. Since the present work is focused on the detection of parabolic curves, in these equations, the parameter must have a value of 1. The most common parameters to identify a parabola are the vertex , the coordinates of the focus, the orientation of the axis of symmetry with respect to the coordinates axes, and the coefficients of (1).

On the other hand, the equation in the Euclidean space that represents a parabolic curve with directrix parallel to the -axis can be defined as follows:and if the directrix is parallel to the -axis, the equation is modified as follows:where the parameter is used to modify a parabola in two different aspects: the aperture direction and its magnitude. For large values of , the aperture is increased, and if is positive, the parabola of (2) opens up or to the right, while in (3) the parabola opens to the positive -axis, as it is shown in Figures 1(a) and 1(b), respectively. Hence, a parabolic shape is completely determined in the Euclidean space by defining the set of parameters .