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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

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 [

Circular Hough transform (CHT) is based on the Hough transform principle and it has been adapted for the detection of circles [

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

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 [

Some other approaches have been proposed for parabola detection. Salehin et al. [

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 [

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) [

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 [

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) [

On the other hand, the equation in the Euclidean space that represents a parabolic curve with directrix parallel to the

Parametric curves described by (

To detect parabolic shapes in images using the Hough transform algorithm, all the pixels with intensity different to zero and with coordinates

Example of a parabola accumulator using the Hough transform.

The main disadvantages of the HT are the computational time it takes to determine the best parameter values and the selection of the optimal peak in the accumulator, where the most commonly strategy used to find it is the local maxima method.

The Estimation of Distribution Algorithms (EDAs) represent an extension to the field of evolutionary computation (EC). EDAs are useful to solve problems in the discrete and continuous domain by using some statistical information of potential solutions, also called individuals [

By using these explicit probabilistic models, EDAs are able to solve optimization problems to cope with high level of epistasis. The principal advantages of EDAs over genetic algorithms are the absence of multiple parameters to be tuned and the expressiveness and transparency of the probabilistic model that guides the search process. EDAs have been proven to be better suited to some applications than GAs, while achieving competitive and robust results in the majority of tackled problems [

This section describes the proposed method to detect parabolic shapes in images using the population-based method of UMDA along with the objective function to evaluate the fitness for each potential solution. The most representative steps of the proposed methodology for parabola detection are represented in Algorithm

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(2) EvaluatePopulation (Population)

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(12) EvaluatePopulation(Offspring)

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(16) Break

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A parametric equation that describes a parabola is required to represent a potential solution for finding parabolic shapes in images. In the Cartesian coordinate system, a parabola can be described by using its general form as follows:

The whole set of potential pixels in the input image are listed by their relative position to an origin, and they are labeled with an index

Finally, the coordinate values of the vertex and the aperture of the parabola can be computed by using the following:

To illustrate the representation of a potential solution in an optimization model fitting process, Table

Example of an individual with 3 indexes where each index represents an 8-bit pixel position.

Individual | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Index | Index | Index | |||||||||||||||||||||

0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |

Since the proposed method uses the indexes to form all the individuals, the UMDA method can easily eliminate unfeasible solutions by using a function to measure the quality of an individual.

To evaluate the fitness of an individual, a binary image

The resulting image

Hadamard product between virtual and input images.

The fitness function used to assess the quality of potential solutions is the number of pixels resulting of the Hadamard product. Given that the binary image of the virtual shape is initialized with all pixels on black (intensity value of zero), with more matching pixels in

In this section, the proposed method for parabola detection is applied on synthetic images and medical images of the retina and human plantar arch. In order to assess the proposed method, it is compared with the Hough transform by applying the algorithm of Sanchez found in the MATLAB central [

The computational implementations are performed by using the MATLAB version 2013b, on a computer with an Intel Core i5, 4 GB of RAM, and 2.4 GHz processor. Moreover, computational experiments using UMDA were performed using 30 runs in order to perform a statistical analysis of the stochastic process applying the parameter values presented in Table

UMDA parameters for all the computational experiments.

Parameter | Value |
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Number of individuals | 10 |

Selection rate | 0.6 |

Maximum number of generations | 30 |

Comparative analysis of execution time using the binary image (Figure

Method | Execution time (s) | Number of iterations | ||
---|---|---|---|---|

Proposed method | Minimum | 1.7141 | Minimum | 15 |

Maximum | 3.5928 | Maximum | 28 | |

Mean | 2.3756 | Mean | 19.55 | |

Median | 2.1934 | Median | 19 | |

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Hough transform [ | 8.8205 per pixel | — | ||

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MIPAV [ | 37.68 | — | ||

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RANSAC [ | 16.35 | 5000 |

Comparative analysis of execution time using the skeleton of the binary image (Figure

Method | Execution time (s) | Number of iterations | ||
---|---|---|---|---|

Proposed method | Minimum | 0.2820 | Minimum | 13 |

Maximum | 3.2988 | Maximum | 29 | |

Mean | 1.4163 | Mean | 17.33 | |

Median | 1.4326 | Median | 18 | |

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Hough transform [ | 8.8205 per pixel | — | ||

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MIPAV [ | 36.47 | — | ||

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RANSAC [ | 14.86 | 5000 |

These parameter settings were determined based on the solutions that give the best tradeoff between precision (lowest RMSE) and computational time using 30 trials to determine the best set of parameters. Moreover, different works were taken into account such as [

The first experiment was performed by using synthetic images like the one in Figure

Parabola detection on synthetic image using the proposed method.

Table

To ensure that the algorithm is robust and results are consistent with different input conditions, the complexity of the image was increased by adding “salt and pepper” noise. The test was performed by obtaining the skeleton of Figure

In order to quantify the fitness of detected parabolas, ten synthetic images with parabolas were generated using (

A test by adding “salt and pepper” noise over the range

Average computational time and standard deviation (in seconds) over 30 runs of UMDA per percentage noise for the skeleton image (Figure

Figure

Comparative RMSE for synthetic and real images shown in Figure

Algorithm | Parameters ( | Time (s) | RMSD | uRMSD | Matched points |
---|---|---|---|---|---|

Proposed method | ( | 4.464 | 9.330 | 64.230 | 1439 |

RANSAC [ | ( | 12.993 | 8.453 | 211.137 | 1625 |

Proposed method | ( | 15.490 | 13.895 | 426.277 | 2752 |

Proposed method | ( | 4.022 | 9.641 | 77.995 | 1939 |

RANSAC [ | ( | 27.099 | 13.932 | 340.910 | 1258 |

Proposed method | ( | 4.357 | 9.712 | 182.037 | 4178 |

Parabola detection performed by the proposed approach and MIPAV software.

Comparative images between RANSAC and the proposed method (noise added). (a) Retinal image process by the proposed algorithm, (b) retinal image processed by RANSAC, (c) synthetic image processed by the proposed method, and (d, e, f) images from (a, b, c) with

The detection of parametric objects has been applied in different areas of engineering. In medical imaging, the detection of parabolas in retinal fundus images has a particular importance, since the form of the retinal vessels can be approximated to the parabola parametric form. In the previous work, the standard model, the Hough transform, has been used to find the parabola that fits the best on the retinal images. For instance, in the work reported by Oloumi et al. [

In the tests reported in the present work, the proposed method was used to approximate the retinal vessels to a parabola. Figure

Results of parabola detection using the proposed method over a subset of retinal fundus images from the DRIVE database.

Table

Comparative analysis of the average execution time using the DRIVE database of retinal fundus images.

Method | Execution time (s) | Number of iterations | ||
---|---|---|---|---|

Proposed method | Minimum | 1.2280 | Minimum | 5 |

Maximum | 6.3285 | Maximum | 22 | |

Mean | 4.5838 | Mean | 16.28 | |

Median | 5.1588 | Median | 19 | |

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Hough transform [ | 8.8205 per pixel | — | ||

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MIPAV [ | 43.65 | — | ||

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RANSAC [ | 16.66 | 5000 |

Given that the skeleton of an image passes through the center section of a set of pixels, if the proposed algorithm is applied to the skeleton of the retinal images, it is then expected for the algorithm to achieve a lower execution time than in the binary image. In Table

Comparative analysis of execution time using the skeleton of the retinal fundus images (DRIVE database).

Method | Execution time (s) | Number of iterations | ||
---|---|---|---|---|

Proposed method | Minimum | 0.9550 | Minimum | 4 |

Maximum | 5.7105 | Maximum | 20 | |

Mean | 3.3706 | Mean | 12 | |

Median | 2.7343 | Median | 10 | |

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Hough transform [ | 8.8205 per pixel | — | ||

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MIPAV [ | 41.87 | — | ||

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RANSAC [ | 12.99 | 5000 |

Since the proposed method takes three pixels to compute a parabola, it can be assumed that the number of iterations to achieve the best result is not related to the size of the image. This assumption is validated with the results shown in Tables

The form of the plantar arch in humans can conduct to several conditions if the contact area with the floor, given by the plantar arch, is not of the correct size. In clinical practice, the procedure for providing a diagnosis of flatfoot degree is by visual inspection of the specialist. Therefore, the proposed method in this work can be applied to plantar arch images to approximate the heel and the plantar arch to parabolas; hence, these parameters could be used to assist a medical diagnostic. The image database used in this test was created by the authors and approved by a specialist (Dr. Carlos Reséndiz Ramírez). This database is composed of 80 images of size

Let

The first step of the method consists in smoothing the image by applying a mean filter, that is, convolving the image with a filter mask:

This filter removes part of the spurious pixels, improving the probability of finding the curves of interest. After the smoothing step, gradients are computed on the image in order to find the edges of the footprint. Canny edge detection is the algorithm used to perform this step. The resulting contour image

The

In order to address the plantar arch issue, the foot image was divided into three sections from the fingers to the heel, where each section contains 33.33% of the image. The sections of interest for this study are only those two that include the plantar arch and the heel.

Table

Comparative analysis of execution time using the database of human plantar arch images.

Method | Execution time (s) | Number of iterations | ||
---|---|---|---|---|

Proposed method | Minimum | 4.2118 | Minimum | 3 |

Maximum | 7.1688 | Maximum | 26 | |

Mean | 6.18 | Mean | 13.51 | |

Median | 6.40 | Median | 14.15 | |

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Hough transform [ | 8.8205 per pixel | — | ||

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MIPAV [ | 63.78 | — | ||

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RANSAC [ | 22.54 | 5000 |

(a) Subset of human plantar arch images. (b) Preprocessing step applied to the images in (a). (c) Results of parabola detection (plantar arch and heel) over the edge images. (d) Quantitative analysis of the detected parabolas from the images in (a).

In addition, the comparative results in Table

Finally, Figure

Evolution of the average fitness through generations using the database of human plantar arch images.

In this paper, a new method based on the Estimation of Distribution Algorithms (EDAs) has been proposed to detect parabolic shapes. The method computes the constant values of the generic parabola equation by selecting three random pixels from the input image. The proposed method was evaluated in terms of computational time and compared with freely available implementations of the parabola Hough transform. According to experimental results, the average time of the proposed method is significantly better (

The authors declare that they have no competing interests.

This work has been supported by the Mexican National Council on Science and Technology under Grant no. 429450/265881 and Cátedras-CONACYT no. 3150-3097. The authors would like to thank the doctors Carlos Reséndiz Ramírez and Hilda Alejandra Sierra-Hernandez for providing their clinical advice and expertise. Moreover, a special recognition is due to the bachelor student Joao Manuel Calvillo-Rodriguez for his support to collect the database of images.