# Algorithmic Analysis of Vesselness and Blobness for Detecting Retinopathies Based on Fractional Gaussian Filters

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Materials and Methods

#### 3.1. Fractional Order Gaussian Filters

#### 3.1.1. Model Parameters Selection

#### 3.1.2. Differential Evolution (DE) Algorithm

#### 3.1.3. Objective Function

#### 3.2. Kittler Thresholding Method

#### 3.3. Blood Vessels and Fovea Correction

#### 3.3.1. Fovea Detection

#### 3.3.2. Blood Vessels Detection

#### 3.4. Candidate Lesion Classification

## 4. Numerical Results

^{®}Core™i5-6200U CPU @ 2.3–2.4 GHz, and 8 GB RAM. The algorithms were coded and evaluated in MATLAB

^{®}R2016a running on Microsoft

^{®}Windows™10 OS. The design of optimal filters for improving contrast as well as enhancing features are complex tasks, but fundamental in medical imaging. Such a complexity relies on a high number of parameters for tuning and a close correlation among them. The adverse effects of this correlation are reduced by simultaneously adjusting all filter parameters. For this purpose, the DE optimization algorithm was implemented, and its parameters ${C}_{R}$, ${F}_{1}$, and ${F}_{2}$ were analyzed by combining exhaustively of their values. The chosen values that provide the best results were ${C}_{R}=0.5$, ${F}_{1}=0.5$, and ${F}_{2}=0.01$. Figure 9 depicts the objective function value (error) evolution during the tuning process.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DE | Differential Evolution |

SVM | Support Vector Machines |

MESSIDOR | Methods for Evaluating Segmentation and Indexing Techniques Dedicated to |

Retinal Ophthalmology | |

DR | Diabetic Retinopathy |

AVR | Arteriovenous Ratio |

kNN | k-Nearest Neighbors |

KDE | Kernel Density Estimation |

CIE-LAB | Commission Internationale d’Éclairage ${L}^{*}{a}^{*}{b}^{*}$ |

FOGF | Fractional Order Gaussian Filter |

FI | Fundus Images |

OD | Optic Disk |

FOV | Field of View |

$NV$ | NeoVascularization |

H | retinal Hemorrhages |

$mA$ | MicroAneurysm |

MER | Macula Edema Risk |

RD | Retinopathy Degree |

$MCE$ | MaCula to Exudate distance |

DoG | Difference of Gaussian filters |

${N}_{x}(\mu ,\sigma )$ | Normal Gaussian Distribution |

${D}_{x}^{\nu}f\left(x\right)$ | $\nu $-th Fractional Derivative of $f\left(x\right)$ |

${\mathbb{Z}}_{++}$ | Positive Integers and zero |

$\underset{x}{arg\; min}f$ | A point x in the domain of $f\left(x\right)$ where the function is minimized |

$f\circ s$ | Opening of $f\left(x\right)$ by a structuring element s |

$f\oplus s$ | Dilation of $f\left(x\right)$ by a structuring element s |

$f\ominus s$ | Erosion of $f\left(x\right)$ by a structuring element s |

$f\u2022s$ | Closing of $f\left(x\right)$ by a structuring element s |

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**Figure 1.**Internal characteristic elements of a fundus image: macula, fovea, optic disk, and blood vessels.

**Figure 2.**Characteristics elements associated with retinopathies. These elements are indicated with black arrows and dashed circles.

**Figure 4.**FOGF kernels using the base Gaussian function $f\left(\mathbf{x}\right)={N}_{x}(0.0,1.0)$ for the four fractional derivative orders $\nu \in \{0.1,0.4,1.3,1.9\}$.

**Figure 8.**Blood vessels detection: (

**a**) response of the Frangi filter, (

**b**) Kittler thresholding, and segmentation using (

**c**) a fixed threshold $T=0.01$ and (

**d**) a fixed threshold $T=0.01$ and a length filter of 100 pixels, $|\overline{{V}_{l}}|=100$.

**Figure 10.**Microaneurysms and hemorrhages detection: (

**a**) original image; (

**b**) FOGF tuned to detect microaneurysms (${D}_{x}^{0.98}{N}_{x}(0,2.72)\phantom{\rule{0.277778em}{0ex}}\forall x\in [-10.5,10.5],\phantom{\rule{0.277778em}{0ex}}k=22$); (

**c**) retinal image response from an FOGF; and (

**d**) microaneurysms, hemorrhages, fovea, and blood vessels detected (binary output image).

**Figure 12.**Hemorrhages detection using proposed method: (

**a**) original image, (

**b**) elements detected as microaneurysms and hemorrhages, and (

**c**) final detection obtained by thresholding.

**Table 1.**Specific constraints to evaluate the Retinopathy Degree (RD) and Macular Edema Risk (MER), by employing the ocurrence frequency of microaneurysms ($mA$), hemorrhages (H), and neovascularization ($NV$), as well as the MaCula to Exudate distance (${\widehat{MCE}}_{\mathrm{min}}$) and the optic disk diameter (${D}_{\mathrm{OD}}$) [25].

Level | Retinopathy Degree (RD) | Macular Edema Risk (MER) |
---|---|---|

0 | $(mA=0)\cap (H=0)$ | Non-visible exudates |

1 | $(0<mA\le 5)\cap (H=0)$ | ${\widehat{MCE}}_{\mathrm{min}}>{D}_{\mathrm{OD}}$ |

2 | $(5<mA\le 15)\cup (H<5)\cap (NV<0)$ | ${\widehat{MCE}}_{\mathrm{min}}\le {D}_{\mathrm{OD}}$ |

3 | $(mA\ge 15)\cup (H\ge 5)\cup (NV=1)$ | – |

Metric | Binary | Output Value |
---|---|---|

Description | Metric | (Avg. ± St. Dev.) |

Accuracy | Acc | 0.9995 ± 0.0004 |

Balanced Accuracy | BAcc | 0.8909 ± 0.0927 |

Sensitivity | Sen | 0.7820 ± 0.1853 |

Specificity | Spe | 0.9998 ± 0.0001 |

Computing Time | ${T}_{c}$ | 15.4170 ± 2.7757 [s] |

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## Share and Cite

**MDPI and ACS Style**

Estudillo-Ayala, M.d.J.; Aguirre-Ramos, H.; Avina-Cervantes, J.G.; Cruz-Duarte, J.M.; Cruz-Aceves, I.; Ruiz-Pinales, J.
Algorithmic Analysis of Vesselness and Blobness for Detecting Retinopathies Based on Fractional Gaussian Filters. *Mathematics* **2020**, *8*, 744.
https://doi.org/10.3390/math8050744

**AMA Style**

Estudillo-Ayala MdJ, Aguirre-Ramos H, Avina-Cervantes JG, Cruz-Duarte JM, Cruz-Aceves I, Ruiz-Pinales J.
Algorithmic Analysis of Vesselness and Blobness for Detecting Retinopathies Based on Fractional Gaussian Filters. *Mathematics*. 2020; 8(5):744.
https://doi.org/10.3390/math8050744

**Chicago/Turabian Style**

Estudillo-Ayala, Maria de Jesus, Hugo Aguirre-Ramos, Juan Gabriel Avina-Cervantes, Jorge Mario Cruz-Duarte, Ivan Cruz-Aceves, and Jose Ruiz-Pinales.
2020. "Algorithmic Analysis of Vesselness and Blobness for Detecting Retinopathies Based on Fractional Gaussian Filters" *Mathematics* 8, no. 5: 744.
https://doi.org/10.3390/math8050744