# Systematic Review of Aggregation Functions Applied to Image Edge Detection

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## Abstract

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## 1. Introduction

**1.**- Fill this gap in the literature of systematic reviews on the topic;
**2.**- Summarize the existing technology regarding methods that make use of those functions in digital image processing, more specifically regarding edge detection;
**3.**- Identify the gaps in the detection approach using aggregation or pre-aggregation functions, proposing themes for future work.

**Preliminaries**), the most widespread edge detection methods to date are presented chronologically. Section 3 (

**Materials and Methods**) presents the search terms, where inclusion, delimitation, and exclusion criteria are discussed, as well as the number of papers found and admitted for review. Then, in Section 4 (

**Results and Discussion**), we present the aforementioned research question, summarizing the methods reviewed and performing a qualitative assessment that compares the main techniques and approaches. Finally, Section 5 is the

**Conclusions**.

## 2. Preliminaries

#### 2.1. Aggregation and Pre-Aggregation Functions

#### 2.2. Gradient-Based Methods

**(i)**- Computation of the magnitude of the gradient $|\nabla I|$ and its orientation $\eta $, using a kernel $3\times 3$, (Table 1), with a first-order, vertical and horizontal derivative filter such as steerable Gaussian, oriented anisotropic Gaussian kernels, or a combination of two half-Gaussian kernels;
**(ii)**- Non-maximum suppression operation for thinner edges: selection of pixels with a local maximum gradient magnitude along the direction $\eta $ of the gradient, which is perpendicular to the orientation edge;
**(iii)**- Determination of the thresholds of the fine contours to obtain the contour map.

#### 2.3. Region-Based Segmentation Methods

#### 2.4. Methods Based on Machine Learning

#### 2.5. Fuzzy-Logic-Based Methods

## 3. Materials and Methods

#### 3.1. Systematic Literature Review Scope

#### 3.2. Definition of Criteria, Search in Indexing Databases, and Obtaining Primary Research

**(i)**- “preaggregation” OR “pre-aggregation" OR “aggregation” OR “fusion” OR “Sugeno” OR “Type-2” that appear in the abstract OR;
**(ii)**- The terms of the item
**(i)**joined to the terms “Fuzzy” and “Logic” concatenated with “Type-2” with the exception of “Sugeno” as terms that appear in the title; **(iii)**- The terms of the items
**(i)**and**(ii)**in conjunction with the terms “image” AND “segmentation” OR “image” AND “processing” OR “edge” AND “detection” as terms that appear in the abstract.

## 4. Results and Discussion

Found Papers | Papers Included | Total Number of Reviewed Papers |
---|---|---|

[44,51,80,81,82,83,84,86,87,89,90,93,94,95,96,97,98] | [45,46,47,48,85,88,91] | 24 |

#### 4.1. Summary of Methods

#### 4.1.1. Multiple Descriptors Extraction and Aggregation

**Definition 1**

#### 4.1.2. Based on the Aggregation of Distance Functions and FCM

- D is the number of data points;
- N is the number of clusters;
- m is a fuzzy partition matrix exponent for controlling the degree of fuzzy overlap, with m > 1. Fuzzy overlap refers to how fuzzy the boundaries between clusters are; that is, the number of data points that have significant membership in more than one cluster;
- ${\mu}_{i,j}$ is the degree of membership of ${x}_{i}$ in the jth cluster. For a given data point ${x}_{i}$, the sum of the membership values for all clusters is one;
- ${d}_{i,j}^{2}$ is a measure of distance, which, in general, can vary according to the proposed approach, but is classically given by $\left|\right|{x}_{i}-{c}_{j}{\left|\right|}_{A}^{2}$, where ${\left|\right|.\left|\right|}_{A}$ is a norm.

**(i)**- The ${d}_{S}$ intensity of pixels ${p}_{1}$ and ${p}_{1}$, given by:$${d}_{S}({p}_{1},{p}_{2})=\frac{1}{255}|{s}_{2}-{s}_{1}|$$
**(ii)**- The average pixel in an eight-connected neighborhood, calculated as$${d}_{N}({p}_{1},{p}_{2})=\frac{1}{255}|{n}_{2}-{n}_{1}|.$$

#### 4.1.3. Based on Fuzzy Set Theory: Type-2 Fuzzy and Neutrosophic Set

**(i)**- Multiple descriptors extraction and aggregation;
**(ii)**- Based on the aggregation of distance functions and FCM;
**(iii)**- Based on fuzzy set theory: type-2 fuzzy and neutrosophic sets.

**(i)**- These are methods that take direct inspiration from the way that the human visual system works, seeking to simulate, through feature extraction or the determination of local or global descriptors, the visual stimuli and the processing of information for the recognition of some pattern through aggregation or pre-aggregation functions.
**(ii)**- Functions that make use of weighing vectors and that enable a better modeling of nonlinear behavior, such as the Choquet integral, would perform better.
**(iii)**- Depth information is essential, especially when working with images with complex backgrounds or low contrast, the latter being an important feature in the performance of the models.
**(iv)**- Combining detection methods with aggregation methods or information fusion makes it possible to reduce information uncertainty and minimize redundancy, improve reliability, and maximize the information relevant to a task.
**(v)**- Works that link extraction and detection via clustering functions have not been found.

**(i)**- They are based on the importance of distance functions as decision criteria in data clustering algorithms;
**(ii)**- Aggregation and pre-aggregation functions are used to construct new distance functions from others that represent some dimension relevant to the problem;
**(iii)**- The papers are limited to application to a single clustering algorithm and do not make the performance of other methods clear.

## 5. Conclusions

**(i)**- Multiple descriptors extraction and aggregation;
**(ii)**- Based on the aggregation of distance functions and FCM;
**(iii)**- Based on fuzzy set theory: type-2 fuzzy and neutrosophic sets.

- (i)
- Fill an existing gap in the literature of systematic reviews on edge detection using clustering functions;
- (ii)
- Summarize the existing technology regarding methods that make use of these functions in edge detection;
- (iii)
- Identify the gaps in the detection approach using aggregation or pre-aggregation functions, proposing topics for future work and fulfilling the initial objectives of the research.

**(i)**- The application of the method of constructing distance functions by clustering functions in different data clustering techniques, such as DBscan, region growth, K-means, and others;
**(ii)**- Applying non-average aggregation functions for edge detection;
**(iii)**- The development of further work involving the modeling of W ganglion cells; in this sense, work with depth information, such as lidar sensors and others;
**(iv)**- Exploring the combination of aggregation functions, both in feature extraction and in information fusion;
**(v)**- The direct use of classical detectors in an ensemble fused by aggregation functions, also taking into account the fusion of descriptors and other visual cues, thus ensuring the participation of primitive shapes and the influence of contrast, color, and depth information, among others discussed here.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Table 1.**The gradient magnitude and orientation calculation for a scalar image I, where ${I}_{\theta}$ represents the derivative of the image using the first-order filter in orientation $\theta $ in radians [60].

Operator Type | Fixed Operator | Filter-Oriented | Half-Gaussian Kernels |
---|---|---|---|

Magnitude of gradient | $|\nabla I|=\sqrt{{I}_{0}^{2}+{I}_{\pi /2}^{2}}$ | $|\nabla I|={max}_{\theta \in [0,\pi [}\left|{I}_{\theta}\right|$ | $|\nabla I|={max}_{\theta \in [0,2\pi [}|{I}_{\theta}|-{min}_{\theta \in [0,2\pi [}\left|{I}_{\theta}\right|$ |

Gradient direction | $\eta =arctan\frac{{I}_{\pi /2}}{{I}_{0}}$ | $\eta ={arg\; max}_{\theta \in [0,\pi [}\left|{I}_{\theta}\right|+\frac{\pi}{2}$ | $\eta =\left({arg\; max}_{\theta \in [0,2\pi [}{I}_{\theta}+{arg\; min}_{\theta \in [0,2\pi [}{I}_{\theta}\right)/2$ |

Multiple Descriptors Extraction and Aggregation | Based on Aggregation of Distance Functions and FCM | Based on Fuzzy Theory: Type-2 Fuzzy and Neutrosophic Set |
---|---|---|

[44,45,47,80,81,82,83,84,85,86,87,93,95,96,97,98] | [46,48,51,52,91] | [88,89,90,94] |

Ref. | Aggregation | Measures | Clustering Algorithm | |
---|---|---|---|---|

[91] | OWA | ${d}_{S}({p}_{1},{p}_{2})=\frac{1}{255}|{s}_{2}-{s}_{1}|$ e ${d}_{N}({p}_{1},{p}_{2})=\frac{1}{255}|{n}_{2}-{n}_{1}|$ | ^{1} | |

[48] | GQ-AM | ${C}_{\omega}={\tau}^{\omega}.{t}^{\omega}$ | ^{2} | |

[51] | EPP and EWAMP | ${d}_{\lambda ,\omega}({p}_{1},{p}_{2})$ | ^{3} | FCM |

[46] | OWA and PP and WAMP | ${d}_{\alpha}{\omega}_{\left[5\right]},{\lambda}_{\left[5\right]}({p}_{i,j},{p}_{k,n})$ | ^{3} | |

[52] | AOOCC | ${d}_{{\alpha}_{r},{\alpha}_{g},{\alpha}_{b}}(({r}_{1},{g}_{1},{b}_{1}),({r}_{2},{g}_{2},{b}_{2}))$; ${d}_{r}(({r}_{1},{g}_{1},{b}_{1}),({r}_{2},{g}_{2},{b}_{2}))$; ${d}_{g}(({r}_{1},{g}_{1},{b}_{1}),({r}_{2},{g}_{2},{b}_{2}))$ and, ${d}_{b}(({r}_{1},{g}_{1},{b}_{1}),({r}_{2},{g}_{2},{b}_{2}))$ | ^{4} |

^{1}S

_{i,j}represents the gray level of the pixel and ni,j the average gray levels in the eight-connected neighborhood;

^{2}τ and t described in Equations (25) and (26), where F

_{i}are the normalized color components, D

_{i}is a similarity descriptor, and K a coefficient of adjustment;

^{3}Equation (16), where ω and λ are adjustment parameters;

^{4}Equations (12)–(15).

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**MDPI and ACS Style**

Amorim, M.; Dimuro, G.; Borges, E.; Dalmazo, B.L.; Marco-Detchart, C.; Lucca, G.; Bustince, H.
Systematic Review of Aggregation Functions Applied to Image Edge Detection. *Axioms* **2023**, *12*, 330.
https://doi.org/10.3390/axioms12040330

**AMA Style**

Amorim M, Dimuro G, Borges E, Dalmazo BL, Marco-Detchart C, Lucca G, Bustince H.
Systematic Review of Aggregation Functions Applied to Image Edge Detection. *Axioms*. 2023; 12(4):330.
https://doi.org/10.3390/axioms12040330

**Chicago/Turabian Style**

Amorim, Miqueias, Gracaliz Dimuro, Eduardo Borges, Bruno L. Dalmazo, Cedric Marco-Detchart, Giancarlo Lucca, and Humberto Bustince.
2023. "Systematic Review of Aggregation Functions Applied to Image Edge Detection" *Axioms* 12, no. 4: 330.
https://doi.org/10.3390/axioms12040330