### 1. Introduction

### 2. Fuzzy Clustering

**n**patterns or objects

**X={x**

_{1}**, x**

_{2}**,…, x**

_{n}**}**, where each

**x**

_{i}**ε R**

**is a feature vector consisting of**

^{d}**d**real-valued measurements describing the features of the object represented by

**x**

**[12] and its objective consists of assigning the elements to only one group.**

_{i}**X**into non-overlapping, non-empty partitions

**G**

_{1}**,…, G**

**, whereas, in fuzzy clustering algorithms the goal is to partition the dataset**

_{c}**X**into partitions that allow the data object to belong in a certain (possibly null) degree to every fuzzy cluster [12].

**U = [u**

_{ij}**]**

**, as shown in Eq. (1), where,**

_{c x n}**u**

_{ij}**∈ [0, 1]**indicates the membership of the

**i**

**element to the**

^{th}**j**

**group of the fuzzy cluster. The fuzzy partitioning of the data follows the following conditions:**

^{th}**c**is the cluster,

**n**is data of the image (pixel), and

**u**

**represents the membership degree of a pixel**

_{ij}**i**in group

**j,**and

**c**

**is the center of the group**

_{j}**j**, we have:

### 3. Fourier Transform

### 3.1 Mathematically

**f(x,y)**be a function of two variables representing the intensity of an image, with:

**FT**that

**F(v**

_{x}**,v**

_{y}**)**is generally a complex number, even if

**f(x,y)**is a real number. Thus,

**F(v**

_{x}**,v**

_{y}**)**has amplitude and a phase. One can choose to represent either one or the other, and we were only interested that the amplitude [16,17]. And reciprocally the inverse Fourier transform is as follows:

### 3.2 Adjustment

### 3.3 Filtering and Convolution

**(u, v)**→

**(ω, θ)**, then the value of

**F(ωcosθ, ωsinθ)**for a pair

**(ω, θ)**gives the amplitude of a sinusoid complex in pulsation

**ω**in the direction

**θ**(Fig. 3).

**θ**and decreases steadily with

**ω**. If we decrease the amplitude of the high frequency (low-pass filtering according to

**ω**for all values of

**θ**), the image appears blurred and the edges are less sharp. If instead, we increase the amplitude at high frequencies, the contours are enhanced, but the image seems noisier (it has a larger grain) [17].

### 4. Harmony Search Algorithm

Playing any famous pitch exactly from his/her memory.

Playing a pitch similar to a memorized pitch.

Composing a new pitch from a possible range of pitches.

Harmony Memory Consideration (HMC): A decision value is picked from a stored solution in the Harmony Memory (HM) with the probability of the Harmony Memory Consideration Rate (HMCR).

Pitch Adjustment (PA): Decides whether the chosen decision value from the HM can be adjusted to the neighboring values with a probability of HMCR × Pitch Adjustment Rate (PAR).

Random Search (RS): A decision value picked randomly from a defined range with a probability of (1-HMCR).

**Step 1. Initialize the optimization problem and HS algorithm parameters.**Initialize the parameter of the HS algorithm, the number of decision variables, values range, HMS (i.e., the number of vectors in the solution of harmony memory), HMCR where HMCR ∈ [0, 1], PAR where PAR ∈ [0, 1], and the stopping criteria (i.e., the number of iterations [IT]).**Step 2. Initialize the harmony memory.**Initialize the HM by generating solution vectors from the decision variable (the entire search space of the problem to be optimized).**Step 3. Improvise a New Harmony.**Improvise a new solution vector using HM, HMCR, PAR, RS, and the three HS algorithm improvisation rules.**Step 4. Update the harmony memory.**If the new improvised solution vector is better than the worst solution vector in the HM, replace the worst solution with the new improvisation (in terms of the objective function value).**Step 5. Check the stopping criterion.**Check for convergence. If convergence or stopping criteria have not been met go back to Step 3, otherwise take out the best solution in the HM as the optimal solution vector of the function.

### 5. Clustering by Fuzzy Harmony Search–Fourier Transform

### 5.1 Image Representation

### 5.2 Fourier Transform Application

### 5.3 Initialization of Parameters

### 5.4 Initialization of Harmony Memory

**a**

**of solution vectors is between 0 to 255.) with their respective values of the objective function [4,24], as explained in Section 5.5.**

_{k}### 5.5 FHS-FT Process

*x’=(x*

_{1}

*’, x*

_{2}

*’,⋯, x*

_{cxd}**is generated by considering three rules of improvising a New Harmony (memory consideration, pitch adjustment, and random) until the stopping criterion is reached, as shown in Fig. 4.**

*’)*

*x’=(x*

_{1}

^{1}

*, x*

_{1}

^{2}

*,⋯, x*

_{cxd}

^{HMS}**[24]. It uses the HMCR parameter, which varies between 0 and 1, and is the probability of selecting one value from HM; whereas (1-HMCR), is the probability of randomly selecting from the possible range. The selection procedure for HMCR is shown in Eq. (10) [10,19]:**

*)***is a bandwidth distance that is arbitrarily used to improve the performance of the HS, and in this paper it is set to**

*bw*

*bw***= 0,0001 MaxValeur (n)**; where

**MaxValeur (n) = 255**is the maximum value that can take a pixel and

**rand ()**is a uniform random number between 0 and 1.

**is replaced with**

*xi’*

*xi’ ← xi’***±**

*rand (0,1)***×**

**with probability PAR, while doing nothing with probability (1-PAR) [10,13,19].**

*bw*### 5.6 Objective Function

*x*

*is the*

_{k}*k*th data point,

*v*

*are cluster prototypes (cluster centers),*

_{i}*c*is the number of clusters,

*v*is the grand mean of all data

*x*

_{k}*, u*

*is the membership value of data*

_{ik}*x*

*of class*

_{k}*c*

*, and*

_{i}*|c*

_{i}*|*is the total amount of data belonging to cluster

*i.*

### 6. Experimental Results and Analysis

### 6.1 Image Used

### 6.2 Experiences and Discussion

*Jm*as the membership value of data of class

*u**, while the HS used the index*

_{ij}*Fs*, the FHS algorithm used the index

*Fs*with consideration the membership of data of class

*u**(fuzzy logic), and the FHS-FT algorithm used the index*

_{ij}*Fs*, fuzzy logic, and the Fourier transform.