A train roof image stitching method based on differential evolution algorithm

Abstract

The present invention discloses a train roof image stitching method based on a differential evolution algorithm, comprising the following steps: collecting a number of train roof fragment images; aligning and stitching the roof images of each train compartment using a differential evolution algorithm, so that the stitched train roof image coincides with the train roof image in a static state. The differential evolution algorithm is used to optimize the image scaling and preliminary displacement to achieve a high degree of overlap between the stitched train roof image and the template train roof in a static state. The present invention introduces the structural similarity of the improved image as a fitness function, and selects a rectangular image landmark with obvious features as the area for calculating the structural similarity, which significantly improves the algorithm rate.

Description

Technical Field

[0001] The present invention relates to the field of intelligent image stitching, and mainly relates to a train roof image stitching method based on a differential evolution algorithm. Background Art

[0002] In 1997, Rainer Storn and Kenneth Price proposed the differential evolution algorithm based on the evolutionary ideas such as genetic algorithms. As a new and efficient heuristic parallel search technology, the differential evolution algorithm has the advantages of fast convergence, few control parameters, simple settings, and robust optimization results. Therefore, it has been widely used in engineering, mathematics, computer science, physics and other fields, making extraordinary contributions.

[0003] Image stitching includes three steps: image preprocessing, image alignment, and image fusion. This is an image processing technology that fuses two or more images with overlapping areas using a certain method to obtain a wider field of view or higher resolution. In the modeling of the train roof, the first thing to do is to collect plane images. The traditional method is to use an area array camera or a line array camera to collect roof images of the moving locomotive. The general image acquisition method is the locomotive passing type, that is, when the train moves through, a high-speed camera placed directly above the roof shoots downward. Due to the long size of the train, the camera cannot capture a complete roof image at one time, so the train roof is usually divided into multiple shots to obtain train segment images. In addition, due to the variable acceleration of the train itself, the scaling factor of the captured train segments is not consistent, which makes it difficult to stitch the captured train segments and overlap them with the template train roof in a stationary state.

[0004] In this case, the use of image stitching algorithms is essential. There are two main branches of image stitching technology, direct pixel-to-pixel methods and feature-based stitching algorithms. Feature-based image stitching technology has better robustness in scale, direction, and projection reduction. Among them, point features mainly include feature points extracted by scale-invariant feature transform (SIFT) (DGLowe. Distinctive image features from scale-invariant keypoints [J]. International Journal of Computer Vision, 2004, 60 (2): 91-110), etc.; line features include features extracted by Susan operators (SMSmith, JM Brady. SUSAN-a new approach to low level image processing [J]. International journal of computer vision. 1997, 23 (1): 45-78), etc.; surface features include some regional features, etc. However, these common feature extraction algorithms will have problems when stitching a large number of train segments, that is, due to the high number of pixels in the picture, the calculation amount is too large and the time consumption is too long, making it difficult to achieve real-time performance. At the same time, most of the area on the train roof is empty, and the number of extractable feature points is not enough, resulting in insufficient image alignment accuracy. In order to solve this problem, the present invention proposes a train roof image stitching technology based on a differential evolution algorithm, which can achieve a high degree of overlap between the stitched train roof image and the template train roof in a static state, and has a certain degree of real-time performance. Summary of the invention

[0005] In order to solve the above problems, the present invention proposes a train roof image stitching method based on a differential evolution algorithm, which can achieve a high degree of overlap between the stitched train roof image and the template train roof in a static state, and has a certain real-time performance. The present invention uses the differential evolution algorithm to establish a model of the train roof collected images, and the differential evolution algorithm uses the structural similarity indicator as the fitness.

[0006] The present invention is achieved by at least one of the following technical solutions.

[0007] A train roof image stitching method based on a differential evolution algorithm comprises the following steps: collecting a number of train roof segment images; aligning and stitching the roof images of each train compartment using a differential evolution algorithm, so that the stitched train roof images overlap with the train roof images in a static state.

[0008] Furthermore, the stitching is to use the train roof image in a stationary state as a template image, stitch the collected train roof images and align them with the template image.

[0009] Furthermore, the step of stitching the collected train roof images and aligning them with the template image comprises the following steps:

[0010] A rectangular area is selected in each carriage of the template image and marked as a landmark. The upper, middle and lower parts of the landmark are respectively cut into square areas with the same side length and landmark width, and then compared with the corresponding areas of the spliced ​​roof image. The structural similarity of the square area images is calculated and added to obtain the fitness value. The individual with the highest fitness value and its corresponding roof image scaling ratio and displacement are recorded.

[0011] Furthermore, the differential evolution algorithm comprises the following steps:

[0012] 21) After initialization, the population begins to evolve with a period of G. For the i-th individual X in the g-th generation population i After randomly generating the scaling factor F, a differential mutation operation is performed to generate the corresponding mutant individual V i ;

[0013] 22) Define the crossover rate and randomly generate a value. If the value is greater than the crossover rate, then for the i-th individual X i and variant individual V i Implement crossover operation to generate test individuals U i ; If not, test individual U i With the variant individual V i Keep consistent and go to step 23);

[0014] 23) For the experimental individual U i Evaluation is performed, and the evaluation index is the structural similarity of the image; if the fitness value is higher than the individual X i (g) The value of fitness, then replace X in the population i (g); if not, go to step 24);

[0015] 24) Define the constant cycle and mutation rate l, for the i-th individual X in the g-th generation population i (g), if g is divisible by cycle, and the randomly generated value is less than the mutation rate l, then the corresponding mutant individual V is generated i , for V i For evaluation, if its fitness value is higher than that of individual X i (g) The value of fitness, then replace X in the population i(g); if not, go to step 25);

[0016] 25) If the end condition is met, terminate; otherwise, go to step 21) to execute.

[0017] Furthermore, the initialization includes variables that need to be optimized, and the variables include the image scaling ratio r and the image initial displacement s; assuming that the number of initialized individuals is n in the population, the initial values ​​are randomly generated.

[0018] Furthermore, the population after initialization is as follows:

[0019]

[0020] Among them, X i (0) is the i-th individual of the initialized population, j represents the j-th parameter of the individual; x i，1 That is, the scaling ratio r of the top image of the carriage of the i-th individual k , x i，2 and x i，3 They represent the displacement from the left boundary of the template image to the left boundary of the roof image and the displacement from the upper boundary of the template image to the upper boundary of the roof image respectively;

[0021] Initial Value in and are the upper and lower bounds of the j-th parameter respectively, and rand(0,1) represents a random number in the interval [0,1].

[0022] Furthermore, the differential evolution algorithm uses SSIM as the fitness function:

[0023]

[0024] where μ x is the mean value of image x, μ y is the mean value of image y, δ x 2 is the variance of x, δ y 2 is the variance of y, δ xy is the covariance of x and y, c 1 =(k 1 L) 2 , c 2 =(k 2 L) 2 , is a constant used to maintain stability, and L is the dynamic range of pixel values.

[0025] Furthermore, the variant individual V i The generation method is as follows:

[0026]

[0027] where x i，j (g) represents the i-th individual X in the g-th generation i The jth parameter to be optimized, M j is a fixed constant. If the calculated parameter value v i，j Greater than the upper limit of this parameter Or less than the lower limit set by this parameter rand(0, 1) represents a random number in the interval [0, 1].

[0028] Furthermore, the formula for the crossover operation is expressed as:

[0029]

[0030] where x i，j (g) represents the i-th individual X in the g-th generation i The jth parameter to be optimized, v i，j Represents x i，j (g) CR is the crossover probability constant of an individual and its value range is [0, 1].

[0031] Furthermore, for the experimental individual U i The evaluation is expressed as:

[0032]

[0033] Among them, f represents the fitness function with the structural similarity of the image as the evaluation index, U i represents the experimental individual obtained after the execution of mutation and crossover operations, X i (g) represents the i-th individual X in the g-th generation i .

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The differential evolution algorithm is applied to the train roof image stitching, and the landmark areas with obvious features are marked for comparison. The SSIM index is used to measure the structural similarity between images, so that the moving train roof image can be aligned with the stationary train roof image with a certain real-time performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 The present invention is a flow chart of a train roof image stitching method based on a differential evolution algorithm according to an embodiment;

[0037] Figure 2 The SSIM value in Example 2 is a curve diagram showing the change with the increase of the number of cycles;

[0038] Figure 3 This is a flow chart of Example 2 in which the differential evolution algorithm is applied to the roof image of the first carriage. DETAILED DESCRIPTION

[0039] The method of the present invention is further described below in conjunction with the accompanying drawings.

[0040] Example 1

[0041] A train roof image stitching method based on a differential evolution algorithm comprises the following steps: collecting a number of train roof segment images; aligning and stitching the roof images of each train compartment using a differential evolution algorithm, so that the stitched train roof images overlap with the train roof images in a static state.

[0042] This embodiment models the model on the roof of a train. The problem to be solved by stitching images collected on the roof of a train is specifically described as follows:

[0043] In order to establish a model of the train roof, a locomotive passing method is used to collect plane images, that is, shooting is performed when the train passes. A high-speed camera is placed on the roof of the train. Due to the long size of the train, the train roof is usually photographed in multiple times to obtain train segment images. Secondly, since the acceleration of the train at different times is not the same, the real-time train roof data generated, that is, the collection of train roof segment images, cannot guarantee the same scaling ratio compared to the stationary train roof. The goal of the present invention is to splice the captured images of the train roof through a differential evolution algorithm, and to achieve that the spliced ​​train roof image is basically consistent with the stationary train roof image.

[0044] To achieve this goal, SSIM (Structure Similarity, SSIM) is used as the fitness function of the differential evolution algorithm. Given two images x and y, the calculation formula of the improved SSIM is as follows:

[0045]

[0046] where μ x is the mean value of x, μ y is the mean value of y, δ x 2 is the variance of x, δ y 2 is the variance of y, δ xy is the covariance of x and y. 1 =(k 1 L) 2 , c 2 =(k 2 L) 2 , is a constant used to maintain stability. L is the dynamic range of pixel values, which is 256. 1 =0.01, k 2 = 0.03. This function describes the structural similarity between image x and image y. The larger the function value, the higher the structural similarity between image x and image y, and the higher the alignment degree.

[0047] In the differential evolution algorithm, the fitness function directly affects the convergence speed of the algorithm and whether the optimal solution can be found, so it needs to be selected carefully. Experiments have shown that this fitness function can converge the fitness value quickly. The larger the SSIM value, the higher the structural similarity of the two images, that is, the higher the degree of image alignment.

[0048] The image of the train roof in a stationary state is used as the template image. Figure 1 A flowchart of the present invention is given to realize the basic alignment of the train roof collection pictures with the template picture after splicing. The specific implementation method of the whole algorithm is described step by step based on the contents of the flowchart:

[0049] 1. Initialization

[0050] In the present invention, since the collected data set is a collection of train roof segment images, and there is a problem that the train's acceleration at different times is different, each segment image has its own independent scaling ratio. In order to roughly align the train roof image with the template image, it is assumed that the scaling ratio of the top image of the same carriage is r k are the same. A train has eight fixed carriages, so 1≤k≤8. For each carriage, the top image T k Differential evolution algorithms are used for alignment. In addition to scaling, each carriage top image will also have a displacement s compared to the template image. k =[left k , up k ], where left k is the distance between the left edge of the template image and the left edge of the roof image, up k It is the distance between the upper boundary of the template image and the upper boundary of the roof image (the range of the roof image captured by default is larger than the template image).

[0051] Put together the top fragments of the first carriage to get T 1 Since the train was traveling at high speed when the camera was taken, and the length of the roof segment image captured by the camera in the direction of the train's travel was very small, the width of the train roof can be regarded as a constant value w. The first carriage T 1 Roof length h 1 Then there is a scaling factor r 1 , and s 1 Both are parameters that need to be optimized, s1 =[left 1 , up 1 ], that is, the displacement difference between the top image of the first carriage and the template image, where left 1 is the distance between the left edge of the top image of the first carriage and the left edge of the roof image, up 1 It is the distance between the upper boundary of the first carriage top image and the upper boundary of the roof image.

[0052] After the data set is processed, the population is initialized to n individuals and the initial values ​​of the parameters to be optimized are given. The initialized population is as follows:

[0053]

[0054] Among them, X i (0) is the i-th individual of the initialized population, and j is the j-th parameter of the individual. i，1 That is, the scaling ratio r of the top image of the carriage of the i-th individual k , x i，2 and x i，3 They represent the displacement from the left boundary of the template image to the left boundary of the roof image and the displacement from the upper boundary of the template image to the upper boundary of the roof image respectively.

[0055] Initial Value in and are the upper and lower bounds of the jth parameter, respectively, and rand(0,1) represents a random number in the interval [0,1]. In the present invention,

[0056] In addition, the fitness function SSIM is needed to measure the structural similarity between the car roof image and the template image when evaluating the initialized population. The present invention does not use the entire roof image to match the template image, because it is too time-consuming to do so; if a part of the roof image is randomly intercepted and compared with the template image, the result will be very error-prone, because there are multiple repeated areas on the top of the train, and it is difficult to accurately find its corresponding part on the template image. Therefore, after comprehensive consideration, a rectangular area with obvious features is selected in each car of the template image (most of the area on the roof of the train is empty, so it is necessary to select car connections, roof vents and other areas with obvious features that can be used for positioning), marked as landmarks, and then compared with the corresponding areas of the spliced ​​roof image, such as the front and rear of the car.

[0057] After finding the landmark on the template image, since the fitness function SSIM requires that the comparison area must be two square areas of the same size, the upper, middle, and lower parts of the landmark will be cut into square areas with the same side length and landmark width. Then, the corresponding area is also cropped in the top image of the first carriage. Note that when considering image scaling, the side length of this area should be multiplied by the scaling ratio r 1 For each individual X in the population i (0), calculate the structural similarity of the three square area images, X i The fitness value x of (0) i，4 (0) is the sum of the three calculated SSIM values. The individual with the highest fitness value and its corresponding roof image scaling ratio and displacement are recorded.

[0058] Image alignment is achieved by optimizing the image scaling and initial image displacement. On the one hand, a rectangular image landmark with obvious features is selected as the area for calculating structural similarity, which improves the calculation speed. On the other hand, each individual in the population undergoes self-mutation after a fixed number of optimization cycles, reducing the number of cycles to find the optimal solution.

[0059] 2. Genetic manipulation

[0060] After the population is initialized, it begins to evolve with a total number of cycles G, and the current generation is g. Like the genetic algorithm, the differential evolution algorithm also performs three genetic operations: mutation, crossover, and selection.

[0061] The differential evolution algorithm achieves individual mutation through differential strategy. i , randomly select three different i Individual X a , X b , X c , (1≤a<b<c≤n), X b and X c The vector difference is scaled with X a Perform vector synthesis. The specific formula for mutation is as follows:

[0062] V i =X a (g)+F(X b (g)-X c (g)) (3)

[0063] Where V i is the mutant individual produced, X a (g) is the evolutionary individual X of the gth generation a , X b(g) is the evolutionary individual X of the gth generation b , X c (g) is the evolutionary individual X of the gth generation c , F is the scaling factor, and is randomly generated in the range of [0, 2] each time the mutation occurs. The corresponding mutant individual V i The three parameter values ​​of are calculated by the above formula through three parent individuals. If the calculated parameter value v to be optimized i，j Greater than the upper limit of this parameter Or less than the lower limit set by this parameter Will replace v with a randomly generated value within a certain range i，j , the specific formula is as follows:

[0064]

[0065] where x i，j (g) represents the i-th individual X in the g-th generation i The jth parameter x to be optimized i，j In this embodiment, there is a certain probability that the evolution of the population will perform a crossover operation, thereby obtaining the experimental individual U i Generate a random integer random in the interval [1, 3], then the formula for the crossover operation can be expressed as:

[0066]

[0067] Where CR is the individual crossover probability constant, and its value range is [0, 1]. j≠random ensures that the variant individual V i At least one parameter will be preserved.

[0068] Get the experimental individual U i After that, calculate its fitness value and compare it with X i (g) Compare and select the individuals with higher fitness values, i.e., better individuals, according to the greedy selection strategy. The selection method is as follows:

[0069]

[0070] Among them, f represents the fitness function with the structural similarity of the image as the evaluation index, U i represents the experimental individual obtained after the execution of mutation and crossover operations, X i (g) represents the i-th individual X in the g-th generation i .

[0071] In the present invention, as the population evolves, after every cycle of cycle, i.e., when g%cycle=0, there is a certain probability l that an individual will mutate again and be selected after completing the operations of mutation, crossover and selection, l is called the mutation rate. i (g), variant individual V i The generation method is as follows:

[0072]

[0073] The random parameter generation of the variant individual is only related to one individual in the population, where M j is a fixed value, whose value is M 1 =0.05, M 2 =50,M 3 = 10. If the calculated parameter value v i，j Greater than the upper limit of this parameter Or less than the lower limit set by this parameter The processing method is the same as shown above, replacing v with a randomly generated value within a certain range i，j , see formula (4) for details. Calculate the fitness value of the variant individual and compare it with X i Compare and select the individual with higher fitness value.

[0074] Determine whether all G cycles of evolution have been completed. If not, enter the next cycle. If yes, end this part of the algorithm.

[0075] 3. Optimize alignment

[0076] Assuming the same carriage top image scaling ratio r k On the same basis, the above steps can achieve the approximate alignment of the train roof image and the template image. However, in fact, the inconsistency of the scaling ratio within the same carriage roof image cannot be ignored, which will greatly affect the alignment operation. Therefore, based on the observation of the image, in the present invention, the rectangular air supply duct on the top of the carriage is regarded as a segmentation element. Since there are two rectangular air supply ducts on the top of each carriage, we use the left and right boundaries of the rectangular air supply duct as the dividing line to divide the top of each carriage into five rectangular images of unequal sizes, which are combined into a new data set. The same operation is also applied to the template image. For each rectangular image in the data set, find its corresponding template image area and adjust the size to be consistent with the corresponding area. The final results show that the train roof image spliced ​​after the completion of this step has a high degree of overlap with the train roof image in a static state.

[0077] Example 2

[0078] This example illustrates the process of applying the differential evolution algorithm to the roof image of the first compartment of a moving train. After the splicing of the captured clips, the upper and lower parts of the image do not belong to the compartment roof, which needs to be cut off first; and the image pixels are too large, and the efficiency of traversing the entire image pixel points is too low. Therefore, in this example, the part of the front of the train is selected as the landmark.

[0079] The differential evolution algorithm is used to align the roof image of the first carriage of the moving train with the template image. There are three parameters to be optimized: the overall scaling ratio r of the carriage roof image 1 , the distance between the left edge of the template image and the left edge of the car roof image left 1 , and the distance up between the upper boundary of the template image and the upper boundary of the car roof image 1 .

[0080] The population size is initialized to 15, and the values ​​of the three parameters to be optimized are initialized within a certain range. The differential evolution algorithm will execute a total of 100 cycles to ensure that its fitness value finally converges. In each cycle, the landmark in the car roof image and the corresponding area of ​​the template image will be used to calculate the fitness value, that is, SSIM, and the individual with the highest fitness value will be kept and recorded at all times. The final result is r 1 =9.00272028e-01, left 1 =2.76640000e+04, up 1 =6.04000000e+02, SSIM=2.78975188e+00. The function graph of SSIM and the number of cycles is as follows Figure 2 shown.

[0081] So far, the car roof image and the template image have been roughly aligned. However, due to the long length of a car, there are also different scaling factors inside the car. In order to solve this problem, the car is further subdivided into five parts (left side of the left vent, left vent, between the left and right vents, right vent, right side of the right vent), and aligned with the corresponding areas of the template image again. Finally, OpenCV is used to subtract the two images to achieve basic alignment of the car roof image and the template image. The specific process is as follows Figure 3 .

[0082] Example 3

[0083] This embodiment describes a process of applying a differential evolution algorithm to an image of a roof of a second carriage of a moving train.

[0084] The overall process of the differential evolution algorithm is consistent with that of Example 2, except for the selection of the landmark area. For the first carriage in Example 2, the front of the car is a distinct feature that distinguishes it from other areas, while for the second carriage, the landmark selected will be the connection between the carriages. The final result is r 2 =9.58745940e-01, left 2 =2.78720000e+04, up 2 =6.04000000e+02, SSIM=2.58707416e+00.

[0085] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific implementation methods described. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A train roof image stitching method based on differential evolution algorithm, characterized by: The following steps are involved: Collect a number of train roof segment images; align and splice the roof images of each carriage of the train using a differential evolution algorithm, so that the spliced ​​train roof images overlap with the train roof images in a static state; The stitching is to use the train roof image in a stationary state as a template image, stitch the collected train roof image and align it with the template image; The step of stitching the collected train roof images and aligning them with the template image comprises the following steps: A rectangular area is selected in each carriage of the template image and marked as a landmark. The upper, middle and lower parts of the landmark are respectively cut into square areas with the same side length and landmark width, and then compared with the corresponding areas of the spliced ​​roof image. The structural similarity of the square area images is calculated and added to obtain the fitness value. The individual with the highest fitness value and its corresponding roof image scaling ratio and displacement are recorded. The differential evolution algorithm comprises the following steps: 21) After initialization, the population begins to evolve with a period of G. For the i-th individual X of the g-th generation population i After randomly generating the scaling factor F, a differential mutation operation is performed to generate the corresponding mutant individual V i ; 22) Define the crossover rate and randomly generate a value. If the value is greater than the crossover rate, then for the i-th individual X i and variant individual V i Implement crossover operation to generate test individuals U i ; If not, test individual U i With the variant individual V i Keep consistent and go to step 23); 23) For the experimental individual U i Evaluation is performed, and the evaluation index is the structural similarity of the image; if the fitness value is higher than the individual X i (g) The value of fitness, then replace X in the population i (g); if not, go to step 24); 24) Define the constant cycle and mutation rate l, for the i-th individual X in the g-th generation population i (g), if g is divisible by cycle, and the randomly generated value is less than the mutation rate l, then the corresponding mutant individual V is generated i , for V i For evaluation, if its fitness value is higher than that of individual X i (g) The value of fitness, then replace X in the population i (g); if not, go to step 25); 25) If the end condition is met, the process terminates, otherwise go to step 21) for execution; The differential evolution algorithm uses SSIM as the fitness function: where μ x is the mean value of image x, μ y is the mean value of image y, δ x 2 is the variance of x, δ y 2 is the variance of y, δ xy is the covariance of x and y, c1=(k1L) 2 , c2=(k2L) 2 , k1=0.01, k2=0.03, is a constant used to maintain stability, and L is the dynamic range of pixel values.

2. The train roof image stitching method based on differential evolution algorithm according to claim 1 is characterized by: Initialization includes variables that need to be optimized, including image scaling ratio r and image initial displacement s. Assume that the number of initialized individuals is a population of n, and the initial values ​​are randomly generated.

3. The train roof image stitching method based on differential evolution algorithm according to claim 2 is characterized by: The population after initialization is as follows: Among them, X i (0) is the i-th individual of the initialized population, j represents the j-th parameter of the individual; x i,1 That is, the scaling ratio r of the top image of the carriage of the i-th individual k , x i,2 and x i,3 They represent the displacement from the left boundary of the template image to the left boundary of the roof image and the displacement from the upper boundary of the template image to the upper boundary of the roof image respectively; Initial Value in and are the upper and lower bounds of the j-th parameter respectively, and rand(0,1) represents a random number in the interval [0,1].

4. The train roof image stitching method based on differential evolution algorithm according to claim 1 is characterized by: Mutant Individual V i The generation method is as follows: where x i,j (g) represents the i-th individual X in the g-th generation i The jth parameter to be optimized, M j is a fixed constant, and the calculated parameter value v i,j Greater than the upper limit of this parameter Or less than the lower limit set by this parameter rand(0,1) represents a random number in the interval [0,1].

5. The train roof image stitching method based on differential evolution algorithm according to claim 1 is characterized by: The formula for the crossover operation is: where x i,j (g) represents the i-th individual X in the g-th generation i The jth parameter to be optimized, v i,j Represents x i,j (g) Parameter value, CR is the individual crossover probability constant, and its value range is [0,1].

6. The train roof image stitching method based on differential evolution algorithm according to claim 1 is characterized by: For the experimental individual U i The evaluation is expressed as: Among them, f represents the fitness function with the structural similarity of the image as the evaluation index, U i represents the experimental individual obtained after the execution of mutation and crossover operations, X i (g) represents the i-th individual X in the g-th generation i .

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