Open Access Published by De Gruyter November 28, 2023

Multilevel thresholding image segmentation algorithm based on Mumford–Shah model

Xiancai Kang and Chuangli Hua

From the journal Journal of Intelligent Systems

https://doi.org/10.1515/jisys-2022-0290

Abstract

Image segmentation is one of the important tasks of computer vision and computer image processing, and the purpose of image segmentation is to achieve the extraction and recognition of the target image region. The classical Mumford–Shah (MSh) image segmentation model is used to achieve the segmentation of images. With the goal to get the best segmentation effect on images by minimizing the MSh energy generalization function, a level set strategy is developed, and a model with global information infinite curve evolution is utilized. However, considering the low efficiency of this model for processing level set curves and the general quality of image segmentation. A multi-layer threshold search scheme is proposed to achieve rapid convergence of the target image level set curve. The experimental results showed that the multi-level thresholding image segmentation algorithm based on the MSh model can significantly improve the segmentation effect of images and reduce the segmentation time. The suggested MSK method outperforms the MPO algorithm, SSA algorithm, and EMO algorithm in the picture segmentation convergence time test, respectively, in terms of runtime efficiency by 356, 289, and 71%. Additionally, it performs superbly in both threshold searches and picture quality tests. The research topic has significant reference value for the study of contemporary computer vision imaging technologies.

Keywords: Mumford; Shah model; multilevel thresholding; image segmentation; convergence

1 Introduction

With the continuous development of computer vision processing technology, digital processing techniques have a wide range of applications in image processing. Common digital image processing techniques include image compression, image segmentation, and image identification. Image segmentation is mainly based on the pixel characteristics of the image to divide the target image into homogeneous, non-overlapping image regions to achieve the analysis and processing of the image [1,2]. However, image segmentation has been a difficult task in computer vision processing and is the focus of the current development of digital image processing technology. Edge segmentation, region segmentation, threshold segmentation, and particular theoretical combinations of image segmentation are some of the popular segmentation processing approaches used today [3]. One of them, threshold segmentation, has the most applications in the field of image segmentation due in large part to the threshold image segmentation method’s ease of use, calculation simplicity, and broad variety of applications. The Mumford–Shah (MSh) model belongs to the more classical region image segmentation model, which is an image segmentation model method evolved on the basis of variational image segmentation. However, this model has the problem of inconsistency between contour lines and image contours, which leads to slow convergence of image segmentation. To increase the precision and segmentation performance of the image segmentation algorithm, the MSh model approach and the threshold segmentation method are merged [4,5]. The research’s findings are highly relevant to the advancement and development of computer vision technology.

2 Related work

The segmentation of images is one of the key components of computer vision processing methods. Researchers both domestically and overseas have conducted a significant amount of research on picture segmentation algorithms. Wang and Shi [6] combined the advantages of contour wave transform and adaptive fuzzy Markov random field model to propose a new segmentation algorithm to achieve accurate continuous segmentation of synthetic aperture radar images. The results showed that the algorithm achieved better results in noise suppression, target region smoothing, and accurate continuous segmentation of fuzzy textures. Lin et al. [7] found that the traditional spectral clustering algorithm leads to loss of image information and degrades the segmentation performance. To overcome the problems of traditional spectral clustering, an image segmentation algorithm based on super pixel clustering was proposed. The similarity matrix is used to provide input information for the spectral clustering algorithm to obtain the final image segmentation results. The results show that the algorithm can effectively improve the performance of image segmentation compared to the traditional spectral clustering algorithm. To segment apples under various colors and lighting conditions during the growing phase, Wang et al. [8] researched a color-independent segmentation technique. To eliminate the backdrop and extract apples, the algorithm combines the saliency and contour elements of the image. The methodology outperformed six other methodologies, according to the data. Bhandari and Rahul [9] proposed a new Masi entropy image segmentation method based on context-sensitive energy profile. A new stochastic metaheuristic optimization algorithm was introduced. It is used to simplify the extensive exploration problem of finding the optimal threshold and to improve the image quality. The results showed that MSA provided higher performance in terms of threshold quality and low computational cost, unlike other metaheuristic algorithms used for thresholding operations. Optimal Path Snake (OPS), a novel adaptive technique with no parameters for calculating the total energy of an active contour model with automatic initialization and ending conditions, was proposed by Filho et al. [10]. The outcomes demonstrated that OPS was a technique for picture segmentation with promise, producing positive outcomes for both discrete cosine transform and high definition technology. Manju and Lenin Fred [11] created an efficient segmentation method to separate background images, text, and graphics from composite images for independent compression. The method was implemented on MATLAB workbench and the results were analyzed. For supercomplex Chebyshev orthogonal moments paired with fractional order chaotic scrambling, Tao and Qian [12] proposed an image hash authentication technique. The findings demonstrated the good performance of the suggested approach. An enhanced partial differential equation function and circular segmentation mechanism were utilized to process the initial image and output the secondary image. Shubham and Bhandari [13] proposed a new multilevel threshold criterion for color satellite images based on Masi entropy. The algorithm is based on Masi entropy and deals with additive/non-extended information through the cohesive entropy parameter r′ and the results showed that the proposed Masi entropy-based algorithm had better performance for normal and color satellite image segmentation. Experiments were conducted on various color test images to specify the efficiency of the algorithm. Many fidelity parameters were calculated for segmentation purposes. Previous algorithms for segmenting 2D pictures, according to Zhu et al. [14], modeled the color, position, or higher spectral information. However, there is no full segmentation solution to eliminate the blurring in the bokeh and occlusion boundary regions due to the limitations of the Gaussian imaging principle in typical cameras. Light in light space is taken into consideration as the essential component of picture pixels, and light field super pixels are offered to clear up any confusion. In terms of traditional evaluation measures, the experimental findings demonstrated the benefit above the current state-of-the-art.

With the continuous development of computer artificial intelligence technology, neural grid algorithms have a wide range of applications in the field of computer vision. A variable filter size residual learning convolutional neural network with batch normalization layer was proposed as a result of Zhao et al.’s [15] discovery of several irksome distortions and artifacts in lossy compressed films. We use the model to analyze chromaticity and brightness of images, in contrast to earlier techniques. The outcomes demonstrated that the method worked better than already available equivalent methods. Chen et al. [16] found that deep neural networks achieved significant results in improving accuracy. Binary neural networks were then utilized to learn differentiated binary descriptors to improve parallax. However, CNNs are usually over-parameterized and contain a large number of redundant filters or parameters. A unified algorithm was then proposed to efficiently compress CNNs of in air handwritten chinese character recognition with little loss of accuracy. The results showed that the evaluation of other benchmark datasets (including ICDAR-2013 and MNIST) further demonstrated the effectiveness of the method [17]. CNNs are computationally demanding and have a large memory requirement, according to Gamanayake et al. [18]. Then, solutions for the aforementioned issues, including weight pruning, filter pruning, and quantization, were presented for network compression techniques. Using both datasets and the aforementioned hardware architecture, a novel greedy strategy dubbed clustering pruning was presented. The results demonstrated that our method outperformed the conventional filter pruning method. Boogaard et al. [19] studied the internode length of cucumber plants and proposed a method to estimate internode length and internode development over time. The results showed that the method was able to measure the internode length of cucumber plants with higher accuracy and greater temporal resolution. In order to increase the precision of small-scale human motion detection in video and the computational effectiveness of large-scale datasets, Gao et al. [20] suggested a multidimensional data model for motion recognition and motion capture in video pictures based on a deep learning framework. The gradient histogram can be used to identify the human body. The findings indicate that the algorithm’s average classification accuracy is 85.79%. The algorithm operates at a speed of 20 frames per second.

The domestic research in this area shows that neural network methods have many uses in the field of computer vision, greatly enhancing the computer’s ability to handle picture data. Additionally, the use of neural network algorithms for picture segmentation will advance computer technology in the area of image processing.

3 Multi-level thresholding image segmentation algorithm based on MSh model

3.1 Multi-level image thresholding processing and expression

Image segmentation has been the focus and difficulty of research in the field of computer vision. While ensuring that the feature pictures inside the divided areas have consistent properties, although there are noticeable changes across regions, image segmentation algorithms separate the image features into several regions. The original image is represented by I and the features within different image regions have consistency requirements; then H ( ⋅ ) is utilized as a measurement requirement. And process the original image through segmentation technology, dividing the image into multiple sub regions, and the sub-regions of image division are represented by W 1 , W 2 , W 3 … W n , then the image region has completeness characteristics as shown in equation (1).

(1) I = ⋃ i = 1 n W i , i = 1 , 2 , … n .

The image is divided and each region is characterized by independence, and each region does not overlap with each other, as seen in equation (2).

(2) W i ∩ W j = ∅ , ( i ≠ j ) .

In equation (2), i and j both represent arbitrary numbers. Additionally, consistency and difference are features of the picture segmentation region. Traditional image segmentation methods use digital images, atlases, and theoretical understanding of mathematics to divide up images. The principle of threshold image segmentation is to use the grayscale difference between images to calculate the grayscale threshold of the image, and compare the grayscale of the pixels in the graph with the threshold value and divide the appropriate area. If there are multiple targets in the image processing, multiple thresholding is required for segmentation. The threshold segmentation method has regional thresholding and local thresholding depending on the image division [21]. Global thresholding mainly refers to pixel targets with grayscale values greater than the threshold of T and less than T as background, while local thresholding is not unique for T threshold selection. Therefore, the full-threshold segmentation technique is usually used to have a better image segmentation effect, as shown in Figure 1.

Figure 1

Full threshold segmentation histogram.

The choice of the threshold value is the essential component of the threshold segmentation method. The article uses the Kapor entropy method, which has more obvious benefits for multi-threshold processing and performs better for noisy images than the Otsu and minimum error methods. The Kapor entropy method is also called the maximum entropy method; by using the Shannon entropy, the grayscale histogram is obtained to achieve the search for the optimal threshold value. The Shannon definition representation is shown in equation (3).

(3) H = − ∫ − ∞ + ∞ log p ( v ) ⋅ p ( v ) d v .

In equation (3), v denotes the grayscale value of the image pixel, and p ( v ) denotes the probability density function. Grayscale value range grayscale values [ 0 , 1 , . . . , S − 1 ] , whose image entropy is represented as seen in equation (4).

(4) H = − ∑ i = 0 S − 1 ln p i ⋅ p i .

If the gray level of the image is assumed to be d, the gray level is taken as [ 0 , 1 , . . . , S − 1 ] , the number of pixels is set as n i , and the total number of pixels is represented by N , then ∑ i = 0 S − 1 p i = 1 . The source image is divided into foreground and background in the segmentation; C 0 represents the foreground, C 1 represents the background, and the percentages of pixels in the foreground and background are represented by R 0 and R 1 , respectively. The information entropy of the foreground is shown in equation (5).

(5) h 0 = − ∑ i = 0 t p i R 1 ⋅ ln p i R 0 .

The information entropy of the background is shown in equation (6).

(6) h 1 = − ∑ i = t + 1 S − 1 p i R 1 ⋅ ln p i R 1 .

Recording to equations (5) and (6), then the target information entropy background region entropy is represented as demonstrated in equation (7).

(7) H ( t 0 , t 1 ) = h 0 + h 1 .

The optimal entropy value is obtained by maximizing the entropy method as shown in equation (8).

(8) H ( t opt ) = Max 0 ≤ t ≤ S − 1 { H ( t 0 , t 1 ) } .

The information entropy representation of the single-threshold extension to multi-threshold processing, which segments the image into n classes, is shown in equation (9).

(9) h 0 = − ∑ i = 0 t 0 p i R 0 ⋅ ln p i R 0 .

The information entropy of C n − 1 for multiple thresholds is shown in equation (10).

(10) h j = − ∑ i = t j − 1 + 1 t t p i R j ⋅ ln p i R j , ( j = 1 , … , n − 1 ) .

The background entropy and the target entropy sum are delineated in equation (11).

(11) H ( t 0 , t 1 , … , t n − 1 ) = ∑ j = 0 n − 1 h j .

Top representatives multiple threshold groups denoted by T opt , then the following conditions are satisfied, as addressed in equation (12).

(12) H ( t opt ) = H ( t 0 opt , t 1 opt , … , t n − 1 opt ) = Max 0 ≤ t 0 < t 1 < ⋯ < n − 1 ≤ S − 1 { H ( t 0 , t 1 , … , t n − 1 ) } .

The core of multi-threshold image segmentation lies in the selection of thresholds, and the flow of the image thresholding method is seen in Figure 2.

Figure 2

Schematic diagram of image threshold segmentation process.

3.2 Multilevel thresholding image segmentation algorithm based on MSh model

MSh is a traditional image segmentation model in the field of image vision and maintains the segmented image’s smooth target. Due to its numerical implementation and adaptability in image segmentation processing, this model is frequently employed in the field nowadays. The mathematical expression of this model is shown in equation (13).

(13) E ( u , C ) = λ ∫ Ω ( u 0 − u ) 2 d x d y + a ∫ Ω / C ∣ ∇ u ∣ 2 d x d y + μ Length ( C ) .

In equation (13), u 0 represents the function of segmented smoothing approximating u , u represents the original image, C represents the set of edges in the image region, and Length ( C ) is the curve length μ . The parameters λ and a are non-negative penalty parameters, where the larger the a is, the smoother the image, and the larger the μ is, the shorter the curve C will be. The MSh principle of image segmentation is that for a given graph u can approximate the data pair ( u , C ) , and by controlling the length of the contour curve, the first image is processed. In image processing, the approximation function is needed to make the image as smooth as possible outside of removing the image contour lines. Therefore, the MSh model for image segmentation solving needs to be transformed into solving for the minimum energy parametric function value as seen in equation (14).

(14) min u , C E ( u , C ) .

To realize the image segmentation process more conveniently, a simplified image segmentation model is proposed based on the original MSh model, which is simplified to C − V model, where the original image I ( x , y ) is divided into two regions of background and target under the processing of curve C , and the gray scale values between the two regions are C 0 and C b , respectively. To find the best segmentation curve, the segmented image is represented as seen in equation (15).

(15) I ( x , y ) = c 0 , ( x , y ) ∈ inside ( C ) c b , ( x , y ) ∈ inside ( C ) .

Then the energy generalization function of the simplified MSh model is obtained as seen in equation (16).

(16) F ( C ) = F 0 ( C ) + F b ( C ) = ∫ inside ( C ) ∣ I − c 0 ∣ 2 d x d y + ∫ outside ( C ) ∣ I − C b ∣ 2 d x d y .

In equation (13), C represents any closed curve. If the curve C is not in the two boundaries of the image, the energy inverse function will not be able to obtain the minimum value. Therefore, three cases where the curve is in the boundary region need to be taken into account to obtain the minimum value of the F ( C ) function, as shown in Figure 3.

Figure 3

Schematic diagram of curve evolution segmentation model.

The solution to equation (13) above is not unique, to make the solution of the Equation unique, the length of the curve of positive regularity toward the solution, and the area of the internal region of Figure 4 is added, then C − V model image generalized function as seen in equation (17).

(17) E CV ( C , C 0 , C b ) = μ L ( C ) + v A ( C ) + λ 0 ∫ inside ( C ) ∣ I − C 0 ∣ 2 d x d y + ∫ outside ( C ) ∣ I − C b ∣ 2 d x d y .

Figure 4

Average results of multi algorithm optimization under standard function.

The C − V model can achieve global optimization of the MSh model by using an initial closed contour line to achieve target detection, and also can effectively achieve the image denoising effect, which makes the MSh model have a better image segmentation effect. However, in the image segmentation process, the accuracy of the MSh model will be degraded when there are multiple targets with segmentation and the distance between the targets is far, or the cavity area between the targets is thick. Therefore, the multi-level threshold segmentation method is used to solve the multi-target segmentation problem of the MSh model, and the multi-level threshold Kapur entropy image segmentation algorithm (MSK) based on the MSh model is proposed. The MSh model is used to determine the ideal threshold and convert the problem into a Rui-objective function optimization problem. By extending the multi-target segmentation to the multi-threshold processing case, where the number of thresholds increases as the number of targets processed increases, the difficulty of the algorithm can be decreased while simultaneously improving the processing accuracy of the MSK algorithm.

In the MSh model for multilevel thresholding image segmentation, the objective function of the image will be partitioned into different target regions for the entropy calculation process. The MSh model for multiple objectives is represented in equation (18).

(18) x i = ( x 1 , x 2 , x 3 … x n ) .

In the MSh model for multiple objectives, the objective processing problem is represented as the processing of the multi-threshold problem; then the mathematical dimension to be solved is represented by d , which can also be understood as the number of thresholds to find. Then the final algorithm needs to continuously iterate to get the convergence result near the optimal position and get the optimal solution of the algorithm x best . The workflow of the MSK algorithm is as follows: set the corresponding parameter values of MSh model and Kapur entropy image segmentation method, select the original target image input image, adopt equation (8) as the objective function, and use the MSh model to achieve the solution of the extreme value of the objective function x best and derive the image data after x best segmentation to obtain the final image segmentation processing results.

4 Experimental performance testing and analysis

To verify the performance effect of the proposed MSK algorithm in image segmentation, the performance of the algorithm will be tested by image, data comparison, and analysis. The experimental test platform is Windows 10; the computer memory is 16 GB; the processor speed is 3.5 Ghz; and the performance test of all the image data is finished using MATLAB and associated image test sets. Hunter, Cameraman, BSDS, Lena, and other picture data sets are among them. These data sets have a very high recognition rate in picture segmentation tests and are frequently used in the field of computer vision image processing. Table 1 displays the relevant parameter settings.

Table 1

Parameter settings of the experiment

Parameter type	Preset value
Target quantity	40
Iterations	200
Lower bound	0
Upper bound	225
Maximum elimination threshold	10
Nonnegative penalty parameter a	0.04
Nonnegative penalty parameter μ	2
Nonnegative penalty parameter λ	1

In addition, in order to better explore the conversion performance of the MSK algorithm, a comparison was made between the MSK algorithm, Kapor algorithm, and MSh algorithm in the MATLAB environment. Figure 4 shows the conversion results of image curves under variable algorithms.

The mean value outcomes of various algorithms for selecting the best under the standard function are shown in Figure 4. The results show that the convergence speed and declining trend of the fusion-based MSK method are much faster. The convergence curves of the MSK and MSh algorithms converge in less than 200 iterations and drop to a value close to 0, which meets the function accuracy requirement. However, the MSh curve fluctuates more, while the MSK algorithm converges more gently in 160 iterations and the MSh algorithm converges in 172 iterations. The overall curve of the Kapor algorithm fluctuates more gently, but the convergence accuracy and the convergence speed are worse, and it converges after 203 iterations with a convergence value of 5. Therefore, the MSK algorithm proposed in the multi-algorithm performance test has excellent convergence performance and convergence accuracy. To further test the performance of each algorithm, 500 classical image data points in the BSDS dataset were selected for the algorithm image segmentation performance test. In the test, the selected images were grayscale processed, and the data sizes of the images were 512 × 512, 256 × 256, and 481 × 321. Figure 5 shows the original image data.

Figure 5

Original image of BSD data set selected for image segmentation experiment. (a) Couple, (b) cameraman, (c) tree, and (d) pepper.

Figure 5 shows the original image of BSD data set selected in the image segmentation experiment. The histogram corresponding to the image in the data set obtained by the Kapor entropy method is shown in Figure 6.

Figure 6

The square curve corresponding to the image in the dataset. (a) Couple, (b) cameraman, (c) tree, and (d) pepper.

Figure 6 shows the histograms corresponding to the dataset. Four images in the BSD dataset are selected as the test objects, and Kapor entropy is tested on the images based on the histogram data. And all four images have unique corresponding grayscale histograms. The variety properties of histograms make the use of conventional segmentation challenging. In order to guarantee the quality of the images, the MSh model with multi-level thresholding is utilized for image segmentation. Also in Figure 7, the locations of the thresholds taken by the images are marked with striking co-colored lines. The m is used to indicate the size of the threshold value set for the image, and the threshold equivalence is taken in the range of having three ranges of 2, 3, and 5. To better compare the effect of the proposed algorithm, image segmentation will be compared by the MFO algorithm, SSA algorithm, and EMO algorithm with the proposed MSK algorithm. The number of thresholds obtained by multiple algorithms is shown in Table 2.

Figure 7

Iterative operation time of image segmentation for different calculations. (a) Couple, and (b) cameraman.

Table 2

Number of thresholds obtained for multiple algorithms

Image data	Couple (m)			Cameraman (m)			Tree (m)			Pepper (m)
Image data	2	3	5	2	3	5	2	3	5	2	3	5
MFO	76	45	33	123	43	40	90	81	68	104	94	17
	140	100	76	198	102	96	176	143	106	165	150	82
	—	151	114	—	196	145	—	198	143	—	210	127
	—	—	161	—	—	192	—	—	180	—	—	171
	—	—	235	—	—	222	—	—	220	—	—	216
SSA	75	45	33	124	43	41	90	80	68	105	94	16
	141	99	75	198	102	95	175	142	104	165	150	81
	—	150	114	—	196	144	—	197	143	—	210	126
	—	—	159	—	—	193	—	—	180	—	—	171
	—	—	235	—	—	223	—	—	222	—	—	217
EMO	76	46	35	124	41	42	91	81	68	103	94	16
	142	101	76	198	103	95	176	142	104	165	150	80
	—	153	117	—	199	145	—	196	143	—	211	127
	—	—	162	—	—	193	—	—	181	—	—	169
	—	—	234	—	—	224	—	—	220	—	—	218
MSK	76	45	30	124	40	41	90	80	68	104	94	17
	142	101	32	198	103	96	177	141	106	165	150	81
	—	152	118	—	198	145	—	196	143	—	212	127
	—	—	166	—	—	192	—	—	180	—	—	170
	—	—	234	—	—	224	—	—	221	—	—	218

Table 2 shows the optimal threshold values obtained for different number of thresholds under multiple algorithms. The table data show that all algorithms can successfully extract the image objective function values; however, the objective function values of the suggested MSK approach are more precise. In image Cameraman, the corresponding optimal thresholds are 198, 198, and 224 when the number of thresholds is 2, 3, and 5, respectively, and the image segmentation quality of all four algorithms is improved with the increase of the number of thresholds. A detailed comparison shows that the EMO algorithm and the proposed MSK algorithm have better objective function values, while the MFO and SSA algorithms are slightly lower, but the overall difference is not significant. As can be seen, while processing picture data, the MSK method has a superior threshold processing effect and the processing effect is closer to the target quality, which has a remarkable impact on image segmentation in the context of flat images. At the same time, comparative tests were conducted on the image segmentation quality under different thresholds, as shown in Table 3.

Table 3

PSNR value obtained from image segmentation with different threshold values under multiple algorithms

Image data	Peak signal to noise ratio
Image data	m	MFO	SSA	EMO	MSK
Couple	2	14.5889	14.5900	14.59	14.5901
	3	17.1121	17.1148	17.2154	17.2155
	5	20.3245	20.2456	20.3153	20.3361
Cameraman	2	13.9198	13.9200	13.9201	13.9202
	3	14.4618	14.4622	14.4621	14.4620
	5	20.2434	20.6454	20.7731	21.0712
Tree	2	15.1881	15.1895	15.1865	15.1869
	3	18.5023	18.5095	18.5095	18.5117
	5	21.1545	21.0045	21.0548	21.1685
Pepper	2	15.0101	15.0106	15.0104	15.0813
	3	15.9667	16.0512	15.9664	16.0565
	5	19.2184	18.9466	18.6847	19.3608

Table 3 shows the peak signal to noise ratio (PSNR) values obtained for image segmentation with different threshold values under multiple algorithms. As seen from the image segmentation quality evaluation, all four algorithms can obtain satisfactory PSNR values with different threshold values. Except for the MFO approach, which requires a smaller value, the rest of the thresholds can satisfy the criteria for image segmentation evaluation at the threshold values of 2 and 3. Additionally, the proposed MSK algorithm may obtain the greatest evaluation value under various threshold values from the table’s total data. In the Tree image, the PSNR values of the MSK algorithm are 15.1869, 18.5117, and 21.1685 for the threshold values of 2, 3, and 5, respectively, which can start the best PSNR values. To guarantee that the image can obtain better segmentation quality, the image segmentation process running time will be tested in order to better verify the speed of MSK algorithm. The experimental test’s iterative test settings are the same, and to determine the final runtime when the multi-algorithm converges, the algorithm operation is halted after the value of the algorithm’s fitness function has been tested continuously for 18 iterations and has not changed. The test image is Cameraman and the experimental results are shown in Figure 7.

Figure 7 shows the iterative operation time of image segmentation under different calculations. In Figure 7(a), the experimental test image is Couple, and the overall operation efficiency of MPO algorithm and SSA algorithm is not high. The best performance of the operation efficiency is the MSK algorithm, followed by the EMO algorithm. In Figure 7(b), the experimental test image is Cameraman, and the MPO algorithm takes the longest time to reach convergence, and the running times of the MFO algorithm are 3.54, 5.26, and 8.65 s for threshold values of 2, 3, and 5, respectively. The best performance in terms of operational efficiency is the MSK algorithm, which takes 0.85, 2.15, and 3.46 s for threshold values of 2, 3, and 5, respectively. It can be seen that the proposed MSK algorithm has excellent operational efficiency and takes the shortest time in image segmentation. The number of thresholds affects how long each algorithm takes to segment an image, but the suggested MSK algorithm outperforms the MPO, SSA, and EMO algorithms in terms of running time efficiency by 356, 289, and 71%, respectively. The final segmentation results of MSK algorithm images with different threshold values are shown in Figure 8.

Figure 8

Image segmentation results of MSK algorithm under different threshold values. (a) Couple, (b) cameraman, (c) tree, and (d) pepper.

Figure 8 shows the image segmentation results of MSK algorithm under different threshold values. From the overall image segmentation results, the image segmentation can achieve better results when the number of thresholds is taken as 5. Although the convergence time will also increase as the number of thresholds rises, the segmentation accuracy of the image will continue to advance and retain more features. Finally, contrast the study technique’s image segmentation results with those of the method used in the literature [21]. In accordance with Table 4, the effect of image segmentation improves as the signal to noise ratio (SNR) value increases.

Table 4

Comparison of SNR values of several image segmentation methods (dB)

Image name	MPO	SSA	EMO	Mozaffari and Won-Sook [21]	MSK
Couple	25	28	23	29	35
Cameraman	27	32	27	32	36
Tree	21	27	31	30	37
Pepper	24	26	31	29	37

Table 4 shows the image segmentation contrast results of several segmentation methods. The results showed that MSK had the best image segmentation quality among the four kinds of images. The SNR values in Couple, Cameraman, Tree, and Pepper are 35, 36, 37, and 37, respectively, which can effectively reflect the quality of the original image. The method used in Mozaffari and Won-Sook [21] performs second in terms of image processing performance, with good results in Cameraman and Tree, but not as good as MSK. The proposed MSK algorithm performs superbly during the image segmentation process and receives favorable performance ratings for image segmentation effectiveness, quality, and other factors, fully satisfying the application criteria for image segmentation. The suggested MSK algorithm performs superbly during the image segmentation process and receives favorable performance ratings for both image segmentation efficiency and quality, satisfying the application criteria in the field of image segmentation.

5 Conclusion

Image segmentation is a crucial step in the computer vision process and serves as the foundation for comprehending and analyzing images. For obtaining the globally optimal segmentation of images using the reduced MSh energy generalization function, the MSh image segmentation model is used, as well as a technique called evolution of infinite curves with global information. Considering the difficulties of the traditional MSh image segmentation model in segmenting complex multiple targets, a multi-level thresholding image segmentation algorithm based on the MSh model is proposed, which uses multi-level thresholding to achieve target search and fast convergence of the target image level set curves. The experimental results showed that the proposed MSK algorithm had the best objective function value with the corresponding optimal thresholds of 198, 198, and 224 when the number of thresholds is 2, 3, and 5 in the image Cameraman optimal threshold search, respectively. The segmentation quality and convergence reach time of the image are also tested, and the MSK algorithm has excellent performance. It is obvious that the suggested MSK method satisfies the criteria for image segmentation. However, there are issues with the research’s substance. The proposed MSK image segmentation method is only applicable to flat images, and the method needs to be improved in a later stage to adapt to more complex 3D image data. The MSK algorithm has excellent threshold processing ability, but the testing dataset is small.

Funding Information: Authors state no funding involved.
Author contributions: Xiancai Kang – Writing original manuscript; Chuangli Hua – Review and editing manuscript.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2022-12-09

Revised: 2023-04-28

Accepted: 2023-06-23

Published Online: 2023-11-28