An edge detection algorithm for neutrosophic set based on adaptive threshold

doi:10.21203/rs.3.rs-1860703/v1

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An edge detection algorithm for neutrosophic set based on adaptive threshold

https://doi.org/10.21203/rs.3.rs-1860703/v1

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In image processing, edge detection is extremely important especially in feature detection. An edge detection algorithm for neutrosophic set based on adaptive threshold is proposed to effectively filter out the large amount of noise in the image and improve the detection speed and accuracy. The algorithm uses side window filtering algorithm to filter out noise based on neutrosophic set; Furthermore, it modifies the subset segmentation algorithm to split the image into three subsets of T, F, I; Then, an adaptive threshold extraction algorithm is proposed to segment the subset information; Finally, the segmentation information is merged to obtain edge features. In order to verify the effectiveness of the algorithm, an experiment is compared with the neutrosophic set edge detection algorithm based on maximum entropy (NMNE) under different noise conditions. An experimental comparison between Canny and Sober is also carried out additionally. Experimental results show that the algorithm has good anti-noise performance and improves the detection speed and accuracy of the neutrosophic set edge detection algorithm based on maximum entropy (NMNE). The algorithm proposed in this paper can provide an effective reference for mid- and high-level vision tasks.

Image processing

Edge detection

Neutrosophic set

Adaptive threshold

The aim of edge detection is to extract object boundaries and perceptually significant edges from natural images, thereby preserving key points of the image and ignoring unexpected details, especially important for various mid-to-high-level vision tasks (such as image segmentation, object detection and recognition, image classification). With the development of technology, the demand for edge detection is getting higher and higher. The required edge information is more abundant and more accurate in the areas of intelligent driving, medical detection, and industrial applications[1].

Over time, edge detection algorithms can be divided into two categories. One is traditional edge detection algorithms, such as Canny, Sober, Laplacian edge detection [2–4]; The second is modern edge detection algorithms, such as SUSAN edge detection algorithm, ant colony optimization, neural network detection algorithm (fast-rcnn), watershed algorithm based on neutrosophic sets, etc [5–8]. There is no balance between detection accuracy and detection speed although the detection accuracy is constantly improved. In addition, images are susceptible to various types of noise during collecting image process. Therefore, it is more important to apply new methods and concepts.

In image processing, traditional fuzzy theory is limited to some applications. For example, fuzzy sets not only fail to effectively process noisy images, but are still sensitive to noise, and have defects in processing spatial uncertainty [9]. Therefore, people found that the edge of image is actually an uncertain information. It is in an uncertain state on both the object and the background. Smarandache proposed the concept of intelligence set (NS), which can effectively solve the problem of uncertainty [10]. In NS, real, uncertain and false information can be independent, and this assumption is important in many applications. Therefore, someone introduced True(T), Uncertain(I) and False(F) subsets into image processing, such as edge detection and image segmentation, and achieved good results [11, 12]. However, there are only a few studies on edge detection based on NS. SudanJha et al. proposed a neutrosophic image segmentation with Dice Coefficients. The neutrosophic sets and Dice Coefficients were fused to ensure correct assessment of missing data uncertainty and image segmentation uncertainty [13]. Ahmed M.Anter et al. proposed a hybrid segmentation method for CT liver tumor based on neutrosophic set, fast fuzzy C-means and adaptive watershed algorithm [14]. Ming Zhang et al. proposed a watershed-based CIMC image segmentation algorithm. The watershed algorithm was used for segmentation based on the neutral set [8]. Furthermore, EserSert proposed a new and improved mid-intelligence set segmentation based on this research [15]. However, there are few researches on edge detection. Therefore, Eser and Derya proposed a maximum entropy-based neutrosophic edge detection algorithm (NMNE) [16]. But there are common problems, the algorithm is complex and sensitive to noise.

Therefore, this paper proposes an edge detection algorithm based on adaptive threshold for neutrosophic set (P_NMNE), which effectively solves the problem of high complexity and sensitivity to noise. The proposed algorithm is compared with NMNE to reveal its effectiveness. Furthermore, the anti-noise performance of the proposed algorithm is compared with Canny edge and Sobel edge detection methods. The experimental results prove that the proposed algorithm has the advantages of low complexity and high anti-noise performance. The work is organized as follows: Following the introduction, Section 2 describes the theory of neutrosophic sets, and based on it, an edge detection algorithm with adaptive threshold is proposed; Section 3 conducts edge detection experiments on images in multiple domains, comparing the P_NMNE algorithm with other algorithms to verify the accuracy and performance improvement of the proposed algorithm. The conclusion is given in Section 4 to end the paper.

2.1. The concept of neutrosophic set (NS)

Briefly introduce the concept of the neutrosophic sets, which referred as NS below. In the NS, it supposes that E represents an event and -E is the opposite event of E. N-E is defined as neither E nor -E. For example, if E = large size, -E = small size, and N-E is medium, which is neither large nor small size. The basic idea of neutrosophic theory is that any point of view has a certain degree of authenticity, uncertainty and falsehood [17]. Therefore, the neutrosophic theory introduces T, I and F, which represent the authenticity, uncertainty and falseness of events, respectively [11–16]. In this paper, the image is divided into T, F, I, where T represents the object information of the image, F represents the background information, and I represents the other information such as color, noise, edge and so on. Figure 1 shows the entire image processing process, which can be roughly divided into:

Step 1: Obtain the original image and perform grayscale processing.

Step 2: Perform denoising processing.

Step 3: T, F, I subset segmentation.

Step 4: Adaptive thresholding to segment the subset.

Step 5: Obtain edge feature information by fusing segmentation subset information.

2.2. Proposed method

2.2.1 Preprocessing

The images to be processed are generally noisy in the real environment, so some image preprocessing work is needed. Firstly, the original image is converted to a grayscale image, and then filtered for denoising. At present, the widely used and effective image smoothing methods mainly include: mean filtering, Gaussian filtering, etc, but they are easy to cause loss of details.

However, the recently emerged Side Window Filtering (SWF) algorithm can better preserve edge details while de-tuning noise [18]. The filtering divides any kind of filter into 8 directions, and then adaptively selects the best direction, as shown in Fig. 2. Therefore, this paper compares the mean filtering and Gaussian filtering experimentally. Finally, the edge window mean filtering algorithm is used, as shown in Fig. 3–4.

2.2.2 Acquisition of T and F

In this paper, T subset mainly contains detected object information, F subset mainly contains background information, and the fuzzy boundary is mainly in I. The entropy method is used to obtain subsets of T and F, but its obvious disadvantage is that the algorithm is complex, the calculation amount is large, and the calculation process is somewhat cumbersome [16]. This paper improves it and proposes a more efficient subset acquisition algorithm to solve the problem of complexity and time-consuming. According to Function1, the image can be converted into T and F subsets, as shown in Fig. 5 and Fig. 6 (i.e. T subset and F subset).

\((T,F)=function1(img)\)(1)

The definition details of the equation are as follows:

\(T=\frac{{\left( {img - \hbox{min} \left( {img} \right)} \right)}}{{\left( {\hbox{max} \left( {img} \right) - \hbox{min} \left( {img} \right)} \right)}}\)(2)
\(F=1 - T\)(3)

Among them, img represents the pixel value of the image, min(img) represents the minimum value of the image pixel, and max(img) represents the maximum value of the image pixel value.

2.2.3 Acquisition of I

The acquisition of subset I plays an important role in image edge detection in this paper. One of the most important parameters is homogeneity (ho), which is associated with local information. The value of this parameter is increased in various situations in the image, such as edge regions, color transitions, and background transitions [13]. Thus, the I subset can be obtained according to function2, as shown in Fig. 7.

\(I=function2(img)\)(4)

The detailed steps are as follows:

(1) Select the 8-connected area (3⊆3), calculate the mean Localmean[i,j] of the 8-connected area, and calculate the mean square error of the 8-connected area, save it as devia[i,j].

\(devia[i,j]=\sqrt {\sum\limits_{{m=i - (\alpha - 1)/2}}^{{i+(\alpha - 1)/2}} {\sum\limits_{{n=j - (\alpha - 1)/2}}^{{j+(\alpha - 1)/2}} {{{\left( {img[m,n] - Localmean[i,j]} \right)}^2}} } }\)(5)

(2) Calculate the discontinuity of the image according to the Gaussian kernel. where \({G_x}\), \({G_y}\) are the horizontal and vertical discontinuities of the image.

\(dis[i,j]=\sqrt {{G_x}{{[i,j]}^2}+{G_y}{{[i,j]}^2}}\)(6)

(3) The image subset I is calculated according to the formula.

\(I(i,j)=\frac{{devia(i,j)}}{{devia\hbox{max} }}\cdot \frac{{dis(i,j)}}{{dis\hbox{max} }}\)(7)

Among them, img is the image pixel value, devia is the average difference value of the 8-connected areas, and dis is the image discontinuity.

2.2.4 Obtaining edge information

Traditional edge detection, such as Canny, uses fixed threshold segmentation to obtain edge information. Although adaptive threshold edge detection algorithms have emerged in recent years, there is no simple and effective threshold extraction algorithm. For this reason, an adaptive threshold extraction algorithm based on maximum algorithm is proposed. According to the subset, its gray histograms and the probability values between the pixels are obtained. The pixel value of the image is between 0-256 [19], and the pixel value of subsets in this paper is between 0–1. In this paper, L represents the range of pixel values, n represents the number of pixels in this interval, and N represents the size of the image (\(x \times y\)) Therefore, this article can obtain the segmentation threshold according to Function3, as shown in Fig. 8 for the image histogram of the subset:

\((opt,opf,opi)=Function3(F,T,I)\)(8)

The definition details of the equation are as follows:

\({\partial _{\hbox{max} }}=\frac{{{{\left( {\eta \times {\omega _k} - {\eta _k}} \right)}^2}}}{{{\omega _k} \times \left( {1 - {\omega _k}} \right)}}\)(9)
\({p_i}=\frac{{{n_i}}}{N},{\omega _k}=\sum\limits_{{i=1}}^{k} {{p_i},\eta =\sum\limits_{{i=1}}^{k} {i \times p} }\)(10)

The value of k is [L0, L1, L2...L255]. When the maximum value is obtained, the corresponding L value is opt. Similarly, opf and opi can be obtained, where opt is the segmentation threshold of T subset, opf is the segmentation threshold of F subset, and opi is the segmentation threshold of I subset. The values obtained in this case are 0.50390625, 0.49609375, and 0.25390625, respectively.

As introduced above, the T subset contains object information, the F subset contains background information, and the I subset contains edge information. Therefore the T subset is used to obtain the body information of the object, and the F subset is reversely divided to obtain the object information and I subsets are divided to obtain information such as edges. The specific designed edge extraction algorithm is shown in Funtion4, and the detection results are shown in Fig. 9.

\(Edge=Function4(T,F,I,opt,opf,opi)\)(11)

The definition details of the equation are as follows:

\(\begin{gathered} If(F \geqslant opf\parallel T \leqslant opt)and(I \geqslant opi): \hfill \\ Edge=255 \hfill \\ else \hfill \\ Edge=0 \hfill \\ \end{gathered}\)(12)

In order to verify the effectiveness of the proposed algorithm in this paper, a computer configured with: Intel 2.6 GHz CPU, 4 GB memory, Microsoft Windows 10 64-bit operating system, and anaconda 3 (python 3.6) software is installed as the experimental platform to conduct improved experimental tests with the NMNE algorithm on images in multiple fields. On this basis, this paper also conducts an experimental comparison of Canny and Sober. The experimental results show that, compared with NMNE algorithm, P_NMNE algorithm has greatly reduced time consumption and improved anti-noise performance, especially in the salt and pepper noise.

3.1. Anti-noise performance experiment

Since the side body sheet metal parts of automobiles have rich edge information, they are used as the object of this experiment. During the experiment, Canny, Sober, NMNE and this improved NMNE algorithm were used for experimental verification, which can be divided into two parts: one is to compare the experiments when there is no noise, as shown in Fig. 10; The orther is to add different noises for experimental comparison, as shown in Fig. 11 and Fig. 12.

3.2. Time complexity experiment

In order to highlight the main innovation of this algorithm, the time complexity is O(1), and the processing speed is greatly improved compared to the NMNE algorithm. Therefore, the detection speed comparison experiment is conducted in the multi-domain images, as shown in Table 1 and Fig. 15.

Table 1

Data comparison
Image number	Image size	Gaussian noise			Pepper-salt noise
Image number	Image size	NMNE	P_NMNE	Performance boost	NMNE	P_NMNE	Performance boost
0	(254,500)	128.87	6.81	94.67%	223.71	6.94	96.90%
1	(256,256)	88.49	6.5	92.65%	91.31	7.09	92.24%
2	(128,128)	17.41	7.04	14.11%	12.53	7.62	39.19%
3	(293,332)	73.49	10.01	86.38%	76.03	10.10	86.72%
4	(285,369)	16.88	10.93	13.25%	13.47	11.74	12.84%
5	(209,358)	77.90	12.03	84.56%	39.90	13.00	67.42%
6	(178,178)	29.08	13.08	55.02%	26.85	14.23	47.00%
7	(252,254)	73.77	15.41	79.11%	40.49	16.25	59.87%
8	(252,252)	17.11	10.21	40.30%	31.23	18.4	41.08%
9	(136,180)	31.38	18.9	40.60%	19.54	19.78	44.83%
10	(288,288)	76.59	21.83	71.50%	64.48	23.81	63.07%
11	(240,239)	44.32	24.36	45.04%	75.95	19.01	74.97%
Performance boost				59.77%			60.51%

From the results, it is not difficult to see that the algorithm in this paper has a great improvement in processing conditions at any scale, with a minimum improvement of 12.84% and a maximum improvement of 96.96%. The average performance is improved by 59.77% under Gaussian noise image processing. Under the image processing of salt and pepper noise, the performance is improved by 60.51% on average. The whole show superiority.

In this study, an adaptive threshold-based edge detection algorithm for neutrosomatic sets is proposed, which can better locate the edge and suppress noise. And the method is fast in detection speed. This algorithm improves the image subset segmentation algorithm, and proposes an adaptive threshold extraction algorithm to segment the subset image. Finally, the edge features are obtained by fusing the segmentation information, and compared with Canny, Sober, and NMNE. The experimental results show that the proposed algorithm solves the problem that the traditional Canny, Sober and NMNE edge detection algorithms are sensitive to noise, and effectively suppresses the noise in the image. Among them, the algorithm has good suppression effect on salt and pepper noise, relatively weak on Gaussian noise. It protects the edges with rich details and slow changes, and has high edge location accuracy. In addition, the problem of high time complexity of NMNE is solved, and the image subset segmentation algorithm and the adaptive threshold extraction algorithm are improved, which greatly reduces the processing time of the NMNE algorithm.

Acknowledgments

The authors would like to thank the support of Guangxi Innovation Driven Development Project (No. 2020AA28010AA).

Data Availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

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No competing interests reported.

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An edge detection algorithm for neutrosophic set based on adaptive threshold

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Abstract

Figures

1. Introduction

2. Theoretical Knowledge And Proposed Algorithms

2.1. The concept of neutrosophic set (NS)

2.2. Proposed method

2.2.1 Preprocessing

2.2.2 Acquisition of T and F

2.2.3 Acquisition of I

2.2.4 Obtaining edge information

3. Experiment And Result Comparison

3.1. Anti-noise performance experiment

3.2. Time complexity experiment

4. Conclusions

Declarations

References

Additional Declarations

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