Image edge detection algorithm based on fuzzy set

2.1 Related work

In order to overcome the interference of the sky background on the enhancement of infrared image and highlight the target in the image, Su J proposed an infrared image enhancement method based on pulse coupled neural network (PCNN) segmentation and fuzzy set theory, which uses PCNN to image It is divided into sky background area and target area, and the image is blurred by adaptive fuzzy technology. The Y-enhancement method based on the ridge distribution is used to enhance the target area reflectance image acquired by the variational viewpoint, and the enhanced reflectance image is merged with the illumination image, and the local average of the target area is assigned to the sky background area. The enhanced image is then acquired by the reconnaissance plane [9]. He J proposed an improved rough fuzzy C-means clustering algorithm. The algorithm designed a parameter selection strategy to adaptively adjust the weighting parameters according to the distribution characteristics of each cluster instead of manually selecting a constant parameter. This online decision making approach allows the formed prototype to be closed to the ideal location. Experimental results on synthetic data sets, real data sets and image segmentation problems demonstrate the effectiveness of the proposed adaptive parameter selection strategy. With the introduction of adaptive parameter selection strategy, the improved rough set clustering algorithm is superior to similar algorithms in some cases [10].

2.2 Edge detection

(1) Edge detection concept

Both the image imaging process and the formation of digital images determine that the image edges contain a large amount of information resources, but the edges contain multiple types. Edge detection is a very challenging technique, and the following basic concepts should be understood before studying image edge detection.

1) Digital image edge: Refers to a collection of gray scale variations in a digital image with a particular pattern. This form of analysis can be performed directly using a one-dimensional function (discrete signal). In fact, for this one-dimensional feature, we detect every point in the image in each direction. The gray scale variation mode of interest can be selected according to the needs of use. For example, if you want to detect the boundary of a target in an image, we can select the step mode as the specific gray scale mode to detect the image. Actually detected result: The edge of the image may correspond to the boundary of the target or may not correspond to the boundary of the target. However, regardless of how the image is processed, the range of boundary points of the search target will be greatly reduced.

2) Digital image edge detection: refers to the positioning, direction and metric of gray scale changes in a digital image with a specific pattern. When the cause of the gray scale variation of the image is not considered, the task of edge detection is to detect all edge points of the gray scale change exceeding certain intensity. In practical applications, we need to further infer the edge detection results without considering physical reasons, based on specific purposes and prior knowledge. Determine if this gray scale change is caused by an event we really care about.

3) Edge line: The edge point of the gradation change mode form in which the gray level change appears in one direction at a certain point of the image, and these edge points are aggregated to form a continuous line. Edge points of the gray-scale varying image are formed in gray scale changes in one direction at a specific point of the image, and these edge points are gathered to form a continuous line.

4) Corner point: The edge point of the gradation change pattern expressed in a certain direction at an arbitrary point of the image. These edge points are discontinuous in the image, that is, some isolated points.

(2) Edge detection step

The general procedure of the edge detection algorithm filters the original image to obtain a smooth image and enhances the edges of the image using various algorithms. The obtained image is an image of 256 gray levels. Only the gray-scale mutation area remains in the image. Finally, after the threshold segmentation, the 256-level image is changed to a 2-value image, the edge variation is clearly displayed, and the edge image is obtained.

Figure 1 shows the edge detection step of a gray scale image.

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Fig. 1

Edge detection steps.

1) Edge positioning: Edge positioning is processing an edge image obtained by edge extraction, and it is desirable to obtain a binary image having a single pixel width. Commonly used techniques are threshold processing and zero crossing methods. Threshold methods include zero-threshold methods and hysteresis threshold methods, which are simple but produce edges that are typically not single-pixel wide. In order to generate a single-pixel wide-edge image, Canny first introduced a non-maximum suppression method in the edge localization algorithm, and later extended the non-maximum suppression method and introduced the LBE parameter. Zero-crossing positioning methods are also widely used. Marr locates the edge by looking up the point set of the Laplacian transform of the function to change the symbol. In addition, Kalitzin uses symbolic combination methods to locate the edge of the step and the edge of the ridge, and also achieves better positioning.

2) Edge links: Edges resulting from edge positioning typically have some small edge segments due to interference such as noise. In order to form meaningful edges for further recognition or feature extraction, the edges of the links need to be linked. Edge linking also has some difficulties because the shape of the target object in the image is usually unknown in advance. Edge Links After more than two decades of historical research, a variety of linking algorithms have been developed, usually divided into two types: partial edge links and global edge links.

A local edge link is a link operation performed in a local area, including two steps of marking and linking, where a pixel label is a unique label assigned to a set of eight connected pixels. After completing the pixel marking, the next step is to link the pixels belonging to the same label to form a continuous edge.

A global edge link is a link operation that occurs throughout the domain. The Hough transform is a global edge link that uses the global properties of the image to directly detect the target contour. Under the condition that the shape of the region is known in advance, the Hough transform can be used to conveniently obtain the boundary curve and link the discontinuous points. The Hough transform is less affected by noise and curve discontinuities, and the target of the image space is detected by determining a reference point in the parameter space. The Huff transform cannot be used for detection when the curve or target contour to be detected is not easily analyzed. To solve this problem, some scholars have proposed a generalized Hough transform that uses a table to establish the relationship between a curve or contour point and a reference point.

(3) Difficulties in edge detection

The edges of the image are representations of grayscale discontinuities and are a collection of sharply varying gray dots in the image. The process of grayscale variation in an image can be described in terms of a physical process that causes a change in grayscale in the image. For example, geometrically induced edges of images formed by depth discontinuities, surface orientations, colors, and textures are different; these scene characteristics are often mixed together, making the resulting edges more difficult to interpret due to certain factors. Moreover, in actual situations, noise data is often mixed in the image data. Therefore, a requirement for the edge detection method is to be able to detect the exact position of the edge and to suppress uncorrelated details and noise, and generally use a differential method to locate the edge.

The numerical differentiation of the signal is a “morbid” problem. A small change in the input signal causes a large change in the output signal. If f (x) is the input signal, it is assumed that a small change occurs in the f (x) due to the influence of noise. Among them, ɛ ⪡ 1, the signal after introducing noise is represented by f^′(x), as shown in equation (1). f ′ ( x ) = f ( x ) + ɛ sin wx(1)

To find the derivatives on both sides of equation (2), then: df ′ ( x ) dx = df ( x ) dx + ɛ w cos wx(2)

It is known from the formula (2) that if the w is large enough, that is, the noise is high frequency noise; the differential output of the signal f (x) is seriously affected, thereby affecting the result of the edge detection. In order to regularize the difference, it is necessary to smooth the image first. However, image smoothing results in loss of information and moves the main structure of the image plane. In addition, if the differential operators used are different, the same image will have different edges, so noise cancellation and edge localization are two contradictory parts. This is two difficulties in edge detection.

In practical applications, the best compromise between these two aspects should be based on specific requirements. Smoothing of the image is achieved by convolving the image with a filter. In filter theory, regularized parameters are also called “scales.” Take the Gaussian function as an example: g ( x ) = 1 2 π σ e - x 2 2 σ 2(3)

Where σ > 0 is the filtering scale? The scale σ determines the degree of compromise between noise cancellation and edge location.

The Canny edge detection method is a method of detecting edges using local extrema. Canny derives the optimal edge detection operator Canny operator from the three criteria that the edge detection operator should satisfy. For the first time, Canny expressed the standard in mathematical form and then used the optimized numerical method to get the best edge detection template. This operator is currently the most perfect edge detection algorithm in theory. Canny’s three criteria for evaluating edge detection performance are:

(1) Good signal to noise ratio criteria. The probability of misjudgement of non-edge points as edge points is as low as possible. The probability of misjudged edge points as non-edge points should also be as low as possible.

(2) Good positioning performance criteria. The detected edge points are most likely to fall in the center of the real edge.

(3) Single edge response criteria. A single edge has a unique response, the probability of multiple responses from a single edge is low, and the response to false edges should be suppressed to the utmost extent.

In actual operation, the first derivative of the Gaussian function is selected as the suboptimal detection operator of the step edge, and the two-dimensional image is separately convoluted by using the template in several directions. Then take the most likely edge direction. Algorithm steps for the Canny operator to detect edges:

(1) The image is smoothed by a Gaussian function to remove noise in the image.

(2) Calculate the local gradient and edge direction at each pixel. Sobel operators, Roberts operators, etc. can be used to implement the calculations. The edge point is defined as the point at which the intensity is locally largest in the direction of the gradient.

(3) “Non-maximum suppression” of the gradient. The edge points determined in (2) will cause ridges to appear in the gradient amplitude image, algorithmically track the tops of all ridges, and set all pixels that are not at the top of the ridge to zero to give a thin line in the output.

(4) Double thresholding and edge connection. The ridge pixels use two thresholds and perform thresholding processing, wherein ridge pixels having a larger value are referred to as strong edge pixels, and ridge pixels between them are referred to as weak edge pixels. Since the edge array aperture is obtained with a high threshold, it contains fewer false edges, but it also loses some useful edge information. The edge array has a lower threshold and retains more information. Thus, based on the edge array, the edge array can be used to complement the connections, and finally the edge image is obtained.

In the case of two-dimensional space, the directionality of the Canny operator makes the edge detection and edge positioning performance better than the LOG operator. At the same time, it has better anti-noise performance, and can generate edge point direction and edge point intensity direction information for subsequent processing.

2.4 Fuzzy mathematics

Fuzzy mathematics is a new development direction of mathematical science after classical mathematics and statistical mathematics. It is also a branch of mathematics, which is contrary to the characteristics of traditional mathematics. The typical feature of fuzzy mathematics is ambiguity, but It also has the commonality of mathematics, that is, it is well-organized and meticulous. Even if it describes the ambiguity, it will be clearly described. Fuzzy sets, also called fuzzy subsets or fuzzy sets, are the basis of fuzzy mathematics; membership functions are often regarded as equal to fuzzy sets, and membership degree is the basic idea of fuzzy mathematics.

(1) Fuzzy set

The classical set is uniquely determined by its feature function. For example, it is known that set A is mapped by the following features: Φ A : X → { 0 , 1 } , x → Φ A ( x ) = { 1 , x ∈ A 0 , x ∉ A(4)

It can be seen that the degree of membership of x to set A is only 0 and 1 states, 1 means that x belongs to A, and 0 means that x does not belong to A. For example, set A = {all fruits}, if x = watermelon, then x belongs to A; if x = ant, then x does not belong to A. However, in the real world, not everything is in this state. For example, Cordyceps sinensis, Nepenthes, have both the characteristics of plants and the characteristics of animals. So how do we distinguish between animals or plants? It is obviously more difficult to use classical mathematics. In order to solve the same problems as the examples described above, the concept of fuzzy mathematics came into being. If we use fuzzy mathematics, it is much simpler to distinguish them. Below we give a brief introduction to fuzzy set knowledge.

Suppose U represents a domain, we call a fuzzy set on U, if A satisfies: μ A : U → [ 0 , 1 ] x → μ A ( x ) ∈ [ 0 , 1 ](5)

Where μ _A represents the mapping, which is the membership function of A, and μ _A represents the membership degree of x to A. If the membership degree of A is equal to 0.5, then x is called the transit point of A, and the point is the most ambiguous; When the membership of x to A is equal to 0, this case corresponds to the non-dependent in the classical set; when the membership of x to A is equal to 1, it corresponds to the belonging in the classical set.

(2) Membership function of fuzzy sets

As the basis of fuzzy sets, the membership function is an important concept both in application and in theory. However, in many cases, the degree of membership cannot be directly given, because the reasons for the ambiguity are various, which brings great challenges to the determination of the membership function. The determination of membership function is objective, but different people may have different understandings of the same fuzzy concept. Therefore, the determination of membership function has certain subjectivity. From this point of view, the concept of fuzzy mathematical membership function is it is objective and subjective. The general membership function is determined by experience or peer, or by experts and authorities.

(3) Fuzzy statistical method

There is more than one way to determine the membership function. The most basic one is fuzzy statistics. The following introduces the basic idea of fuzzy statistical methods: μ₀ is a certain element on the domain U, A^* is a variable clear set on the domain U, the basic idea of the fuzzy statistical method is to do whether μ₀ belongs to A^* make a clear judgment. However, for different researchers, the A^* may have different boundaries for the clear set, but these clear sets correspond to the same fuzzy set with the same probability and statistics. When the number of trials tends to be arbitrarily large, the probability can be replaced by frequency.

Each time an experiment is performed, it is determined whether the μ₀ belongs to A^*. If n experiments are performed, the number of times μ₀ belongs to A^* can be obtained, and the subordinate frequency of μ₀ to A can be obtained.

Subordinate Frequency of μ 0 to A = The Number of “ μ 0 Belongs to A *″ n(6)

It is proved by experiments that the membership frequency shows stability with the increase of n. This is called the membership frequency stability. As n tends to positive infinity, the membership frequency tends to a value. We call this value a frequency stability value. Also known as the membership of μ₀ to A:

μ ( A ) ( μ 0 ) = lim n → ∞ The Number of “ μ 0 Belongs to A *″ n(7)

(4) Binary comparison ordering method

The binary comparison ordering method compares the degree to which all the elements in the domain belong to the fuzzy set, and then processes them to obtain the membership degree of each element.

Suppose the domain is U = {u₁, u₂, . . . , u _n }, and A is a subset of the domain U. For any u _i , u _j , r _ij is used to indicate the degree to which u _i has a priority with respect to A than u _j , and r _ij is not equal to r _ji , and has the following restrictions: 0 ⩽ r ij ⩽ 1(8) r ii = 1 ( i = 1 , 2 , . . . , n )(9) r ij + r ji = 1 ( i , j = 1 , 2 , . . . , n )(10)

Then the square matrix R can be obtained, and for each row, the minimum value is the membership degree of u _i to A, that is, μ A ( u i ) = min j = 1 n r ij(11)

Then there is:

A = { min r 1 j , min r 2 j , . . . , min r nj } = { μ A ( u 1 ) , μ A ( u 2 ) , . . . , μ A ( u n ) }(12)

3.1 Experimental platform

MATLAB is a commercial mathematics software with an interactive working environment designed for advanced computing. It includes data simulation, data analysis, data view, code writing, numerical simulation and other functions. It is mainly used for data visualization and visualization. MATLAB has a convenient window interface, which integrates various functions such as matrix, numerical value, simulation, model establishment and other functions to adapt to different needs. It provides great information for engineering problem research, application data modeling and scientific calculation numerical analysis. Convenience, this interactive design interface is convenient for users to use, making it an effective tool for solving various problems such as engineering problems and numerical analysis.

The object processed by MATLAB is a matrix, which has convenient operations on the matrix. The engineering model suitable for the specific problem is established by simulation and the data is simulated. The programmed function can be called by coding, compared with other high-level languages. It has more efficient computing power, and many of its functions are designed to solve mathematical problems, so it has more processing power and analysis level in mathematical research. The more convenient visual interface can be used to analyze and debug the program intuitively, and can interface with other high-level languages. It has a wide range of applications in image processing, engineering model building, and control engineering. MATLAB has been published for many years, and it has undergone many revisions and performance enhancements, making it an efficient platform for dealing with various data analysis model creation problems.

3.2 Edge detection effect requirements

Various noises such as Gaussian noise, salt and pepper noise may occur due to various factors. Noise brings many difficulties to image processing. Image filtering requires that noise outside the image be removed while maintaining image detail. Linear filters are the most commonly used when noise is Gaussian noise and are easy to analyze and implement. However, when the noise is salt and pepper noise, the filtering effect of the linear filter is very poor. The traditional median filter can reduce the salt and pepper noise in the image, but the effect is not ideal, that is, the completely dispersed noise is removed. Noise close to each other will be preserved, so when the salt and pepper noise is more serious, the filtering effect is significantly deteriorated.

In general, the edge detection of images has the following requirements:

(1) Has good positioning and does not generate incorrect positioning;

(2) Excellent positioning accuracy, which can accurately position the edge points so as not to cause offset;

(3) It has a certain anti-interference ability to noise;

(4) The obtained edge is preferably a single pixel wide and the edges have been refined;

The above requirements are often contradictory and it is difficult to find an edge detection method that satisfies all requirements. The currently widely accepted evaluation method is to first look at the visual effect of the edge image and then evaluate its performance.

4.1 Image enhancement algorithm based on fuzzy set

(1) Simulation of image enhancement algorithm for fuzzy sets

In Fig. 2, the overall gray value and contrast are relatively low. In order to improve the overall gray scale, it needs to be enhanced by a suitable algorithm. First, after the global histogram is reluctantly processed, the result is shown in Fig. 2(b). From the results, the overall gray level of the original image is improved, but the gray level of the image is very severe, and the high gray scale is concentrated. There are many pixels inside, which makes the image too bright and the contrast of the image is not significantly improved, and the noise in the image is obviously enhanced. Finally, the original image is enhanced by the algorithm of this paper. The obtained result is Fig. 2(c). From the visual effect, the overall gray level of the original image is greatly improved, and the balance is evenly distributed throughout the gray scale. The contrast is greatly improved and the noise in the image is well suppressed. The data comparison is shown in Table 1 and Fig. 3.

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Fig. 2

Image enhancement results of different methods.

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Fig. 3

Data comparison chart.

(2) Analysis of image enhancement algorithms based on fuzzy sets

In order to make the contrast independent of the spatial frequency and spatial distribution of the image content, the root mean square is used here and the contrast is calculated as follows:

C σ = σ I w × h = 1 w × h · ∑ I w × h ( I ( x , y ) - 1 w × h · ∑ I w × h I ( x , y ) ) 2(13)

Where I_w×h represents a window of size w × h and I (x, y) represents the pixels of the window.

In order to verify the feasibility and superiority of the algorithm, they are enhanced by a new algorithm. After comparing the results, it can be seen that the proposed algorithm is not only feasible, but also has more advantages than other image reluctance methods. The method of this paper can not only enhance the contrast of the image, but also effectively suppress the interference of noise, and has very good use effect.

(1) Simulation analysis

At present, the commonly used edge detection operators include the robust operator, the Sobull operator, the canny operator and the Pal-King operator. However, these algorithms have some shortcomings, such as: the image detected by the Pal-King operator, the edge connectivity is poor, and the low gray level information loss is serious; the wide value selection of the canny operator needs to be obtained by experience and multiple experiments. Lack of adaptability. Aiming at these problems, this paper presents an improved image edge detection algorithm based on fuzzy enhancement and maximum inter-class variance for canny operator. The algorithm is divided into two parts, the first part is image blur enhancement, and the second part is image. Edge detection is performed on an enhanced basis. Here we use an image to detect the effectiveness of the algorithm. In order to prove that the algorithm is more advantageous than the traditional algorithm, we use different algorithms to perform edge detection on the same image, compare and analyze the results, and use objective data. Show the superiority of this algorithm.

Figure 4 shows the results of edge detection using the Sobel operator, Pal-King operator, canny operator, Roberts’s algorithm and the algorithm of the enhanced graph. Comparing the results of the five edge detection algorithms, we can see that the edge detection is more serious with the Sobel operator for edge detection. The edge lines are thicker and the continuity is poor.As a result of the detection, it can be seen that the edge image distortion is also serious, and although the edge lines are finer, the continuity is also worse. The image of the traditional canny operator edge detection has its high and low thresholds obtained through multiple experiments, which are 0.6 and 0.3 respectively. It can be seen that the edge detection effect is obviously more effective than the previous algorithm, not only the outline of the object is clear, the edge line is thin, and the line continuity is also good. The algorithm in this paper is clearer and more complete than the previous algorithm, and the edge location is more accurate. The statistics of experimental data are shown in Table 2 and Fig. 5.

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Fig. 4

Edge detection effect.

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Fig. 5

Experimental data statistics.

(2) Analysis of detection results

In this paper, the method of the degree of linear connection of the reaction edges is used to judge the merits of the results obtained in Fig. 4. Where M represents the number of detected edge points, N represents the number of pixels in the 8-neighbor single connection, and N/M represents the degree of edge linear connection. The smaller the N/M is, the better the edge extraction effect is.

Using the algorithm of this paper and other traditional algorithms to deal with a low grayscale and low contrast image, the results show that the proposed algorithm is reasonable and effective, and has certain advantages.