# Edge Location Method for Multidimensional Image Based on Edge Symmetry Algorithm

Authors: Chen Li
Journal: Security and Communication Networks
 (2021)

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## Abstract

The most basic feature of an image is edge, which is the junction of one attribute area and another attribute area in the image. It is the most uncertain place in the image and the place where the image information is most concentrated. The edge of an image contains rich information. So, the edge location plays an important role in image processing, and its positioning method directly affects the image effect. In order to further improve the accuracy of edge location for multidimensional image, an edge location method for multidimensional image based on edge symmetry is proposed. The method first detects and counts the edges of multidimensional image, sets the region of interest, preprocesses the image with the Gauss filter, detects the vertical edges of the filtered image, and superposes the vertical gradient values of each pixel in the vertical direction to obtain candidate image regions. The symmetry axis position of the candidate image region is analyzed, and its symmetry intensity is measured. Then, the symmetry of vertical gradient projection in the candidate image region is analyzed to verify whether the candidate region is a real edge region. The multidimensional pulse coupled neural network (PCNN) model is used to synthesize the real edge region after edge symmetry processing, and the result of edge location of the multidimensional image is obtained. The results show that the method has strong antinoise ability, clear edge contour, and precise location.


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## Body

## 1. Introduction

The most basic feature of an image is edge, which is the junction of one attribute area and another attribute area in the image. It is the place where the regional attributes mutate. It is the most uncertain place in the image and the place where the image information is most concentrated. The edge of an image contains rich information. Edge widely exists between objects and background, objects and objects, and primitives and primitives, so it is an important feature of image segmentation [1]. There are three kinds of common image edges. The first one is step edge, which is from one gray level to another gray level much higher than it. The second one is roof edge, which gradually increases to a certain degree and then decreases. Another is line edge, whose gray level changes from one level to another and then returns. Edge location is one of the most basic contents in image processing and recognition. An image is an information system, and a lot of information is provided by its contour edge. Therefore, edge location plays an important role in image processing, and its positioning method directly affects the image effect [2]. Multidimensional images are not susceptible to external environmental impact and have more feature information. The recognition accuracy is not affected by single feature change, and it has natural symmetry [3]. Symmetry is a common feature of many natural objects and artificial phenomena. It is widely used in image processing to describe the shape features of objects. Symmetry detection plays an important role in object recognition and location, multidimensional object reconstruction, and other fields [4].Image edge location methods mainly include the spatial moment location method and the subpixel location method, each of which has its own advantages and disadvantages. Spatial moment location is a common method in a fixed background. It can generally provide complete edge data, but its location effect depends on the merits of background model updating algorithm. It is particularly sensitive to scene changes, such as illumination and interference from external unrelated events [5]. The subpixel location method has strong adaptability to scene changes, but, generally, it cannot locate all the relevant edge pixels completely. It produces holes in the image entity, and it is easy to miss the location of some image edges [6]. In view of the shortcomings of the above two methods, this paper proposes an edge location method for the multidimensional image based on edge symmetry algorithm to improve the accuracy of edge location for the multidimensional image. This model was defined and tested for pictures in order to assess the performance of the suggested model. One of the test parameters was the number of iterations. The result is evaluated using the Pratt quality factor value. The suggested model’s output is compared to the performance of the two approaches described above.


## 2. Edge Symmetry Algorithm for Locating Multidimensional Image’s Edges

### 2.1. Edge Detection and Statistics

Firstly, the image is preprocessed to improve the image quality; secondly, the edge gradient in the edge region of the multidimensional image is calculated by using the vertical direction of the Sobel operator [7]; then, the candidate region is determined according to the characteristics of large jump and the large number of edge changes between the edge and the background.


#### 2.1.1. Image Preprocessing

(1)
Region of interest: in order to simultaneously detect a large range of image regions and reduce the computational complexity, the algorithm sets multiple regions of interest according to the number of pixels occupied by the image. The resolution of the image is 720 × 288, and a region of interest with a size of 200 × 50 is set in the three regions at the bottom of the image.(2)
Image preprocessing: in order to improve the accuracy of location, the image in the region of interest is preprocessed [8]. The color image is transformed into the gray image, and the noise is filtered by Gauss smoothing. The Gauss filter is a kind of linear smoothing filter which chooses weights according to the shape of Gauss function (i.e., normal distribution function). Gauss smoothing filter is very effective for removing noise which obeys normal distribution.One-dimensional zero-mean Gauss function is(1)gx=e−x2/2σ2,where the Gaussian distribution parameter σ determines the width of the Gaussian filter. For image edge processing, two-dimensional zero-mean discrete Gauss function is often used as the smoothing filter:(2)Gx,y=Ae−x2+y2/2σ2=Ae−r2/2σ2.Sampling and quantifying the continuous Gauss distribution above and normalizing the template, the discrete template is obtained:(3)G3=116121242121.The Gauss filter can solve the problem of spatial distance weighting and pixel gradient. The pixel gradient reflects the image edge characteristics, which is very helpful for edge location of the multidimensional image.


#### 2.1.2. Vertical Edge Detection

Vertical edge detection is applied to the preprocessed image. The essence of edge detection is to find the fast-changing region of brightness in the image, that is, the region whose first derivative of brightness is larger in magnitude than the specified range value. Because the vertical edge of the image has the strongest symmetry, in order to eliminate interference and reduce the amount of calculation, the proposed algorithm only processes the vertical component of the edge [9]. The vertical mask of the Sobel operator is used to digitally approximate the first derivative; that is, the vertical gradient of the image is calculated by formula (1) and presented as(4)Gx,y=fx+1,y−1+2fx+1,y+fx+1,y+1−fx−1,y−1+2fx−1,y+fx−1,y+1,where fx,y is the gray value of the pixel x,y and Gx,y is the approximate first derivative of the point in the vertical direction, i.e., the vertical gradient value.


#### 2.1.3. Gradient Statistics

The vertical gradient values of each pixel are projected in the vertical direction:(5)Bi=∑j=0H−1Gi,j,where H is the height of the detection block, B is the vertical superposition projection of the detection block, and i is the column position of the vertical superposition projection.The average value of vertical gradient projection is(6)B¯=∑i=0W−1BiW,where W is the width of the region of interest and B is the mean value.The mean value of vertical gradient projection is greater due to the large leap between edge and background, the significant number of edge modifications, and the quantity of vertical edges. In addition, the edge of the image is not susceptible to external environmental impact. Shadows, lights, and other interference factors cannot form a strong vertical gradient value, which will not have a great impact on the accuracy of location [10]. Therefore, when the mean value of vertical gradient projection exceeds a certain threshold, it can be determined as a candidate image region.


### 2.2. Symmetry Analysis

After edge identification and statistics, the symmetry of potential picture areas is examined. The image’s symmetry axis direction has been established as the vertical direction. It is required to establish the symmetrical axis’s position and quantify its symmetrical intensity [11]. The symmetry of the candidate image region’s vertical gradient projection is examined to determine whether the candidate region is a true edge region.


#### 2.2.1. Symmetry Analysis

The gradient vertical projection value obtained in the previous section is regarded as one-dimensional functiongx, assuming that its symmetric axis coordinate is XSW/2≤XS≤W−W/2 and W is the width of the symmetric region, which can be selected according to the width of the image [8]. Let u=X−XS; gu is a function with XS as the origin of the coordinate. Since an arbitrary function fx can be expressed as the sum of an odd function and an even function, as mentioned in formula (7), the odd function component and even function component of the definable function gx=gXs+u are, respectively,(7)Ou,Xs=gXs+u−gXs−u2,Eu,Xs=gXs+u−gXs−u2,−W/2≤u≤W/2.The even function components are normalized to make the mean value as 0 by using(8)Enu,Xs=Eu,Xs−1W∫−w/2+w/2Eu,Xsdu.Then, the symmetry is measured by comparing the energy of odd component function with that of an even component function:(9)SXS=∫−w/2+w/2Enu,Xs2du−∫−w/2+w/2Ou,Xs2du∫−w/2+w/2Enu,Xs2du+∫−w/2+w/2Ou,Xs2du,−1≤SXS≤1,where S=1, and it is completely symmetric, and when S=−1, it is completely asymmetric.


#### 2.2.2. Verification of Candidate Region

According to the size of the candidate region, the symmetric axis is searched by changing theXs value of the symmetric coordinate axis in a certain range. In the search interval, Xs is taken as the coordinate of the symmetrical axis of the candidate region when the maximum value of the symmetrical evaluation function SXS appears. When SXS〉0.5 is located on the symmetrical axis, it is considered to be in line with the candidate region.In addition, the vertical projection value of the edge should be the largest on both sides of the image edge, so locating the maximum value on both sides of the symmetric axis of the even function componentEu,Xs can determine the approximate left and right boundaries of the image edge. Since the edge width of image satisfies certain constraints [12–15], it can be further verified whether it is a real image edge.


### 2.3. PCNN Edge Location of Multidimensional Image

The real edge region processed by edge symmetry is synthesized by PCNN for edge location. The PCNN model plays an important role in image denoising, smoothing, segmentation, edge extraction, and feature extraction. As multidimensional images provide richer target information than two-dimensional images, the edge location of the multidimensional image has attracted more and more attention [16–19]. At present, PCNN can only deal with two-dimensional images directly, which have great limitations. This paper introduces vector matrix and multidimensional convolution ⊗ to extend PCNN, which is called the multidimensional PCNN model for short, and the application of PCNN is extended so that PCNN can directly realize multidimensional image’s edge location [20–23].Figure1(a) represents the process of two-dimensional PCNN, and Figure 1(b) represents the multidimensional PCNN. It can be seen from Figure 1 that a multidimensional PCNN is equivalent to the edge location of PCNN image with three real edge regions. The multidimensional PCNN model satisfies the following formulas:(10)Fijn=sij,(11)Lijn=Yijn−1⊗W,(12)Uijn=Fijn−11+βLijn,(13)Yijn=1,UijnEijn,0,other,(14)Eijn=exp−αEEijn−1+νEYijn−1,where S, F, L, U, Y, and W are vector matrices, sij is the external input stimulus signal (here is the gradient vector Gija,Gijb,Gijc of the input image at i,j, Fijn is the nth feedback input of the i,jth image, Lijn is the nth connection input of thei,jth image, Uijn is the internal active item of the image, Eij is the dynamic threshold, Yij is the output sequence of PCNN, W is the intensity constant of adjacent connected images in the connection domain, νE is the intrinsic potential in Eijn, αE is the attenuation time constant of Eijn, n is the iteration number, which can be selected according to actual needs, and ⊗ is the multidimensional convolution; if convolution is the multidimensional convolution and conv2 is the two-dimensional convolution, then YconvnW=YAconv2WA,YBconv2WB,YCconv2WC.Figure 1
(a)Two-dimensional and (b) multidimensional PCNN processing process.
(a)(b)According to the principle of vector gradient and the definition of multidimensional PCNN, we can design an edge location algorithm for the multidimensional image based on the real-edge region obtained from edge symmetry. The flowchart of the algorithm is shown in Figure2.(1)
Vector gradient operator DV is used to compute the gradient of the real-edge region processed by the symmetry of the input edge, which reflects the change of color information in the neighborhood of the pixel and provides the basis for PCNN edge location [24, 25].(2)
Initialization of PNCC:Eij is the maximum value in gradient graph, which can suppress the ignition of small gradient pixels, and only large gradient can ignite. Yij is all zero.(3)
The vector gradient obtained in step 1 is input into the multidimensional PNCC model, and the thresholdEij is attenuated to determine whether the value of the internal activity item is greater than the threshold value. If the value is greater than the threshold value, Y is set to 1, otherwise 0 is set.(4)
If the number of iterations is less than the prescribed maximum number of iterations, all the pixels will return to step 3 to continue iteration after judging, otherwise the iteration will end [26–28].(5)
Finally, the best processing effect is selected from the real edge regions and obtained when the number of iterations is small, and the final multidimensional image’s edges are obtained by adding the three channel results of the multidimensional PCNN model. Since the threshold is attenuated from the maximum gradient value, when the iteration times are small, the small gradient is suppressed [29]. Only those large gradient values will ignite first, the sharp change of pixel points, namely, edge points, can be detected.Figure 2
Algorithm flowchart.Because the convolution in formula (11) is multidimensional convolution and PCNN is vector operation, there is no need for three channels to run separately. Parallel operation reduces the operation time. Finally, the edge subgraphs of three channels are added together to obtain the final edge location results. The location results are determined by the output of all channels, which can locate the edges of multidimensional images more accurately.


## 2.1. Edge Detection and Statistics

Firstly, the image is preprocessed to improve the image quality; secondly, the edge gradient in the edge region of the multidimensional image is calculated by using the vertical direction of the Sobel operator [7]; then, the candidate region is determined according to the characteristics of large jump and the large number of edge changes between the edge and the background.


### 2.1.1. Image Preprocessing

(1)
Region of interest: in order to simultaneously detect a large range of image regions and reduce the computational complexity, the algorithm sets multiple regions of interest according to the number of pixels occupied by the image. The resolution of the image is 720 × 288, and a region of interest with a size of 200 × 50 is set in the three regions at the bottom of the image.(2)
Image preprocessing: in order to improve the accuracy of location, the image in the region of interest is preprocessed [8]. The color image is transformed into the gray image, and the noise is filtered by Gauss smoothing. The Gauss filter is a kind of linear smoothing filter which chooses weights according to the shape of Gauss function (i.e., normal distribution function). Gauss smoothing filter is very effective for removing noise which obeys normal distribution.One-dimensional zero-mean Gauss function is(1)gx=e−x2/2σ2,where the Gaussian distribution parameter σ determines the width of the Gaussian filter. For image edge processing, two-dimensional zero-mean discrete Gauss function is often used as the smoothing filter:(2)Gx,y=Ae−x2+y2/2σ2=Ae−r2/2σ2.Sampling and quantifying the continuous Gauss distribution above and normalizing the template, the discrete template is obtained:(3)G3=116121242121.The Gauss filter can solve the problem of spatial distance weighting and pixel gradient. The pixel gradient reflects the image edge characteristics, which is very helpful for edge location of the multidimensional image.


### 2.1.2. Vertical Edge Detection

Vertical edge detection is applied to the preprocessed image. The essence of edge detection is to find the fast-changing region of brightness in the image, that is, the region whose first derivative of brightness is larger in magnitude than the specified range value. Because the vertical edge of the image has the strongest symmetry, in order to eliminate interference and reduce the amount of calculation, the proposed algorithm only processes the vertical component of the edge [9]. The vertical mask of the Sobel operator is used to digitally approximate the first derivative; that is, the vertical gradient of the image is calculated by formula (1) and presented as(4)Gx,y=fx+1,y−1+2fx+1,y+fx+1,y+1−fx−1,y−1+2fx−1,y+fx−1,y+1,where fx,y is the gray value of the pixel x,y and Gx,y is the approximate first derivative of the point in the vertical direction, i.e., the vertical gradient value.


### 2.1.3. Gradient Statistics

The vertical gradient values of each pixel are projected in the vertical direction:(5)Bi=∑j=0H−1Gi,j,where H is the height of the detection block, B is the vertical superposition projection of the detection block, and i is the column position of the vertical superposition projection.The average value of vertical gradient projection is(6)B¯=∑i=0W−1BiW,where W is the width of the region of interest and B is the mean value.The mean value of vertical gradient projection is greater due to the large leap between edge and background, the significant number of edge modifications, and the quantity of vertical edges. In addition, the edge of the image is not susceptible to external environmental impact. Shadows, lights, and other interference factors cannot form a strong vertical gradient value, which will not have a great impact on the accuracy of location [10]. Therefore, when the mean value of vertical gradient projection exceeds a certain threshold, it can be determined as a candidate image region.


## 2.1.1. Image Preprocessing

(1)
Region of interest: in order to simultaneously detect a large range of image regions and reduce the computational complexity, the algorithm sets multiple regions of interest according to the number of pixels occupied by the image. The resolution of the image is 720 × 288, and a region of interest with a size of 200 × 50 is set in the three regions at the bottom of the image.(2)
Image preprocessing: in order to improve the accuracy of location, the image in the region of interest is preprocessed [8]. The color image is transformed into the gray image, and the noise is filtered by Gauss smoothing. The Gauss filter is a kind of linear smoothing filter which chooses weights according to the shape of Gauss function (i.e., normal distribution function). Gauss smoothing filter is very effective for removing noise which obeys normal distribution.One-dimensional zero-mean Gauss function is(1)gx=e−x2/2σ2,where the Gaussian distribution parameter σ determines the width of the Gaussian filter. For image edge processing, two-dimensional zero-mean discrete Gauss function is often used as the smoothing filter:(2)Gx,y=Ae−x2+y2/2σ2=Ae−r2/2σ2.Sampling and quantifying the continuous Gauss distribution above and normalizing the template, the discrete template is obtained:(3)G3=116121242121.The Gauss filter can solve the problem of spatial distance weighting and pixel gradient. The pixel gradient reflects the image edge characteristics, which is very helpful for edge location of the multidimensional image.


## 2.1.2. Vertical Edge Detection

Vertical edge detection is applied to the preprocessed image. The essence of edge detection is to find the fast-changing region of brightness in the image, that is, the region whose first derivative of brightness is larger in magnitude than the specified range value. Because the vertical edge of the image has the strongest symmetry, in order to eliminate interference and reduce the amount of calculation, the proposed algorithm only processes the vertical component of the edge [9]. The vertical mask of the Sobel operator is used to digitally approximate the first derivative; that is, the vertical gradient of the image is calculated by formula (1) and presented as(4)Gx,y=fx+1,y−1+2fx+1,y+fx+1,y+1−fx−1,y−1+2fx−1,y+fx−1,y+1,where fx,y is the gray value of the pixel x,y and Gx,y is the approximate first derivative of the point in the vertical direction, i.e., the vertical gradient value.


## 2.1.3. Gradient Statistics

The vertical gradient values of each pixel are projected in the vertical direction:(5)Bi=∑j=0H−1Gi,j,where H is the height of the detection block, B is the vertical superposition projection of the detection block, and i is the column position of the vertical superposition projection.The average value of vertical gradient projection is(6)B¯=∑i=0W−1BiW,where W is the width of the region of interest and B is the mean value.The mean value of vertical gradient projection is greater due to the large leap between edge and background, the significant number of edge modifications, and the quantity of vertical edges. In addition, the edge of the image is not susceptible to external environmental impact. Shadows, lights, and other interference factors cannot form a strong vertical gradient value, which will not have a great impact on the accuracy of location [10]. Therefore, when the mean value of vertical gradient projection exceeds a certain threshold, it can be determined as a candidate image region.


## 2.2. Symmetry Analysis

After edge identification and statistics, the symmetry of potential picture areas is examined. The image’s symmetry axis direction has been established as the vertical direction. It is required to establish the symmetrical axis’s position and quantify its symmetrical intensity [11]. The symmetry of the candidate image region’s vertical gradient projection is examined to determine whether the candidate region is a true edge region.


### 2.2.1. Symmetry Analysis

The gradient vertical projection value obtained in the previous section is regarded as one-dimensional functiongx, assuming that its symmetric axis coordinate is XSW/2≤XS≤W−W/2 and W is the width of the symmetric region, which can be selected according to the width of the image [8]. Let u=X−XS; gu is a function with XS as the origin of the coordinate. Since an arbitrary function fx can be expressed as the sum of an odd function and an even function, as mentioned in formula (7), the odd function component and even function component of the definable function gx=gXs+u are, respectively,(7)Ou,Xs=gXs+u−gXs−u2,Eu,Xs=gXs+u−gXs−u2,−W/2≤u≤W/2.The even function components are normalized to make the mean value as 0 by using(8)Enu,Xs=Eu,Xs−1W∫−w/2+w/2Eu,Xsdu.Then, the symmetry is measured by comparing the energy of odd component function with that of an even component function:(9)SXS=∫−w/2+w/2Enu,Xs2du−∫−w/2+w/2Ou,Xs2du∫−w/2+w/2Enu,Xs2du+∫−w/2+w/2Ou,Xs2du,−1≤SXS≤1,where S=1, and it is completely symmetric, and when S=−1, it is completely asymmetric.


### 2.2.2. Verification of Candidate Region

According to the size of the candidate region, the symmetric axis is searched by changing theXs value of the symmetric coordinate axis in a certain range. In the search interval, Xs is taken as the coordinate of the symmetrical axis of the candidate region when the maximum value of the symmetrical evaluation function SXS appears. When SXS〉0.5 is located on the symmetrical axis, it is considered to be in line with the candidate region.In addition, the vertical projection value of the edge should be the largest on both sides of the image edge, so locating the maximum value on both sides of the symmetric axis of the even function componentEu,Xs can determine the approximate left and right boundaries of the image edge. Since the edge width of image satisfies certain constraints [12–15], it can be further verified whether it is a real image edge.


## 2.2.1. Symmetry Analysis

The gradient vertical projection value obtained in the previous section is regarded as one-dimensional functiongx, assuming that its symmetric axis coordinate is XSW/2≤XS≤W−W/2 and W is the width of the symmetric region, which can be selected according to the width of the image [8]. Let u=X−XS; gu is a function with XS as the origin of the coordinate. Since an arbitrary function fx can be expressed as the sum of an odd function and an even function, as mentioned in formula (7), the odd function component and even function component of the definable function gx=gXs+u are, respectively,(7)Ou,Xs=gXs+u−gXs−u2,Eu,Xs=gXs+u−gXs−u2,−W/2≤u≤W/2.The even function components are normalized to make the mean value as 0 by using(8)Enu,Xs=Eu,Xs−1W∫−w/2+w/2Eu,Xsdu.Then, the symmetry is measured by comparing the energy of odd component function with that of an even component function:(9)SXS=∫−w/2+w/2Enu,Xs2du−∫−w/2+w/2Ou,Xs2du∫−w/2+w/2Enu,Xs2du+∫−w/2+w/2Ou,Xs2du,−1≤SXS≤1,where S=1, and it is completely symmetric, and when S=−1, it is completely asymmetric.


## 2.2.2. Verification of Candidate Region

According to the size of the candidate region, the symmetric axis is searched by changing theXs value of the symmetric coordinate axis in a certain range. In the search interval, Xs is taken as the coordinate of the symmetrical axis of the candidate region when the maximum value of the symmetrical evaluation function SXS appears. When SXS〉0.5 is located on the symmetrical axis, it is considered to be in line with the candidate region.In addition, the vertical projection value of the edge should be the largest on both sides of the image edge, so locating the maximum value on both sides of the symmetric axis of the even function componentEu,Xs can determine the approximate left and right boundaries of the image edge. Since the edge width of image satisfies certain constraints [12–15], it can be further verified whether it is a real image edge.


## 2.3. PCNN Edge Location of Multidimensional Image

The real edge region processed by edge symmetry is synthesized by PCNN for edge location. The PCNN model plays an important role in image denoising, smoothing, segmentation, edge extraction, and feature extraction. As multidimensional images provide richer target information than two-dimensional images, the edge location of the multidimensional image has attracted more and more attention [16–19]. At present, PCNN can only deal with two-dimensional images directly, which have great limitations. This paper introduces vector matrix and multidimensional convolution ⊗ to extend PCNN, which is called the multidimensional PCNN model for short, and the application of PCNN is extended so that PCNN can directly realize multidimensional image’s edge location [20–23].Figure1(a) represents the process of two-dimensional PCNN, and Figure 1(b) represents the multidimensional PCNN. It can be seen from Figure 1 that a multidimensional PCNN is equivalent to the edge location of PCNN image with three real edge regions. The multidimensional PCNN model satisfies the following formulas:(10)Fijn=sij,(11)Lijn=Yijn−1⊗W,(12)Uijn=Fijn−11+βLijn,(13)Yijn=1,UijnEijn,0,other,(14)Eijn=exp−αEEijn−1+νEYijn−1,where S, F, L, U, Y, and W are vector matrices, sij is the external input stimulus signal (here is the gradient vector Gija,Gijb,Gijc of the input image at i,j, Fijn is the nth feedback input of the i,jth image, Lijn is the nth connection input of thei,jth image, Uijn is the internal active item of the image, Eij is the dynamic threshold, Yij is the output sequence of PCNN, W is the intensity constant of adjacent connected images in the connection domain, νE is the intrinsic potential in Eijn, αE is the attenuation time constant of Eijn, n is the iteration number, which can be selected according to actual needs, and ⊗ is the multidimensional convolution; if convolution is the multidimensional convolution and conv2 is the two-dimensional convolution, then YconvnW=YAconv2WA,YBconv2WB,YCconv2WC.Figure 1
(a)Two-dimensional and (b) multidimensional PCNN processing process.
(a)(b)According to the principle of vector gradient and the definition of multidimensional PCNN, we can design an edge location algorithm for the multidimensional image based on the real-edge region obtained from edge symmetry. The flowchart of the algorithm is shown in Figure2.(1)
Vector gradient operator DV is used to compute the gradient of the real-edge region processed by the symmetry of the input edge, which reflects the change of color information in the neighborhood of the pixel and provides the basis for PCNN edge location [24, 25].(2)
Initialization of PNCC:Eij is the maximum value in gradient graph, which can suppress the ignition of small gradient pixels, and only large gradient can ignite. Yij is all zero.(3)
The vector gradient obtained in step 1 is input into the multidimensional PNCC model, and the thresholdEij is attenuated to determine whether the value of the internal activity item is greater than the threshold value. If the value is greater than the threshold value, Y is set to 1, otherwise 0 is set.(4)
If the number of iterations is less than the prescribed maximum number of iterations, all the pixels will return to step 3 to continue iteration after judging, otherwise the iteration will end [26–28].(5)
Finally, the best processing effect is selected from the real edge regions and obtained when the number of iterations is small, and the final multidimensional image’s edges are obtained by adding the three channel results of the multidimensional PCNN model. Since the threshold is attenuated from the maximum gradient value, when the iteration times are small, the small gradient is suppressed [29]. Only those large gradient values will ignite first, the sharp change of pixel points, namely, edge points, can be detected.Figure 2
Algorithm flowchart.Because the convolution in formula (11) is multidimensional convolution and PCNN is vector operation, there is no need for three channels to run separately. Parallel operation reduces the operation time. Finally, the edge subgraphs of three channels are added together to obtain the final edge location results. The location results are determined by the output of all channels, which can locate the edges of multidimensional images more accurately.


## 3. Results

In order to verify the accuracy of edge location for multidimensional images, the proposed method is compared with the subpixel method and the spatial distance method.The Pratt quality factor is a representative objective evaluation index of edge detection results and a more comprehensive evaluation parameter. Therefore, the Pratt quality factor is selected to objectively evaluate the performance of image’s edge location of different edge location methods. At the same time, the vertical standard map is used to test. Noise and signal-to-noise ratio are added to analyze different methods under different iterations. The result of edge location of the multidimensional image is shown in Figures3 and 4.Figure 3
Evaluation results,N = 1.Figure 4
Evaluation results,N = 3.Figure3 represents the variation of quality factor with respect to the signal-to-noise ratio (SNR) for single iteration, i.e., N = 1. The results show that the Pratt quality factor value of the proposed method is higher than that of the other two methods. It is clear from Figure 3 that the edge location effect of the proposed method is better than that of the other two methods, but the effect is not obvious.Figure4 represents the Pratt quality factor vs. signal-to-noise ratio (SNR) for three number of iterations, i.e., N = 3. The Pratt quality factor for the proposed method is superior to the other two methods. The infrared image House and Lena are selected to simulate and compare the proposed method with the subpixel method and the space distance method. The simulation results are shown as follows.As can be seen in Figures5–14, the image edge image localized by the proposed method is better than the image edge localized by the nonenhanced edge image, the subpixel method, and the spatial moment method. The image processed by the proposed method not only has clearer layers between regions but also enhances the gray contrast on both sides of the edge. The simulation results also show that the edge location of this method is more precise.Figure 5
House original.Figure 6
Edge diagram without enhancement (N = 0).Figure 7
The method edge location graph in this paper.Figure 8
Subpixel method edge map.Figure 9
Spatial moment method edge location map.Figure 10
Original map of Lena.Figure 11
Edge diagram without enhancement (N = 0).Figure 12
The method edge location diagram in this paper.Figure 13
Subpixel method edge map.Figure 14
Spatial moment method edge mapping.In order to verify the edge location effect of this method in multidimensional images, two multiband remote-sensing images are selected for edge location, and the location results are shown in the figures. Figure15 is the IKONOS satellite image, with the size of 544 × 342, and membership function parameters of a=23, b=159, and T=0.85. Figure 16 is the IKONOS pseudocolor remote-sensing image, with the size of 600 × 450, and membership function parameters of a=11, b=173, and T=0.79. The subpixel method, the spatial moment method, and the proposed method describe the results of edge location in Figure 15, by using Figures 17–19. The subpixel method, the spatial moment method, and the proposed method describe the results of edge location in Figure 16, by using Figures 20–22.Figure 15
Original image of IKONOS satellite image.Figure 16
Original image of IKONOS false color remote-sensing images.Figure 17
Subpixel method edge positioning.Figure 18
Spatial moment method edge positioning.Figure 19
Edge positioning of this method.Figure 20
Subpixel method edge positioning.Figure 21
Spatial moment method edge positioning.Figure 22
Edge positioning of this method.According to the above figures, we can see that the multidimensional image’s edges located by the subpixel method and the spatial moment method contain many false edges and secondary edges, and the effect is not ideal. However, the method in this paper is accurate and effective in multidimensional image’s edge location.In order to verify the location accuracy of the proposed method, the test image is selected as the multidimensional image of the microgear scanned by the multidimensional profiler. Considering the smoothing effect of the actual optical system and sensors on the image, the multidimensional image of the microgear is smoothed by 3 × 3 pixel neighborhood filter, and then, the standard multidimensional test image for experiment is generated. Multidimensional profilometer is a precision-measuring instrument. Its dimension measuring accuracy can reach the level of nm. In this paper, the measured data are taken as actual data. Tables1–3 are the results of edge location of multidimensional image of microgears by three methods in simulation experiments.Table 1
The method of edge positioning of the multidimensional image of the microgear.
Label pointActual distance (μm)This method distance (μm)Error against actual value (μm)D17497501D27527502D37417454D47717732D57617654D6195419562D7195219493D8198719861D9196119643D10197419722Table 2
Subpixel method for edge positioning of microgear multidimensional image.
Label pointActual distance (μm)This method distance (μm)Error against actual value (μm)D174973316D275273913D374172714D477175912D576174615D61954194113D71952194012D81987197512D91961194615D101974196212Table 3
Spatial moment method for edge positioning of microgear multidimensional image.
Label pointActual distance (μm)This method distance (μm)Error against actual value (μm)D174976718D275277119D374176221D477178716D576177918D61954197016D71952196917D81987200518D91961198019D101974199420By comparing and analyzing Tables1–3, the average error of the proposed method is 2.4 μm when locating the multidimensional image edge of the microgear, while the average error of the subpixel method and the spatial moment method is 18.2 μm and 13.4 μm when locating the multidimensional image edge. Compared with the subpixel method and the spatial moment method, the proposed method is more accurate than the subpixel method and the spatial moment method by 15.8 μm and 11 μm. It can be seen that the proposed method can locate the multidimensional image edge accurately.


## 4. Discussion

In view of the above analysis and discussion, this paper proposes an edge location method for the multidimensional image based on edge symmetry, which can process the multidimensional image with strong antinoise and can locate the multidimensional image’s edge more precisely and clearly, and the effect is obvious. The analysis is from the following three aspects:(1)
In this paper, a Gauss filter is introduced to preprocess multidimensional images. The problem of spatial distance weighting and pixel gradient is well solved by the Gauss filter. The pixel gradient reflects the image edge features, which is very helpful for multidimensional image’s edge location.(2)
In this paper, we analyze the symmetric axis position of candidate image region and measure its symmetry intensity. Then, the vertical gradient projection symmetry of the candidate image region is analyzed to verify whether the candidate region is a real edge region. The symmetry analysis of the multidimensional image is carried out. The accuracy of image’s edge location and robustness of the algorithm are effectively improved by means of the verification of vertical gradient projection’s mean value, symmetry detection, and width constraint.(3)
After introducing vector matrix and multidimensional convolution into PCNN, this paper generalizes PCNN, which is called the multidimensional PCNN model for short, and expands the application of PCNN so that PCNN can directly realize multidimensional image’s edge location. The multidimensional PCNN model is used to synthesize the real-edge region after edge symmetry processing, and the result of edge location of the multidimensional image is obtained.


## 5. Conclusions

The proposed method uses the Gauss filter to preprocess the multidimensional image, which can not only suppress noise but also better preserve image edge and high-frequency detail information so that the edge image outline is clear, the edge position is precise, and the error is small, which is of great benefit to image location research. Moreover, the real edge region after edge symmetry processing is synthesized by using the multidimensional PCNN model to obtain accurate edge location results of the multidimensional image. The experimental results show that the proposed method is superior to the subpixel method and the spatial distance method in terms of noise immunity and multidimensional image’s location accuracy and clarity. The location accuracy of the multidimensional image of microgears by using the proposed method is 2.4μm on an average, which is more accurate than the subpixel method and the spatial moment method with 15.8 μm and 11 μm. The results for the proposed method are superior to the conventional methods; also, as the number of iterations increases, the quality of results also increases. It can be seen that the proposed method has clear edge location and high accuracy in the multidimensional image.



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