Design of an optimized fuzzy system for edge detection in images

2.1.1 Laplacian edge detection filter

The Laplacian filter operator is known as a 2-D derivative operator. Using a Laplacian filter is to highlight and enhance fine details. Like the threaded edges. As you know, the Laplacian filter is a linear non-rotating actuator. The Laplacian filter can be expressed separately for all of dimensions. Partial quadratic derivatives in the x and y directions can be obtained with (1) [15]. Mask matrix of Laplacian filter has the 3×3 dimension that is shown in Fig. 1. ∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 ∂ 2 f ∂ x 2 = f ( x + 1 , y ) + f ( x - 1 , y ) - 2 f ( x , y ) ∂ 2 f ∂ y 2 = f ( x , y + 1 ) + f ( x , y - 1 ) - 2 f ( x , y )(1)

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Fig. 1

The Two Samples of Laplacian kernels.

Laplacian operator is a second-order minimum agent with convergence properties in border detection that plays a significant role in image processing techniques. But anyway, there are two drawbacks. First, Laplacian cannot obtain accurate edge direction information; Secondly, because of the sensitivity to noise, the border often appears with a width of two pixels. Therefore, the Laplacein operator is used only to recognize the image pixels, as the Laplacian operator is in the black or light region [16].

Laplace filter itself increases the noise in the image and detects the noise as an edge. Therefore, to use this filter, first use an anti-noise filter (Gaussian filter) on the image and then apply the Laplacein filter. Also, the use of a Laplacian filter as a result of the median filter, prevents this problem. This most of the noise is reduced in the middle filtration process [15]. Figure 2, represents the main image and Laplacian Filtered image, and this filter is applied to the original picture in this paper.

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Fig. 2

(a) Original Image and (b) Laplacian Filtered Image.

The Prewitt agent is a first-class differential agent in edge recognition technique. Using the variation in the amount of gray between the pixels above and below, and the left and right neighbor pixels, the edge can be recognized with a short amount in the edge, and some pseudo-edges can be eliminated, which also making the noise smoother. The general rule is for using two main patterns in the image pixels to doing of the convolution in the image. Main patterns distinguish the horizontal edge and the vertical edge. The Prewitt agent is an essential technique for border recognition due to computational simplicity and ease of implementation with hardware. So, this filter has been used a lot. Although, the custom Prewitt operator has a weak efficacy on busy images and apparent deviations from the trial results [17].

The Prewitt mask in this method, is a discrete operator and uses two 3×3 mask matrices for getting the approximate values in horizontal and vertical orientations. Figure 3 shows this mask matrix [17].

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Fig. 3

Mask Matrix in Prewitt Filter.

Horizontal derivative and vertical derivative approximate as (2) [9]. g x = ( z 3 + z 6 + z 9 ) - ( z 1 + z 4 + z 7 ) g y = ( z 7 + z 8 + z 9 ) - ( z 1 + z 2 + z 3 )(2)

The method of detection of the Prewitt operator can easily lead to incorrect detection of border pixels, since the gray amount of many pixels is high. If the small border is recognized, the picture’s border is removed [17]. Figure 4 shows this matter.

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Fig. 4

(a) Original Image and (b) Prewitt Filtered Image.

This filter approximates the slope of a picture via digital derivation. The Robert mask matric, along with the whole picture is convolved using Robert horizontal and vertical masks to show the edge of the recognized picture, in the x and y directions, correspondingly [18]. Figure 5 shows this mask matrix.

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Fig. 5

Mask Matrixs in Robert Filter.

Horizontal derivative approximates and vertical derivative approximate as (3) [18]: g x = ( z 4 - z 1 ) g y = ( z 3 - z 2 )(3)

In the use of diagonal edge, we need 2-D kernels, and the Roberts cross-gradient agents are one of the closest efforts to applying it. The agents are based on fulfilment the diagonal differences [19]. Robert produced a dark background with the least edging information [20]. Figure 6 shows this matter.

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Fig. 6

(a) Original Image and (b) Prewitt Filtered Image.

This filter (Sobel operator), a widespread applied edge detection algorithm, was first introduced by Erwin Sobel at the Stanford Artificial Intelligence Laboratory in 1968. This operator uses the second brightness level change to recognize the edges inside the picture. The gradient vector in a situation (x, y) is presented as (4). In (4), G _x and G _y are the gradient of pixels in the x and y senses, correspondingly [21]. ∇ f = [ G x G y ] = [ df dx df dy ](4) The altitude and phase ∇f are defined by (5): | ∇ f | = G x 2 + G y 2 ∠ ( ∇ f ) = tan - 1 G x G y(5)

| ∇ f| is the largest size of conversion of f(x, y) per unique interval in orientation of gradient vector at the given position (x, y). ∠ (∇ f) is given as orientation of the gradient vector. Figure 7 presents Sobel mask matrices that are presented in (6) to illustrate the G _x and G _y vectors [22, 23].

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Fig. 7

Mask Matrix in Sobel Filter: (a) Mask matrix for x-direction and (b) Mask matrix for the y-direction.

The G _x and G _y are described as follows [23]. The mask matrix related to them is shown in Fig. 7. G x = { f ( x + 1 , y - 1 ) + 2 f ( x + 1 , y ) + f ( x + 1 , y + 1 ) } - { f ( x - 1 , y - 1 ) + 2 f ( x - 1 , y ) + f ( x - 1 , y + 1 ) } G y = { f ( x + 1 , y - 1 ) + 2 f ( x , y + 1 ) + f ( x + 1 , y + 1 ) } - { f ( x - 1 , y - 1 ) + 2 f ( x , y - 1 ) + f ( x + 1 , y - 1 ) }(6)

Two filters are applied to the image, and the results are combined to give us a binary image of the edges [19]. Figure 8 shows this matter.

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Fig. 8

(a) Original Image and (b) Sobel Filtered Image.