information Article
Edge Detection Method Based on General Type-2 Fuzzy Logic Applied to Color Images Claudia I. Gonzalez, Patricia Melin *
ID
and Oscar Castillo
ID
Tijuana Institute of Technology, Tijuana 22379, Mexico;
[email protected] (C.I.G.);
[email protected] (O.C.) * Correspondence:
[email protected]; Tel.: +52-664-6236318 Received: 1 July 2017; Accepted: 25 August 2017; Published: 28 August 2017
Abstract: This paper presents a new general type-2 fuzzy logic method for edge detection applied to color format images. The proposed algorithm combines the methodology based on the image gradients and general type-2 fuzzy logic theory to provide a powerful edge detection method. General type-2 fuzzy inference systems are approximated using the α-planes approach. The edge detection method is tested on a database of color images with the idea of illustrating the advantage of applying the fuzzy edge detection approach on color images against grayscale format images, and also when the images are corrupted by noise. This paper compares the proposed method based on general type-2 fuzzy logic with other edge detection algorithms, such as ones based on type-1 and interval type-2 fuzzy systems. Simulation results show that edge detection based on a general type-2 fuzzy system outperforms the other methods because of its ability to handle the intrinsic uncertainty in this problem. Keywords: α-planes; general type-2 fuzzy sets; general type-2 fuzzy sets; interval type-2 fuzzy sets; type-1 fuzzy sets; image processing; color edge detection; computer vision
1. Introduction Edge detection is a fundamental technique in many computer vision and image processing applications, such as pattern recognition, medical image processing, object recognition, and motion analysis. In the past few decades most research has concentrated on designing edge detector algorithms for grayscale images; however, color image processing applications have recently been receiving increasing attention because color images provide more information about the objects in a scene than grayscale images and this information can be used to improve the performance of image processing systems [1]. Edge detection in color images is computationally more complex than in grayscale images, but there are many advantages to using color images. For example, the increase in the amount of information can lead to more accurate object location and the possibility of processing images that are more complex [2]. Various edge detection techniques have been proposed over the years, but the common approach is to apply the first or second derivative. Based on this, the edge detection can be classified as gradient edge detectors (first derivative), Laplacian method (second derivative), or Gaussian edge detectors [3]. In the fuzzy system area, there are edge detection methods that have been developed for grayscale images [4–8]. In Gonzalez et al. [9], an improved method for processing edge detection based on general type-2 fuzzy sets (GT2 FSs) applied to grayscale images is proposed. According with the results presented in [9], the author demonstrated that the edge detection approach based on GT2 FSs outperformed the results obtained by the methods based on interval type-2 fuzzy sets (IT2 FSs), type-1 fuzzy sets (T1 FSs), and of course the traditional edge detection methods. The contribution of this paper is focused on exploiting the benefits of using digital images in color format and combining this process with GT2 FSs, which has shown satisfactory results in image Information 2017, 8, 104; doi:10.3390/info8030104
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processing applications; especially, when the image is corrupted by noise, this kind of technique is helpful to handle the uncertainty, so we are proposing a novel method of edge detection based on GT2 FSs to be applied to color images. The process is designed using Lab, HSV, and RGB to show the advantages of using one kind of color format with respect to the others. The inputs and outputs of the fuzzy system are built with Gaussian membership functions (MFs), which have demonstrated good results in these kinds of applications. Moreover, we present a comparative analysis of the results achieved with this edge detection approach versus the results achieved by using TI and IT2 FSs. The paper is organized as follows. In Section 2 important definitions about GT2 FSs are presented. Section 3 describes the color image edge detector using the gradient approach. Section 4 shows the color image edge detector based on GT2 FSs. In Section 5 the simulation results are presented and explained. Finally, Section 6 offers some conclusions about the results. 2. Fuzzy Logic Systems Fuzzy sets theory was introduced by Lofti A. Zadeh in 1965 [10]. Fuzzy sets provide a way to model the uncertainty associated with vagueness, imprecision, and lack of information regarding a problem. The uncertainty arises from ignorance due to a lack of knowledge, from randomness, or from vagueness, like the fuzziness existing in human language [10,11]. In recent years we have witnessed a rapid growth in a number of applications of fuzzy logic in various areas, such as pattern recognition, neural networks, expert systems, artificial intelligence, theory control, automata theory, computer science, decision making, medical diagnosis, and robotics, to name a few. In digital image processing, which is the area of interest of this paper, there are some applications, like in edge detection [5,6,8], noise filtering [12,13], feature extraction [14], classification and clustering [2], and it also has been applied in video processing such as in background/foreground separation, and in 3D computer vision [15–17]. Advances in fuzzy sets theory have made it possible to research and apply more complex forms of fuzzy systems. In addition, great progress has been made in transitioning from traditional type-1 FS to interval type-2 FS [18–21], and recently there has been a growing interest in general type-2 FS [22–25]. In what follows, we introduce the basic concepts and properties about general type-2 FS and the α-planes representation, which are important because the main contribution of this paper is based on the use of these concepts. 2.1. Definition of General Type-2 Fuzzy Sets e can be represented by Equation (1) [22,24]: A general type-2 fuzzy set (GT2 FS) A, e = ( x, u), µ e ( x, u) ∀ x ∈ X, ∀u ∈ [0, 1] , A A
(1)
e x. The 3D membership function is usually where X is the universe for the primary variable of A, e is continuous it denoted by µ Ae ( x, u), where x e X and u e U ⊆ [0, 1] and 0 ≤ µ Ae ( x, u) ≤ 1. If A can be expressed as Z Z e A= µ Ae ( x, u)/( x, u) , (2) x∈X
u∈[0,1]
RR
e can be where denotes the union over all admissible x and u. For discrete universes of discourse, A denoted by Equation (3): e = ∑ ∑ µ e ( x, u)/( x, u) . A (3) x∈X u∈[0,1] A In Equations (1)–(3) u is called the secondary variable, and has the domain U = [0, 1] at each e is called an interval type-2 fuzzy set x e X. When µ Ae ( x, u) = 1 for ∀ x ∈ X and ∀u ∈ U then A (IT2 FS).
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At each point of x, say x = x 0 , the 2D plane whose axes are u and µ Ae ( x, u) is called a vertical slice of µ Ae ( x, u). A secondary membership function is a vertical slice of µ Ae ( x, u), i.e., Z µ Ae x = x 0 , u ≡ µ Ae x 0 =
u∈ J u0 ⊆[0,1]
f x (u)/u.
(4)
x
In Equation (4) Jxu0 is the subset of U that is the support of µ Ae ( x 0 ), and is called the primary e The amplitude of the secondary membership function f x (u), is called the secondary membership of A. grade or secondary set. e i.e., The two-dimensional support of µ Ae ( x, u) is called the footprint of uncertainty (FOU) of A, e) = ( x, u) ∈ X × [0, 1] µ e ( x, u) > 0 . FOU ( A A
(5)
e) can also be expressed as the union of all primary memberships, i.e., FOU ( A e) = ∪ x∈X Jxu . FOU ( A
(6)
2.2. General Type-2 Fuzzy Sets Representation In order to limit the complexity of general type-2 fuzzy logic, several efforts have been proposed; for example, Wagner and Hagras [1,26] have introduced the zSlices representation, and Mendel and Liu [23,27,28] have put forward a representation based on the α-plane concept, both of which enable the representation of, and computation with, general type-2 fuzzy sets. e denoted by A eα , is the union of all primary memberships functions An α-plane for the GT2 FS A, e of A whose secondary grades are greater than or equal to α (0 ≤ α ≤ 1) [24]. The equation of the α-plane is represented by Equation (7): Z e α = ( x, u), µ e ( x, u) ≥ α ∀ x ∈ X, ∀u ∈ ∀u ⊆ [0, 1] = A A
Z ∀ x ∈ X ∀u∈[0, 1]
{( x, u)| f x (u) ≥ α}.
(7)
3. Edge Detection Process Based on the Gradient Approach The edge detection process in a digital image consists in identifying the pixels, where the brightness changes dramatically or has discontinuities with respect with their neighbor pixels. There exist some approaches to perform the edge detection process; the majority are based on image gradients, which are calculated with the first derivative of an image. The most important and most used methods based on the image gradients are from Kirsch [29], Sobel [30], Prewitt [31], and Canny [32]. In these techniques the Euclidean distance is used to calculate the image gradients in conjunction with a kernel [33]. The edge detection method presented in this paper consists of calculating the image gradients with the Euclidean distance, but without using any kernel. This process consists of obtaining four image gradients based on a 3 × 3 matrix to indicate the edge direction ( Di , for i = 1 . . . 4) (Figure 1). Each matrix position (Pi ) of Figure 1 is represented in Figure 2, where f represents the image, x-axis the columns and y-axis the rows. According with these positions, the Euclidean distance is applied to calculate the gradients Di using the expressions illustrated in Figure 3a–d and the gradient magnitude or the output gradient is obtained with the aggregation of the four gradients ( FE = D1 + D2 + D3 + D4).
Each matrix position (P ) of Figure 1 is represented in Figure 2, where represents the image, x-axis the columns and y-axis the rows. According with these positions, the Euclidean distance is applied to calculate the gradients D using the expressions illustrated in Figure 3a–d and the gradient magnitude or the output gradient is obtained with the aggregation of the four gradients Information 4 of 15 ( = 12017, + 8,2 104 + 3 + 4).
Information 2017, 8, 104 Figure 1. 3 × 3 matrix to indicate direction of the four gradients Di . Information 2017, 8, 104 Figure 1. 3 × 3 matrix to indicate direction
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Figure 2. 2. Matrix Matrix position. Figure position.
Figure the gradients gradientsD1, D1,D2, D2,D3, D3,and andD4. D4. Figure3.3.Positions Positionsto to calculate calculate the
Figure 3. Positions to calculate the gradients D1, D2, D3, and D4. Gradient Approach Edge Images Gradient Approach EdgeDetection DetectionApplied Appliedon onColor Color Format Format Images
the literature we commonly findedge edge format images In In the literature we commonly find detection methods appliedon ongrayscale grayscale format images Gradient Approach Edge Detection Applied on Color Formatmethods Images applied (two-dimensionalimages); images);this thisisisbecause because the the computational computational complexity but thethe image (two-dimensional complexityisisreduced, reduced, but image In the literature we commonly find edge detection methods applied on grayscale format images could information when is captured in color format it is changed to grayscale to be could loselose information when it is itcaptured in color format and itand is changed to grayscale to be processed. (two-dimensional images); this isisbecause the the computational complexity is reduced, but the image processed. The aim of paper to perform edgeprocess detection process using The aim of this paper is this to perform the edge detection using images in images color. in color. could lose information when it is captured in color format and it is changed to grayscale to be There exist several models represent color images; most used could mention There exist several models to to represent color images; of of thethe most used wewe could mention thethe HSV processed. Thethe aimLab of this paper isRGB to perform the CMY edge color detection process using images color. HSV the model, model, themodel, model, the model, the model, YUV model, thein HIS color model, Lab model, the RGB the CMY color model, the YUV the HIS color model, Theretheexist several models to represent color images; of thedetection most used we could mention the CMYK color[33–35], model [33–35], etc. In this paper the detection edge method is applied on the themodel, CMYK color model etc. In this paper the edge method is applied on the HSV, HSV model, the Lab the RGB model, theforCMY model, thevalue YUV(brightness); model, the itHIS color HSV, Lab, and RGBmodel, color models. HSV stands hue, color saturation, and is the model, CMYK colorin model [33–35], etc.applications. In this paperLab the color edge space detection applied on the most the common model computer vision is a method three-axisis color system HSV, and RGB color for color; hue, saturation, and (brightness); it is the withLab, the dimensions L formodels. lightnessHSV and astands and b for working with thevalue Lab color space includes all of the colors in thein spectrum, as vision well as applications. colors outside Lab of human In the RGBcolor model, an most common model computer color perception. space is a three-axis system image of three image planes, in each of the primary colors: red, green and with the consists dimensions L forindependent lightness and a and b for one color; working with the Lab color space includes
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Lab, and RGB color models. HSV stands for hue, saturation, and value (brightness); it is the most common model in computer vision applications. Lab color space is a three-axis color system with the dimensions L for lightness and a and b for color; working with the Lab color space includes all of the colors in the spectrum, as well as colors outside of human perception. In the RGB model, an image consists of three independent image planes, one in each of the primary colors: red, green and blue; so, this model can be considered the fundamental color representation in computers, digital cameras, and scanners. To apply the gradient approach on a color format image (Lab, HSV, and RGB), we need to calculate the four gradients (Steps 5 to 8), and after that the gradient magnitude (Step 11) for each channel and finally the output edge (Step 16). Considering that the Lab, HSV, and RGB color formats, the gradients are calculated for three channels. The step to process this is described in the following pseudocode: Information 2017, 8, 104
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Pseudocode to calculate the gradients for each channel of a color format image ( f ). Input. Theformats, color format image are f r,c,dcalculated where (rfor the rows size ofThe thestep image, columns size andin(dthe ) isthree (c) the this ) the the gradients channels. to process is described following channel number; d =pseudocode: 1, 2, 3. Output. The gradients ( D1d , D2d , D3d , and D4d ), the gradient magnitude ( Gd ) for each channel of the image (f), and the output edgeto(Edge). Pseudocode calculate the gradients for each channel of a color format image ( ). ) is(the Input.the TheD1 color where size to of obtain the image, 1: Calculate D2d , image D3d , and d) rows channel ( Gd() ) the columns size and , , D4 d , format d for (each ( ) the channel number; d = 1, 2, 3. 2: for d = 1 to 3 Output. The gradients ( 1 , 2 , 3 , 4 ), the gradient magnitude ( ) for each channel of the 3: for x = 2 to r − 1 image (f), and the output edge (Edge). 4: for y1:= 2q to c − 1 Calculate the 1 , 2 , 3 , 4 for each ( ) channel to obtain ( )
5:
6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:
D1d2:= for( df (=x,1 to y)3− f ( x, y − 1))2 + ( f ( x, y) − f ( x, y + 1))2 q 3: for x = 2 to − 1 D2d = ( f ( x, y) − f ( x − 1, y))2 + ( f ( x, y) − f ( x + 1, y))2 4: q for y = 2 to − 1 2 ( ((f (, x,) − ( , f (+ 1))1, y + 1)) D3d5:= 1( f = x −(1,, y − −1)) 1))2 ++ y) − x+ ( x, y() (−, f)(− q 2 = ( ( , ) − ( − 1, )) 2 + ( ( , ) − ( + 1, )) 6: D4d = ( f ( x, y) − f ( x + 1, y − 1)) + ( f ( x, y) − f ( x − 1, y + 1))2 3 = ( ( , ) − ( − 1, − 1)) + ( ( , ) − ( + 1, + 1)) 7: end 4 = ( ( , ) − ( + 1, − 1)) + ( ( , ) − ( − 1, + 1)) 8: end 9: end Gd = 10:D1 end d + D2d + D3d + D4d 11: = Gradient 1 + 2 G+ + 4 between {0, 1} Normalize the d in3values 12:G Normalize the Gradient in values {0, 1} Gd = − min min G /max ( ( )) (max ( Gbetween d d d )) − min ( min ( Gd )), where min and max represent the ( ) − ( ) , where min and max represent the ( ) 13: = minimum − maximum and pixel value of Gd , respectively , respectively maximum and minimum pixel value of end 14: end Calculate the output (Edge) 15: Calculate the edge output edge (Edge) 3 Edge16:= ∑dd= =1=G d∑
An example of the of output theprevious previous pseudocode is illustrated ( Edge An example the output ( ) achieved ) achievedby by the pseudocode is illustrated in Figure 4.in Figure 4.
(a)
(b)
Figure 4. Output color edge detection based on the gradient approach. (a) Input image; (b) edge
Figure 4. detection Outputoutput. color edge detection based on the gradient approach. (a) Input image; (b) edge detection output. 4. Edge Detection Process Based on the Gradient Approach and GT2 FS This section presents an edge detection method based on the gradient approach combined with GT2 FSs to be applied on color format images (Lab, HSV, and RGB). The motivation of using GT FSs in the proposed edge detection method is to provide the ability to handle higher degrees of uncertainty in the edge detection process for color format images.
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4. Edge Detection Process Based on the Gradient Approach and GT2 FS This section presents an edge detection method based on the gradient approach combined with GT2 FSs to be applied on color format images (Lab, HSV, and RGB). The motivation of using GT FSs in the proposed edge detection method is to provide the ability to handle higher degrees of uncertainty in the edge detection process for color format images. The methodology to develop the proposed method consists of three steps. At first, the input color image ( f ) is separated into (d, for d = 1, . . . , 3) channels. Secondly, the gradient approach is applied in each single channel to obtain the four gradients ( D1d + D2d + D3d + D4d ); these gradients are calculated with the Steps 5 to 8 of the pseudocode presented in Section 3. Finally, the 12 (four for each channel) gradients previously obtained are identified as ( D11 , D21 , D31 , D41 , D12 , D22 , D32 , D42 , D13 , D23 , D33 , D43 ) and these are used as the inputs to the GT2 FSs. The GT2 fuzzy system is a singleton Mamdani type, which was approximated using α-planes [24,27], where each α-plane centroid was computed by the Karnik–Mendel algorithm [19]. In more detail, the GT2 FSs was designed with 12 inputs (D11 , D21 , D31 , D41 , D12 , D22 , D32 , D42 , D13 , D23 , D33 , D43 ) and three outputs (G1 , G2 and G3 ) to generate the gradient magnitude for each image channel. According to the pseudocode presented in Section 3, the ( G1 , G2 and G3 ) are calculated with the aggregation (Step 11) of the four gradients obtained in each channel. In the proposed method, the idea is to eliminate this step, and the G1 , G2 , and G3 values are inferred directly with a GT2 fuzzy system. The inputs and outputs are fuzzified using GT2 Gaussian membership functions with uncertain mean (gausmgausstype2), each input has three linguistic values (low, middle, and high) to determine the grade that the evaluated gradient correspond to be the output edge. Each output is granulated in two linguistic values (background and edge) to produce the gradient magnitude for each color image channel and the combination of these generates the output color edge detection. The parameters required for the GT2 MFs (gausmgausstype2) are expressed in Equation (8), and these are calculated depending of the image gradient values, i.e., considering the image of Figure 4a, the parameters are obtained with Equations (9) to (14). e( x, u) = gausmgausstype2( x, u, [σx , m1 , m2 ]) µ
(8)
low = min(Di )
(9)
high = max(Di )
(10)
medium = low + (high − low)/2
(11)
σ = high/4
(12)
m1 = high
(13)
m2 = m1 + (m1 *FOU), where FOU is in (0, 1)
(14)
In Figure 5 an example of the membership functions for the D11 , D21 , D31 , and D41 inputs are illustrated, where the parameters were calculated with the previous equations. These inputs correspond to the first channel of the color image. The output for the first channel ( G1 ) is shown in Figure 6. Another important part of a fuzzy inference system is the fuzzy rules; for this proposed method the fuzzy rules were designed considering different combinations of the gradients ( D11 , D21 , D31 , D41 , D12 , D22 , D32 , D42 , D13 , D23 , D33 , D43 ) obtained from the different channels of the color images to infer the gradient magnitude (outputs) for each image channel ( G1 , G2 , G3 ). In our approach, the fuzzy rules were separately combined according to the color image channel, based on experimental results and human experts in this kind of applications, with the idea of reaching the best accuracy edge detection with a minimum number of rules. After several tests,
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Information 2017, 8, 104 of 15 the best performance was obtained by the set of nine fuzzy rules presented in Table 1; these fuzzy 7rules Informationthe 2017, 8, 104 represent knowledge about the edge detection process.
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Figure GT2 input membership function forthe thevariables variablesD1 2 2,1 ,, 3D3 3, and , and .4 1. . Figure 5.5.5. GT2 input Figure GT2 inputmembership membershipfunction function for for variables 1 11,, , D2 1 , and4 D4
Figure 6. GT2 output membership function for the variable . Figure 6. 6. GT2 GT2 output output membership membership function function for for the the variable variable G1 .. Figure Table 1. Knowledge base of nine fuzzy rules for edge detection process. Table 1. Knowledge base of nine fuzzy rules for edge detection process. process.
Fuzzy Rules
Rules 1. If (RD1 is HIGH) or (RD2 is HIGH) or (RD3 Fuzzy isFuzzy HIGH) or (RD4 is HIGH) then (GR is EDGE) Rules 1. If (RD1 is HIGH) or (RD2 is HIGH) or (RD3 is HIGH) or (RD4 is (RD4 HIGH) then (GR is EDGE) 2. If (RD1 is MIDDLE) or (RD2 is MIDDLE) or (RD3 is MIDDLE) is MIDDLE) then (GR is EDGE) 1. If (RD1 is HIGH) or (RD2 is HIGH) or (RD3 is HIGH) or (RD4 isorHIGH) then (GR is EDGE) 2. 2. If (RD1 isis MIDDLE) or (RD2 is MIDDLE) or (RD3 is MIDDLE) or (RD4 is MIDDLE) then is EDGE) 3. If If (RD1 (RD1 is LOW) and (RD2 is LOW) and (RD3 is LOW) and (RD4 is LOW) then (GR is BACKGROUND) MIDDLE) or (RD2 is MIDDLE) or (RD3 is MIDDLE) or (RD4 is MIDDLE) then (GR(GR is EDGE) 4. If If (RD1 (GD1 HIGH) or (GD2 is HIGH) or (GD3 is HIGH) or (GD4 is HIGH) then (GG is EDGE) 3. 3. If (RD1 isisis LOW) and (RD2 is LOW) and (RD3 is LOW) and (RD4 is LOW) then (GR is BACKGROUND) LOW) and (RD2 is LOW) and (RD3 is LOW) and (RD4 is LOW) then (GR is BACKGROUND) 5. If If (GD1 (GD1 is MIDDLE) or (GD2 is MIDDLE) or is (GD3 is MIDDLE) (GD4 is MIDDLE) then (GG is EDGE) HIGH) or(GD2 (GD2 is HIGH) HIGH) or (GD3 oror(GD4 then (GG is EDGE) 4. 4. If (GD1 isis HIGH) or is (GD3 isHIGH) HIGH) (GD4isor isHIGH) HIGH) then (GG is EDGE) MIDDLE) or(GD2 (GD2isis isLOW) MIDDLE) or isisMIDDLE) oror (GD4 is is MIDDLE) (GG(GG is EDGE) 6. If If (GD1 (GD1 is LOW) and and (GD3 is LOW) and (GD4 is(GD4 LOW) then (GGthen is BACKGROUND) 5. 5. If (GD1 isis MIDDLE) or (GD2 MIDDLE) or(GD3 (GD3 MIDDLE) MIDDLE) then is EDGE) LOW) and (GD2isis isHIGH) LOW) or and (GD3 isisLOW) (GD4 is is LOW) then (GG is BACKGROUND) 7. If If (GD1 (BD1isisLOW) HIGH) or (BD2 (BD3 is HIGH) orand (BD4 is HIGH) then (GB is(GG EDGE) 6. 6. If (GD1 and (GD2 LOW) and (GD3 LOW) and (GD4 LOW) then is BACKGROUND) 7. is HIGH) or (BD2 is HIGH) or (BD3 HIGH) or (BD4 isorHIGH) then (GB is EDGE) 8. If If (BD1 (BD1 MIDDLE) or (BD2 is MIDDLE) orisis (BD3 is MIDDLE) is MIDDLE) then (GB is EDGE) 7. If (BD1 is is HIGH) or (BD2 is HIGH) or (BD3 HIGH) or (BD4 is (BD4 HIGH) then (GB is EDGE) 8. If (BD1 is MIDDLE) or (BD2 is MIDDLE) or (BD3 is MIDDLE) or (BD4 is MIDDLE) then (GB is EDGE) 9. If (BD1 is LOW) and (BD2 is LOW) and (BD3 is LOW) and (BD4 is LOW) then (GB is BACKGROUND) 8. 9. If If (BD1 (BD2isisLOW) MIDDLE) or (BD3 is MIDDLE) (BD4 isthen MIDDLE) then (GB is EDGE) (BD1isisMIDDLE) LOW) andor(BD2 and (BD3 is LOW) and (BD4or is LOW) (GB is BACKGROUND) 9. If (BD1 is LOW) and (BD2 is LOW) and (BD3 is LOW) and (BD4 is LOW) then (GB is BACKGROUND)
5. Experimental Results 5. Experimental Results 5. Experimental Results This section presents the edge detection accuracy and noise robustness achieved by the This section presents thethe edge detection accuracy robustnessinachieved bywhen the proposed proposed method based on gradient technique andand GT2noise FSs (explained Section 4) this is This section presents the edge detection accuracy and noise robustness achieved by the method based on the gradient technique and GT2 FSsformats. (explained Section 4) when is applied applied on color images in the Lab, HSV, and RGB Theinexperiments were this applied on the on proposed method based on the gradient technique and GT2 FSs (explained in Section 4) when this is color images in the Lab, HSV, RGB formats. The experiments were applied on isthe image database image database presented in and Table 2 and the performance of the proposed method compared with applied on color images in the Lab, HSV, and RGB formats. The experiments were applied on the two other fuzzy edge detection approaches (T1 and IT2 FSs), and with the two classic methods presented in Table 2 and the performance of the proposed method is compared with two other fuzzy image database presented in Table 2 and the performance of the proposed method is compared with (Canny and Sobel). edge detection approaches (T1 and IT2 FSs), and with the two classic methods (Canny and Sobel). two other fuzzy edge detection approaches (T1 and IT2 FSs), and with the two classic methods (Canny and Sobel).
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Table 2. Synthetic color image database. Table 2. Synthetic color image database. Table 2. Synthetic color image database.
Table 2. Synthetic color image database. Table 2. Synthetic color image database. Synthetic Images Reference Images Table 2.Images Synthetic color image database. Table 2. Synthetic color image database. Table 2. Synthetic color image database. Table 2. Synthetic color image database. Synthetic Reference Images Synthetic Images Reference Images
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Image Number Resolution Image Number Resolution Image Number Synthetic Images Reference Images Resolution Image Number Synthetic Images Reference Images Resolution Table 2. Synthetic color image database. Table 2. Synthetic color image database. Type of file: PNG Type of file: PNG filefile Image Number Synthetic Images Reference Images Resolution Image NumberSynthetic Synthetic Images Reference Images Resolution Image Number Synthetic Images Reference Images Resolution Image Number Images Reference Images Resolution Type offile: file: PNG file Type of PNG file Type of file: PNG file Type of file: PNG file Width: 420 pixels Type of file: PNG filefile Width: 420 pixels Type of file: PNG Image Number Synthetic Images Reference Images Type Resolution of file: PNG file Image Number Synthetic Images Reference Images Resolution Width: 420 pixels Width: 420 pixels Width: 420 pixels Width: 420 pixels Width: 420 pixels 1 1 1 Width: 420 pixels Type of file: PNG Width: 420 pixels Type of file: PNG filefile 1 11 1 Width: 420 pixels Width: 420 pixels Height: 182 pixels Height: 182 pixels Height: 182 pixels 1 1 Height: pixels Height: 182 pixels Height: 182 pixels Height: 182182 pixels Height: pixels Height: 182 182 pixels 1 1 Height: 182 pixels Height: pixels Type of182 file: PNG Type of file: PNG filefile TypeType of file: file Type file: PNG Type of file: PNG ofofPNG file: PNG filefile Width: 143 pixels Width: 143 pixels Type of file: PNG file Type of file: PNG Type of PNG file file Width: 143file: pixels Width: 143 pixels Width: 143 pixels Width: 143 pixels Width: 143 pixels Width: 143 pixels Type of file: PNG Type of file: PNG filefile Width: 143 pixels 2 2 22 2 2 2 Width: 143 pixels Width: 143 pixels Height: 241 pixels Height: 241 pixels Height: 241 pixels 2 2 Height: 241241 pixels Height: pixels Height: pixels Height: 241241 pixels Height: pixels Height: 241 241 pixels 2 2 Height: 241 pixels Height: 241 pixels Type of file: PNG TypeType of file: file ofofPNG file: PNG file Type file: PNG filefile Type of file: PNG Type of file: PNG filefile Width: 219 pixels Width: 219 pixels Width: 219 pixels Width: 219 pixels Type ofoffile: Type file:PNG PNG file Type of file: PNG file file Width: pixels Width: 219219 pixels Width: 219 pixels Width: 219 pixels Type of file: PNG file Type of file: PNG file 3 33 3 Width: 219 pixels 3 3 3 Height: 219 pixels Height: 219 pixels Width: 219 pixels Width: 219 pixels Height: 219 pixels Height: 219 pixels 3 3 Height: pixels Height: 219219 pixels Height: 219 pixels Height: pixels Height: 219 219 pixels 3 3 Type of file: PNG file Type of file: PNG file Height: 219 pixels Height: pixels Type of219 file: PNG Type of file: PNG filefile Width: 307 pixels Width: 307 pixels Type of file: PNG Type of file: PNG filefile Width: 307 pixels Width: 307 pixels Type of file: PNG Type of file: PNG file file 4 4 Width: 307 pixels Width: 307file: pixels Type of PNG file 4 4 Width: 307 pixels Width: 307 pixels Type of file: PNG Type of file: PNG filefile Height: 183 pixels Height: 183 pixels 4 4 Height: 183 pixels Height: 183 pixels Width: 307 pixels Width: 307 pixels Width: 307 pixels 44 4 Height: pixels Height: 183183 pixels Height: PNG file PNG file Height: 183 183 pixels 4 4 Height: 183pixels pixels PNG file PNG file Width: 310 pixels Width: 310 pixels Height: 183 pixels Height: 183 pixels PNG file PNG file Width: 310 pixels Width: 310 pixels PNG file 5 PNG file 5 Width: pixels Width: 310310 pixels Height: 207 pixels Height: 207 pixels Width: 310 pixels PNG file Width: 310 pixels PNG file 5 5 PNG file 5 5 Width: pixels Width: 310 pixels Height: 207 pixels Height: 207310 pixels 5 5 Height: pixels Height: 207207 pixels Width: 310 pixels 5 5 5 Height: pixels Height: 207 207 pixels The performance is measured by using the figure of merit of Pratt of Merit; FOM)FOM) [36]; [36]; The performance is measured by using the figure of merit of (Figure Pratt (Figure of Merit; Height: pixels Height: 207207 pixels Height: 207 pixels wherewhere the FOM is a useful metricmetric for assessing the quality of edge detectors, whichwhich consists of the FOM is a useful for assessing the quality of edge detectors, consists of The performance is measured by using the figure of merit of Pratt (Figure of Merit; FOM) [36]; The performance is measured by using the figure of merit of Pratt (Figure of Merit; FOM) [36]; evaluating the distance between two images—the first represents the output edges obtained by the evaluating the distance between two images—the first represents the output edges obtained by the The performance is measured by using the figure of merit of Pratt (Figure of Merit; FOM) [36]; The performance is measured by using the figure of quality meritofofof Pratt (Figure of which Merit; FOM) [36]; where the FOM ameasured useful metric assessing the edge detectors, which consists of where FOM is (GT2 aisis useful metric for assessing the quality edge detectors, consists of The performance is measured byfor using figure of of Pratt (Figure of Merit; FOM) [36]; proposed method fuzzy edge detector) and the second is amerit reference image or of theMerit; ground truth Thethe performance by using the the figure of merit of (Figure FOM) [36]; proposed method (GT2 fuzzy edge detector) and the second is aPratt reference image or the ground truth where the FOM is a useful metric for assessing the quality of edge detectors, which consists of where the FOM is a useful metric for assessing the quality of edge detectors, which consists of evaluating the distance between two images—the first represents the output edges obtained by the evaluating the distance between two images—the first represents the output edges obtained by the (inThe Table 2). The varies in the [0, 1],quality where 1Pratt isedge the value, i.e., the (inperformance Table 2).output measure varies inrange the range [0, 1], where 1 (Figure isoptimal the optimal value, i.e.,[36]; the where the FOM isuseful aisoutput useful metric for assessing the of detectors, which consists of where the FOM is aThe metric forby assessing the quality ofof edge detectors, which consists of[36]; The performance ismeasure measured using the figure of merit of Pratt (Figure ofobtained Merit; FOM) measured by using the figure of merit of Merit; FOM) evaluating the distance between two images—the first represents the output edges obtained by the evaluating the distance between two images—the first represents the output edges by the The performance is measured byrespect using the figure of merit of Pratt (Figure of Merit; FOM) [36]; proposed method (GT2 fuzzy edge the second isreference athe reference image orwhich the ground proposed method (GT2 fuzzy edge detector) andand the second is aof image or obtained the ground output edges areis similar with respect todetector) the reference image. output edges are similar with to the reference image. evaluating the distance between two images—the first represents the output edges obtained bytruth evaluating the distance two images—the first represents output edges bytruth the where the FOM abetween useful metric for assessing the quality edge detectors, consists of where the FOM ais useful metric for assessing the quality edge detectors, consists ofthe proposed method (GT2 fuzzy edge detector) and the second aof reference image or value, the ground truth proposed method (GT2 fuzzy edge detector) and the second is1], aisreference image orwhich the ground truth (in Table 2). The output measure varies in the range [0, where 1 is the optimal value, i.e., the (in Table 2). The output measure varies in the range [0, 1], where 1 is the optimal i.e., the where proposed the FOM is a useful metric for assessing the quality of edge detectors, which consists of proposed method (GT2 fuzzy edge detector) the second a the reference image orobtained the ground truth method (GT2 fuzzy edge detector) andand thefirst second is a is reference image or the ground truth evaluating the distance between two images—the first represents output edges obtained by the evaluating the distance between two images—the represents output edges by the (in Table 2). The output measure inSynthetic the range [0, 1], where is the optimal value, i.e., the (in Table 2). The output measure varies inthe the range [0, 1], where 1the is1 the optimal value, i.e., the Fuzzy Edge Detection Method Applied onvaries thethe Synthetic Color Images Fuzzy Edge Detection Method Applied on the Color Images output edges are similar with respect to reference image. output edges are similar with respect to reference image. evaluating the distance between two images—the first represents the output edges obtained by the (in Table 2). The output measure varies in the range [0, is 1],aiswhere isimage the optimal value, i.e., the (in Table 2). The output measure varies in the range [0,second 1], where 1 is1the optimal i.e.,truth the proposed method (GT2 fuzzy edge detector) and the a reference image orvalue, the ground truth proposed method (GT2 fuzzy edge detector) and the second reference or the ground output edges similar with respect to the reference image. output edges areare similar with respect to the reference image. In the first test, the fuzzy edge detection methods based on1], T1, FSs were applied the first test, the fuzzy edge detection methods based on T1, IT2, and GT2 FSs were edges are similar with respect to reference image. output edges are similar with respect tovaries the reference image. (in Table 2). The output measure in the range [0, 1 the isGT2 the optimal value, i.e., (inoutput Table 2).In The output measure varies inthe the range [0, 1], where 1and is optimal value, i.e., thethe truth proposed method (GT2 fuzzy edge detector) and the second iswhere aIT2, reference image or theapplied ground Fuzzy Edge Detection Method Applied on Synthetic Color Images Fuzzy Edge Detection Method Applied onin theTable Synthetic Color Images on the image database presented in Table 2.the These images were converted to theto Lab, and RGB on the image database presented 2.reference These images were converted theHSV, Lab, HSV, and RGB output edges are similar with respect to the image. output edges are similar with respect tothe the reference Fuzzy Edge Detection Method Applied onthe the Synthetic Color Images 1 is the optimal value, i.e., the output (in Table 2). The output measure varies range [0,image. 1], where Fuzzy Edge Detection Method Applied onin Synthetic Color Images color formats in order to analyze the advantage of methods using one format color over the others. color formats in to analyze the advantage of Color using one color over the others. Fuzzy Edge Detection Method Applied on the Synthetic Images Fuzzy Edge Detection Method Applied ondetection the Synthetic Color Images In the first test, the fuzzy edge detection based on IT2, and GT2 FSs were applied In the first test, theorder fuzzy edge methods based onformat T1,T1, IT2, and GT2 FSs were applied edges are similar with respect to the reference image. In Table 3test, the FOM values achieved by the T1the FSs for the Lab, HSV, and RGB formats are Infirst Table 3the the FOM values achieved by T1edge FSs edge for the Lab, HSV, and RGB formats are InEdge the test, the fuzzy edge detection methods based on T1, IT2, and GT2 FSs were applied Inthe the first fuzzy edge detection methods based onconverted T1, IT2, and GT2 FSs were applied Fuzzy Detection Method Applied on the Synthetic Color Images Fuzzy Edge Detection Method Applied on the Synthetic Color Images on image database presented in Table 2. These images were converted to the Lab, HSV, and RGB on the image database presented in Table 2. These images were to the Lab, HSV, and RGB In the fuzzy edge detection methods based on T1, were applied In the firstfirst test,test, the the fuzzy edge detection methods based on T1, IT2,IT2, andand GT2GT2 FSs FSs were applied presented. presented. on the image database presented in Table 2. These images were converted to the Lab, HSV, and RGB on the image database presented in Table 2. These images were converted to the Lab, HSV, and RGB color formats in order to analyze the advantage of using one format color over the others. color formats in order to analyze the advantage of using one format color over the others. onIn the image database presented in Table 2. methods These images were converted toGT2 the Lab, HSV, and RGB on the image database presented inedge Table 2. These images were converted to the Lab, HSV, and RGB In the first test, the fuzzy detection methods based on T1, IT2, and GT2 FSs were applied the first test, the fuzzy edge detection based on T1, IT2, and FSs were applied Fuzzy Edge Detection Method Applied on the Synthetic Color Images color formats in order to analyze the advantage of using one format color over the others. color formats in order to analyze the advantage of using one format color over the others. In Table 3inthe FOM values by the T1 FSs edge for Lab, and RGB formats are Inimage Table 3 the FOM values achieved by theThese T1 FSs edge forformat thethe Lab, HSV, and RGB formats are formats order to analyze the advantage of using one color over the others. color formats in order to analyze the advantage of using one format color over others. on database presented in Table 2. images were tothe the Lab, HSV, and RGB oncolor the database presented inachieved Table 2. These were converted toHSV, the Lab, HSV, and RGB Inimage Table 3 the FOM values achieved by the T1 FSs edge for the Lab, HSV, and RGB formats are Inthe Table 3 the FOM values achieved by the T1images FSs edge for theconverted Lab, HSV, and RGB formats are presented. In presented. the first test, the fuzzy edge detection methods based on T1, IT2, and GT2 FSs were applied on Informats Table 3order FOM achieved by the T1using FSs edge for the Lab, HSV, RGB formats Informats Table 3 in the FOM achieved by the T1 edge for the Lab, HSV, and RGB formats are are color inthe order to values analyze the advantage of one format color over the others. color tovalues analyze the advantage of FSs using one format color over theand others. presented. presented. presented. presented. the image database invalues Table 2. These images were converted to HSV, theand Lab, HSV, andare RGB In Table 3 the FOM achieved the edge for Lab, and RGB formats are color In Table 3presented the FOM values achieved by by the T1 T1 FSsFSs edge for thethe Lab, HSV, RGB formats formatspresented. inpresented. order to analyze the advantage of using one format color over the others.
In Table 3 the FOM values achieved by the T1 FSs edge for the Lab, HSV, and RGB formats are presented.
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Table 3. T1 FSs edge detection applied on Lab, HSV, and RGB images. T1 FSs Edge Detection Image Number
1 2 3 4 5 Mean
FOM Lab
HSV
RGB
0.9430 0.9504 0.9485 0.9493 0.9428 0.9468
0.9380 0.9372 0.9403 0.9453 0.9353 0.9392
0.9393 0.9478 0.9464 0.9491 0.9415 0.9448
The results achieved by the IT2 fuzzy edge detection for the Lab, HSV, and RGB color format images are shown in Table 4. Table 4. IT2 FSs edge detection applied on Lab, HSV, and RGB images. IT2 FSs Edge Detection Image Number
1 2 3 4 5 Mean
FOM Lab
HSV
RGB
0.9437 0.9510 0.9491 0.9496 0.9430 0.9473
0.9401 0.9478 0.9464 0.9491 0.9415 0.9450
0.9401 0.9479 0.9470 0.9493 0.9417 0.9452
The FOM values achieved by the proposed method based on GT2 FSs are presented in Table 5. Table 5. GT2 FSs edge detection applied on Lab, HSV, and RGB color format images. GT2 FSs Edge Detection Image Number
1 2 3 4 5 Mean
FOM Lab
HSV
RGB
0.9457 0.9523 0.9497 0.9503 0.9434 0.9483
0.9416 0.9481 0.9472 0.9497 0.9419 0.9457
0.9416 0.9481 0.9473 0.9497 0.9420 0.9457
In Table 6, a summary of Tables 3–5 is presented and we also include the results achieved by the Canny and Sobel edge detection methods. According to these results, we can conclude that the edge detection method based on IT2 FSs achieved better FOM values than the T1 FSs for the three different color formats. Nevertheless, the proposed method based on GT2 FSs improved most of the results in comparison with the IT2 FSs, T1 FSs, Canny, and Sobel methodologies; additionally, we can notice that the FOM values obtained by the Sobel operator are the lowest, followed by those of the Canny method. An interesting finding in Table 6 is that, according to the FOM values, four of the five methods achieved better results when the Lab color format was used; this means that the Lab format provides more information about the image, which is a good point to consider in image processing applications, especially those that work with the edge detection process. In Figure 7, we present a plot of the FOM values achieved by the three different fuzzy edge detection methods when these are applied over
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the Lab, HSV, and RGB formats; we can see the advantage of using the Lab color format in this kind applied over the Lab, HSV, and RGB formats; we can see the advantage of using the Lab color format of application. in this kind of application. Table 6. Results achieved by T1FSs, IT2FSs, and GT2FSs edge detectors applied on Lab, HSV, and RGB Table 6. Results achieved by T1FSs, IT2FSs, and GT2FSs edge detectors applied on Lab, HSV, and format color images. RGB format color images. Fuzzy Edge Detector
Fuzzy Edge Detector Lab Lab T1 FSs 0.9468 T1 FSs 0.9468 IT2 FSs 0.9473 IT2 FSs 0.9473 GT2 FSs 0.9483 GT2 FSs 0.9483 Canny 0.8113 Canny 0.8113 Sobel 0.5343 Sobel 0.5343
FOM
FOM HSV HSV 0.9392 0.9392 0.9450 0.9450 0.9457 0.9457 0.9264 0.9264 0.4762 0.4762
RGBRGB 0.9448 0.9448 0.9452 0.9452 0.9457 0.9457 0.8021 0.8021 0.4396 0.4396
Figure 7. byby thethe fuzzy edgeedge detectors in Lab, RGB 7. FOM FOMvalues valuesachieved achieved fuzzy detectors in HSV, Lab, and HSV, andcolor RGBformat color images. format images.
In order to evaluate the ability of the fuzzy techniques to handle the uncertainty (noise) of the In of order to2,evaluate thecorrupted ability ofusing the fuzzy techniques to handle of 7–9 the images Table they were Gaussian noise levels (20, 30,the 40,uncertainty and 50 dBi).(noise) In Tables images of Table 2, they were corrupted using by Gaussian levels (20,FSs 30,edge 40, and 50 dBi).after In Tables we present the FOM average value achieved the T1,noise IT2, and GT2 detection these 7–9 we present the FOM average value achieved by the T1, IT2, and GT2 FSs edge detection after were applied on Lab, HSV, and RGB images. theseFrom wereTable applied on Lab, HSV, and RGB images. 7 (T1 FSs) we see that for images with Gaussian noise of 20 dBi, the FOM value (0.9367) From Table 7 (T1color FSs)format. we seeFor that30for Gaussian of 0.9485) 20 dBi, were the FOM was better in the HSV andimages 40 dBi with the FOM valuesnoise (0.9489, bettervalue with (0.9367) was better in the HSV color format. For 30 and 40 dBi the FOM values (0.9489, 0.9485) the Lab color format; for 50 dBi the FOM result (0.9457) was higher using the RGB color format. were better with the Lab color format; for 50 dBi the FOM result (0.9457) was higher using the RGB color format. Table 7. FOM values achieved by T1 FSs in images with Gaussian noise. Table 7. FOM values achieved byT1T1FSs FSs in images with Gaussian noise.
T1 FSs FOM 20 dBi 30 dBi FOM40 dBi 50 dBi Color Format 20 dBi 0.9489 30 dBi 400.9485 dBi 50 dBi Lab 0.8494 0.8972 HSV Lab 0.93670.8494 0.9393 0.9392 0.9391 0.9489 0.9485 0.8972 RGB 0.8844 0.9461 0.9458 0.9457 HSV 0.9367 0.9393 0.9392 0.9391 RGB 0.8844 0.9461 0.9458 0.9457 From Table 8 (IT2 FSs) we observe that for the images with 20 dBi of Gaussian noise, the HSV colorFrom format has 8a (IT2 better FOM (0.9385); however, for 30, 40,20and the FOM values Table FSs) wevalue observe that for the images with dBi50ofdBi Gaussian noise, the were HSV higher in the has Lab aformat. color format better FOM value (0.9385); however, for 30, 40, and 50 dBi the FOM values were higher in the Lab format. Color Format
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Table 8. FOM values achieved by IT2 FSs in images with Gaussian noise. IT2 FSs Color Format Lab HSV RGB
FOM 20 dBi
30 dBi
40 dBi
50 dBi
0.8597 0.9385 0.9081
0.9492 0.9394 0.9466
0.9485 0.9393 0.9458
0.9483 0.9393 0.9456
In Table 9, the simulation results after applying the GT2 FS edge detection on images with Gaussian noise are presented. In these results we notice that the FOM value (0.9392) was better when the HSV color format was used under a noise level of 20 dBi; for 40 and 50 dBi the FOM values (0.9491, 0.9489) were better using the Lab color format; in this test the RGB format has higher FOM for a noise level of 30 dBi (0.9563). Table 9. FOM values achieved by GT2 FSs in images with Gaussian noise. GT2 FSs Color Format Lab HSV RGB
FOM 20 dBi
30 dBi
40 dBi
50 dBi
0.8600 0.9392 0.9158
0.9496 0.9395 0.9563
0.9491 0.9393 0.9470
0.9489 0.9392 0.9460
Finally, we present a comparative analysis in which the results are classified by color format. In Tables 10–12 we present the results achieved by the three fuzzy edge detection methods and two traditional methods (Canny and Sobel) when these are applied on Lab, HSV, and RGB color format images, respectively. According to the results of Table 10, the FOM values were better when the GT2 FS was applied for the different noise levels (20, 30, 40, and 50 dBi). Moreover, in Table 11 (HSV) and Table 12 (RGB) we notice that the GT2 FSs also improved the results, obtaining the best FOM accuracy. Additionally, we notice that the Sobel method achieved the lowest results for all the cases, followed by Canny; this means that the traditional methods do not have additional parameters to handle uncertainty, which is the advantage of using fuzzy techniques in this kind of application. In Figure 8, we present a plot to illustrate the FOM values by color format obtained by the five edge detection methods when these are applied in images with noise; in this figure we cannot observe clearly the difference, and for this reason we calculate the mean for each format color. For Lab color we obtained a FOM mean of 0.8126, for HSV a FOM mean of 0.8385, and for RGB a mean value of 0.8333. Therefore, for these tests we conclude that when the images are corrupted by noise, the HSV color format represents a good option, followed by the RGB color format. Table 10. T1, IT2, and GT2 FS edge detection applied on Lab color format images with Gaussian noise. Lab Fuzzy Edge Detection T1 FSs IT2 FSs GT2 FSs Canny Sobel
FOM 20 dBi
30 dBi
40 dBi
50 dBi
0.8494 0.8597 0.8600 0.8306 0.4824
0.9489 0.9492 0.9496 0.8126 0.4824
0.9485 0.9485 0.9491 0.8113 0.4824
0.8972 0.9483 0.9489 0.8110 0.4824
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Table 11. T1, IT2, and GT2 FS edge detection applied on HSV color format images with Gaussian noise. HSV Fuzzy Edge Detection
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T1 FSs IT2 FSs GT2 FSs Canny Sobel
FOM 20 dBi
30 dBi
40 dBi
50 dBi
0.9367 0.9385 0.9392 0.8175 0.4762
0.9393 0.9394 0.9395 0.9261 0.4762
0.9392 0.9393 0.9393 0.9264 0.4762
0.9391 0.9393 0.9392 0.9264 0.4762
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Table 11. T1, T1,IT2, IT2,and andGT2 GT2FSFS edge detection applied on HSV format images with Gaussian Table 12. edge detection applied on RGB colorcolor format images with Gaussian noise. noise. RGB HSV Fuzzy Edge Detection Fuzzy Edge Detection T1T1 FSsFSs IT2IT2 FSsFSs GT2 FSsFSs GT2 Canny Canny Sobel
Sobel
FOM FOM 20 20 dBidBi 30 30dBi dBi
0.9367 0.8844 0.9081 0.9385 0.9158 0.9392 0.8302 0.8175 0.5343 0.4762
0.9393 0.9461 0.9466 0.9394 0.9563 0.9395 0.8230 0.9261 0.5343 0.4762
40 dBi dBi 40 0.9392 0.9458 0.9458 0.9393 0.9470 0.9393 0.8220 0.9264 0.5343 0.4762
dBi 5050dBi 0.9391 0.9457 0.9456 0.9393 0.9460 0.9392 0.8212 0.9264 0.5343 0.4762
Figure 8. 8. Comparison Comparison of of results results of of the the FOM FOM values values for for the the different different color color formats. formats. Figure
Table 12. T1,edge IT2, and GT2 FS edge detection applied RGB color format images with13. Gaussian The visual detections for T1, IT2, and GT2onFSs are presented in Table Of course, noise. we cannot clearly appreciate the difference in the detected edges and for this reason graphically
the FOM measure was applied, but tests supportRGB the conclusion that GT2 FSs can outperform IT2 and T1 in edge detection process. FOM Fuzzy Edge Detection 20 dBi 30 dBi 40 dBi 50 dBi T1 FSs 0.8844 0.9461 0.9458 0.9457 IT2 FSs 0.9081 0.9466 0.9458 0.9456 GT2 FSs 0.9158 0.9563 0.9470 0.9460 Canny 0.8302 0.8230 0.8220 0.8212 Sobel 0.5343 0.5343 0.5343 0.5343
The visual edge detections for T1, IT2, and GT2 FSs are presented in Table 13. Of course, graphically we cannot clearly appreciate the difference in the detected edges and for this reason the FOM measure was applied, but tests support the conclusion that GT2 FSs can outperform IT2 and T1 in edge detection process.
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Table 13. Edge detection output using T1, IT2, and GT2 FSs. Table 13.13. Edge detection output using T1,T1, IT2, and GT2 FSs. Table Edge detection output using IT2, and GT2 FSs. Table 13. Edge detection output using T1, IT2, and GT2 output using T1, IT2, and GT2 FSs.FSs. Table 13. Edge detection output using T1, IT2, and GT2 FSs. Table 13. Edge detection output using T1, IT2, and GT2 FSs.
T1 FSs T1T1 FSs FSs T1 T1 FSsFSs T1 FSs
IT2IT2 FSs IT2 FSs IT2 IT2 FSsFSs IT2 FSs
GT2 FSs GT2 GT2 FSs GT2 GT2 FSsFSs GT2 FSs
6. Conclusions Conclusions 6. 6. 6. Conclusions 6. Conclusions Conclusions 6. Conclusions In In this paper we presented anedge edge detection approach based ongradient the gradient technique this paper presented an edge detection approach based on the gradient technique In this paper we presented an edge detection approach based on the gradient technique In this paper we presented an edge detection approach based on the gradient technique In this paper we presented an edge detection approach based on the gradient technique In this paper wewe presented an detection approach based on the technique combined combined with three fuzzy approaches (T1, IT2, and GT2 FSs), where thethe main aim was to show thethe combined with three fuzzy approaches (T1, IT2, and GT2 FSs), where main aim was to combined with three fuzzy approaches (T1, IT2, and GT2 FSs), where the aim was to show combined with three fuzzy approaches (T1, IT2, and GT2 FSs), where the main aim was to show show the combined with three fuzzy approaches IT2, and GT2 FSs), where the main aim was to show the with three fuzzy approaches (T1, IT2, (T1, and GT2 FSs), where the main aim was tomain show the advantage ofthe advantage of using GT2 FSs in image processing systems and analyze their performance with advantage of using GT2 FSs in image processing systems and analyze their performance with advantage of using GT2 FSs in image processing systems and analyze their performance with advantage of using GT2 FSs in image processing systems and analyze their performance with advantage of using GT2 FSs in image processing systems and analyze their performance with using GT2 FSs in image processing systems and analyze their performance with respect to the higher respect to the the higher uncertainty levels. Inconclude general, we can conclude that the edge detection based on respect to higher In we can conclude that the edge detection based on respect to higher uncertainty levels. In general, we can conclude that the edge detection on respect to the higher uncertainty levels. In general, we can conclude that the edge detection respect to the the higher uncertainty levels. In general, general, we can conclude that the edge detection based on on uncertainty levels. Inuncertainty general, welevels. can that the edge detection based on GT2 Fsbased has abased good GT2 Fs hason good control onon theon image gradients in images images with noise and without noise for the GT2 Fs has aa good control the image gradients in with noise and without noise for the GT2 Fs a good control the image gradients in images with noise and without noise for GT2 Fs has good control on the image gradients in images images with noise and without noise for thethe GT2 Fs has aa has good control on the image gradients in with noise and without noise for the control the image gradients in images with noise and without noise for the different color formats. different color formats. different color formats. different color formats. different color formats. different color formats. On the other hand, different color format images were analyzed (Lab, HSV, and RGB) and, On the other hand, different color format images were analyzed (Lab, HSV, and RGB) and, OnOn the other hand, different color format images were analyzed (Lab, HSV, and RGB) and, On hand, different format images were analyzed (Lab, HSV, and RGB) and, On the other hand, different color format images were analyzed (Lab, HSV, and RGB) and, the other hand, different color format images were (Lab, HSV, and RGB) and, On the other hand, different color format images were analyzed (Lab, HSV, and RGB) and, according tothe theother FOM metric, based oncolor the simulation results inanalyzed images without noise the fuzzy edge according to the FOM metric, based on the simulation results in images without noise the fuzzy edge according to the FOM metric, based on the simulation results in images without noise the fuzzy edge according to the FOM metric, based on the simulation results in images without noise the fuzzy edge according to the FOM metric, based on the simulation results in images without noise the fuzzy edge according to the FOM metric, based on the simulation results in images without noise the fuzzy edge detection approaches were better for images in Lab color format, followed by the RGB format; the detection approaches were better forfor images in Lab Lab color format, followed byby the RGB format; the detection approaches were better images in Lab color format, followed the RGB format; the detection approaches were better for images in color format, followed by the RGB format; the detection approaches were better for images in Lab color format, followed by the RGB format; detection approaches were better for images informat. Lab color format, followed by the RGB format; thethe lowest performance was from the HSV color In images with noise, the performance was lowest performance was from theworst HSV color format. In In images with noise, the performance was lowest performance was from the HSV color format. images with noise, the performance was lowest performance was from the HSV format. In images with noise, the performance was lowest performance was from the HSV color format. In images with noise, the performance was lowest performance from the HSV color format. In images with noise, the performance was better for the HSVwas format and for thecolor Lab format. This represents an important and helpful better for the HSV format and worst for the Lab format. This represents an important and helpful better for the HSV format and worst for the Lab format. This represents an important and helpful better for the HSV format and worst for the Lab format. This represents an important and helpful better for the HSV format and worst for the Lab format. This represents an important and helpful better for the HSV format and worst for the Lab format. This represents an important and helpful characteristic to be considered when we are working with this kind of application. characteristic to be be considered when wewe arewe working with this kind of application. characteristic to be when are working with this kind of characteristic to considered we are working with this kind application. characteristic to considered be when are working with this kind of application. characteristic to be considered when we are working with this kind of application. application. In future work theconsidered idea when is to improve the fuzzy system using aof metaheuristic to optimize the main In future work the idea is to improve the fuzzy system using a metaheuristic to course optimize the In future work the idea is to improve the fuzzy system using a metaheuristic to the In future work the idea is to improve the fuzzy system using a metaheuristic to optimize In future work the idea is to improve the fuzzy system using a metaheuristic to optimize optimize thethe In future work the idea is to improve the fuzzy system using a metaheuristic to optimize components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of thethe fuzzy main components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of course main components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of course main components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of course main components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of course main components of the GT2 FSs, such as the type of MF, the parameters of the MF, and of course rules; in addition, the next goal will be improving the GT2 FSs method to reduce the computing time the fuzzy rules; in addition, the next goal will be improving the GT2 FSs method to reduce thethe thethe fuzzy rules; in in addition, the next goal will be improving thethe GT2 FSs method to to reduce thethe the fuzzy rules; in addition, the next goal will be improving the GT2 FSs method to reduce the fuzzy rules; in addition, the next will be improving the GT2 FSs method to reduce the fuzzy rules; addition, the next goal will be improving GT2 FSs method reduce the fuzzy rules; in addition, the next goal will be improving the GT2 FSs method to reduce the for achieving faster results, which is goal required in this kind of area for real-world application. computing time for achieving faster results, which is required required in in this kind of of area for real-world computing time for achieving faster results, which is this kind area for real-world computing time for achieving faster results, which is required in this kind of area for real-world computing time for achieving faster results, which is required required in this kind of area for real-world computing time for achieving faster results, which is in this kind of area for real-world application. application. application. application. application.
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Author Contributions: Claudia I. Gonzalez proposed the idea of color images and implemented the software, as well as made the experiments. Patricia Melin proposed the general idea of type 2 edge detection and verified the correctness of implementation and results. Oscar Castillo verified the correctness of type-2 fuzzy logic implementation and the writing of the paper. Conflicts of Interest: The authors declare no conflict of interest.
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