Edge detection is an important task in image processing. There are a lot of approaches in this area: Sobel, Canny operators and others. One of the perspective techniques in image processing is the use of fuzzy logic and fuzzy sets theory. They allow us to increase processing quality by representing information in its fuzzy form. Most of the existing fuzzy image processing methods switch to fuzzy sets on very late stages, so this leads to some useful information loss. In this paper a novel method of edge detection based on fuzzy image representation and fuzzy pixels is proposed. With this approach we convert the image to fuzzy form on the first step. Different approaches to this conversion are described. Several membership functions for fuzzy pixel description and requirements for their form and view are given. A novel approach to edge detection based on Sobel operator and fuzzy image representation is proposed. Experimental testing of developed method was performed on remote sensing images. Comparison of result with Sobel, Prewitt, Roberts and Canny operators is presented. Developed method selected more details (edges) rather then Sobel, Prewitt and Roberts operators, but less than Canny operator. This is because the selected convolution kernel (Sobel) has size 3x3. There are also used only simple functions of estimating the real intensities of pixels. Later, to increase quality it is necessary to use more complex masks of size 5x5 and 7x7 or median filters. Developed approach showed its workability in solving image processing problems. The proposed fuzzy model in the future can be extended to use higher level fuzzy sets (Type-2 FS and others).
0.8 0.6 0.4 0.2 150 160 170 180 190 200 210 150 160 170 180 190 200 210 (a) (b)
Computations that does not take into account these details, soon could accumulate big error related to source continuous image. Methods of fuzzy sets theory allow us to save the uncertainty till the latest stages of image processing and analysis. To do it we should switch to fuzzy image representation. U(F, x, y) where μ(F(x, y)) – membership function of a pixel with coordinates (x, y) to intensity level F(x, y). Graphically this can be represented as shown on Fig. 1.
There are a lot of membership functions known. One must select those which satisfy the following conditions These include triangular, trapezoidal, bell, Gauss-like, π functions and others. In the simplest case we will be using π-function which is based on s-function: 1 0.8 0.6 0.4 0.2 U(F, x, y) = μ(F(x, y)), lim μ(F(x, y)) = 0, l→∞ L−1 Z
μ(F(x, y))dl > 0, 0
l ∈ [0; L − 1]. π(l) = s(l, c − b, c − 2b , c), 1 − s(l, c, c + b2 , c + b), l ≤ c, l ≥ c, 0, s(l) = 2 cl−−aa 2, 1 − 2 cl−−ca 2, 1, l ≤ a, a ≤ l ≤ b, b ≤ l ≤ c, l ≥ c, where b = a+2c , l – intensity level. Form of π-function is shown on Fig. 1.
To pick μ(F(x, y)) = π(F(x, y)) function, that is, pick b and c parameters, it is necessary to extract some additional information from the image. Firstly, let us simplify this task reducing the selection to single parameter Ic — center of membership function π(Ic) = 1. In this case π-function lies symmetrically on this point. To set slope inclination, we must choose necessary width of w section, where π(x) > 0. Then let us use the following equation to find parameters of function π: b = w 2 , c = Ic.
Value of w is chosen empirically, e.g. w = 60. With small value of w slope will be very big, and small intensity deviations will make the pixel insignificant (π < 0.5).
Parameter Ic can be selected differently. In our case π(Ic) = 1 means real intensity of the pixel, and π(F(x, y)) , 1 shows interference, noise, errors in quantization and similar errors. Value of Ic can be computed using neighbor of point (x, y), that is shown on Fig. 2. Let us consider some possible approaches: 1. average between horizontal pixels 2. average between vertical pixels
Ic = (P4 + P8)/2,
Ic = (P2 + P6)/2, Ic = (P2 + P4 + P6 + P8)/4, -1 0 1 (a) -1 0 1 -1 0 1 , |∇π| = 1 − qG2xμ + Gy2μ. |∇Gπ| = |∇G||∇π|.
-1 -2 -1 0 0 0
In more complex case to get Ic value one could use one of the existing smoothing methods like median filters, approximations etc.
Most of the edge detection operators uses gradient operator that has modulo |∇G| and direction θ where Gx = Mx ∗ G, Gy = My ∗ G – the result of convolution operator with horizontal and vertical matrices.
Let us consider possibility of the use of fuzzy pixels in Sobel and Prewitt edge detection operator (Fig. 3, 4). To do this we must process the membership function values π(F(x, y)) in addition to normal intensity levels. Calculations of Gxμ and Gyμ can be done analogously. The difference here is in gradient computations of π-function.
We take complementary value because after squaring, sum and square root operations from values on interval [0; 1], the result is near to 0. Final value of gradient will be
Threshold value of gradient can be selected manually or with different binarization techniques (e.g. with Otsu method [
Let us consider transition to fuzzy pixels. Test remote sensing image has size 160×160. Simple averaging procedures described above were applied to it. Result images are shown on Fig. 5, according membership functions are shown on Fig. 6 and the values are put in Table 1. Comparison of different methods is shown on Fig. 7.
Proposed method extracted more details (edges) than Sobel, Prewitt and Roberts operators, but less than Canny operator. This is because the selected convolution kernel (Sobel) has size 3x3. Also we used only simple functions during real pixel intensity estimation. To increase quality of the results more complex masks (5x5, 7x7) could be used and median filters also.
On processed images we can see that proposed method found a lot of ”islands”, which in the original image are green plantings.
Rivers were marked well. Presence of the big number of details after the use of Canny operator in some cases could bring some
issues during further steps, so additional filtering may be applied. In proposed method number of details less than after Canny
operator and this could be useful. Later, selected edges could be used in segmentation and object detection algorithms [
Proposed algorithm showed their applicability in edge detection task during image processing. Main feature is the use of fuzzy image representation based on fuzzy pixels. This approach is very perspective because it saves the uncertainty much better rather other existing algorithms. In the future this model could be extended to type-2 and higher fuzzy sets and also to other kinds of fuzzy sets. 1 0.8 0.6 0.4 0.2 0 (a) Sobel operator (b) Prewitt operator