A bilinear fast fuzzy enhancement algorithm for image boundary detection

Abstract

Through fuzzy membership function, the fuzzy algorithm of image boundary detection based on power function can transform ordinary space into generalized fuzzy space. However, the algorithm has a large amount of operation and slow speed, and it will lose the boundary information of some low gray value in the image, thus the quality of the image boundary detection is poor. Therefore, a bilinear fast enhancement fuzzy algorithm for image boundary detection is proposed in this paper. Based on the defined generalized fuzzy set GFS and the generalized fuzzy operator LGFO, the linear left half trapezoid fuzzy distribution function is first used as the generalized membership transformation of the image.The general space of grayscale image is transformed into generalized fuzzy space, and then boundary detection algorithm based on bilinear fast image enhancement is used to transform color image into gray scale and transform to generalized fuzzy set. The generalized fuzzy operator LGFO is used to enhance the contrast of the generalized fuzzy sets. The generalized fuzzy set after the enhancement is transformed into an ordinary fuzzy subset. The boundary extraction is carried out for the ordinary fuzzy subset after processing, and the image boundary detection is realized. The experimental results show that the proposed algorithm greatly improves the speed and quality of image boundary detection.

Keywords

Image boundary detection bilinear fast fuzzy enhancement LGFO generalized fuzzy space

1 Introduction

In 1965, L.A. Zadeh put forward fuzzy set theory. It is a software calculation tool to describe imprecise or uncertain problems, and it is used to study the problem of imperfections and inaccuracies in information system [1]. The generalized fuzzy set was proposed by Chen Wufan in 1995. Using the generalized fuzzy set theory, he proposed a new algorithm for image boundary detection, that is, the generalized fuzzy operator method (GFO).

In the field of image processing, boundary detection is a very important branch. It is often used in the fields of image segmentation and image analysis [2]. At present, there are many methods for image boundary detection, which are basically based on the theory of gray scale discontinuity.Usually a good image system should have a good match with the visual mechanism. So we want to use models and methods that can describe the visual characteristics of people. Generalized fuzzy set theory is an effective soft computing tool in artificial intelligence, pattern recognition and image processing and analysis. Generalized fuzzy set theory has been applied to image boundary detection successfully. The image boundary detection technology based on the generalized fuzzy set theory has also received more and more attentions [3]. A fuzzy algorithm for image boundary detection has been proposed by Pal et al.This method has achieved good results in X image’s boundary detection, but the algorithm uses fuzzy membership function in the form of power function to transform the ordinary space into generalized fuzzy space, with large amount of computation, slow speed [4], and the boundary information of part of the low gray value will be loss in image [5]. At the same time, when the algorithm is enhanced for image processing, the transform is also based on the nonlinear transformation of power function [6]. In this paper, the generalized fuzzy set (GFS) theory is used, to give a newlinear generalized fuzzy operator (LGFO) for image contrast in the fuzzy enhancement area, so that gives the image boundary detection based on bilinear fast fuzzy enhancement algorithm, greatly improving the speed and quality of boundary detection.

2 Methods

2.1 Generalized fuzzy set GFS and generalized fuzzy operator LGFO

Define 1. Discourse domain U. In order to study the problem, for a specific problem, usually object to discuss is limited in a certain range, all the discussed objects are called as the discourse domain [7].

Define 2. In the discourse domain U, a set of A is as ordinary fuzzy set, it is a set represented by a membership function μ_A (u) : U → [0, 1] on U. Where, μ_A (u) is the degree of element u belonging to the fuzzy set U, a general fuzzy set A is recoded as μ_A (u) or A (u).

Define 3. In the discourse domain U, a set A is called a generalized fuzzy set GFS, which is a set represented by a generalized membership function μ_GA (u) : U → [- 1, 1] on U. Among them, μ_GA (u) ∈ [- 1, 0) is known as the generalized membership that u did not belong to the set A on U; μ_GA (u) ∈ [0, 1) is called the generalized membership that u belongs to the set A on U; μ_GA (u) =0 is called generalized fuzzy cutoff point of A on U [8], and a generalized fuzzy set A is recorded as μ_GA (u) or GA (u).

Define 4. The generalized fuzzy operator LGFO is a linear transformation which can produce an ordinary fuzzy set A′ by its function in the generalized fuzzy set A [9], that is, A′ = LGFO (A), LGFO is defined as:

$\begin{matrix} μ_{A^{'}} (x) = LGFO (μ_{A} (x)) \\ = {\begin{matrix} - \frac{r}{r + 2 t} μ (x) + \frac{2 t}{r + 2 t}, \\ - 1 \leq μ (x) < - \frac{1}{2} (r + 2 t + 1) + \frac{t}{r} \\ - \frac{r}{r - 2 t} μ (x) - \frac{2 t}{r - 2 t}, \\ - \frac{1}{2} (r + 2 t + 1) + \frac{t}{r} \leq μ (x) \\ < - \frac{1}{2} r - t - \frac{r}{r + 2 t} μ (x), \\ - \frac{1}{2} r - t \leq μ (x) < 0 \\ \frac{r}{t + 2 t} μ (x), 0 \leq μ (x) < \frac{1}{2} r + t \\ \frac{r}{r - 2 t} μ (x) - \frac{2 rt}{r - 2 t}, \\ \frac{1}{2} r + t \leq μ (x) < \frac{1}{2} (r + 2 t + 1) - \frac{t}{r} \\ \frac{r}{r + 2 t} μ (x) + \frac{2 t}{r + 2 t}, \\ \frac{1}{2} (r + 2 t + 1) - \frac{t}{r} \leq μ (x) \leq 1 \end{matrix} \end{matrix}$ (1)

r ∈ (0, 1) and t ∈ (0, r / 2) are adjustable parameters, and according to definition 4, LGFO is proved to have the following properties:

The Property 1.For any -1 ≤ μ_A (x) ≤1, formula (1) is a linear continuous piecewise function.

The Property 2.For any -1 ≤ μ_A (x) ≤1, 0 ≤ LGFO (μ_A (x)) ≤1, then the GFS is transformed into a common fuzzy set [10].

The Property 3.When -1 ≤ μ_A (x) ≤0 orr ≤ μ_A (x) ≤1, $μ_{A} (x) \leq μ_{A}^{'} (x)$ , which adds the value of the image pixels in the region -1 ≤ μ_A (x) ≤0 or ther ≤ μ_A (x) ≤1.

The Property 4.When 0 ≤ μ_A (x) ≤ r, $μ_{A}^{'} (x) \leq μ_{A} (x)$ , which reduces the value of the image pixels in the region 0 ≤ μ_A (x) ≤ r.

According to the property 1–4, we can see that the formula (1) can play a role of enhancing the contrast of the image region [11].

The following Fig. 1 is the curve of $μ_{A} (x) \sim μ_{A}^{'} (x)$ when r = 0.3, f = 0.1. The point line divides the $μ_{A} (x) \sim μ_{A}^{'} (x)$ plane into two parts, in the upper left triangle area $μ_{A} (x) \leq μ_{A}^{'} (x)$ , and in the triangle area in the lower right corner $μ_{A} (x) \geq μ_{A}^{'} (x)$ . The solid line is the $μ_{A} (x) \sim μ_{A}^{'} (x)$ curve, and the pixel value in the corresponding region of the segmented curve crossing the upper left triangle area is enhanced [12]. On the other hand, the pixel values in the corresponding region of the segmented curve crossing the right lower triangle area are reduced [13], and the dotted lines have demarcated the segmentation point coordinates of the fuzzy enhancement. It is obvious that the experimental curve is in agreement with the theoretical analysis, and then the property 4–7 is verified.

Fig.1

The curve of $μ_{A} (x) \sim μ_{A}^{'} (x)$ .

2.2 Linear generalized membership transformation

In order to make the image enhancement of the gray image $I = (x_{if}) (\begin{matrix} x_{if} \in {0, 1, \dots, 255}, \\ i \in {1, 2, \dots, m}, \\ j \in {1, 2, \dots, n} \end{matrix})$ [14], we need to transform the image from the gray space of the image to the generalized fuzzy space P = (p_ij [- 1, 1]) first. In this paper, the linear left half trapezoid fuzzy distribution function is used as the generalized membership transformation of the image [15].

The linear generalized membership transformation LT (x_ij) of the image is as follows: $p_{ij} = LT (x_{ij}) = \frac{x_{ij} - M}{x_{max} - M}$ (2)

Where, 0 < M ≤ (x_max - x_min)/ - 2 is the adjustable parameter, x_max and x_min are the maximum and the minimum of the pixel value respectively. It is not difficult to prove that the generalized fuzzy set P = (p_ij) after the P = (p_ij) transformation satisfies P = (p_ij).

2.3 Bilinear fast image enhancement for boundary detection algorithm

To sum up, for image I, the latest algorithm of a bilinear fast fuzzy enhancement for image boundary detection is as follows:

The gray level of the color image.

If the image I is a color image, that is, I = (R_ij, G_ij, B_ij), (R_ij, G_ij, B_ij)∈ { 0, 1, ⋯, 255 }, i∈ { 1, 2, ⋯, m } and j∈ { 1, 2, ⋯, n }, the RGB bitmap is needed to be converted to a gray scale I = (x_ij). The specific method is x_ij = 0.29900 • R_ij + 0.58700 • G_ij + 0.11400 • B_ij. Where, x_ij∈ { 0, 1, ⋯, 255 }, i∈ { 1, 2, ⋯, m }, j∈ { 1, 2, ⋯, n } which represents the gray value of each pixel; R_ij, G_ij, and B_ij represent the red, green and blue color components of each pixel, respectively.

Generalized fuzzification of grayscale images.

For the gray image $\begin{matrix} I = (x_{ij}) (x_{ij} \in {0, 1, \dots, 255}, \\ i \in {1, 2, \dots, m}, \\ j \in {1, 2, \dots, n},) \end{matrix}$ the gray image I = (x_ij) is transformed to the generalized fuzzy set P = (p_ij) ∈ [- 1, 1] by using the nonlinear generalized membership transformation LT (x_ij), shown in the formula (2).

The enhancement processing of the generalized fuzzy set.

For the generalized fuzzy set P = (p_ij), the generalized fuzzy set P = (p_ij) ∈ [- 1, 1] is transformed to the ordinary fuzzy set $P^{'} = (p_{ij}^{'}) \in [0, 1]$ by using the linear generalized fuzzy operator LGFO shown in the formula (1). In fact, this process plays the role of fuzzy enhancement [16].

The gray level of the ordinary fuzzy set.

For the ordinary fuzzy set $P^{'} = (p_{ij}^{'})$ , the ordinary fuzzy set $P^{'} = (p_{ij}^{'})$ is transformed into the enhanced gray image $\begin{matrix} I^{'} = ({x^{'}}_{ij}), ({x^{'}}_{ij} \in {0, 1, \dots, 255}, \\ i \in {1, 2, \dots, m}, j \in {1, 2, \dots, n}) \end{matrix}$ by using the inverse of the linear generalized membership transformation as follows [17]. Where, the linear generalized subordinate inverter $RLT (p_{ij}^{'})$ is as follows: $\begin{matrix} x_{ij}^{'} & = & RLT (p_{ij}^{'}) \\ = & {LT}^{- 1} (p_{ij}^{'}) = p_{ij}^{'} \cdot (x_{ma \bar{x}} t) + t \end{matrix}$

The boundary extraction of the image. Using “Min” or “Max” to extract the boundary of the image [18], the accurate detection of the image boundary is realized [19].

The flow chart of the boundary detection algorithm based on bilinear fast image enhancement is shown in Fig. 2.

Fig.2

Flow chart of boundary detection algorithm based on bilinear fast image enhancement.

3 Results

3.1 Result detection by the proposed algorithm

A pair of color image with 1024 pixel×768 pixel is selected for the experiment. Because the size of the original image is too large, it is not suitable for image processing, it is reduced to 128 pixels×96 pixels. Figure 3 shows the original color images, the grayimages and boundary contourimagesfor the experiment. Table 1 enumerates the influence of different parameters r and f of the linear generalized fuzzy enhancement operator LGFOin the proposed algorithm on the experimental results. Figures 4–17 gives corresponding experimental results.

Fig.3

Color image and its gray image and boundary contour image.

Fig.4

NO.1 experimental results.

Fig.5

NO.2 experimental results.

Fig.6

NO.3 experimental results.

Fig.7

NO.4 experimental results.

Fig.8

NO.5 experimental results.

Fig.9

NO.6 experimental results.

Fig.10

NO.7 experimental results.

Fig.11

NO.8 experimental results.

Fig.12

NO.9 experimental results.

Fig.13

NO.10 experimental results.

Fig.14

NO.11 experimental results.

Fig.15

NO.12 experimental results.

Fig.16

NO.13 experimental results.

Fig.17

NO.14 experimental results.

Table 1

Experimental results using different paraments r and f in LGFO, respectively

r	f	D_best and its Iteration number	Q_max	σ	R	Number of experimental image and figure in Table 2	r
0.3	0.05	45	10	0.0857	2.7798	0.2382	0.3
0.3	0.1	70	15	0.0919	2.6686	0.2451	0.3
0.3	0.15	0	1	0.1177	1.4826	0.1746	0.3
0.3	0.2	40	9	0.1362	1.2708	0.1731	0.3
0.3	0.1	20	5	0.0845	2.7798	0.235	0.3
0.3	0.2	0	1	0.1083	1.7791	0.1927	0.3
0.3	0.3	Null	Null	Null	Null	No.7	0.3
0.3	0.4	60	13	0.1387	1.2708	0.1763	0.3
0.3	0.1	30	7	0.0869	2.426	0.2109	0.3
0.3	0.2	55	12	0.0862	2.1554	0.1859	0.3
0.3	0.3	5	2	0.0979	1.7791	0.1742	0.3
0.3	0.4	15	4	0.0981	1.5698	0.154	0.3
0.3	0.45	0	1	0.0966	1.4826	0.1431	0.3
0.3	0.5	25	6	0.1266	1.4045	0.1707	0.3

Note: x_min = 0, x_max = 255, 0 < D ≤ 127.5.

As shown in Table 1, when the different values of parameter r and f of the linear generalized fuzzy enhancement operator LGFO are selected, the best D value D_best, the maximum Q value Q_max, the image noise standard deviation σ value and the image detectable boundary degree R value will be obtained. As for the selection of r and f, which is more reasonable, it is illustrated by the experimental results in Figs. 4–17, including: enhanced images and its boundary contour images, $μ_{A} (x) \sim μ_{A}^{'} (x)$ curve and D ∼ Q curve. As shown in Figs. 4–17, the effect of enhanced image and its boundary contour images is better. No.1, No.2, No.5 and No.9 correspond to the values of r = 0.3, f = 0.05, r = 0.3, f = 0.1, r = 0.6, f = 0.1 and r = 0.9, f = 0.1, respectively. It is verified that the value of f must be taken in the interval $(0, \frac{r}{2})$ . At the same time, it is found that the Q_max value is near 0.09. When the value of f is $\frac{r}{2}$ , that is r = 0.3, f = 0.15 (No. 3), r = 0.6, f = 0.3 (No. 7) and r = 0.9, f = 0.45 (No. 13), resulting in the meaningless of $μ_{X}^{'}$ segment. All the $μ_{A} (x) \sim μ_{A}^{'} (x)$ curves are incomplete, which leads to the fact that algorithm is unable to run the policy, such as No.7. For No.3 and No.13, only the initial D value (D = 0) corresponding to the random effect is generated, which is meaningless, but at the same time, it can also see that the enhanced image and its boundary contour has been seriously distorted [20 –23]. As long as $f > \frac{r}{2}$ , namely, r = 0.3, f = 0.2 (No. 4), r = 0.6, f = 0.4 (No. 8) and r = 0.9, f = 0.5 (No. 14), $μ_{A} (x) \sim μ_{A}^{'} (x)$ curve shows the phenomenon of inversion and over the range, resulting in poor effect of enhanced image and its boundary contour image. In addition, under the condition of r = 0.6, f = 0.2 (No. 6), r = 0.9, f = 0.2 (No. 10), r = 0.9, f = 0.3 (No. 11), r = 0.9 and f = 0.4 (No. 12), though $f < \frac{r}{2}$ , the $μ_{A} (x) \sim μ_{A}^{'} (x)$ curve in the -rf₈ ≤ μ_A (x) < 0 segment all appear to be over value, especially the No.11 and No.12 of which image’s effect is common. In No.1, No.2, No.5 and No.9, by comparison, the best quality is No.2, namely in the time of r = 0.3 and f = 0.1, The $μ_{A} (x) \sim μ_{A}^{'} (x)$ curve has the largest transformation amplitude at each subsection, and the curve segment is obvious. The enhanced image contrast is strong, the level is outstanding, the boundary contour is clearer and the details are more abundant. Especially in the weak contrast area except the mountain and sky, the contrast is enhanced obviously. The corresponding detail regions of the boundary contour images are extracted and displayed. To sum up, the selection of parameters r and f should be satisfied:

The value near 0.3 is selected for r;

The value near 0.1 is selected for f.

Another linear generalized membership transformation function is given, which is defined as: $μ_{ij} = LT (x_{ij}) = 1 - \frac{x_{max} - x_{ij}}{D}$ (3)

In the formula, $\frac{x_{max} - x_{ij}}{2} < D \leq x_{max}$ is adaptive adjustable parameter.

3.2 Comparison of performance indexes between the proposed algorithm and other algorithms

The performance of the proposed algorithm is compared with the performance of other algorithms, as shown in Table 2. Among them, the test image is gray scale image with the pixel of 1024×768×256, and the computer is configured as PIV 1.6 GHz/256 M/32 M/60 G, and the software is IL6.0 (because the speed IDL6.0 of running program is 1–4 times faster than that of Mat-Lab6.5, IDL6.0 is selected). Analysis Table 2 can see that compared with the other three algorithms, the proposed algorithm has the highest speed and can extract the detail boundary of image, and there is no loss of grayscale information.

Table 2
Comparison table of performance indicators of this algorithm and other algorithms

Gorithm Pal gorithm Chen Wufan algorithm Wang Hui algorithm Algorithm in this paper

Algorithm speed(s) 25.265381 24.645826 13.263682 5.375

Extracting detail boundary no no no yes (r, M, t)

Neglecting the details of the boundary yes yes yes no

Is it linear Homogeneous nonlinearity Homogeneous nonlinearity linear + Nonlinearity + linear linear + linear + linear

Gray information loss loss loss loss Non destructive

Gorithm	Pal gorithm	Chen Wufan algorithm	Wang Hui algorithm	Algorithm in this paper
Algorithm speed(s)	25.265381	24.645826	13.263682	5.375
Extracting detail boundary	no	no	no	yes (r, M, t)
Neglecting the details of the boundary	yes	yes	yes	no
Is it linear	Homogeneous nonlinearity	Homogeneous nonlinearity	linear + Nonlinearity + linear	linear + linear + linear
Gray information loss	loss	loss	loss	Non destructive

4 Conclusions

In order to solve the disadvantages of traditional image boundary detection algorithm based on power function, such as the large amount of computation and the poor quality, a bilinear fast enhancement fuzzy algorithm for image boundary detection is proposed in this paper. First, the ordinary space of the gray image is converted into a generalized fuzzy space, and then the boundary detection algorithm is enhanced by the bilinear fast image enhancement. By using the generalized fuzzy operator LGFO, the generalized fuzzy set GFS is carried out the regional contrast enhancement. The enhanced generalized fuzzy sets are transformed into ordinary fuzzy subsets, and the boundary extraction of the processed ordinary fuzzy subsets greatly enhances the efficiency and quality of image boundary detection, and has important application value.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.51778050) and the Technology Research and Development project of China Railway (No.2017G002-H).

References

, Geng

and Qiu

, A cluster boundary detection algorithm based on shadowed set, Intelligent Data Analysis20(1) (2016), 29–45.

Fang

and Wang

, Lane boundary detection algorithm based on vector fuzzy connectedness, Cognitive Computation9(5) (2017), 1–12.

Gómez

, Yáñez

, Guada

, et al., Fuzzy image segmentation based upon hierarchical clustering, Knowlboundary-Based Systems87(C) (2015), 26–37.

Nabizadeh

, John

and Wright

, Histogram-based gravitational optimization algorithm on single MR modality for automatic brain lesion detection and segmentation, Expert Systems with Applications41(17) (2014), 7820–7836.

Liu

, Tang

, Yang

, et al., Defects’ geometric feature recognition based on infrared image boundary detection, Infrared Physics & Technology67 (2014), 387–390.

Feng

and Wu

H.N.

, Hybrid robust boundary and fuzzy control for disturbance attenuation of nonlinear coupled ODE-beam systems with application to a flexible spacecraft, IEEE Transactions on Fuzzy Systems25(5) (2017), 1293–1305.

Yang

, Zhang

K.P.

and Liu

H.E.

, Online regulation of high speed train trajectory control based on T-S fuzzy bilinear model, IEEE Transactions on Intelligent Transportation Systems17(6) (2016), 1496–1508.

Zhao

X.L.

and Wang

Y.G.

, Optimization and simulation of image restoration in virtual reality, Computer Simulation34(4) (2017), 440–443.

Yang

and Jo

K.H.

, Output feedback fault-tolerant control for a class of discrete-time fuzzy bilinear systems, International Journal of Control Automation & Systems14(2) (2016), 486–494.

10.

Yang

Y.U.

and Wei

, Fault-tolerant output feedback control for a class of multiple input fuzzy bilinear systems, Chinese Journal of Engineering Mathematics172(6) (2015), 247–253.

11.

and Lu

, Fuzzy varying coefficient bilinear regression of yield series, Journal of Data Analysis & Information Processing03(3) (2015), 43–54.

12.

Saoudi

, Chadli

and Braiek

N.B.

, Fault tolerant control for uncertain fuzzy bilinear models, Ifac Papersonline48(21) (2015), 1256–1261.

13.

Wang

R.H.

, Low-resolution outdoor multi-barrier architectural image layout calibration simulation, Computer Simulation34(6) (2017), 236–240.

14.

Liu

, Xu

and Yan

, Exact subspace segmentation and outlier detection by low-rank representation, Mathematics16(1) (2014), 409–421.

15.

Yang

, Han

, Pan

, Xing

and Li

, Optimum surface roughness prediction for titanium alloy by adopting response surface methodology, Results in Physics7 (2017), 1046–1050.

16.

Wang

, Zhang

D.Q.

, Su

, Dong

J.W.

and Tan

S.K.

, Assessing hydrological effects and performance of low impact development practices based on future scenarios modeling, Journal of Cleaner Production179 (2018), 12–23.

17.

Ismail

, Kanwal

and Shahbaz

M.Q.

, Generalized ratio-product-type estimator for variance using auxiliary information in simple random sampling, Kuwait Journal of Science45(1) (2018), 79–88.

18.

Pongnu

and Pochai

, Numerical simulation of groundwater measurement using alternating direction methods, Journal of Interdisciplinary Mathematics20(2) (2017), 513–541.

19.

Al-Hassan

Q.M.

, On some interesting properties of a special type of tridiagonal matrices, Journal of Discrete Mathematical Sciences and Cryptography20(2) (2017), 493–502.

20.

Dipierro

and Valdinoci

, (Non) local and (non) linear free boundary problems, Discrete and Continuous Dynamical Systems-Series S11(3SI) (2018), 465–476.

21.

López Ureña

and Beneito

R.M.D.

, High-accuracy approximation of piecewise smooth functions using the truncation and encode approach, Applied Mathematics and Nonlinear Sciences2 (2017), 367–384.

22.

Vajravelu

, Sreenadh

and Saravana

, Influence of velocity slip and temperature jump conditions on the peristaltic flow of a Jeffrey fluid in contact with a Newtonian fluid, Applied Mathematics and Nonlinear Sciences2 (2017), 429–442.