Minimum square distance thresholding method applying gray-gradient co-occurrence matrix

Abstract

In image thresholding segmentation, gray level of pixels is the basic element to describe images. Besides, the gradient information of pixels is also a key feature to represent image space distribution. Therefore, the co-occurrence probability of gray and gradient of pixels is an effective information to describe image. In this paper, gray-gradient asymmetrical co-occurrence matrix is constructed, uniformity probability of image region is produced, and a minimum square distance criterion function based on gray-gradient co-occurrence matrix is proposed to measure the deviation between original and binary images. Comparing with gray-gray asymmetrical co-occurrence matrix and relative entropy-based symmetrical co-occurrence matrix method, the proposed method can obtain more complete segmentation results, especially for small-size object extraction. The peak signal to noise ratio probability also shows the better segmentation performance of our proposed method.

Keywords

Thresholding method gray-gradient co-occurrence matrix minimum square distance

1. Introduction

Image segmentation is an important step in objects identifying, image analysis and understanding [1]. It is one of the most difficult tasks in image processing , and is still very challenging and difficult to accomplish [2]. The thresholding segmentation method is a classic technology for object extraction. It utilizes the characteristic information of pixels to find the threshold value, which separate the pixels into several classes of object and background [3]. Commonly, the threshold segmentation algorithms are based on clustering, object properties, spatial and positional information, entropy, histogram and shape algorithms [4]. Among of them, the algorithms considering the information of spatial and domain of pixels can more fully employ the features of images to search the accurate threshold value [5].

In image features information, the co-occurrence probability of pixels and its neighborhood pixels is important as a component to build thresholding algorithm [6]. According to the construction of application model, the co-occurrence matrix can be classified into the asymmetric and symmetric [7]. The gray-gradient co-occurrence probability can construct an asymmetric matrix. Sen and Pal [8] used a linear gradient operator to generate a gradient histogram, explored the specific properties of the probability density function to determine the region of interest in the gradient histogram, and obtain the threshold value based on the standard histogram thresholding technique. Shima et al. [9] designed a more accurate gradient operator for edge information extraction. Mignote [10] studied the specification of gradient histograms.

For the existing asymmetric co-occurrence matrix threshold methods, the statistics of ordinary gray scale and spatial information is mainly based on the gray distribution features [11], and the description for the image edge information is limited [12]. Gradient information is often used in image processing to describe the variation of image gray scale in space, can reflect the edge information of images. By designing the reasonable gradient operators [13, 14], the gradient model can be established to improve the accuracy of edge acquisition [15]. The thresholding methods combining with gradient information can quantitatively describe the relationship between selected threshold and image space composition, is an effective thresholding segmentation and edge detection technology [16]. In this paper, we selected Laplacian operator to abstract the gradient information, conformed a two-demention histogram using gray-gradient co-occurrence probability, and defined a uniformity probability of the binarized image region, and then constructed a minimum squared distance function as a new thresholding criterion. It can be seen from the experimental results, the proposed thresholding method is effective and has prominent performance, especially for images of extracted object is much smaller the background.

2. Gray-gradient co-occurrence matrix

2.1 Image gradient

For image $f(x,y)$ , the gradient on coordinate $(x,y)$ is set to be:

$\displaystyle\nabla\vec{f}=\left[{{\begin{array}[]{*{20}c}{G_{x}}\hfill\\ {G_{y}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{\frac{\partial f}{% \partial x}}\hfill\\ {\frac{\partial f}{\partial y}}\hfill\\ \end{array}}}\right]$ (1)

The amplitude of gradient is denoted as:

$\displaystyle\nabla f=\textit{mag}\left({\nabla\vec{f}}\right)=\sqrt{G_{x}^{2}% +G_{y}^{2}}=\sqrt{\left({\frac{\partial f}{\partial x}}\right)^{2}+\left({% \frac{\partial f}{\partial y}}\right)^{2}}$ (2)

The vector angle is as follows:

$\displaystyle a(x,y)=\arctan\frac{G_{y}}{G_{x}}$ (3)

In generally, the gradient is expressed in the form of differential operator and then implemented using a fast convolution function.

2.2 The Laplace operator

Currently, the commonly used gradient operators are Roberts, Sobel, Prewitt and Laplacian, among of them, Roberts, Sobel and Prewitt are first-order differential operators, and Laplacian is second-order differential operators. The experimental results of the four operators show that, for the noise-free and noisy images, the Laplacian operator is better in keeping the integrity of edge information. Therefore, in this paper, we apply the Laplacian operator to compute the gradient value.

For a two-dimensional function $f(x,y)$ , the second-order derivative are:

$\displaystyle\frac{\partial^{2}f}{\partial x^{2}}=f\left({x+1,y}\right)\!+\!f% \left({x-1,y}\right)-2f\left({x,y}\right)$ (4) $\displaystyle\frac{\partial^{2}f}{\partial y^{2}}=f\left({x,y+1}\right)\!+\!f(% x,y-1)-2f(x,y)$ (5)

The difference formula of the Laplacian operator is

$\displaystyle\nabla^{2}f(x,y)=\frac{\partial^{2}f}{\partial x^{2}}+\frac{% \partial^{2}f}{\partial y^{2}}=\left[{f(x+1,y)+f(x-1,y)-2f(x,y)}\right]+\left[% {f(x,y+1)+f(x,y-1)-2f(x,y)}\right]=f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)-4f(x,y)$ (6)

2.3 Gray-gradient co-occurrence matrix

Assuming an image $G$ of $M\times N$ size, the gray level is from 0 to $L-1(L=256)$ . The normalized gray and gradient image are respectively represented by $F(x,y)$ and $H\left({x,y}\right)$ , and the matrix element is defined as the total pixel number of $F(x,y)$ with gray value $i$ and $H\left({x,y}\right)$ with gradient value $j$ [17].

The normalization gray image $F\left({x,y}\right)$ :

$\displaystyle F(x,y)=\textit{INT}\left({f\left({x,y}\right)\times{N_{f}}% \mathord{\left/{\vphantom{{N_{f}}{f_{\max}}}}\right.\kern-1.2pt}{f_{\max}}}% \right)+1$ (7)

Where, $\textit{INT}(\cdot)$ denotes the integer operation. $f_{\max}$ is the maximum gray value of the image, and $N_{f}$ is the maximum gray value of normalization.

Similarly, the normalized gradient image $H\left({x,y}\right)$ :

$\displaystyle H(x,y)=\textit{INT}\left({h(x,y)\times{N_{h}}\mathord{\left/{% \vphantom{{N_{h}}{h_{\max}}}}\right.\kern-1.2pt}{h_{\max}}}\right)+1$ (8)

$h_{\max}$ is the maximum gradient value, and $N_{h}$ is the maximum gradient value in normalization. Here, $N_{f}=256,N_{h}=64$ .

Base on the normalized gray image $F(x,y)$ and gradient image $H\left({x,y}\right)$ , the co-occurrence matrix $C_{m}$ is constructed, its element $c_{ij}$ is the frequency of the pair $F(x,y)=i$ and $H(x,y)=j$ , the probability of the matrix $C_{m}$ is denoted as $p_{ij}$ :

$\displaystyle p_{ij}={c_{ij}}\mathord{\left/{\vphantom{{c_{ij}}{\sum\limits_{i% }{\sum\limits_{j}{c_{ij}}}}}}\right.\kern-1.2pt}{\sum\limits_{i}{\sum\limits_{% j}{c_{ij}}}}$ (9)

Where, $0\leqslant i\leqslant N_{f}-1,0\leqslant j\leqslant N_{h}-1.$

$\displaystyle\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{j=0}^{N_{h}-1}{p_{ij}=1}}$ (10)

If the pair of gray and gradient $(s,t)$ segment the image into four regions of A, B, C and D, as shown in Fig. 1, the abscissa represents the gradient value of image pixels and the ordinate represents the gray value of pixels.

Figure 1.

Region of gray-gradient co-occurrence matrix $C_{m}$ .

Region A includes the pixels belonging to the object but not the edge, B includes the pixels belonging to the background but not the edge, C is the pixels belonging to both the object and edge, and D is both the background and edge. Inside of the object and background, the gray level is relatively uniform, and the gradient value is zero or lower. In the edge of object and background, the gradient value is larger. The probability of four regions of matrix $C_{m}$ are:

$\displaystyle P_{A}(s,t)=\sum\limits_{i=0}^{s}{\sum\limits_{j=0}^{t}{p_{ij}}}$ (11) $\displaystyle P_{B}(s,t)=\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=0}^{t}{p% _{ij}}}$ (12) $\displaystyle P_{C}(s,t)=\sum\limits_{i=0}^{s}{\sum\limits_{j=t+1}^{{}_{N_{h}-% 1}}{p_{ij}}}$ (13) $\displaystyle P_{D}(s,t)=\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=t+1}^{N_% {h}-1}{p_{ij}}}$ (14)

3. Uniformity probability

If the image $G$ is segmented by threshold pair $(s,t)$ , we can obtain a binary image $G^{\ast}$ , then get four regions by $(s,t)$ , and extract the uniform value of probability information, which is defined as $p_{ij}^{\Delta(A)}$ , $p_{ij}^{\Delta(B)}$ , $p_{ij}^{\Delta(C)}$ and $p_{ij}^{\Delta(D)}$ :

$\displaystyle p_{ij}^{\Delta(A)}=q_{A}(s,t)=\frac{P_{A}(s,t)}{(s+1)\times(t+1)% },0\leqslant i\leqslant s,0\leqslant j\leqslant t$ (15) $\displaystyle p_{ij}^{\Delta(B)}=q_{B}(s,t)=\frac{P_{B}(s,t)}{(N_{f}\!-\!s\!-% \!1)\times(t+1)},s+1\leqslant i\leqslant N_{f}-1,0\leqslant j\leqslant t$ (16) $\displaystyle p_{ij}^{\Delta(C)}=q_{C}(s,t)=\frac{P_{C}(s,t)}{(s+1)\times(N_{h% }-t-1)},0\leqslant i\leqslant s,t+1\leqslant j\leqslant N_{h}-1$ (17) $\displaystyle p_{ij}^{\Delta(D)}=q_{D}(s,t)=\frac{P_{D}(s,t)}{(N_{f}\!-\!s\!-% \!1)\times(N_{h}\!-\!t\!-\!1)},s+1\leqslant i\leqslant N_{f}-1,t+1\leqslant j% \leqslant N_{h}-1$ (18)

The probability $p_{ij}^{\Delta(A)}$ can describe the uniformity within object, and $p_{ij}^{\Delta(B)}$ present the uniformity of background, $p_{ij}^{\Delta(C)}$ and $p_{ij}^{\Delta(D)}$ reflect the variety of edge parts.

4. Minimum square distance thresholding method

When selecting the threshold value of the image, we hope that the deviation between the original image and the binary image can be minimum, that means the optimal matching of the two images need to be obtained. According to the optimal match property, we can established a thresholding criterion function, which is defined as minimum square distance criterion.

4.1 Minimum square distance threshold method

We supposed the co-occurrence probability of original image is $p$ , and the probability of binary image is $p^{\Delta}$ , the apply the Euclidean distance to measure the deviation between the two images, which is defined as:

$\displaystyle F(p;p^{\Delta})=\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{j=0}^{N% _{h}-1}{(p_{ij}-p_{ij}^{\Delta})^{2}}}=\sum\limits_{i=0}^{s}\sum\limits_{j=0}^% {t}(p_{ij}-q_{A})^{2}+\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=0}^{t}{(p_{% ij}-q_{B})^{2}}}+\sum\limits_{i=0}^{s}{\sum\limits_{j=t+1}^{N_{h}-1}{(p_{ij}-q% _{C})^{2}}}+\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=t+1}^{N_{h}-1}{(p_{ij% }-q_{D})^{2}}}$ (19)

Here,

$\displaystyle\sum\limits_{i=0}^{s}\sum\limits_{j=0}^{t}(p_{ij}-q_{A})^{2}=\sum% \limits_{i=0}^{s}\sum\limits_{j=0}^{t}\left(p_{ij}^{2}-2p_{ij}\frac{p_{A}(s,t)% }{(s+1)(t+1)}+\left({\frac{p_{A}(s,t)}{(s+1)(t+1)}}\right)^{2}\right)=\sum% \limits_{i=0}^{s}{\sum\limits_{j=0}^{t}{p_{ij}^{2}-\frac{p_{A}^{2}(s,t)}{(s+1)% (t+1)}}}$ (20) $\displaystyle\sum\limits_{i=s+1}^{N_{f}-1}\sum\limits_{j=0}^{t}(p_{ij}-q_{B})^% {2}=\sum\limits_{i=s+1}^{N_{f}-1}\sum\limits_{j=0}^{t}p_{ij}^{2}-\frac{p_{B}^{% 2}(s,t)}{(N_{f}-s-1)(t+1)}$ (21) $\displaystyle\sum\limits_{i=0}^{s}\sum\limits_{j=t+1}^{N_{h}-1}(p_{ij}-q_{C})^% {2}=\sum\limits_{i=0}^{s}{\sum\limits_{j=t+1}^{N_{h}-1}{p_{ij}^{2}}}-\frac{P_{% C}^{2}(s,t)}{(s+1)(N_{h}-t-1)}$ (22) $\displaystyle\sum\limits_{i=s+1}^{N_{f}-1}\sum\limits_{j=t+1}^{N_{h}-1}(p_{ij}% -q_{D})^{2}=\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=t+1}^{N_{h}-1}{p_{ij}% ^{2}}}-\frac{p_{D}^{2}(s,t)}{(N_{f}-s-1)(N_{h}-t-1)}$ (23)

According to the above formula, we can deduce that:

$\displaystyle F\left({p;p^{\Delta}}\right)=\sum\limits_{i=0}^{N_{f}-1}{\sum% \limits_{j=0}^{N_{h}-1}{p_{ij}^{2}}}-\frac{P_{A}^{2}(s,t)}{(s+1)\times(t+1)}-% \frac{P_{B}^{2}(s,t)}{(N_{f}-s-1)\times(t+1)}-\frac{P_{C}^{2}(s,t)}{(s+1)% \times(N_{h}-t-1)}-\frac{P_{D}^{2}(s,t)}{(N_{f}-s-1)\times(N_{h}-t-1)}$ (24)

In Eq. (24), the first term $\sum_{i=0}^{N_{\text{f}}{-1}}{\sum_{j=0}^{N_{h}-1}{p_{ij}^{2}}}$ is a constant, then the Eq. (24) can be simplified, the new distance measurement criterion $F_{\textit{total}}(p;p^{\Delta})$ is:

$\displaystyle F_{\textit{total}}(p;p^{\Delta})=\frac{P_{A}^{2}(s,t)}{(s+1)% \times(t+1)}+\frac{P_{B}^{2}(s,t)}{(N_{f}-s-1)\times(t+1)}+\frac{P_{C}^{2}(s,t% )}{(s+1)\times(N_{h}-t-1)}+\frac{P_{D}^{2}(s,t)}{(N_{f}-s-1)\times(N_{h}-t-1)}$ (25)

The maximum value of $F_{\textit{total}}(p;p^{\Delta})$ is equal to the minimum value of $F\left({p;p^{\Delta}}\right)$ .

4.2 Interpretation of vector correlation coefficient

The vector correlation coefficient can also interpret the criterion function $F_{\textit{total}}(p;p^{\Delta})$ . The co-occurrence matrix of the original image $({p_{ij}})_{N_{f}\times N_{h}}$ and the threshold image $({p_{ij}^{\Delta}})_{N_{f}\times N_{h}}$ can both constitute a vector with $N_{f}\times N_{h}$ dimension, then the correlation coefficient $r(s,t)$ of the vector can be calculated:

$\displaystyle r(s,t)=\frac{\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{j=0}^{N_{h% }-1}{p_{ij}p_{ij}^{\Delta}}}}{\sqrt{\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{j% =0}^{N_{h}-1}{p_{ij}^{2}}}}\cdot\sqrt{\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_% {j=0}^{N_{h}-1}{(p_{ij}^{\Delta})^{2}}}}}=\frac{\sum\limits_{i=0}^{s}{\sum% \limits_{j=0}^{t}{\frac{p_{ij}P_{A}(s,t)}{(s+1)(t+1)}+\sum\limits_{i=s+1}^{N_{% f}-1}{\sum\limits_{j=0}^{t}{\frac{p_{ij}P_{B}(s,t)}{(N_{f}-s-1)(t+1)}}}}}}{% \sqrt{\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{j=0}^{N_{h}-1}{p_{ij}^{2}}}}% \sqrt{\frac{P_{A}^{2}(s,t)}{(s+1)(t+1)}+\frac{P_{B}^{2}(s,t)}{(N_{f}-s-1)(t+1)% }+\frac{P_{C}^{2}(s,t)}{(s+1)(N_{h}-t-1)}+\frac{P_{D}^{2}(s,t)}{(N_{f}-s-1)(N_% {h}-t-1)}}}+\frac{\sum\limits_{i=0}^{s}{\sum\limits_{j=t+1}^{N_{h}-1}{\frac{p_% {ij}P_{C}(s,t)}{(s+1)(N_{h}-t-1)}+\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j% =t+1}^{N_{h}-1}{\frac{p_{ij}P_{D}(s,t)}{(N_{f}-s-1)(N_{h}-t-1)}}}}}}{\sqrt{% \sum\limits_{i=0}^{M-1}{\sum\limits_{j=0}^{N-1}{p_{ij}^{2}}}}\sqrt{\frac{P_{A}% ^{2}(s,t)}{(s+1)(t+1)}+\frac{P_{B}^{2}(s,t)}{(N_{f}-s-1)(t+1)}+\frac{P_{C}^{2}% (s,t)}{(s+1)(N_{h}-t-1)}+\frac{P_{D}^{2}(s,t)}{(N_{f}-s-1)(N_{h}-t-1)}}}=\frac% {\sqrt{\frac{P_{A}^{2}(s,t)}{(s+1)(t+1)}+\frac{P_{B}^{2}(s,t)}{(s+1)(N_{h}-t-1% )}+\frac{P_{C}^{2}(s,t)}{(s+1)(N_{h}-t-1)}+\frac{P_{D}^{2}(s,t)}{(N_{f}-s-1)(N% _{h}-t-1)}}}{\sqrt{\sum\limits_{i=0}^{N_{f}-1}{\sum\limits_{i=0}^{N_{h}-1}{p_{% ij}^{2}}}}}$ (26)

Due to $\sqrt{\sum_{i=0}^{N_{f}-1}{\sum_{j=0}^{N_{h}-1}{p_{ij}^{2}}}}$ is a constant, it is equivalent to maximize $r(s,t)$ and maximize $F_{\textit{total}}(p;p^{\Delta})$ .

4.3 Advanced minimum square distance measurement

In the definition of $p_{ij}^{\Delta}$ , it is unreasonable to be non-zero at the position of $p_{ij}=0$ . In order to make $p_{ij}^{\Delta}$ more reasonable, we can only consider the information at the position of $p_{ij}\neq 0$ [18]. Therefore:

$\displaystyle q_{A}(s,t)=\frac{P_{A}(s,t)}{\sum\limits_{i=0}^{s}{\sum\limits_{% j=0}^{t}{m_{ij}}}}$ (27) $\displaystyle q_{B}(s,t)=\frac{P_{B}(s,t)}{\sum\limits_{i=s+1}^{N_{f}-1}{\sum% \limits_{j=0}^{t}{m_{ij}}}}$ (28) $\displaystyle q_{C}(s,t)=\frac{P_{C}(s,t)}{\sum\limits_{i=0}^{s}{\sum\limits_{% j=t+1}^{N_{h}-1}{m_{ij}}}}$ (29) $\displaystyle q_{D}(s,t)=\frac{P_{D}(s,t)}{\sum\limits_{i=s+1}^{N_{f}-1}{\sum% \limits_{j=t+1}^{N_{h}-1}{m_{ij}}}}$ (30)

Here,

$\displaystyle m_{ij}=\left\{\begin{array}[]{ll}1,&\textit{if}\ p_{ij}\neq 0\\ 0,&\textit{if}\ p_{ij}=0\end{array}\right.$ (31)

Then,

$\displaystyle F_{\textit{total}}(p;p^{\Delta})=\frac{P_{A}^{2}(s,t)}{\sum% \limits_{i=0}^{s}{\sum\limits_{j=0}^{t}{m_{ij}}}}+\frac{P_{B}^{2}(s,t)}{\sum% \limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=0}^{t}{m_{ij}}}}+\frac{P_{C}^{2}(s,t)% }{\sum\limits_{i=0}^{s}{\sum\limits_{j=t+1}^{N_{h}-1}{m_{ij}}}}+\frac{P_{D}^{2% }(s,t)}{\sum\limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=t+1}^{N_{h}-1}{m_{ij}}}}$ (32)

The global squared distance threshold criterion $F_{\textit{total}}\left({p;p^{\Delta}}\right)$ takes into account the information for all four regions A, B, C, and D of the image [19]. If only the interior region information of object and background is considered, the local distance threshold criterion $F_{\textit{local}}(p;p^{\Delta})$ can be obtained:

$\displaystyle F_{\textit{local}}(p;p^{\Delta})=\frac{P_{A}^{2}(s,t)}{\sum% \limits_{i=0}^{s}{\sum\limits_{j=0}^{t}{m_{ij}}}}+\frac{P_{B}^{2}(s,t)}{\sum% \limits_{i=s+1}^{N_{f}-1}{\sum\limits_{j=0}^{t}{m_{ij}}}}$ (33)

If the edge region information of object and background is considered, the joint distance threshold criterion $F_{\textit{joint}}(p;p^{\Delta})$ is obtained:

$\displaystyle F_{joint}(p;p^{\Delta})=\frac{P_{C}^{2}(s,t)}{\sum\limits_{i=0}^% {s}{\sum\limits_{j=t+1}^{N_{h}-1}{m_{ij}}}}+\frac{P_{D}^{2}(s,t)}{\sum\limits_% {i=s+1}^{N_{f}-1}{\sum\limits_{j=t+1}^{N_{h}-1}{m_{ij}}}}$ (34)

Then, the optimal threshold value $s^{\ast}$ is taken by maximizing the above criterion function.

5. Thresholding results and analysis

5.1 Experimental results

In order to compare the proposed method with other methods, we selected some image for thresholding segmentation test. They are Infrared, Circle, Splash, Point and Rice grains image, their size are 360 $\times$ 240, 256 $\times$ 256, 512 $\times$ 512, 214 $\times$ 159 and 256 $\times$ 256, respectively. Figure 2 are the original images, Fig. 3 are their two-dimension (2_D) gray-gradient histograms. Figures 4–8(a) are the results of our gray-gradient local distance thresholding method (denoted as GGLD), Figs 4–8(b) are the results of our gray-gradient joint distance method (denoted as GGJD), Figs 4–8(c) show the results of our gray-gradient total distance method (denoted as GGTD). We also listed the results of minimum squares distance based on gray-gray asymmetrical co-occurrence matrix [20] (denoted as SDGG) in Fig. 4–8(d), and relative entropy method for comparing [21] (denoted as RESM), shown in Figs 4–8(e).

Figure 2.

Original images.

Figure 3.

2_D gray-gradient histograms.

Figure 4.

The results of Infrared image.

From the thresholding results, we can find that our proposed local distance (GGLD) and total distance (GGTD) method obtained the best object extraction results for the five images. Especially, the character segmented in Circle image is clearer using GGLD and GGTD method than other methods. The misclassification pixels of GGLD and GGTD method are the least in Infrared, Splash and Point images. In Rice grains image, our GGLD, GGJD and GGTD methods can keep the complete information of Rice grains.

Table 1 shows the thresholds of five methods for the five images. It can be seen that, the segmentation threshold values of GGLD and GGTD method are same, and they are very different with GGJD, GGSD and RECM method.

Table 1

The threshold value of five methods for the five images

[height=0.8cm,width=2.4cm]ImageMethod	GGLD	GGJD	GGTD	GGSD	RECM
Infrared	112	66	112	106	38
Circle	27	204	27	13	39
Splash	143	139	143	133	82
Point	167	122	167	157	135
Rice grains	101	103	101	40	119

Figure 5.

The results of Circle image.

Figure 6.

The results of Splash image.

Figure 7.

The results of Point image.

Figure 8.

The results of Rice grains image.

5.2 Performance evaluation

Furthermore, for testing the the segmentation performance of our proposed method quantitatively, we chosen five representative images from a database of non-destructive testing (NDT) to evaluate the performance [5], their size are 131 $\times$ 232, 93 $\times$ 81, 166 $\times$ 60, 132 $\times$ 60 and 271 $\times$ 56 respectively, shown in Fig. 9(a)–(e). We also displayed their ground-truth images [5], shown in Figs 10(a)–(e), they are downloaded from the website of Mehmet Sezgin [22]. Figures 11–13 are the results of five test images using five methods, and Table 2 listed the corresponding thresholding value. From the results we can find that, our proposed GGLD and GGTD methods also obtain the best results in object extraction.

Figure 9.

The results of Test image1.

Figure 10.

The results of Test image2.

Figure 11.

The results of Test image3.

Figure 12.

The results of Test image4.

Figure 13.

The results of Test image5.

Due to the peak signal to noise ratio (PSNR) [23] gives the similarity of a thresholded image against a reference image, we apply PSNR to measure the performance of methods. It is given by Eq. (34),

$\displaystyle\textit{PSNR(x,y)}=20\log_{10}\left({\frac{255}{\sqrt{}\textit{% RMSE(x,y)}}}\right)$ (35)

Here, $(x,y)$ is the pixel of image.

Table 3 shows the value of PSNR of the five methods. We can see that the value of our proposed GGLD and GGTD methods are highest in five test images.

Table 2

The results of five test images

[height=0.8cm,width=2.4cm]ImageMethod	GGLD	GGJD	GGTD	GGSD	RECM
Test image1	137	95	137	111	94
Test image2	166	146	166	140	133
Test image3	135	80	135	117	81
Test image4	124	60	124	111	88
Test image5	167	103	167	90	102

Table 3

The PSNR of five methods

[height=0.8cm,width=2.4cm]ImageMethod	GGLD	GGJD	GGTD	GGSD	RECM
Test image1	39.3349	2.0602	39.3349	13.1544	2.4556
Test image2	19.4415	13.5435	19.4415	11.8046	9.6296
Test image3	33.9620	1.5081	33.9620	17.7544	2.3244
Test image4	18.8704	0.9242	18.8704	18.3975	4.0079
Test image5	26.3709	1.5137	26.3709	0.9687	1.4453

6. Conclusion

In this paper, the gray-gradient asymmetrical co-occurrence matrix was constructed, the uniformity probability of image region is presented to represent the spatial distribution. Base on gray-gradient co-occurrence matrix, utilizing uniformity probability information, we proposed square distance function as image thresholding criterion, and formed local, joint and total square distance discriminants. Further, we also explored the explanation of vector correlation coefficient, proved the validity of our methods.

Experimental results of real images and standard test images show that, comparing with the gray-gray asymmetrical co-occurrence matrix (GGSD) and relative entropy-based symmetrical co-occurrence matrix (RECM) methods, our proposed local and total square distance (GGLD and GGTD) methods can obtain the best extraction results and highest PSNR value.

Footnotes

Acknowledgments

This work is supported by the National Science Foundation of China (No. 61571361,61671377), and the Science Plan Foundation of the Education Bureau of Shaanxi Province (No. 15JK1682).

References

Breve

, Interactive image segmentation using label propagation through complex networks, Expert Systems With Applications 123 (2019), 18–33.

Avendiab

M.R.

Kheradvarb

and Jafarkhani

, A combined deep-learning and deformable-model approach to fully automatic segmentation of the left ventricle in cardiac MRI, Medical Image Analysis 30 (2016), 108–119.

Sang

Lin

Z.L.

and Acton

S.T.

, Learning automata for image segmentation, Pattern Recognition Letters 74 (2016), 46–52.

Xiang

, Image segmentation for whole tomato plant recognition at night, Computers and Electronics in Agriculture 154 (2018), 434–442.

Sezgin

and Sankur

, Survey over image thresholding techniques and quantitative performance evaluation, Journal of Electronic Imaging 13(1) (2004), 146–168.

Liang

Kaneko

Hashimoto

et al., Co-occurrence probability-based pixel pairs background model for robust objects detection in dynamic scenes, Pattern Recognition 48 (2015), 1370–1386.

Chanda

Chaudhuri

B.B.

and Majumder

D.D.

, On image enhancement and threshold selection using the gray-level co-occurrence matrix, Pattern recognition Letters 3(2) (1985), 243–251.

Sen

and Pal

S.K.

, Gradient histogram: Thresholding in a region of interest for edge detection, Image and Vision Computing (2010), 677–695.

Shima

Saito

and Nakajiama

, Design and evaluation of more accurate gradient operators on hexagonal lattices, IEEE Transactions on Pattern Analysis and Machine Intelligence 32(6) (2010), 961–973.

10.

Mignote

, An energy-based model for the image-histogram specification problem, IEEE Transactions on Image Processing 21(1) (2012), 379–386.

11.

El-Feghi

Adem

Sid-Ahmed

M.A.

et al., Improved co-occurrence matrix as a feature space for relative entropy-based image thresholding. Proceedings of the computer graphics, Imaging and Visualization 49 (2007), 314–320.

12.

Fan

J.L.

and Ren

, Symmetric co-occurrence matrix thresholding method based on square distance, Acta Electronica Sinica 39(10) (2011), 2277–2281.

13.

Fan

J.L.

and Zhang

, A unique relative entropy-based symmetrical co-occurrence matrix thresholding with statistical spatial information, Chinese Journal of Electronics 24(3) (2015), 622–626.

14.

Pare

Bhandari

A.K.

Kumar

and Singh

G.K.

, An optimal color image multilevel thresholding technique using grey-level co-occurrence matrix, Expert Systems With Applications 87 (2017), 335–362.

15.

Mingde

Sun

Z.G.

and Li

Y.S.

, Textural fabric defect detection using adaptive quantized gray-level co-occurrence matrix and support vector description data, Information Technology Journal 11(6) (2012), 673–685.

16.

Seong

H.K.

Shin

Yoo

et al., Reaction-diffusion equation based topology optimization combined with the modified conjugate gradient method, Finite Elements in Analysis & Design 140(2) (2018), 84–95.

17.

Hong

J.G.

, Gray-gradient symbiotic matrix texture analysis, Acta Automatica Sinica 10(1) (1984), 22–25.

18.

Ramac

L.C.

and Varshney

P.K.

, Image thresholding based on ali-silvey distance measures, Pattern Recognition 30(7) (1997), 1161–1174.

19.

Chang

C.I.

Chen

Wang

et al., A relative entropy-based approach to image thresholding, Pattern Recognition 27(9) (1994), 1275–1289.

20.

Zhang

Zhi

Yang

and , Minimum square distance thresholding based on asymmetrical co-occurrence matrix, The 5th Euro-China Conference on Intelligent Data Analysis and Applications 10 (2018), 814–823.

21.

Fan

J.L.

and Zhang

, A unique relative entropy-based symmetrical co-occurrence matrix thresholding with statistical spatial information, Chinese Journal of Electronics 24(3) (2015), 622–626.

22.

Sezgin

, blt_image_references, http://mehmetsezgin.net, 2017.

23.

Akay

, A study on particle swarm optimization and artificial bee colony algorithms for multilevel thresholding, Applied Soft Computing 13 (2013), 3066–3091.