1 Introduction

Change detection aims founding changes occurred on the Earth surface by analyzing remotely sensed images acquired in the same geographical area at two or more different times [5, 13]. Moreover, it is widely used to detect disasters, monitor environmental changes and identify land use/land cover.

Markov random field (MRF) is an image processing method that comprehensively utilizes spectral and spatial information [4]. With the adoption of maximum a posteriori (MAP) criterion, MRF takes full advantage of the spectral characteristic of pixels and the label field characteristic of their neighborhoods, and finally provides the optimal image analysis results by solving the minimum of the sum of characteristic energy of the two [18, 20, 24]. However, in an MRF, because of the inaccuracy in defining the spatial relationship among the pixels in the neighborhood and setting the weights of spatial information, the spatial information cannot be fully and reasonably utilized [2, 14, 15]. Hence, an over-smooth change map will be finally brought without accurate spatial relationships between neighborhoods and weights for the spatial information [3, 17].

In order to solve these problems, this paper proposed an improved MRF method to change detection by focusing on the determination of the weights of spatial information and the definition of the spatial relationship between neighborhood pixels. Firstly, the difference image is obtained using change vector analysis (CVA) method. Secondly, the fuzzy c-means (FCM) clustering algorithm is implemented to the difference image, in which membership degree and centers of changed and unchanged parts are obtained. The linear spatial weights are then computed through diving the difference image into unchanged, uncertainty and changed regions based on the gray values of pixels. Additionally, the spatial attraction model is also introduced to refine spatial relationships between neighborhood pixels to improve the accuracy of spatial information. Finally, the improved MRF is applied to detect changes. To evaluate the effectiveness of proposed method, two experiments were carried on satellite images. The paper is organized as follows. Section 2 shows the related works, and the improved MRF method for change detection is proposed in the Section 3, Section 4 designed some experiments to verify the proposed method. Finally, Section 5 draws conclusions.

2 Related works

In the past three decades, so many change detection methods have been devolved, which can be grouped into three stages. In the first stage, changes are found by comparing the multitemporal images using simple algebra methods, such as image differencing, image ratio, image regression and CVA, etc. [13, 21]. However, simple algebra methods need a threshold to identify changes, and it was usually decided manually. Therefore, many automatic methods for finding the optimal threshold were proposed in the second stage [16]. Nevertheless, a new problem was found that only spectral information is exploited resulting in much noise contained in the change map. In the third stage, more methods were implemented to the difference image generated by image differencing or CVA combing spectral and spatial information, and MRF is one of the most effective methods [2]. But it results in an over-smooth change map that the spatial information cannot be fully and reasonably utilized in the MRF. The edge information was integrated into the spatial energy function of MRF to preserve small structures and edges [3, 12]. But there are still have some shortcomings, on the one hand paper does not provide good results for objects with occlusion/disocclusion the other hand if the object of interest has cast shadows in the scene, the proposed scheme does not yield good results. The method proposed in paper [16] obtains accurate and robust classification maps for different kinds of images.

In order to solve the above problems, lots of researchers did a lot work and gain some results. The spatial attraction was introduced into the spatial energy function to replace the equivalent value in the Kronecker delta function of MRF [5, 23], which reduced over-smooth results. Additionally, a custom-designed Potts model was proposed to improve the accuracy of weights for the spatial information [6]. But the effects of weight for the spatial information and spatial relationships between neighborhood pixels haven’t been considered together. Therefore, this paper proposes an improved MRF by enhancing the accuracy of spatial information to improve the accuracy of change-detection results.

3 Proposed method

To solve the aforementioned problem, a change-detection method based on the improved MRF is proposed. Figure 1 shows the detailed flow chart of proposed method, and the specific procedures are presented as follows.

Fig. 1
figure 1

Flow chart of the proposed method

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3.1 Fuzzy clustering for the difference image

  1. (1)

    The differential image is obtained by processing the remote-sensing images at two time phases through change vector analysis.

    Suppose the images at t 1 and t 2 are denoted as X 1 = {x b1 (i, j)|1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ b ≤ L} and X 2 = {x b2 (i, j)|1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ b ≤ L}, respectively, in which m, n, and L are the numbers of rows, columns, and bands of the image; x b1 (i, j) and x b2 (i, j) denote the gray values of the pixels at row i, column j in the b-th band at two time phases, respectively. The change vector can be calculated by the following equation [11]:

    $$ \varDelta X={X}_1-{X}_2=\left(\begin{array}{l}{x}_1^1\left(i,j\right)-{x}_2^1\left(i,j\right)\\ {}{x}_1^2\left(i,j\right)-{x}_2^2\left(i,j\right)\\ {}\cdots \\ {}{x}_1^L\left(i,j\right)-{x}_2^L\left(i,j\right)\end{array}\right). $$
    (1)

    The change intensity of the pixel can be expressed by the following Euclidean distance:

    $$ \left\Vert \varDelta X\right\Vert =\sqrt{{\displaystyle \sum_{b=1}^L{\left({x}_1^b\left(i,j\right)-{x}_2^b\left(i,j\right)\right)}^2}}. $$
    (2)

    ΔX includes the information on category change of surface features in two images at two time phases, and ‖ΔX‖ denotes the difference of gray level of two images at two time phases. According to Eq. (2), the differential image X = ‖ΔX‖ = {x(i, j)|1 ≤ i ≤ m, 1 ≤ j ≤ n} can be obtained.

  2. (2)

    The differential images are clustered using the FCM algorithm and classified into unchanged and changed categories. The membership degree u ij of the pixel x j to the i-th category, and the centers of these two categories, C 1 and C 2, are calculated.

    Fuzzy clustering was first proposed by Dunn [8] and improved into a classical FCM clustering algorithm by Bezdek [9]. Suppose X = {x 1, x 2, ⋯, x N } is a dataset consisting of N vectors and fuzzily divided into C categories. u ik (1 ≤ k ≤ c) denotes the degree of membership of the datum x i in the dataset X to the k-th category. The fuzzy division results can be described using the matrix U = {u ik }, which meets the following constraint conditions:

    $$ \begin{array}{c}\hfill {u}_{ik}\in \left[0,1\right],\forall i,k\hfill \\ {}\hfill {\displaystyle \sum_{k=1}^c{u}_{ik}}=1,\forall i\hfill \\ {}\hfill 0<{\displaystyle \sum_{i=1}^N{u}_{ik}<N,\forall k}\hfill \end{array}. $$
    (3)

    The FCM clustering algorithm conforms to the principle that the sum-of-squared differences between each sample and the mean value of the located category are minimum. The objective function Jreaches the minimum by iterating and updating the membership matrix U and the clustering center V. Thus, optimal clustering can be achieved. The objective function is expressed as

    $$ J\left(U,V\right)={\displaystyle \sum_{i=1}^N{\displaystyle \sum_{k=1}^c{\left({u}_{ik}\right)}^q}}{\left\Vert {x}_i-{v}_k\right\Vert}^2 $$
    (4)

    where U = {u ik } is the membership matrix that satisfies Eq. (3), V = {v 1, v 2, ⋯, v c } denotes the set of clustering centers, and q ∈ [1, + ∞) denotes the weighted index, which is used to control the fuzzy degree of clustering results. When q = 1, the fuzzy clustering belongs to traditional c-means clustering. In general, when q = 2, the calculation is simple and the results are ideal. When fuzzy clustering is performed on differential images, the sample x i can be replaced by x ij to represent the gray value of the pixel at (i, j) and v k denotes the mean of the k-th category. Thus, Eq. (4) can be expressed as

    $$ J\left(U,V\right)={\displaystyle \sum_{\begin{array}{l}1\le i\le m\\ {}1\le j\le n\end{array}}{\displaystyle \sum_{k=1}^c{\left({u}_k\left(i,j\right)\right)}^2{\left\Vert x\left(i,j\right)-{v}_k\right\Vert}^2}} $$
    (5)

    where u k (i, j) denotes the degree of membership of the pixel (i, j) to the k-th class and ‖ ⋅ ‖ denotes the Euclidean distance. The specific procedure of FCM clustering algorithm is presented as follows:

    1. 1)

      The number of categories in clustering is set as N, and the clustering center V = {v 1, v 2, ⋯, v c } is initialized.

    2. 2)

      The matrix of fuzzy membership is calculated according to the following equation:

      $$ {u}_k\left(i,j\right)={\left[{\displaystyle \sum_{p=1}^c{\left[\frac{\left\Vert x\left(i,j\right)-{v}_k\right\Vert }{\left\Vert x\left(i,j\right)-vp\right\Vert}\right]}^2}\right]}^{-1}. $$
      (6)
    3. 3)

      The clustering center is updated based on the following equation:

      $$ {v}_k=\frac{{\displaystyle {\sum}_{\begin{array}{l}1\le i\le m\\ {}1\le j\le n\end{array}}{u}_k{\left(i,j\right)}^2x\left(i,j\right)}}{{\displaystyle {\sum}_{\begin{array}{l}1\le i\le m\\ {}1\le j\le n\end{array}}{u}_k{\left(i,j\right)}^2}}. $$
      (7)
    4. 4)

      The convergence of Eq. (5) is judged as follows. When ‖V t + 1 − V t‖ < ε (t is the number of iterations and ε > 0 indicates the stopping of computation), Eq. (5) is convergent and the iteration stops; otherwise, steps 2 and 3 are repeated until Eq. (5) is convergent.

Finally, after processing by FCM clustering, the membership degrees of each pixel and the central values of each category can be obtained.

3.2 Markov random field model

Using MAP criterion, MRF is an image processing method that combines spectral and spatial information. From the perspective of field energy, the sum of energy of spectral characteristic field and label field of each pixel should be minimized so that the optimal result can be obtained [10].

Suppose X = {x 1, x 2, ⋯ x n } ⊂ R d denotes the remote-sensing dataset composed of n vectors in d-dimensional Euclidean space (n is the number of pixels and d is the number of bands), L = {l 1, l 2, ⋯ l c } denotes the category label for each pixel and c is the number of categories. Generally, the ultimate category of images can be determined using the MAP criterion, and the formula is

$$ L= \arg \max \left\{P(L)p\left(\left.X\right|L\right)\right\}, $$
(8)

where P(L) denotes the prior probability of a category label in the dataset, and p(X|L) denotes the conditional probability density function of the pixel in the dataset. Based on the MRF model, the maximum posterior probability can be obtained by solving the minimum of energy function U MRF (x i ) as follows:

$$ {U}_{MRF}\left({x}_i\right)={U}_{spectral}\left({x}_i\right)+{U}_{spatial}\left({x}_i\right), $$
(9)

where U spectral (x i ) denotes the energy function of spectral characteristic field of pixel x i , and U spatial (x i ) denotes the spatial energy function of the pixel x i in local neighborhood.

The energy function of spectral characteristic field of pixel x i can be expressed as

$$ {U}_{spectral}\left({x}_i\right)=\frac{1}{2} \ln \left|2\pi {\sigma}_k^2\right|+\frac{1}{2}{\left({x}_i-{\mu}_k\right)}^2{\left({\sigma}_k^2\right)}^{-1}, $$
(10)

where μ k and σ 2 k denote the mean and variance of pixel gray values in thek-th category, respectively. These two values can be calculated by the initial change-detection images generated in FCM clustering.

The spatial label field energy of the pixel x i can be calculated by

$$ {U}_{spatial}\left({x}_i\right)=\beta {\displaystyle \sum_{j\in {N}_i}I\left(l\left({x}_i\right),l\left({x}_j\right)\right)}, $$
(11)
$$ I\left(l\left({x}_i\right),l\left({x}_j\right)\right)=\left\{\begin{array}{l}-1\kern0.5em l\left({x}_i\right)=l\left({x}_j\right)\\ {}0\kern0.99em l\left({x}_i\right)\ne l\left({x}_j\right)\end{array}\right., $$
(12)

where β > 0 denotes the penalty coefficient defined by the same user, which is used to control the effect of the neighboring pixel of pixel x i on x i ; N i denotes the spatial neighborhood (i ∉ N i ) of the pixel x i ; l(x i ) and l(x j ) ((j ∈ N i )) denote the category labels of the pixel x i and its neighboring pixel x j , respectively. As described in MRF theory, the spatial relation between pixels in the label field can be expressed by the neighborhood system N i and defined by the Potts model in Eq. (12).

Change detection is a discrete combinational optimization problem, and the analysis results can be acquired only by the global or local optimization based on the iterative search method [20]. In this paper, iterated conditional mode (ICM) was used to search for the optimal result of Eq. (9) to solve the minimum value of energy function and analyze the remote-sensing images. ICM refers to the process where the local energy is calculated using local conditional probability, and then the analysis results of images with minimum local energy are obtained by updating the labels of images point by point.

3.3 Designed linear spatial weights of MRF

In the MRF, when the label field energy is calculated using the Potts model, the adopted weights β are the same for all the pixels in differential images without any consideration of the distribution of pixel gray values in differential images. In a differential image, the pixels have different gray values and therefore different probability to change. For a pixel, the larger the gray value is, the greater the probability of change is; on the contrary, the smaller the gray value is, the greater is the probability to remain unchanged. For the pixels with intermediate gray level, determining whether the change exists based solely on gray value is difficult. In the traditional Potts model, all the pixels in differential images are set with the same penalty coefficient β. For the pixels with extremely large or extremely small gray values, this setting easily causes an overuse of spatial neighborhood information, which leads to the oversmoothing on the changed regions.

Based on the centers of the two categories calculated using the FCM algorithm, two threshold values, T 1 and T 2, are set. The differential image is then divided into three parts (changed, uncertain, and unchanged), and the corresponding penalty coefficients are calculated. Thus, the contrast-sensitive Potts model is established.

In the traditional MRF, the energy of the spatial label field of the pixel x i can be calculated by [6, 22, 25]

$$ {U}_{spatial}\left({x}_i\right)={\displaystyle \sum_{j\in {N}_i}I\left(l\left({x}_i\right),l\left({x}_j\right)\right)}, $$
(13)
$$ I\left(l\left({x}_i\right),l\left({x}_j\right)\right)=\beta \left\{\begin{array}{l}-1\kern0.5em l\left({x}_i\right)=l\left({x}_j\right)\\ {}0\kern1em l\left({x}_i\right)\ne l\left({x}_j\right)\end{array}\right., $$
(14)

where β > 0 denotes the penalty coefficient defined by the user and is employed to control the effect on the pixel x i from its neighboring pixel; N i denotes the spatial neighborhood (i ∉ N i ) of the pixel x i ; l(x i ) and l(x j )(j ∈ N i ) denote the category labels of pixel x i and the neighboring pixel x j , respectively.

According to the centers of the two categories, C 1 and C 2, the threshold values T 1 and T 2 were set. As shown in Fig. 2, the pixels in the differential image can be divided into three parts, namely, changed, uncertain, and unchanged. The threshold values T 1 and T 2 can be calculated by

$$ \left\{\begin{array}{l}{T}_1={M}_{mid}-{\alpha}_1\left({M}_{mid}-{C}_1\right)\\ {}{T}_2={M}_{mid}+{\alpha}_2\left({C}_2-{M}_{mid}\right)\end{array}\right., $$
(15)

where M mid denotes the central pixel and has the same membership degree as the changed and unchanged categories when FCM is used; and both α 1 and α 2 are constants that are used to adjust the range of the three parts.

Fig. 2
figure 2

Changed, uncertain, and unchanged parts in differential image

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On this basis, three strategies are proposed to calculate the adaptive penalty coefficient β m in the Potts model for the three parts.

$$ {\beta}_m\left(X\left(i,j\right)\right)=\left\{\begin{array}{l}\beta \frac{X\left(i,j\right)-{X}_{\min }}{T_1-{X}_{\min }}\kern3em X\left(i,j\right)<{T}_1\\ {}\beta \kern8.3em {T}_1\le X\left(i,j\right)\le {T}_2\\ {}\beta \frac{X_{\max }-X\left(i,j\right)}{X_{\max }-{T}_2}\kern2.87em X\left(i,j\right)>{T}_2\end{array}\right., $$
(16)

where X min and X max denote the maximum and minimum gray values in the differential image, respectively. By replacing the traditional penalty coefficient β with the adaptive penalty coefficient β m , we can construct a contrast-sensitive Potts model.

Under this condition, when the gray value of the pixel is less than the threshold value T 1, the penalty coefficient β m decreases linearly as the gray value varies from T 1 to X min. Accordingly, the effects of spatial neighborhood information on the pixels with relatively small gray values can be reduced because these pixels are highly likely to remain unchanged. When the gray value of the pixel is larger than T 2, the penalty coefficient β m decreases linearly as the gray value increases from T 2 to X max. In this way, the effects of spatial neighborhood information on the pixels with relatively large gray values can be reduced because these pixels are highly likely to change. Additionally, for the pixels in the uncertain part, we set a constant penalty coefficient β. Thus, the initial change-detection image can be refined by taking full advantage of spatial information.

3.4 Refined spatial relationship between neighborhoods of MRF

Figure 3 presents the label field of spatial neighborhood when the conventional MRF is used for change detection, where i, j denotes the pixel in row i, column j; 0 indicates no change in the pixel; and 1 indicates change in the pixel. The spatial neighborhood relation between various pixels can be characterized by comparing the numbers of pixels of different categories in the neighborhood. Many mixed pixels are included because of the complexity of surface features and the limitations in the spatial resolution of remote-sensing images. Therefore, the spatial neighborhood relation between pixels cannot be accurately expressed by arbitrarily labeling the neighboring pixels as 0 and 1.

Fig. 3
figure 3

Label field of spatial neighborhood in MRF

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The spatial gravity model is used to introduce the membership information to an MRF so that the spatial relationship between neighboring pixels can be identified accurately.

The surface features and the interaction between surface feature and sensor are highly complex, and the spatial resolution of the sensor is limited. Therefore, numerous mixed pixels are included in images. In the traditional Potts model, the spatial neighborhood relation between pixels is usually defined as 0 or 1, which is too absolute and not accurate. This method easily causes the overuse of spatial information and oversmoothing of the change-detection results. The Potts model can be modified by introducing the spatial gravity model to the degree of membership. The spatial neighborhood relation between pixels can be redefined, and Eq. (12) can be rewritten as

$$ I\left(l\left({x}_i\right),l\left({x}_j\right)\right)=\left\{\begin{array}{l}-{w}_{ij}\kern0.8em l\left({x}_i\right)=l\left({x}_j\right)\\ {}0\kern1.99em l\left({x}_i\right)\ne l\left({x}_j\right)\end{array}\right., $$
(17)

where w ij denotes the spatial gravity between the pixel x i and the neighboring pixel x j , and can be calculated by the following equation [5, 23]:

$$ {w}_{ij}=z\left({p}_i\right)\times z\left({p}_i\right)\times \frac{1}{R_{i,j}^2}, $$
(18)

where i is the index value of the central pixel x i ; j ∈ N i {j = 1, 2, ⋯, 8} denotes the pixels in the neighborhood of central pixel x i in a 3*3 pixel window, as shown in Fig. 4a; z denotes the category label of the central pixel x i ; p i and p j denote the membership degrees of the pixel x i and the neighboring pixel x j with regard to the class z, and can be calculated using the FCM clustering algorithm; R ij denotes the spatial distance between the central pixel x i and the neighboring pixel x j , as shown in Fig. 4b.

Fig. 4
figure 4

Spatial neighborhood of central pixel x i

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4 Results and analysis

To verify the feasibility of the proposed method, we selected two groups of remote-sensing images for change detection. Comparisons were made between the proposed method, the multiresolution level set (MLS), MLS with Kittler algorithm (MLSK) [1], FCM, the combination of expectation maximization (EM) and MRF [13], and the combination of FCM and MRF. Three indices are used to evaluate the results: 1) missed detection (MD): the number of changed pixels incorrectly classified as unchanged. The missed detection rate P m is calculated by the ratio P m = MD/N 0 × 100 %, here N 0 is the total number of changed pixels counted in the ground reference map; 2) false alarm (FA): the number of unchanged pixels wrongly detected as changed. The false detection rate P f is described by the ratio P f = FA/N 1 × 100 %, where N 1 is the total number of unchanged pixels counted in the ground reference map; 3) total error (TE): the total number of detection error including both miss and false detection, which is the sum of the MD and the FA. Hence, the total error rate P t is described using P t = (FA + MD)/(N 0 + N 1) × 100 % [7, 19].

4.1 Experiment of Landsat 7 ETM+ data set

In Experiment 1, we used two remote-sensing images of a certain region in Liaoning province collected by Landsat 7 ETM+ satellite sensor in August 2001 and August 2002.

The images with the size of 300 × 280 pixels were selected as the study area, as shown in Fig. 5a and b, corresponding to the true-color remote-sensing images at two time phases. The image at t 1 was registered to the image at t 2, and the relative radiometric normalization was conducted on them by histogram matching. Subsequently, change vector analysis was used to process the images within all bands except thermal infrared band, and the differential images were acquired. The ground reference data were artificially generated by comparing the images at two time phases, as shown in Fig. 5c.

Fig. 5
figure 5

Landsat 7 ETM+ data adopted in Experiment 1

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Figure 6 shows the change-detection images obtained by MLS, MLSK, FCM, EM, MRF. In the detections with multiscale-level set method (MLS) and threshold-based MLS (MLSK), μ = 0.2, which referenced the initial paper [1]. As shown in Fig. 6a and b, the obtained change-detection images using MLS and MLSK are highly similar to the ground reference data; however, speckle noise can be observed in some comparatively large regions of change, e.g., Region A in the figure.

Fig. 6
figure 6

Change-detection results of data in Experiment 1

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When MRF and the proposed method are used, the values of β are set to be 1.8 and 1, respectively, where the parameter was first set as an empirical value 1.6 and then found the optimal value manually. As shown in Region B in Fig. 6c and Region A in Fig. 6d, a great deal of speckle noise exists in the change-detection images when expectation maximization (EM) and FCM algorithms are used because the spatial neighborhood information is not considered, where the weighting exponent of FCM was set to 2. Although the change-detection regions in Fig. 6e and f are more complete than those in the other four images, more details are retained when the proposed method are used, which also eliminates the oversmoothing phenomena of borders in traditional MRF, as shown by Region C in the figures. In conclusion, the change-detection image generated by the proposed method is most similar to the ground reference data.

Table 1 presents the comparisons of the falsely detected pixels, undetected pixels, and total errors for MLS, MLSK, EM, FCM, MRF, and the proposed method. One can conclude that, compared with the other five methods, the total error rate of the proposed method was reduced by 1.2, 1.2, 4.2, 1.4, and 2.5 %, respectively. At least 1000 errors were reduced, and the change-detection results using the proposed method have the highest accuracy. However, since the proposed the proposed method combines FCM and MRF, the calculation time is longer than that for the single FCM and MRF.

Table 1 Accuracy of change detection in Experiment 1
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4.2 Experiment of Landsat 5 TM data set

In Experiment 2, we used the images of a certain region in Alaska, USA collected by the Landsat 5 TM sensor in July 1985 and July 2005.

The images with a size of 400 × 400 pixels were selected in the present study, as shown in Fig. 7a and b, corresponding to the true-color remote-sensing images at two time phases. The image at t 1 was registered to the image at t 2, and the relative radiometric normalization was conducted on the images by histogram matching. The ground reference data were artificially generated by comparing the images at two time phases, as shown in Fig. 7c.

Fig. 7
figure 7

Landsat 5 TM data adopted in Experiment 2

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Figure 8 shows the obtained change-detection results using MLS, MLSK, EM, FCM, MRF, and the proposed method. When MLS and MLSK were used, the parameter μ was set as 0.2. When MRF and the proposed method were used, the parameter β was set as 1.5 and 0.8, respectively.

Fig. 8
figure 8

Change-detection results of data in Experiment 2

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As shown in Fig. 8a and b, numerous change details are lost in the results by MLS and MLSK (see Region A). As shown in Fig. 8c, a significant amount of speckle noise is included (see Region B), mainly because the EM algorithm detects the changes only by setting the threshold and not by utilizing spatial information. Since a large number of mixed pixels existed and the categories of pixels were labeled as those with a greater degree of membership when FCM was used, many change details were undetected, e.g., the results in Region A in Fig. 8d. The proposed method not only generates a more complete homogeneous region but also detects many change details. Comparatively, the single MRF method makes the change-detection results oversmoothed, as shown in Region C in Fig. 8e and f. Conclusively, compared with the change-detection image generated by the other methods, the image generated by the proposed method is closest to the ground reference data.

Table 2 lists the accuracy of change-detection results by MLS, MLSK, EM, FCM, MRF, and the proposed method, respectively. Since the membership information calculated by FCM was introduced to an MRF through the spatial gravity model, the spatial neighborhood relation between pixels can be defined more accurately. The proposed method compared with other methods, its total error rate was reduced by 1.1, 1.0, 4.6, 1.0 and 4.6 %, respectively, reduced at least more than 1500 errors of pixels, can generate the highest accuracy of change detection results. These results verify that the proposed method is feasible.

Table 2 Accuracy of change detection in Experiment 2
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5 Conclusion

By improving the weights for spatial information and spatial relationships between neighborhood pixels of MRF, an improved MRF is proposed to change detection. The results of the two experiments indicate that, compared with MLS, MLSK, and FCM, the changed region detected by the proposed method is more complete. Compared with EM and MRF, the proposed method can be used to remove the speckle noise effectively. Besides, it can avoid the oversmoothing of the changed region obtained by MRF to a certain degree. Overall, compared with MLS, MLSK, EM, FCM, and MRF, the proposed method can generate a change-detection image closer to the ground reference data, achieving higher accuracy of change detection. The experimental results suggest that the designed linear weights for the spatial information and redefined the spatial neighborhood relation between pixels in the label field of MRF are feasible, which can provide more accurate results than those using the Potts model in traditional MRF. Therefore, the proposed method is a more accuracy approach to change detection for remotely sensed images. However, the proposed method generates more accurate results by combing FCM and MRF, it needs more computing time. Hence, when we want to obtain more accurate results, it is better to adopt the proposed one, whereas, when the change maps need producing in shorter time, other methods used in this paper should be exploited. In the future, an adaptive widow of MRF is worth studying to improve its robustness for different kinds of noise.