Remote-Sensing-Based Change Detection Using Change Vector Analysis in Posterior Probability Space: A Context-Sensitive Bayesian Network Approach

Change vector analysis (CVA) and post-classification change detection (PCC) have been the most widely used change detection methods. However, CVA requires sound radiometric correction to achieve optimal performance, and PCC is susceptible to accumulated classification errors. Although change vector analysis in the posterior probability space (CVAPS) was developed to resolve the limitations of PCC and CVA, the uncertainty of remote sensing imagery limits the performance of CVAPS owing to three major problems: 1) mixed pixels; 2) identical ground cover type with different spectra; and 3) different ground cover types with the same spectrum. To address this problem, this article proposes the FCM-CSBN-CVAPS approach under the CVAPS framework. The proposed approach decomposes the mixed pixels into multiple signal classes using the fuzzy C means (FCM) algorithm. Although the mixed pixel problem is less severe in the high-resolution image, the change detection performance is still enhanced because, as a soft clustering algorithm, FCM is less susceptible to cumulative clustering error. Then, a context-sensitive Bayesian network (CSBN) is constructed to establish multiple-to-multiple stochastic linkages between signal pairs and ground cover types by incorporating spatial information to resolve problems 2) and 3) discussed above. Finally, change detection is performed using CVAPS in the posterior probability space. The effectiveness of the proposed approach is evaluated on three bitemporal remote sensing datasets with different spatial sizes and resolutions. The experimental results confirm the effectiveness of FCM-CSBN-CVAPS in addressing the uncertainty problems of change detection and its superiority over other relevant change detection techniques.


I. INTRODUCTION
I N RECENT years, the earth's surface has been significantly resculpted by large-scale human activities, such as the shrinkage of arable land, industrialization, urbanization, deforestation, grassland reclamation, and resource mining [1]. Because of the speed and scale of change in global ground cover, conventional on-site change detection is becoming infeasible both temporally and economically. The alternative-remote-sensing change detection-has been widely used owing to its low cost and high efficiency.
Among the existing change detection methods, change vector analysis (CVA) and post-classification change detection (PCC) are the most widely used change detection methods for decades. CVA [2], introduced in 1980, performs change detection in a simple and straightforward manner. Therefore, it has gained significant attention from the remote sensing community with several extended versions being proposed, such as weighted CVA [3], Markov random field CVA [3], [4], conditional random field CVA [5], and multiscale CVA [6]. In particular, the CVA-HCRF method [5] combines the traditional conditional random field with an object-based technique into the hybrid conditional random field (HCRF). Because of the object term, the HCRF model can retain the homogeneity of changed objects. Recently, Bovolo et al. proposed several change detection approaches based on change magnitude and spectral angular distance, including compressed CVA [7], hierarchical spectral CVA [8], sequential spectral CVA [9], and multiscale morphological compressed CVA, to detect multiple changes in multispectral and hyperspectral remote sensing imagery [10]. Other studies have used the spectral angle mapper (SAM) to facilitate change detection. Carvalho Jr. et al. [11] proposed a similarity approach that utilizes SAM and spectral correction similarity to estimate the change magnitude and measure the change direction. The main advantage of this approach is its insensitivity to variations in illumination. Recently, Zhuang et al. [12] used SAM and CVA to utilize the spectral vector completely to enhance change detection accuracy. CVA based on deep learning has also been applied to remote sensing change detection. Saha et al. [13] proposed unsupervised deep CVA based on a convolutional neural network to incorporate spatial context information into the change detection procedure. Although the aforementioned This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ CVA-based approaches derive change maps by thresholding change magnitude maps and can identify the change type depending on the direction of the change vector, they require accurate spectral radiometric correction to exhibit satisfactory performance. Moreover, different change types often exhibit different magnitudes in the change magnitude map obtained using the CVA-based approaches. Therefore, a single threshold cannot always guarantee the complete detection of different change types. This problem is more severe in the case of high-resolution remote sensing (HRS) images considering even a single change type often exhibits different change magnitudes owing to high spectral variability.
In contrast, PCC-based approaches [14], [15] perform change detection based on classification maps independently derived from multitemporal images, and hence, do not require accurate radiometric correction and effective thresholding. However, their performance is significantly affected by the accumulation of classification errors. To counteract this flaw, Volpi et al. [16] developed a multidate classification method that treats each transition as a separate class and classifies stacked data. However, the proposed approach requires pixels for all possible transitions to be labeled, which is labor-intensive. An alternative solution is to use compound classification, which exploits the temporal correlation between the images. Compound classification can be classified into two categories-cascade strategies [17], [18] and mutual strategies [19]. A cascade strategy classifies a current image based solely on previous images, which does not fully utilize temporal correlation. Alternatively, a mutual strategy classifies each image based on all the input images, which better exploits temporal information. More recent developments in PCC include [19], which combines multitemporal segmentation with compound classification [20] to perform remote sensing change detection, thereby successfully suppressing the salt-and-pepper effect and reducing false alarms (FAs) caused by area transitions and object misalignment. Tong et al. [21] proposed a four-stage change detection approach that utilizes unsupervised analysis of uncertain areas, active learning, improved transfer learning, and PCC to produce results superior to those of state-of-the-art approaches.
Chen et al. [22], [23] proposed another approach to resolve the defects of PCC and CVA by combining CVA with PCC within the change vector analysis in the posterior probability space (CVAPS) approach. This method does not require strict radiometric corrections and can effectively detect change areas of different change types using a single threshold owing to its similarly scaled change magnitude. CVAPS utilizes a support vector machine (SVM) [24], [25], [26] to estimate pixelwise posterior probability vectors for multiple ground cover types in bitemporal remote sensing imagery. The change posterior probability vector is then used to perform change detection. Strict radiometric correction is not mandatory in this case considering the posterior probability vector is obtained based on a specific classifier. Moreover, utilizing the change posterior probability vector relieves the adverse effects of accumulated classification errors.
Although CVAPS produces satisfactory change detection results, it is still affected by the uncertainty induced by the complex interactions between the natural environment and the diverse spectrum characteristics of different remote sensing imaging instruments, resulting in three major problems: 1) identical ground cover types with different spectra; 2) different ground cover types with the identical spectrum; and 3) mixed pixels. Because these problems cannot be treated directly using SVM, it is difficult to estimate the pixelwise posterior probability vector accurately, thereby limiting the subsequent change detection performance.
To address the aforementioned issues, this study proposes an FCM-CSBN-CVAPS approach within the CVAPS framework. First, the proposed approach decomposes mixed pixels into multiple signal classes using the fuzzy C means (FCM) algorithm. Then, a simple Bayesian network (SBN) can be constructed to establish multiple-to-multiple stochastic linkages between signal classes and ground cover types to prevent the occurrence of problems 1) and 2) discussed above. However, HRS images often have high uncertainty owing to the low spectral separability of ground objects and the high complexity of ground structures. Therefore, if spatial information is not considered, SBN may not effectively accommodate HRS imagery. To effectively mitigate the uncertainty inherent in HRS imagery, this study further proposes a context-sensitive Bayesian network (CSBN) to estimate the posterior probability vector of each pixel based on its spectral information and of its contexts, i.e., the pixels within the predetermined neighborhood. The proposed CSBN shares certain similarities with the Hopfield-type neural network [27]. First, both methods conduct change detection by modeling the spatial correlation between neighboring pixels. Second, they do not need to assume the statistical distribution of the image data. However, the Hopfield-type neural network is unsupervised, whereas the CSBN is supervised. Furthermore, the Hopfield-type neural networks obtain a change detection map by iteratively minimizing the energy function and stabilizing the network. However, instead of being trained by optimization techniques, the CSBN is trained by information statistically collected from the training samples.
The effectiveness of the proposed approach is evaluated on three bitemporal remote sensing datasets with different spatial sizes and resolutions. The experimental results confirm the effectiveness of FCM-CSBN-CVAPS in addressing the uncertainty problems of change detection and its superiority over other relevant change detection techniques.
The rest of this article is organized as follows. Section II provides an overview of the general methodology adopted in this study. The proposed change detection approach, FCM-CSBN-CVAPS, is also described in detail in Section II. Section III introduces the experimental datasets and evaluation criteria, and analyzes the experimental results. Finally, Section IV concludes this article.

II. METHODOLOGY
In this section, we describe the mechanism of CVAPS and discuss the adopted change detection procedure and method in detail.

A. Change Vector Analysis in Posterior Probability Space
Let p i,j denote the pixel in the ith row and jth column of a given image. When the mixed pixel p i,j comprises ground cover types L 1 and L 2 , the spectrum of the pixel in the remote sensing images at times t 1 and t 2 changes slightly, causing its estimated posterior probability vectors in the two temporal images to change insignificantly. At time t 1 , the posterior probabilities P(L 1 |p i,j ) and P(L 2 |p i,j ) are 51% and 49%, respectively. Therefore, according to maximum a posteriori (MAP), the pixel p i,j is classified as L 1 . Similarly, at time t 2 , the posterior probabilities P(L 1 |p i,j ) and P(L 2 |p i,j ) are 49% and 51%. Therefore, pixel p i,j is classified as L 2 . Therefore, the PCC method identifies pixel p i,j as the changed pixel, regardless of the slight spectral difference corresponding to it in the bitemporal images. The misjudgment of PCC should be attributed to the classification error caused by the MAP.
To address this problem, Chen et al. [22], [23] proposed a CVAPS framework. The posterior probability vectors of the pixel at times t 1 and t 2 are denoted as ρ 1 and ρ 2 v denote the posterior probabilities P(L v |p i,j ) at times t 1 and t 2 , respectively, and m denotes the number of ground cover types in the bitemporal image. Therefore, the change vector Δρ of the pixel p i,j is defined as The change magnitude of the pixel p i,j is defined as These are used to produce the change magnitude map. Eventually, a specific automatic threshold algorithm binarizes the change magnitude map into a change map. Considering CVAPS compares bitemporal posterior probability maps instead of classification maps, it significantly reduces the cumulative error. Moreover, compared to CVA, CVAPS normalizes intraclass variability and interclass distance. Therefore, the change magnitudes ||Δρ|| of the different change types lie within the same scale. Therefore, CVAPS can utilize a single threshold to determine the changed/unchanged pixels effectively without employing complex multithresholding algorithms.

B. Overview
As shown in Fig. 1, the method comprises four main steps.
Step 1: The pixelwise posterior probability vector from the bitemporal remote sensing images is estimated using the proposed CSBN.
Step 2: Based on Step 1, CVAPS is employed to generate a change magnitude map.
Step 3: An automatic threshold algorithm (Otsu [28]) is employed to binarize the change magnitude map into a change map.
Step 4: Small fragments and holes are removed from the change map to generate the final change map. The adopted procedure utilizes CSBN to incorporate spatial information into the process of estimation of posterior probability vectors and does not require accurate spectral radiometric correction. Instead of using a classification map, CVAPS utilizes posterior probability vectors to perform change detection. Therefore, it is robust with respect to cumulative classification errors. However, the CSBN step is time-consuming, especially when the neighborhood window is large.

C. Proposed Method: FCM-CSBN-CVAPS
For middle-resolution remote sensing imagery, mixed pixels are ubiquitous and often limit the change detection performance. Therefore, it is necessary to perform mixed pixel decomposition. However, for HRS imagery, the unmixing methods are scarce because: 1) the mixed pixel problem in the HSR image is less severe than in low-and middle-resolution imagery, and 2) evaluating the unmixing precision for the HSR imagery is time-consuming and expensive due to high spatial resolution. Therefore, in this study, we used the FCM algorithm to decompose pixels into multiple signal classes. Unlike the end member detected by the unmixing method, the signal class captured by FCM is a cluster of pixels sharing similar spectral characteristics and cannot be interpreted as a specific ground cover type. However, because fuzzy memberships of signal classes are used to represent the spectrum information of each pixel, FCM can still enhance the change detection performance of the proposed method for HSR imagery because of the following reasons.
1) Because FCM is performed based on all pixels in the remote sensing image, the detected signal classes are reliable due to sufficient data. 2) Because the signal classes are related to all pixels in the image through fuzzy membership and CSBN is trained based on fuzzy memberships of signal classes, the classification information obtained from a small number of training pixels can cover all image pixels. Therefore, only a small number of training samples are required and the training efficiency is increased significantly. can establish multiple-to-multiple linkages between signal class pairs and land cover types, which models the identical ground cover type with different spectra and different ground cover types with the same spectrum. 5) Because the proposed FCM-CSBN-CVAPS exploits all fuzzy memberships of signal classes, the change detection performance is less affected by the cumulative clustering error compared to change detection methods based on unsupervised hard clustering algorithm, such as K-means-CSBN-CVAPS. 6) As an unsupervised clustering algorithm, FCM does not require user intervention. Moreover, the experimental results (see Section III-C6) prove that the proposed method is insensitive to the randomness of FCM and training sample selection. Therefore, the proposed FCM-CSBN-CVAPS is user-friendly, even for users without high training skills and expertise. Related research has demonstrated that incorporating spatial information can effectively treat the spectral variability and mitigate the uncertainty of HRS imagery and yield more accurate change detection results [29]. Therefore, a CSBN [31], [32] is constructed to establish multiple-to-multiple stochastic linkages between signal pairs and ground cover types by incorporating pixel-level spatial information. From one perspective, CSBN can link different signal classes, i.e., different spectrums, to one ground cover type and treat the spectral variability. For example, urban regions in HSR imagery may contain buildings, water bodies, and parks with different spectrums. CSBN can link above different spectra (i.e., signal classes) into ground cover type "urban" and treat the spectral variability of urban. From another perspective, CSBN can establish stochastic linkages between similar signal classes, i.e., the similar spectrum, to different ground cover types. For example, a bright pixel may be a part of a technique installation on the rooftop of a skyscraper, a cloud, or a snow area in the mountains. Therefore, CSBN can model identical ground cover types with different spectra and/or different ground cover types with the same spectrum. Moreover, by considering spatial information, CSBN can effectively deal with HRS images with high uncertainty induced by the low spectral separability of ground objects and high complexity of ground structures. Additionally, since CSBN incorporates spatial information into the pixelwise posterior probability vector estimation, it can be used to treat the spectral variability of HSR imagery. Therefore, the performance of HRS image change detection can be enhanced significantly by coupling FCM with CSBN under the CVAPS framework.
1) CSBN Model: SBN is widely used in the remote sensing information mining field. Datcu et al. [30] proposed the KIM system, which couples SBN with the K-means algorithm to retrieve remotely sensed imagery in a semantics-sensitive manner. However, SBN does not consider spatial information in the image retrieval process. Therefore, Li and Bretschneider [31], [32] proposed CSBN with the K-means algorithm to incorporate spatial information in the image retrieval process.
Because CSBN can incorporate spatial information into the posterior probability estimation, CSBN is used in this study to estimate the pixelwise posterior probability. However, K-means cannot be used to capture signal classes in remote sensing images because of mixed pixel problems and cumulative clustering error, which significantly limits the change detection performance.
Therefore, this study couples the adapted CSBN with FCM to incorporate pixelwise spatial information in change detection in the pixelwise posterior probability estimation stage and training stage. In the estimation stage, CSBN generates the posterior probability vector of a pixel p i,j based on its corresponding fuzzy memberships and fuzzy membership of its adjacent pixel p u,v in the predetermined neighborhood. In the training stage, the probability of signal class pair (ω m , ω n ) occurring in ground cover type L v is estimated based on fuzzy memberships of training pixels and their adjacent pixel in the predetermined neighborhood.
As shown in Fig. 2, the four-layer CSBN model is constructed to estimate the posterior probability P(L v |p i,j ) of the pixel p i,j , which belongs to the ground cover type L v . The first layer comprises a pixel p i,j of a remote sensing image; the second layer comprises pixel pairs (p i,j , p u,v ), where p u,v is located within the neighborhood N i,j of pixel p i,j ; the third layer comprises signal class pairs, where each signal class ω m comprises image pixels with similar spectral or textural characteristics; and the fourth layer comprisessignal class pairs, where each signal the ground cover type within the remotely sensed image.
Given p i,j ࢠI, p u,v ࢠN i,j and indices m and n of signal classes ω m and ω n , respectively, P(L v |p i,j ) is computed using the following equation: According to the Bayesian rule, P(L v |(m,n)) can be expressed as Therefore, P(L v |p i,j ) can be rewritten as P((m,n)) is the prior probability of the signal pair (ω m , ω n ), whereas the conditional probability P((m,n)|L v ) models the stochastic linkage between the signal pair (ω m , ω n ) and ground cover type L v , and can be obtained from training samples provided a priori. The prior probability P(L v ) is assumed to follow a uniform distribution. Alternatively, the P(L v ) can be estimated based on the percentage of pixels that belong to ground cover type L v in the available classification map. The probability P((m,n)|(p i,j ,p u,v )) is calculated based on the corresponding fuzzy memberships where P(m|p i,j ) and P(n|p u,v ) represent the extent to which pixels p i,j and p u,v belong to the signal classes ω m and ω n , respectively. They are estimated based on the fuzzy memberships u m (i, j) and u n (u, v) (see Section II-C2). The probability P((p i,j , p u,v )| p i,j ) reflects the importance of the pixel pair (p i,j , p u,v ) when 2) FCM Algorithm: Mixed pixels are pixels that may belong to multiple signal classes. Therefore, this study utilizes FCM to link each image pixel to various signal classes with different fuzzy memberships to improve the precision of subsequent change detection.
Let I = {p i,j |1≤i≤N, 1≤j≤M} be an M • N remotely sensed image, which is fuzzily classified into C signal classes. u k (i, j) (1≤k≤C) represents the fuzzy membership of the pixel p i,j to the signal class ω k . The fuzzy membership set U = {u k (i, j)} satisfies the following constraints: The FCM algorithm iteratively minimizes the cost function of the form . . , ψ n } denotes the set of signal class centers and ψ k denotes the center of the signal class ω k . q determines the fuzziness of clustering results. From (7) and (8), we can see that P (ω k |p i,j ) and u k (i, j) satisfy identical constraints. Therefore, u k (i, j) can be used to estimate P (ω k |p i,j ), which couples CSBN with FCM.
In summary, the combination of FCM and CSBN is achieved from two perspectives. First, the fuzzy memberships calculated using FCM are used to establish the stochastic linkage between the pixel p i,j and the signal class ω m , which enables mixed pixel decomposition, and second, based on the calculated fuzzy memberships and training samples of the ground cover type L v , the conditional probability P((m,n)|L v ) is estimated to establish the linkage between the signal class pair (ω m ,ω n ) and the ground cover type L v . The multiple-to-multiple relationships between signal class pairs and ground cover types are modeled in Section II-C3.
3) Training the CSBN Model Based on Fuzzy Membership: To calculate the posterior probability P(L v |p i,j ), the probability P((m,n)|L v ) must be learned based on a presupplied training set T v , which involves pixels p x,y that belong to L v . The estimation equation is based on the occurrence frequency of the signal pair, (ω m ,ω n ), corresponding to the ground cover type L v , defined as Unlike hard membership, fuzzy memberships u m (x, y) and u n (u, v) take values ranging from 0 to 1. Therefore, it is necessary to sum the fuzzy membership values of the signal pair (ω m ,ω n ), corresponding to all pixel pairs (p x,y , p u,v ) in the training set T v to derive the signal pair frequency SPF v (m,n).

4) Postprocessing Procedure:
The Otsu algorithm can find the optimal value for the global threshold and divide the change magnitude image into changed and unchanged objects. Furthermore, it can minimize the interclass variance of changed and unchanged objects while maximizing the intraclass variance between them. However, for CVA-based methods, different change types may have different change magnitudes. Therefore, the Otsu algorithm may be ineffective in finding the optimal global threshold. In contrast, for the CVAPS-based methods, the change magnitudes of different change types are within the same range. Therefore, because as a global thresholding technique, the Otsu algorithm is more effective for the CVAPS-based methods; this study uses the Otsu algorithm to produce an initial binary change map.
Owing to the noises in remote sensing imagery, the pixellevel change detection methods (e.g., CVA, SVM-CVAPS, K-means-SBN-CVAPS, K-means-CSBN-CVAPS, FCM-SBN-CVAPS, and FCM-CSBN-CVAPS) usually produce initial binary change map with speckle noises. Although MRF and CRF can be used to eliminate speckle noises, the boundary details of the detected change regions are often removed because of the over-smoothing. Therefore, this study removes fragments smaller than an experimentally determined area in the initial binary change map. Then, the morphological close or dilation is used to fill the small holes. Although the postprocessing procedure is simple, it can retain the boundary details of major change regions. Additionally, to guarantee the objectiveness of comparative experiments, CVA, SVM-CVAPS, K-means-SBN-CVAPS, K-means-CSBN-CVAPS, FCM-SBN-CVAPS, and FCM-CSBN-CVAPS adopt the same postprocessing procedure.

III. EXPERIMENTAL VALIDATION
We conducted several experiments on three multispectral datasets to evaluate the effectiveness of the proposed approach in dealing with middle-resolution remote sensing and HRS imagery. We briefly describe the multispectral datasets and evaluate the metrics. Finally, the experimental results are discussed.

A. Datasets
The first dataset was acquired from the Landsat8 satellite. The second and third datasets were obtained from the Shangtang SenseEarth2020 archive [33]. During the preprocessing phase, two images in each dataset are radiometrically corrected and coregistered to guarantee the accuracy of subsequent change detection.
The following bitemporal image pairs are used. HSR images with three visible bands and a spatial resolution of 3 m/pixel. The images contain scenes of buildings, water bodies, roads, wastelands, and woodlands. The manual change map was created based on visual interpretation and is displayed in Fig. 5(c). The total number of changed pixels is 22 260. Datasets DS2 and DS3 are selected from the same archive to verify the superiority of FCM-CSBN-CVAPS over CVA in Section III-C2 considering the latter performs well on DS2 compared to DS3.

B. Evaluation Criteria
The effectiveness of the proposed change detection approaches was quantitatively compared and analyzed based on the following four metrics [34], [35], [36].
1) The FA rate, which represents the percentage of pixels incorrectly classified into changed regions. 2) The missed detection (MD) rate, which represents the percentage of pixels incorrectly classified into unchanged regions.
3) The overall accuracy (OA), which is the ratio of the total number of correctly detected pixels to the total number of pixels. 4) The Kappa coefficients. All compared algorithms were run on a computer with a 3.61 GHz CPU and 32 GB random access memory. The computational time is compared in Section III-C9.    Fig. 4(m)], considering FCM is less susceptible to cumulative clustering error and can make SBN and CSBN estimate posterior probability vectors with high uncertainty. As shown in Fig. 4(n), by incorporating spatial information, FCM-CSBN-CVAPS significantly improves the change detection performance compared to FCM-SBN-CVAPS. The improvement is more significant than the result obtained in dataset DS1 considering the HSR image contains richer spatial information than the middle-resolution image. In particular, the detected change regions exhibit smoother boundaries and fewer holes owing to the incorporation of spatial information.
The results of the compared methods (CVA, CVA-MRF, CVA-HCRF, SVM-CVAPS, K-means-SBN-CVAPS, K-means-CSBN-CVAPS, FCM-SBN-CVAPS, and FCM-CSBN-CVAPS) on dataset DS3 are shown in Fig. 5. Fig. 5(d) and (e) shows the manually marked training regions, where the training samples are randomly selected. As shown in Fig. 5(a) and (b), it is difficult to employ a satisfactory threshold to detect change regions based on pixelwise spectral features for HSR remote sensing imagery. Therefore, the CVA and CVA-MRF methods [ Fig. 5(g)   [ Fig. 5(m)] overestimated the change regions. Unlike the results obtained in dataset DS2, FCM-CSBN-CVAPS slightly outperforms FCM-SBN-CVAPS with smoother boundaries considering the simple spectral and spatial characteristics of the changed buildings make change detection results less influenced by spatial information.
2) Change Magnitude Analysis: Compared to CVA, the proposed approach yields more consistent results on datasets DS2 and DS3. As shown in Tables II and III, the Kappa values  is significantly poorer on DS3 compared to that on DS2. To explain these experimental results, the change magnitude maps of CVA, FCM-SBN-CVAPS, and FCM-CSBN-CVAPS on DS2 and DS3 are shown in Fig. 6 (manually detected change regions are marked using yellow boundaries). Fig. 6(a) and (d) shows that within the manually detected change regions, the change magnitude map of CVA on dataset DS2 is more homogenous than that on DS3. Therefore, when a single threshold is employed, CVA performed significantly worse on DS3 than on DS2. In contrast, as shown in Fig. 6(c) and (f), the change magnitude maps of FCM-CSBN-CVAPS on DS2 and DS3 exhibited similar homogeneity within the manually detected change regions, which makes a single threshold more effective for the accurate detection of change region. Additionally, as shown in Fig. 6(b) and (e), the homogeneity of the change magnitude maps generated by FCM-SBN-CVAPS is intermediary to those generated by CVA and FCM-CSBN-CVAPS within the manually detected change regions, which explains the intermediary performance quality of FCM-SBN-CVAPS. Although the change magnitude maps of FCM-SBN-CVAPS exhibit severe speckle noise, it does not significantly affect the change detection performance owing to the elimination of small fragments and holes in the change map.
Additionally, the single change type exhibits different change magnitudes in the case of CVA owing to the high spectral variability of HSR imagery [see Fig. 6(d)]. Conversely, for FCM-CSBN-CVAPS, the single change type exhibits similar change magnitudes on the HSR imagery despite the high spectral variability [see Fig. 6(f)]. When a single threshold is used, the OA and Kappa values of FCM-CSBN-CVAPS are 9.91% and 0.5815 higher than those of CVA, respectively. Therefore, FCM-CSBN-CVAPS outperforms CVA owing to its uniformly scaled change magnitude map.
3) Entropy Analysis: On all three datasets, the performance of SVM-CVAPS is significantly poorer than those of FCM-SBN-CVAPS and FCM-CSBN-CVAPS, as shown in Tables III-V. In this section, the entropy of the pixelwise posterior probability vector is used to explain the inferior performance of SVM-CVAPS.
As shown in Fig. 7, SVM uses an optimization technique to enhance classification accuracy, considering it is prone to estimating posterior probability vectors with low uncertainty. Therefore, some ground cover types have a significantly higher posterior probability than others, which can easily induce an overestimation of change pixels, and in turn a high FA rate. In contrast, FCM-SBN-CVAPS and FCM-CSBN-CVAPS estimate posterior probability vectors in a statistical manner. Therefore, they are prone to estimating smoothly distributed posterior probability vectors with high uncertainty, leading to low FA rates. Because entropy is positively correlated to the uncertainty of a probability distribution, the entropy maps (see Fig. 8) of FCM-SBN-CVAPS, FCM-CSBN-CVAPS, and SVM-CVAPS are presented to explain the aforementioned differences in performance.
From one perspective (see Fig. 9), low uncertainty (low entropy) indicates that the posterior probability vector has a high discrimination ability for ground cover types and may lead to high change magnitude and high FA. When the estimated posterior probability vector exhibits the lowest uncertainty (zero entropy), all but one ground cover type exhibits zero probability. Under these conditions, CVAPS degenerates into a PCC-based method. As shown in Table I, the average entropy (AE) of    SVM-CVAPS is significantly lower than that of FCM-CSBN-CVAPS on all three datasets, whereas the FA of SVM-CVAPS is considerably higher than that of FCM-CSBN-CVAPS on all three datasets. Although FCM-SBN-CVAPS exhibits slightly higher AE compared to FCM-CSBN-CVAPS on DS1 and DS3, its corresponding FA is significantly higher than that of FCM-CSBN-CVAPS. Thereby indicating that excessively high uncertainty may adversely affect the FA rate, and incorporating spatial information may reduce the uncertainty reasonably, thereby improving the FA rate.
However, from another perspective (see Fig. 9), high uncertainty (high entropy) indicates that the posterior probability vector may lose its ability to discriminate ground cover types and may lead to low change magnitude and high MD. As shown in Table I, FCM-CSBN-CVAPS exhibits AE and MD values that are higher than those of SVM-CVAPS by 1.1339 and 4%, respectively, on DS1, and by 0.7918 and 4.05% on DS3.

4) Entropy Analysis of Fuzzy Membership:
In this study, the fuzzy membership u k (i, j) calculated by FCM is used to estimate the probability P (ω k |p i,j ). Therefore, the uncertainty of the probability vector P (ω k |p i,j ) (k = 1, …, C) can be reflected by the entropy of fuzzy membership u k (i, j) (k = 1, …, C). As shown in Fig. 10, the high fuzzy degree q is positively correlated to high clustering fuzziness, i.e., high fuzzy membership entropy and uncertainty. Because the probability vector P (ω k |p i,j ) (k = 1, …, C) is used by SBN and CSBN to estimate the posterior probability vector P(L v |p i,j ) (v = 1, …, Z), the uncertainty of probability P (ω k |p i,j ) will propagate to the posterior probability P(L v |p i,j ). As shown in Fig. 10, for SBN and CSBN, the high fuzzy degree q is positively correlated with the high AE of the fuzzy membership map, which in turn is positively correlated with the AE of the posterior probabilistic map. Additionally, as shown in Fig. 10, the average posterior probabilistic entropy of CSBN is always lower than that of SBN, thereby indicating that CSBN can reduce uncertainty by incorporating spatial information.

5) Significance of Spatial Information:
The SBN utilizes fuzzy memberships of a single pixel as the basic unit to estimate the pixelwise posterior probability and fails to consider the fuzzy membership similarity of adjacent pixels, which can introduce noises into the posterior probability estimation process owing to the spectral variability problem, especially for HSR images. In contrast, the CSBN considers the spatial information based on the reasonable assumption that pixels within a specific neighborhood have a higher similarity and are prone to take similar fuzzy memberships regarding signal classes. As a result, the CSBN incorporates pixelwise spatial information in the posterior probability estimation stage and training stage and alleviates the uncertainty of the estimated posterior probability vector caused by spectral variability. As shown in Figs. 3-5, incorporating spatial information enhances the change performance of K-means-CSBN-CVAPS and FCM-CSBN-CVAPS compared to K-means-SBN-CVAPS and FCM-SBN-CVAPS. 6) Randomness Analysis: Two random factors may influence the change detection performance of the proposed method. First, the signal class centers captured by the FCM algorithm, and second, the training samples randomly selected from manually determined training regions in bitemporal images. Two random tests were conducted to verify the robustness of the proposed method against the two factors. First, the signal class centers are fixed, and the training samples of each ground cover type are randomly reselected for five times. Second, the training samples are fixed, and the signal class centers are regenerated five times by the FCM. As shown in Table II, in the first test, the FCM-CSBN-CVAPS obtained very low standard deviations of Kappa, which can be compared to the FCM-SBN-CVAPS. The FCM algorithm reduces a large number of pixelwise spectral vectors into a small number of signal classes. Therefore, the performance of FCM-CSBN-CVAPS is less affected by the change in training samples. The test results verified that the proposed method can yield stable change detection performance when trained by different users. Moreover, the proposed method is user-friendly considering users without high training skills and expertise can still obtain satisfactory results. In the second test, the regeneration of signal class centers only slightly affected the standard Kappa deviation of FCM-CSBN-CVAPS on datasets DS1, DS2, and DS3. The test results show that the randomness of FCM does not significantly affect the performance of the proposed method because the fuzziness of FCM considerably mitigates the cumulative clustering error.
Finally, as shown in Table II, because of the significantly lower cumulative clustering error, the FCM-based approaches are considerably more stable than their K-means counterparts in most cases.

7) Parameter Sensitivity Analysis:
Three signal class numbers (10, 30, and 50) and seven different fuzzy degrees, q, (1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0) were considered to evaluate the effects of different parameters on change detection performance. Note that FCM degenerates into K-means when q = 1.0. In aggregate, 1000 training samples of each ground cover type were randomly selected to train FCM-SBN-CVAPS and FCM-CSBN-CVAPS. The experimental results are shown in Fig. 11.
On middle-resolution imagery obtained from dataset DS1, the lowest Kappa (<0.70) of FCM-SBN-CVAPS and FCM-CSBN-CVAPS is obtained when the fuzzy degree q< = 1.5 under all signal class numbers, as shown in Fig. 11. On HSR imagery obtained from dataset DS2, FCM-SBN-CVAPS and FCM-CSBN-CVAPS achieve the lowest Kappa (<0.50) when the fuzzy degree q< = 1.5 for all signal class numbers. The results show that a low fuzzy degree may induce higher cumulative clustering error and low uncertainty of FCM, leading to low uncertainty of the probabilistic map generated by SBN and CSBN, which in turn degrades the performance of FCM-SBN-CVAPS and FCM-CSBN-CVAPS owing to high cumulative classification error. However, on dataset DS3, the Kappa of FCM-CSBN-CVAPS is less affected by the fuzzy degree q than that of FCM-SBN-CVAPS corresponding to all signal class numbers. The results show that the decomposition of mixed pixels of a low fuzzy degree does not significantly affect the change detection performance of FCM-CSBN-CVAPS when the changed objects have simple spectral and spatial characteristics. On dataset DS1, the Kappa values of In general, owing to the incorporation of spatial information, the Kappa of FCM-CSBN-CVAPS is higher than that of FCM-SBN-CVAPS, especially on HSR imagery.
Finally, the performance of FCM-CSBN-CVAPS corresponding to different window sizes is evaluated, as shown in Fig. 12.
On middle-resolution imagery obtained from dataset DS1 [see Fig. 12(a)], the Kappa of FCM-CSBN-CVAPS exhibits an ascending trend with respect to the increasing window size. The highest Kappa value was achieved when the number of signal classes was 30 and the window size was 13. However, on HSR imagery [see Fig. 12(b) and (c)], the Kappa of FCM-CSBN-CVAPS did not exhibit a consistent trend as the window size increased owing to the complex spectral and textural patterns in high-resolution imagery, which increases the dependence of the optimal window size on the characteristics of the images.
Additionally, a larger window size significantly increases the time taken by FCM-CSBN-CVAPS. Therefore, selecting an appropriate window size is essential to optimize the tradeoff between effectiveness and efficiency in real-world applications.

8) Kappa Corresponding to Different Number of Training Samples:
To investigate the effect of the number of training samples on the proposed approach, 1000, 2000, 3000, 4000, and 5000 training samples are randomly selected for each ground cover type. The experimental results are shown in Fig. 13.
The experimental results presented in Fig. 13(a)-(c) indicate that the number of training samples affects the Kappa values of FCM-SBN-CVAPS and FCM-CSBN-CVAPS only to a slight extent, which suggests that the proposed approach yields satisfactory results that correspond to a small number of training samples.
On all three datasets, the Kappa of FCM-CSBN-CVAPS was consistently higher than that of FCM-SBN-CVAPS owing to the incorporation of spatial information. Moreover, as the number of training samples increased, the Kappa coefficient varied slightly, thereby indicating that the proposed approach is robust with respect to overtraining.
In contrast, as shown in Fig. 13(a)-(c), the Kappa of SVM-CVAPS did not consistently increase as the number of training samples increased. In certain cases, the Kappa of SVM-CVAPS decreased even when the number of training samples increased. This can be attributed to two reasons. First, the selection of training samples determines SVM performance to a large degree. In this experiment, the training samples are randomly selected from manually classified remote sensing images, which may adversely affect SVM performance. And second, SVM is designed to yield optimal performance using a small number of training samples-larger numbers of training samples may not be necessarily correlated to better performance.  Tables III-V. As shown in Table III The slightly better performance of FCM-CSBN-CVAPS proves that incorporating spatial information reasonably reduces the estimated posterior probability uncertainty, thereby improving the change detection performance, even on middle-resolution remote sensing imagery.
As shown in Table IV, FCM-CSBN-CVAPS exhibits the lowest FA and MD rates, second highest OA rate, and the highest Kappa coefficient on HRS images. In particular, the OA of FCM-CSBN-CVAPS is 1.6%, 0.26%, 0.56%, and 7.59%, higher than those of CVA, CVA-MRF, CVA-HMRF, and SVM-CVAPS, respectively. The Kappa value of FCM-CSBN-CVAPS is 0.1208, 0.0253, 0.0613, and 0.2745 higher than those of CVA, CVA-MRF, CVA-HMRF, and SVM-CVAPS, respectively. These results show that FCM-CSBN-CVAPS outperforms the CVA-based methods and SVM-CVAPS significantly on HRS imagery. Additionally, the Kappa of K-means-CSBN-CVAPS and K-means-SBN-CVAPS are 0.3073 and 0.5549 lower than that of FCM-CSBN-CVAPS and FCM-SBN-CVAPS, respectively, owing to the high cumulative clustering error and low clustering uncertainty of K-means algorithm. Compared to those of FCM-SBN-CVAPS, the OA and Kappa of FCM-CSBN-CVAPS are 1.2% and 0.0685 higher, respectively. The significantly higher Kappa value of FCM-CSBN-CVAPS proves that incorporating spatial information considerably reduces the uncertainty of posterior probability estimated by CSBN, thereby improving the change detection performance on HRS imagery.
As shown in Table V, FCM-CSBN-CVAPS exhibits the lowest FA rate, second-lowest MD rate, and the highest OA rate and Kappa coefficient on HSR imagery. Due to the reasons identical to datasets DS1 and DS2, the OA and Kappa of FCM-CSBN-CVAPS were 16.85% and 0.5262 higher compared to those of SVM-CVAPS, respectively, and 9.91% and 0.5815, 6.36% and 0.5260, and 5.26% and 0.478 higher than those of CVA, CVA-MRF, and CVA-HMRF, respectively. These results, similar to DS2, show that FCM-CSBN-CVAPS outperforms CVA-based methods and SVM-CVAPS on HRS imagery. Furthermore, similar to DS1 and DS2, the OA of K-means-CSBN-CVAPS and K-means-SBN-CVAPS is 2.76% and 15.57% lower than that of FCM-CSBN-CVAPS and FCM-SBN-CVAPS, respectively, whereas the Kappa of K-means-CSBN-CVAPS and Kmeans-SBN-CVAPS are 0.1367 and 0.4874 lower than that of FCM-CSBN-CVAPS and FCM-SBN-CVAPS, respectively. The OA and Kappa values of FCM-CSBN-CVAPS are higher than those of FCM-SBN-CVAPS by 0.37% and 0.0217, respectively.
Unlike on DS2, FCM-CSBN-CVAPS only exhibits OA and Kappa values that are modestly higher than those of FCM-SBN-CVAPS because the changed objects in DS3 are primarily buildings with simple spectral and spatial characteristics.
As shown in Tables III-V, compared to other change detection methods, the proposed FCM-CSBN-CVAPS was considerably time-consuming owing to the high computation load of spatial information incorporation at the pixelwise posterior probability estimation and training stages. Because the computational complexity of CSBN is significantly higher than that of SBN, the computational time of FCM-CSBN-CVAPS is 60.96, 3649.31, and 1823.18 s longer than that of FCM-SBN-CVAPS on datasets DS1, DS2, and DS3, respectively. Furthermore, compared to SVM-CVAPS, the computational time of FCM-CSBN-CVAPS is significantly longer. However, SVM utilizes the grid search as a parameter optimization method, whose operation time is 791.04, 291.91, and 1624.34 s on DS1, DS2, and DS3, respectively. Therefore, if parameter optimization time is considered, the time cost of SVM-CVAPS is comparable to that of FCM-CSBN-CVAPS.
In summary, based on the data presented in Tables III-V, CVAPS-based approaches consistently outperform PCC-based approaches, thereby confirming the superiority of CVAPS over PCC. Additionally, FCM-CSBN-CVAPS consistently outperforms CVA-based methods (CVA, CVA-MRF, and CVA-HCRF) on DS1, DS2, and DS3. However, instead of being caused by MRF and HMRF models, the inferior performances of CVA-MRF and CVA-HCRF have to be attributed to the heterogeneous change magnitudes of different change types produced by CVA.

IV. CONCLUSION
The uncertainty of remote sensing imagery, induced by the low spectral separability of ground objects and the high complexity of ground structures, often limits remote-sensing-based change detection performance. This article proposed a novel CVAPS approach that decomposes mixed pixels into multiple signal classes and establishes multiple stochastic linkages between signal pairs and ground cover types (FCM-CSBN-CVAPS). It is successfully implemented on mid-and high-resolution images. The FCM-CSBN-CVAPS approach addresses the uncertainty problem by incorporating pixelwise spatial information. The effectiveness and reliability of the proposed method were evaluated on mid-and high-resolution imagery, and the results confirmed that FCM-CSBN-CVAPS outperformed FCM-SBN-CVAPS, SVM-CVAPS, and CVAbased approaches in terms of change detection by incorporating spatial information. Although the proposed method achieved satisfactory results, it has the following drawbacks. First, the fuzzy degree q, the number of signal classes, and the window size for CSBN have to be experimentally determined. Second, because the computational complexity of CSBN is higher than that of the comparison algorithms, the computational time of the proposed method is significantly longer than that of the comparison algorithms. Finally, the CSBN only incorporates the local spatial information into the classification procedure and omits long-range dependencies of local features. Therefore, in future works, we intend to develop an automatic method to determine the optimal fuzzy degree q, signal class number, and window size and reduce the computational cost of FCM-CSBN-CVPAS to enhance its practicability. Finally, because CSBN only incorporates the local spatial information into the classification procedure and omits long-range dependencies of local features, we plan to integrate the proposed framework with a spatial-channel feature-preserving ViT (SCViT) model [38] because the SCViT can obtain superior classification performance by integrating the detailed geometric information of the HSR image and the different channel information contained in the classification token.