*3.3. Change Detection Classification Approach*

In this study, we employed the Hidden Markov Random Field (HMRF) approach to perform our lodging-based change detection classification. The HMRF approach fully utilizes and enhances our previous change detection classification approach described in [17] by improving its robustness and sensitivity to false alarms. The method in [17] employed the Finite Gaussian Mixture (FGM) model for image classification. For example, if a ratio image *XRi* contains *N* dimensional vector of pixels with *I* = {1, 2, ......... , *N*} being the set of pixel indices, then for each pixel *i* in *XRi* a class label *xi* is inferred using the conditional probability as shown in [17]. Each pixel in the FGM is independent from their neighboring pixels, meaning they do not consider the relationship of pixels within its neighboring system. To improve the FGM, we modified the HMRF approach proposed by Zhang, et al. [24] and employed it. By assuming a Gaussian distribution, the HMRF model is given by

$$p(\mathbf{X}\_{\text{Ri}} | \mathbf{x}\_{\text{Ni}}; \theta) = \sum\_{l \in L} f(\mathbf{X}\_{\text{Ri}}; \theta\_l) \, q(l | \mathbf{x}\_{\text{Ni}}) \tag{1}$$

where *q*(*l*| *xNi*) is a conditional probability mass function (pmf) for a class label given that *xNi* are N neighbors for a pixel *xi*. The segmentation process of HMRF requires an initial estimate for the class labels *x*<sup>0</sup> and initial parameters (θ<sup>0</sup> = *mean*(μ0), *variance* σ2 0 ). In this research, we used K-means clustering approach to provide these initial labels and initial parameters. The update for parameter θ<sup>0</sup> was estimated iteratively using our Expectation-maximization (EM) algorithm [17]. The initial labels were updated iteratively using the maximum a posteriori (MAP) algorithm. The updated label is now used to solve for *x*ˆ that minimizes the total posterior energy

$$\hat{\mathbf{x}} = \operatorname\*{argmin}\_{\mathbf{x} \in \chi} \{ \mathsf{U}(X\_{Ri}|\mathbf{x}) + \mathsf{U}(\mathbf{x}) \} \tag{2}$$

where χ is the set of all possible configurations of labels, *x*ˆ is the estimated class label, *U*(*XLR*|*x*) is the likelihood energy, and *U*(*x*) is the prior energy function which is defined by clique potentials. A clique is a subset of nodes in which every node is connected to every other node. The clique potential was defined on pairs of neighboring pixels. We assumed that one pixel has at most eight neighboring pixels. The MAP algorithm iteration stops when Equation (2) converges or when the maximum iteration we set is reached. In this study, we classified each ratio image in *XRis* into three classes namely no change, moderate change, and severe change. Finally, the classified ratio image was filtered using a majority filter with mathematical morphology to remove small isolated misclassified pixels. Majority filter with mathematical morphology works by first replacing the pixels in a neighborhood using majority of their adjacent neighboring pixels and then applying opening by reconstruction followed with closing by reconstruction. The order of first doing opening by reconstruction followed by closing by reconstruction was designed to reduce noise while preserving the geometric details in the image. In this study, we set the kernel size of our majority filter to 3 and the size of our morphological kernel to 5. In the final classification map, we removed the no change class, and we multiplied each changed class (moderate change and severe change) by each crop (corn and soybean) mask derived from the CDL to estimate lodged crops per unit area.
