# Application of Image Segmentation in Surface Water Extraction of Freshwater Lakes using Radar Data
## 1 Introduction
One classical method to select the threshold is to manually pick the smallest valley values between the two peaks of distributions based upon visual inspection by the researcher. The main issue of this method is the bias caused by each individual observer. The solution to offset the researcher's observation bias is to apply computer programming to select a less biased lowest point in the valley, which can be computationally efficient in linear time. However, the intensity histogram presented by radar imaging may not necessarily provide a sharp valley but usually a flat region between the peaks. Thus, it will be less accurate or reasonable to pick the smallest valley value in this case, as the value of the selected point may deviate slightly from the value of its neighboring points in the open intervals next to the selected point. Furthermore, due to the noise in radar detection, the strict convex property is not guaranteed in the valley region between the two peaks. In other words, there may exist multiple local peaks and minimums which are close to each other. In this case, the method of picking the smallest valley value is badly influenced by the noise. The Gaussian Mixture Model is another conventional method for binary classification based on distribution. The distribution of water and nonwater objects in the radar intensity (dB) histogram presents approximately as two Gaussian Distributions with separate means \(\mu_{1}\) and \(\mu_{2}\) and standard deviations \(\sigma_{1}\) and \(\sigma_{2}\)[23]. One of the distributions is the conditional probability of the dB value of the water pixels while the other is the conditional probability of the dB value of the nonwater pixels. The objective of this model is to maximize conditional probability of the prediction \(\hat{y}\) given any dB values (\(x\)). According to the Bayes Theorem, this equates to maximizing the multiplication of the conditional probability of \(x\) over \(\hat{y}\) and the marginal probability of \(\hat{y}\). However, the issue with such formulation of the problem is based on the assumption of the prior distribution of water and land as a Gaussian Distribution. However, such an assumption cannot be directly assumed to be correct for universal cases. Moreover, the distribution parameters \(\mu_{1}\), \(\sigma_{1}\), \(\mu_{2}\), \(\sigma_{2}\) are unknowns. The researcher also needs to identify estimators for these four parameters through the density diagram. Possible solutions for estimation of these unknown parameters can be iterative methods such as Expectation Maximization Method [24], however, it is unstable for two reasons. First, the iteration process is time consuming to reach a satisfied accuracy. Second, it is also likely to be constrained in some local optimum points and thus never reaches global optimal solution [25]. Instead, we propose to use the Otsu Method to solve this thresholding problem. The Otsu Method is an unsupervised method and it was initially designed to select a threshold to separate an object out of its background, through the gray-level histogram of the image [25]. In application, the Otsu Method can be widely extended to work on other density histograms or distributions other than gray-level histogram from images and can also be applied for multi-thresholding problems. The Otsu Method is a better approach for this problem as compared to some conventional solutions because it automatically selects a threshold from two mixed distributions through the density histogram [25]. In addition, the Otsu Method does not require prior knowledge nor assumptions of the distribution of objects [25]. Furthermore, the Otsu Method is equivalent to the K-Means Method but the Otsu Method can provide the global optimal solution, while K - Means Method may be limited to the local optimum point [26]. Although it is computationally complex and heavy because of iterative searching [26], GEE can speed up the Otsu Method with its cloud computing platform.