### 机器视觉代写|图像处理作业代写Image Processing代考|Ice Pixel Detection

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器视觉代写|图像处理作业代写Image Processing代考|THRESHOLDING

The pixels in the same region have similar intensity. Based on that ice is whiter than water, the pixel values are normally very different between ice and water pixels, and thresholding is thus a natural choice to segment ice regions from water regions.
The thresholding method is based on the pixel’s grayscale value. It extracts the objects from the background and converts the grayscale image into a binary image. Assuming that an object is brighter than the background, the object and background pixels have intensity levels grouped into two dominant modes. The threshold $T$ is selected to distinguish the objects from the background. A pixel is marked as “object” if its value is greater than the threshold value and as “background” otherwise, that is:
$$g(x, y)= \begin{cases}1 & \text { if } f(x, y)>T \ 0 & \text { if } f(x, y) \leq T\end{cases}$$
where $g(x, y)$ and $f(x, y)$ are the pixel intensity values located in the $x^{\text {th }}$ row, $y^{\text {th }}$ column of the binary and grayscale image, respectively. This turns the grayscale image into a binary image.

When a constant threshold value is used over the entire image, it is called global thresholding. Otherwise, it is called variable thresholding, which allows the threshold to vary across the image.

## 机器视觉代写|图像处理作业代写Image Processing代考|GLOBAL THRESHOLDING

When the intensity distributions of objects and background pixels in an image are sufficiently distinct, it is possible to use a global threshold applicable for the entire image. The key to using the global thresholding is in how to select the threshold value, for which there are several different methods.

As mentioned in Section 2.2, image histogram is a useful tool for thresholding. If a histogram has a deep and sharp valley (local minima) between two peaks (local maxima), e.g., the bimodal histogram as shown in Figure 3.1, that represent objects and background, respectively, an appropriate value for the threshold will be in the valley between the two peaks in the histogram.

For example, as seen in Figure 3.2, the histogram of the grayscale sea ice image in Figure 3.2(a) clearly has two distinct modes, one for the objects (sea ice) and the other for the background (water). A suitable threshold for separating these two modes can be chosen at the bottom of this valley. As a result, it is easy to choose a threshold $T=125$ that separates them. Then the grayscale image can be converted into the binary image as shown in Figure $3.2(\mathrm{c})$, and the ice concentration is thereby estimated as $41.47 \%$.

This method is very simple. However, it is often difficult to detect the valley bottom precisely, especially when the image histogram is “noisy”, causing many local minima and maxima. Often the objects and background modes in the histogram are not distinct, making it more difficult to determine where the background intensities end and the object intensities begin. Furthermore, in most applications there are usually enough variability between images such that, even if a global thresholding is feasible, an algorithm capable of automatically estimating the threshold value for each image will be most accurate.

## 机器视觉代写|图像处理作业代写Image Processing代考|Otsu thresholding

To automatically select an optimal value for the threshold, Otsu proposed a method from the viewpoint of discriminant analysis; it directly approaches the feasibility of evaluating the “goodness” of the threshold [114].

Let $[0,1,2, \cdots, L-1]$ denote the $L$ intensity levels for a given image with size $M \times N$, and let $n_{i}$ denote the number of pixels with intensity $i$. The total number of pixels in the image, denoted by $n$, is then:
$$n=M \times N=\sum_{i=0}^{L-1} n_{i}$$
To examine the formulation of this method, the histogram is normalized as a discrete probability density function:
$$p_{i}=\frac{n_{i}}{n}, \quad p_{i} \geq 0, \sum_{i=0}^{L-1} p_{i}=1$$
Now suppose that a threshold $t(0<t<L-1)$ is chosen to divide the pixels into two classes $C_{0}$ and $C_{1}$, where $C_{0}$ is the set of pixels with levels $[0,1, \cdots, t]$, and $C_{1}$ is the set of pixels with levels $[t+1, t+2, \cdots, L-1]$. Then the probabilities of class $C_{0}$ occurrence is given by the cumulative sum:
$$P_{0}(t)=P\left(C_{0}\right)=\sum_{i=0}^{l} p_{i}$$
Similarly, the probability of class $C_{1}$ occurrence is given by
$$P_{1}(t)=P\left(C_{1}\right)=\sum_{i=l+1}^{L-1} p_{i}=1-P_{0}(t)$$
The mean intensity of the pixels in class $C_{0}$ is given by:
\begin{aligned} m_{0}(t) &=\sum_{i=0}^{t} i P\left(i \mid C_{0}\right) \ &=\sum_{i=0}^{t} i \frac{P\left(C_{0} \mid i\right) P(i)}{P\left(C_{0}\right)} \ &=\frac{1}{P_{0}(t)} \sum_{i=0}^{t} i p_{i} \end{aligned}
where $P\left(C_{0} \mid i\right)=1, P(i)=p_{i}$, and $P\left(C_{0}\right)=P_{0}(t)$. Similarly, the mean intensity of the pixels in class $C_{1}$ is given by:
\begin{aligned} m_{1}(t) &=\sum_{i=l+1}^{L-1} i P\left(i \mid C_{0}\right) \ &=\frac{1}{P_{1}(t)} \sum_{i=l+1}^{L-1} i p_{i} \end{aligned}

## 机器视觉代写|图像处理作业代写Image Processing代考|THRESHOLDING

G(X,是)={1 如果 F(X,是)>吨 0 如果 F(X,是)≤吨

## 机器视觉代写|图像处理作业代写Image Processing代考|Otsu thresholding

n=米×ñ=∑一世=0大号−1n一世

p一世=n一世n,p一世≥0,∑一世=0大号−1p一世=1

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