# 计算机代写|图像处理代写Image Processing代考|ECE867

#### Doug I. Jones

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## 计算机代写|图像处理代写Image Processing代考|Comparison

The extension principle has been originally defined for functions. The approaches presented in Sect. 2.5.2 deal mainly with operators (set operators, relationships between sets, etc.). Links between extension principle and combination of $\alpha$-cuts using Eq. $2.80$ have been established in [12]. Let $f$ be a function from $\mathcal{U}_1 \times \ldots \times \mathcal{U}_n$ to $\mathcal{V}$, and $R_f$ a set operator defined as:
$$R_f\left(X_1, X_2, \ldots X_n\right)=\left{f\left(x_1, x_2, \ldots x_n\right) \mid x_1 \in X_1, \ldots x_n \in X_n\right},$$
where $X_1, \ldots X_n$ are subsets of $\mathcal{U}_1, \ldots \mathcal{U}_n$. Then the extension of $R_f$ using Eq. $2.80$ coincides with Zadeh’s extension of $f$ (Eq. 2.72).

Other links exist between definitions of Sect. 2.5.2. For instance, if $R_B$ is a crisp relation taking values in ${0,1}$, its extension using Eq. $2.85$ is a value in $[0,1]$ and is equivalent to the two fuzzification procedures given by Eqs. $2.80$ and $2.81$.

Let us take the example of extending union between sets, using the methods of Sect. 2.5.2. Let $\mu$ and $v$ be two fuzzy sets on $\mathcal{U}$. Using the integration over $\alpha$-cuts, we get:
\begin{aligned} (\mu \cup v)(x) & =\int_0^1\left(\mu_\alpha \cup v_\alpha\right)(x) d \alpha \ & =\int_0^{\max [\mu(x), v(x)]} 1 d \alpha \ & =\max [\mu(x), v(x)], \end{aligned}
since $\left(\mu_\alpha \cup v_\alpha\right)(x)=1$ iff $\mu(x) \geq \alpha$ or $v(x) \geq \alpha$. This extension leads exactly to the fuzzy union as defined originally in [34]. Exactly the same result is obtained using other fuzzification methods, e.g., with Eq. $2.80$ or Eq. $2.85$.

## 计算机代写|图像处理代写Image Processing代考|Introducing the Volume of the Overlapping Domain

However, this form is not always adequate for image processing purposes since it does not include any spatial information. This may even lead to counter-intuitive results, as the expression $\sup _{x \in \mathcal{S}} t[\mu(x), \nu(x)]$ only represents the maximum height of the intersection. Although it is generally low for fuzzy sets that have almost disjoint supports, its value does not account for different overlapping situations, as illustrated in Fig. $3.3$ (for sake of clarity, $\mathcal{S}$ is represented in 1D only).

The degree of intersection and of non-intersection can therefore be reformulated in order to better represent the notion of spatial overlapping. Another solution, which may be better for some applications, consists in defining a degree of intersection by considering the fuzzy hypervolume $V_n$ (in a space of dimension $n$ ) of the intersection. This also corresponds to a translation process, in the sense that we have:
$$X \cap Y=\emptyset \Leftrightarrow V_n(X \cap Y)=0 .$$
The hypervolume of a fuzzy set is simply defined using the classical fuzzy cardinality. This provides for a fuzzy set $\mu$ (having bounded support) in the discrete case:
$$V_n(\mu)=\sum_{x \in \mathcal{S}} \mu(x),$$
and in the continuous case:
$$V_n(\mu)=\int_{x \in \mathcal{S}} \mu(x) d x .$$

# 图像处理代考

## 计算机代写|图像处理代写Image Processing代考|Comparison

2.80已在 [12] 中建立。让 $f$ 是一个函数 $\mathcal{U}1 \times \ldots \times \mathcal{U}_n$ 到V，和 $R_f$ 一个集合运算符定义为: 在哪里 $X_1, \ldots X_n$ 是子集 $\mathcal{U}_1, \ldots \mathcal{U}_n$.然后扩展 $R_f$ 使用 方程式。2.80恰逢 Zadeh 的扩展 $f$ (等式 2.72)。 Sect 的定义之间存在其他链㢺。2.5.2. 例如，如果 $R_B$ 是一个清晰的关系，取值 0,1 ，它的扩展使用方程式。 $2.85$ 是一个值 $[0,1]$ 并且等效于方程式给出的两个模糊 化过程。 $2.80$ 和 $2.81$. 让我们以使用 Sect 的方法扩展集合之间的联合为例。 2.5.2. 让 $\mu$ 和 $v$ 是两个模糊集 $\mathcal{U}$. 使用集成 $\alpha$-削减，我们得 到: $$(\mu \cup v)(x)=\int_0^1\left(\mu\alpha \cup v_\alpha\right)(x) d \alpha \quad=\int_0^{\max [\mu(x)}$$

## 计算机代写|图像处理代写Image Processing代考|Introducing the Volume of the Overlapping Domain

$$V_n(\mu)=\int_{x \in \mathcal{S}} \mu(x) d x$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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