## 计算机代写|图像处理代写Image Processing代考|EEE6512

2023年2月3日

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## 计算机代写|图像处理代写Image Processing代考|Combination of Results on α-Cuts

One way to define crisp sets from a fuzzy set consists in taking the $\alpha$-cuts of this set. Conversely, a fuzzy set can be reconstructed from its $\alpha$-cuts, as seen in Sect. 2.2.4. Therefore a class of methods for defining fuzzy operations from crisp ones relies on the application of the crisp operation on each $\alpha$-cut and then combining the results to reconstruct a fuzzy operation by stacking up the $\alpha$-cuts.

Let us denote by $\mu$ the membership functions of a fuzzy set defined on the space $\mathcal{U}$. Let us consider a crisp set function $R_B$ (or operation on sets) taking values in a space $\mathcal{V}$ (e.g., $\mathbb{R})$. The fuzzy equivalent $R$ of $R_B$ is then defined as a function from $\mathcal{F}$ in $\mathcal{V}$ (see, e.g., $[3,6,20])$ :
$$R(\mu)=\int_0^1 R_B\left(\mu_\alpha\right) d \alpha .$$
Other fuzzification equations are possible, such as in [3, 12]:
$$R(\mu)=\sup {\alpha \in[0,1]} \min \left(\alpha, R_B\left(\mu\alpha\right)\right),$$
if the relation takes values in $[0,1]$, or:
$$R(\mu)=\sup {\alpha \in[0,1]}\left(\alpha R_B\left(\mu\alpha\right)\right) .$$
These equations may provide different results. Kut there are also some links hetween them as shown later in this section.

Let us now consider a crisp operation $R_B$ having two arguments (typically a relation between sets). The fuzzy equivalent $R$ of $R_B$, applied to two fuzzy sets $\mu$ and $v$ of $\mathcal{U}$, is defined as a generalization of the previous equations:

$$R(\mu, v)=\int_0^1 R_B\left(\mu_\alpha, v_\alpha\right) d \alpha,$$
or, in this case, by a double integration as:
$$R(\mu, v)=\int_0^1 \int_0^1 R_B\left(\mu_\alpha, v_\beta\right) d \alpha d \beta .$$
The other fuzzification equations ( $2.80$ and $2.81$ ) can also be directly extended to operations on more than one fuzzy set.
The extension to $\mathrm{n}$-ary operators is straightforward.

## 计算机代写|图像处理代写Image Processing代考|Translating Binary Terms into Functional Ones

A last class of methods consists in translating binary equations into their fuzzy equivalent. This approach completely differs from the two previous ones in the sense that it does not use explicitly the crisp relation or operation. Indeed, in the extension principle as well as in $\alpha$-cuts based approaches, the definition of a fuzzy operation is a function of the corresponding crisp operation. Here, a fuzzy operation is given directly by an equation involving fuzzy terms that just mimics crisp equation.

This translation is generally done term by term. For instance, intersection is replaced by a t-norm, union by a t-conorm, sets by fuzzy set membership functions, etc. This translation is particularly straightforward if the binary relationship can be expressed in set theoretical and logical terms. Table $2.3$ summarizes these main crisp concepts involved in set equations, and their fuzzy equivalent.

The many possibilities to translate, for instance, set union using a t-conorm means that many definitions can be obtained from this method, depending on the choice of the fuzzy operators used for translating the crisp corresponding ones.
Let us take a simple example to illustrate this method. According to Zadeh’s original definition, a fuzzy set $\mu$ is said to be included in another fuzzy set $v$ if:
$$\forall x \in \mathcal{U}, \mu(x) \leq v(x) .$$
This is a crisp definition of inclusion of fuzzy sets. We may also suggest that if two sets are imprecisely defined, their inclusion relationship may be imprecise too. Therefore inclusion of fuzzy sets becomes a matter of degree. This degree of inclusion can be obtained using the translation principle.

In the crisp case, the set equation expressing inclusion of a set $X$ in a set $Y$ can be written as follows:
\begin{aligned} X \subseteq Y & \Leftrightarrow X^C \cup Y=\mathcal{U} \ & \Leftrightarrow \forall x \in \mathcal{U}, x \in X^C \cup Y, \end{aligned}
where $X^C$ denotes the set complement of $X$ in $\mathcal{U}$. Using the equivalence of Table $2.3$ for each term, we have:
\begin{aligned} \forall x \in \mathcal{U} & \leftrightarrow \inf _{x \in \mathcal{U}}, \ x \in X^C & \leftrightarrow c[\mu(x)], \ x \in Y & \leftrightarrow v(x), \ X^C \cup Y & \leftrightarrow T[c(\mu), v] . \end{aligned}

# 图像处理代考

## 计算机代写|图像处理代写Image Processing代考|Combination of Results on α-Cuts

$$R(\mu)=\int_0^1 R_B\left(\mu_\alpha\right) d \alpha .$$

$$R(\mu)=\sup \alpha \in[0,1] \min \left(\alpha, R_B(\mu \alpha)\right),$$

$$R(\mu)=\sup \alpha \in[0,1]\left(\alpha R_B(\mu \alpha)\right) .$$

$$R(\mu, v)=\int_0^1 R_B\left(\mu_\alpha, v_\alpha\right) d \alpha$$

$$R(\mu, v)=\int_0^1 \int_0^1 R_B\left(\mu_\alpha, v_\beta\right) d \alpha d \beta$$

-元运算符的扩展很简单。 $\mathrm{n}$

## 计算机代写|图像处理代写Image Processing代考|Translating Binary Terms into Functional Ones

$$\forall x \in \mathcal{U}, \mu(x) \leq v(x)$$包含在集合中的集合方程可以写成如下形式: 其中在中 的补集。对每一项使用表的等效项，我们有： $X Y$
$$\begin{array}{ll} X \subseteq Y \Leftrightarrow X^C \cup Y=\mathcal{U} & \Leftrightarrow \forall x \in \mathcal{U}, x \in X^C \cup \ X^C X \mathcal{U} 2.3 & \ \forall x \in \mathcal{U} \leftrightarrow \inf _{x \in \mathcal{U}}, x \in X^C & \leftrightarrow c[\mu(x)], x \in Y \leftrightarrow \end{array}$$

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## MATLAB代写

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