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

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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

## 计算机代写|图像处理代写Image Processing代考|Basic Definitions of Fuzzy Sets Theory

Let $\mathcal{U}$ be the universe of discourse, i.e., the space of objects of interest. It is a classical (or crisp) set. We denote by $x, y$, etc. its elements (or points). In image processing, $\mathcal{U}$ can typically be the space on which the image is defined (usually $\mathbb{Z}^n$ or $\mathbb{R}^n$, with $n=2,3, \ldots$ ) and will then be denoted by $\mathcal{S}$. Then the elements of $\mathcal{U}=\mathcal{S}$ are the points of the image (pixels, voxels). The universe can also be a set of values taken by some image characteristics, for instance, the scale of gray levels. Then an element $x$ is a value (a gray level). The set $\mathcal{U}$ can also be a set of features,primitives, or objects extracted from the images (e.g., segments, regions, objects), leading to a higher level representation of the image content.

A subset $X$ of $\mathcal{U}$ is defined by its characteristic function $\mu_X$, such that $\mu_X(x)=1$ if $x \in X$ and $\mu_X(x)=0$ if $x \notin X$. The characteristic function $\mu_X$ is a binary function, specifying the crisp membership of each point of $\mathcal{U}$ to $X$.

Fuzzy set theory aims at dealing with gradual membership, accomplished by a rather modest extension of the definition of $\mu$ to take values in $[0,1]$ rather than ${0,1}$. A fuzzy subset of $\mathcal{U}$ is then defined through its membership function $\mu$ from $\mathcal{U}$ into $[0,1] .{ }^1$ For each $x$ of $\mathcal{U}, \mu(x) \in[0,1]$ represents the membership degree of $x$ to the fuzzy subset, i.e., to which extent $x$ belongs to it. Although the correct terminology would be to speak of “fuzzy subset,” commonly, the simpler term “fuzzy set” is used (just as in the case of crisp subsets). We keep this term in the following, for the sake of simplicity.

Various notations are used to designate a fuzzy set. A fuzzy set is completely defined by the set ${(x, \mu(x)), x \in \mathcal{U}}$, which can be noted as $\int_{\mathcal{U}} \mu(x) / x$ or in the discrete finite case $\sum_{i=1}^N \mu\left(x_i\right) / x_i$ where $N$ denotes the cardinality of $\mathcal{U}$.

Since the set of all couples $(x, \mu(x))$ is completely equivalent to the definition of the function $\mu$, we have chosen here to simplify notations and to always use the functional notation $\mu$, a function of $\mathcal{U}$ into $[0,1]$, and $\mu$ will alternatively denote a fuzzy set or its membership function.

The support of a fuzzy set $\mu$ is the set of points that have a strictly positive membership to $\mu$ (it is a crisp set):
$$\operatorname{Supp}(\mu)={x \in \mathcal{U} \mid \mu(x)>0} .$$

## 计算机代写|图像处理代写Image Processing代考|Set Theoretical Operations: Original Definitions

Since fuzzy sets have been introduced by L. Zadeh in [34] in order to generalize sets, the first operations that have been proposed are set theoretical (algebraic) operations. We recall here the original definitions proposed by L. Zadeh. Further operations are defined later, in Sect. 2.3.

The equality of two fuzzy sets is defined by the equality of their membership functions:
$$\mu=v \Leftrightarrow \forall x \in \mathcal{U}, \mu(x)=v(x) .$$
The inclusion of a fuzzy set in another one is defined as an inequality on their membership functions:
$$\mu \subseteq v \Leftrightarrow \forall x \in \mathcal{U}, \mu(x) \leq v(x) .$$
The intersection (respectively, union) between two fuzzy sets is defined as the pointwise minimum (respectively, maximum) of their membership values:
\begin{aligned} & \forall x \in \mathcal{U},(\mu \cap v)(x)=\min [\mu(x), v(x)], \ & \forall x \in \mathcal{U},(\mu \cup v)(x)=\max [\mu(x), v(x)] . \end{aligned}
The complement of a fuzzy set $\mu, \bar{\mu}$, is defined as:
$$\forall x \in \mathcal{U}, \bar{\mu}(x)=1-\mu(x) .$$
The main properties of these definitions are the following:

• They are all consistent with crisp set operations, that is, in the particular case where the membership functions only take values 0 and 1 (i.e., they are crisp sets), these definitions reduce to the classical definitions; note that this property is important since it is the least we can ask to the fuzzy extension of an operation on sets.
• $\mu=v \Leftrightarrow \mu \subseteq v$ and $v \subseteq \mu$.
• The fuzzy complementation is involutive, that is $\overline{(\bar{\mu})}=\mu$.
• Intersection and union are commutative and associative.
• Intersection and union are idempotent and mutually distributive.
• Intersection and union are dual with respect to the complementation: $\overline{(\mu \cap \nu)}=$ $\bar{\mu} \cup \bar{v}, \overline{(\mu \cup v)}=\bar{\mu} \cap \bar{v}$.
• If we consider the empty set $\emptyset$ as a fuzzy set having membership values all equal to 0 , then we have $\mu \cap \emptyset=\emptyset$ and $\mu \cup \emptyset=\mu$, for any fuzzy set $\mu$ defined on $\mathcal{U}$.

# 图像处理代考

## 计算机代写|图像处理代写Image Processing代考|Basic Definitions of Fuzzy Sets Theory

$\operatorname{Supp}(\mu)=x \in \mathcal{U} \mid \mu(x)>0$

## 计算机代写|图像处理代写Image Processing代考|Set Theoretical Operations: Original Definitions

$$\mu=v \Leftrightarrow \forall x \in \mathcal{U}, \mu(x)=v(x) .$$

$$\mu \subseteq v \Leftrightarrow \forall x \in \mathcal{U}, \mu(x) \leq v(x) .$$

$$\forall x \in \mathcal{U},(\mu \cap v)(x)=\min [\mu(x), v(x)], \quad \forall x \in \mathcal{U},(\mu \cup v)(x)=\max [\mu(x), v(x)]$$

$$\forall x \in \mathcal{U}, \bar{\mu}(x)=1-\mu(x)$$

• 它们都与清晰集操作一致，即在隶属函数仅取值 0 和 1 (即它们是清晰集）的特定情况下，这些定 义简化为经典定义；请注意，此属性很重要，因为它是我们可以对集合操作的模喖扩展提出的最少 要求。
• $\mu=v \Leftrightarrow \mu \subseteq v$ 和 $v \subseteq \mu$.
• 模㗅互补是内合的，即 $(\bar{\mu})=\mu$.
• 交集和并集是可交换的和结合的。
• 交集和并集是幂等且互分配的。
• Intersection 和 union 在互补方面是对偶的: $\overline{(\mu \cap \nu)}=\bar{\mu} \cup \bar{v}, \overline{(\mu \cup v)}=\bar{\mu} \cap \bar{v}$.
• 如果我们考虑空集 $\emptyset$ 作为一个隶属度值都等于 0 的模糊集，那么我们有 $\mu \cap \emptyset=\emptyset$ 和 $\mu \cup \emptyset=\mu$, 对 于任何模糊集 $\mu$ 定义于 $\mathcal{U}$.

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