### 机器视觉代写|图像处理作业代写Image Processing代考|SET OPERATIONS ON BINARY IMAGES

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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|SET OPERATIONS ON BINARY IMAGES

Since a binary image is a matrix containing object pixels of value 1 and background pixels of value 0 , it can simply be represented as the set of those coordinate vectors

$(x, y)$ of the pixels that have value of 1 in the binary image, given by:
$$G={(x, y) \mid g(x, y)=1}$$
where $(x, y)$ are pairs of spatial coordinates, $g(x, y)$ is the pixel value $(0$ or 1$)$ at $(x, y)$, and $G$ represents the set of image pixels describing the object of interest. All other image pixels are assigned to the background.

Let $\mathbb{Z}$ be the set of integers. Let the elements of a binary image be represented by a set $A \subseteq \mathbb{Z} \times \mathbb{Z}$, whose elements are 2-dimensional vectors of the form $(x, y)$, which are spatial coordinates. If a set contains no elements, it is called an empty set or a null set, denoted by $\varnothing$. If $\omega=(x, y)$ is an element of $A$, then it is written as:
$$\omega \in A$$
otherwise, it is written as:
$\omega \notin A$
If every element of a set $A$ is also an element of a set $B$, then $A$ is said to be a subset of $B$ and written as:
$$A \subseteq B$$
A set $B$ of pixel coordinates $\omega$ that satisfy a particular condition is written as:
$$B={\omega \mid \text { condition }}$$
The universe set, $\mathbb{U}$, is the set of all elements in a given application. In image processing, the universe is typically defined as the rectangle containing all the pixels in an image.

The complement (or inverse) of $A$, denoted as $A^{c}$, is the set of all elements of $U$ that do not belong to set $A$, given by:
$$A^{c}={\omega \mid \omega \notin A}=\mathbb{U}-A$$
The complement of the binary image $A$ is the binary image that exchanges black and white, that is, 0 -valued pixels set to 1 -valued and 1 -valued pixels set to 0 -valued.
The union of two sets $A$ and $B$, denoted as $A \cup B$, is the set of all elements that belong to either $A, B$, or both, given by:
$$A \cup B={\omega \in A \text { or } \omega \in B}$$
The union of two binary images $A$ and $B$, is a binary image in which the pixels’ values are 1 if the corresponding input pixels’ values are 1 in $A$ or in $B$.

Similarly, the intersection of two sets $A$ and $B$, denoted as $A \cap B$, is the set of all elements that belong to both $A$ and $B$, given by:
$$A \cap B={\omega \in A \text { and } \omega \in B}$$
The intersection of two binary images $A$ and $B$ is a binary image where the pixels’ values are 1 if the corresponding input pixels’ values are 1 in both $A$ and $B$.

## 机器视觉代写|图像处理作业代写Image Processing代考|SET OPERATIONS ON GRAYSCALE IMAGES

When dealing with grayscale images, the set must represent an image with pixels having more than two values. The image intensity value is the third dimension besides the two spatial dimensions $x$ and $y$. A grayscale image can be represented as a binary image in a 3 -dimensional space, with the third dimension representing image intensities. The intensity values can be viewed as heights at each pixel above the $x y$-plane, according to a function $z=g(x, y)$ corresponds to a surface in the 3 dimensional space. Thus, a grayscale image can be represented as a set given by:
$$G={(x, y, z) \mid z=g(x, y)}$$
Because grayscale images are 3-dimensional sets, where the first two dimensions define the spatial coordinates and the third dimension denotes the grayscale intensity value, the preceding set operations for binary images are not applicable for grayscale images. Let the elements of a grayscale image be represented by a set $A \subseteq \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$, whose elements are 3 -dimensional vectors of the form $(x, y, z)$, where the intensity value $z$ is also an integer value within the interval $\left[0,2^{k}-1\right]$ with $k$ defined as the number of bits used to represent $z$. The complement of $A$ is defined as the pairwise differences between a constant and the intensity of every pixel in an image, given by:
$$A^{c}={(x, y, L-z) \mid(x, y, z) \in A}$$
where $L=2^{k}-1$ is a constant. $A^{c}$ is an image of the same size as $A$; however, its pixel intensities have been inverted by substracting them from the constant $L$.

The union of two grayscale sets (images) $A$ and $B$ is defined as the maximum of corresponding pixel pairs, given by:
$$A \cup B=\left{\max _{z}(a, b) \mid a \in A, b \in B\right}$$
The outcome of $A \cup B$ is an image of the same size as these two images, formed from the maximum intensity between pairs of spatially corresponding elements.

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

The logical operations are derived from Boolean algebra, which is a mathematical approach to describe propositions whose outcome would be either TRUE or FALSE. The logical operations consist of three basic operations: NOT, OR, and AND. The terms NOT, OR, and AND are commonly used to denote complementation, union, and intersection, respectively. The NOT operation simply inverts the input value, that is, the output is FALSE if the input is TRUE, and it sets to TRUE if the input is FALSE. The OR operation produces the output TRUE if either one of the inputs is TRUE, and FALSE if and only if all the inputs are FALSE. The AND operation produces the output TRUE if and only if all inputs are TRUE, and FALSE otherwise. Any other logic operator, such as NAND, NOR, and XOR, etc., can be implemented by using only these three operators.

In image processing, the logic operations compare corresponding pixels of input images of the same size and generate an output image of the same size. When dealing with binary images, consisting of only 1 -valued object pixels and 0 -valued background pixels, the TRUE and FALSE states in logic operations correspond directly to the pixel values 1 and 0 , respectively. Hence, the logic operations can be applied in a straight forward manner on binary images using the rules from logical truth tables, as shown in Table $2.1$, to the pixel values from a pair of input images (or a single input image in the case of NOT operation).

## 机器视觉代写|图像处理作业代写Image Processing代考|SET OPERATIONS ON BINARY IMAGES

(X,是)二值图像中值为 1 的像素数，由下式给出：
G=(X,是)∣G(X,是)=1

ω∈一种

ω∉一种

## 机器视觉代写|图像处理作业代写Image Processing代考|SET OPERATIONS ON GRAYSCALE IMAGES

G=(X,是,和)∣和=G(X,是)

A \cup B=\left{\max _{z}(a, b) \mid a \in A, b \in B\right}A \cup B=\left{\max _{z}(a, b) \mid a \in A, b \in B\right}

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