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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|DISTANCE TRANSFORM

Distance transform is an important tool in image processing, and it is normally only applied to binary images that consist of object and background pixels. A distance transform of a binary image specifies the distance from every pixel to the nearest background pixel. In other words, the distance transform converts a binary image into a grayscale image where each object pixel has a value corresponding to the minimum distance from the background. The resulting grayscale image is a so-called distance map.

Assume $f$ is a binary image, in which the pixels with a value of ‘ 0 ‘ indicate the background while the pixels with a value of ‘ 1 ‘ indicate the object. Let $B=$ ${p \mid f(p)=0}$ be the set of background pixels and $O={p \mid f(p)=1}$ be the set of object pixels. The distance transform of a binary image $f, D(p)$, can be given by [39]:
$$D(p)= \begin{cases}0, & \text { if } p \in B \ \min _{q \in B} d(p, q), & \text { if } p \in O\end{cases}$$
where function $d$ is a distance function or metric which is to determine the distance between pixels.

For pixels $p, q$, and $r$ in an image, a distance function $d$ satisfies the following three criteria [128]:

1. Positive definite: $d(p, q) \geq 0(d(p, q)=0$ iff $p=q)$
2. Symmetric: $d(p, q)=d(q, p)$
3. Triangular: $d(p, r) \leq d(p, q)+d(q, r)$There are several types of distance metrics in image processing. The three most important ones are: Euclidean, city-block, and chessboard.

## 机器视觉代写|图像处理作业代写Image Processing代考|PERFORMANCE OF THE DISTANCE METRICS

Figure 2.14 Effects of different distance transforms.
denoted by the operator ‘ $*$ ‘, is defined as:
\begin{aligned} h(x, y) &=\omega(x, y) * f(x, y) \ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \omega(u, v) f(x-u, y-v) \mathrm{d} u \mathrm{~d} v \end{aligned}
In image processing, where an image is represented by a set of pixels, convolution is a local operation that replaces each pixel in an image by a linear combination of its neighbors. The impulse response $\omega(x, y)$ is then referred to as a convolution kernel, and the convolution becomes the calculation of the sum of products of the kernel coefficients with the intensity values in the region encompassed by the kernel. The convolution of a kernel $\omega(x, y)$ of size $m \times n$ with an image $f(x, y)$ is given by:
\begin{aligned} h(x, y) &=\omega(x, y) * f(x, y) \ &=\sum_{s=-\frac{1}{2}}^{\frac{1}{2}} \sum_{t=-\frac{4}{2}}^{\frac{n}{2}} \omega(s, t) f(x-s, y-t) \end{aligned}
For each pixel $(x, y)$ in the image, the convolution value $h(x, y)$ is the weighted sum of the pixels in the neighborhood about $(x, y)$, where the individual weights are the corresponding coefficients in the convolution kernel. This procedure involves translating the convolution kernel to pixel $(x, y)$ in the image, multiplying each pixel in

the neighborhood by a corresponding coefficient in the convolution kernel, and summing the multiplications to obtain the response at each pixel $(x, y)$. Figure $2.15$ gives an example of convolution of an image with a $3 \times 3$ kernel. In this example, the response of the kernel at the center point $(x, y)$ of the $3 \times 3$ image neighborhood is given by:
\begin{aligned} h(x, y)=& \omega(-1,-1) f(x-1, y-1)+\omega(-1,0) f(x-1, y) \ &+\omega(-1,1) f(x-1, y+1)+\omega(0,-1) f(x, y-1) \ &+\omega(0,0) f(x, y)+\omega(0,1) f(x, y+1)+\omega(1,-1) f(x+1, y-1) \ &+\omega(1,0) f(x+1, y)+\omega(1,1) f(x+1, y+1) \end{aligned}

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

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|DISTANCE TRANSFORM

D(p)={0, 如果 p∈乙 分钟q∈乙d(p,q), 如果 p∈这

1. 正定：d(p,q)≥0(d(p,q)=0当且当p=q)
2. 对称：d(p,q)=d(q,p)
3. 三角形：d(p,r)≤d(p,q)+d(q,r)图像处理中有几种类型的距离度量。最重要的三个是：欧几里得、城市街区和棋盘。

## 机器视觉代写|图像处理作业代写Image Processing代考|PERFORMANCE OF THE DISTANCE METRICS

H(X,是)=ω(X,是)∗F(X,是) =∫−∞∞∫−∞∞ω(在,在)F(X−在,是−在)d在 d在

H(X,是)=ω(X,是)∗F(X,是) =∑s=−1212∑吨=−42n2ω(s,吨)F(X−s,是−吨)

H(X,是)=ω(−1,−1)F(X−1,是−1)+ω(−1,0)F(X−1,是) +ω(−1,1)F(X−1,是+1)+ω(0,−1)F(X,是−1) +ω(0,0)F(X,是)+ω(0,1)F(X,是+1)+ω(1,−1)F(X+1,是−1) +ω(1,0)F(X+1,是)+ω(1,1)F(X+1,是+1)

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

(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

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## MATLAB代写

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