### 数学代写|线性代数代写linear algebra代考|MATH1002

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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

## 数学代写|线性代数代写linear algebra代考|Exploration: Digital Images

In order to understand and solve our tomography task (Section 1.2.1), we must first understand the nature of the radiographs that comprise our data. Each radiograph is actually a digitally stored collection of numerical values. It is convenient for us when they are displayed in a pixel arrangement with colors or grayscale. This section explores the nature of pixelized images and provides exercises and questions to help us understand their place in a linear algebra context.

We begin by formalizing the concept of an image with a definition. We will then consider the most familiar examples of images in this section. In subsequent sections we will revisit this definition and explore other examples.

First, let us look at an image from a camera in grayscale. In Figure $2.3$, we see one of the authors learning to sail. When we zoom in on a small patch, we see squares of uniform color. These are the pixels in the image. Each square (or pixel) has an associated intensity or brightness. Intensities are given a corresponding numerical value for storage in computer or camera memory. Brighter pixels are assigned larger numerical values.

Consider the $4 \times 4$ grayscale image in Figure 2.4. This image corresponds to the array of numbers at right, where a black pixel corresponds to intensity 0 and increasingly lighter shades of gray correspond to increasing intensity values. A white pixel (not shown) corresponds to an intensity of $16 .$

A given image can be displayed on different scales; in Figure 2.3, a pixel value of 0 is displayed as black and 255 is displayed as white, while in Figure $2.4$ a pixel value of 0 is displayed as black and 16 is displayed as white. The display scale does not change the underlying pixel values of the image.
Also, the same object may produce different images when imaged with different recording devices, or even when imaged using the same recording device with different calibrations. For example, this is what a smart phone is doing when you touch a portion of the screen to adjust the brightness when you take a picture with it.

Our definition of an image yields a natural way to think about arithmetic operations on images such as multiplication by a scalar and adding two images. For example, suppose we start with the three images A, B, and C in Figure 2.5.

Multiplying Image A by one half results in Image 1 in Figure 2.6. Every intensity value is now half what it previously was, so all pixels have become darker gray (representing their lower intensity). Adding Image 1 to Image $\mathrm{C}$ results in Image 2 (also in Figure 2.6); so Image 2 is created by doing arithmetic on Images A and $\mathrm{C}$.

Caution: Digital images and matrices are both arrays of numbers. However, not all digital images have rectangular geometric configurations like matrices ${ }^{1}$, and even digital images with rectangular configurations are not matrices, since there are operations ${ }^{2}$ that can be performed with matrices that do not make sense for digital images.

## 数学代写|线性代数代写linear algebra代考|Exercises

For some of these exercises you will need access to OCTAVE or MATLAB software. The following exercises refer to images found in Figures $2.5$ and 2.6.

1. Express Image 3 using arithmetic operations on Images A, B, and $C$. (Note that the pixel intensities in Image 3 are all either 4,8 , or 16.)
2. Express Image 4 using arithmetic operations on Images A, B, and C. (Note that the pixel intensities in Image 4 are all either 0 or 16.)
3. Input the following lines of code into the command window of OCTAVE/MATLAB. Note that ending a line with a semicolon suppresses terminal output. If you want to show the result of a computation, delete the semicolon at the end of its line. Briefly describe what each of these lines of code produces.
1. Enter the following lines of code one at a time and state what each does.
2. Write your own lines of code to check your conjectures for producing Images 3 and/or 4 . How close are these to Images 3 and/or 4?
3. We often consider display scales that assign pixels with value 0 to the color black. If a recording device uses such a scale then we do not expect any images it produces to contain pixels with negative values. However, in our definition of an image we do not restrict the pixel values. In this problem you will explore how OCTAVE/MATLAB displays an image with negative pixel values, and you will explore the effects of different gray scale ranges on an image.

Input the image pictured below into OCTAVE/MATLAB. Then display the image using each of the following five grayscale ranges.

## 数学代写|线性代数代写linear algebra代考|Systems of Equations

In Section $2.1$, we considered various $4 \times 4$ images (see page 11 ). We showed that Image 2 could be formed by performing image addition and scalar multiplication on Images $A, B$, and $C$. In particular,
$$(\text { Image } 2)=\left(\frac{1}{2}\right)(\text { Image } A)+(0)(\text { Image } B)+(1)(\text { Image } C)$$
We also posed the question about whether or not Images 3 and 4 can be formed using any arithmetic operations of Images $A, B$, and C. One can definitely determine, by inspection, the answer to these questions. Sometimes, however, trying to answer such questions by inspection can be a very tedious task. In this section, we introduce tools that can be used to answer such questions. In particular, we will discuss the method of elimination, used for solving systems of linear equations. We will also use matrix reduction on an augmented matrix to solve the corresponding system of equations. We will conclude the section with a key connection between the number of solutions to a system of equations and a reduced form of the augmented matrix.

Let $I_{1}$ and $I_{2}$ be images. We say that $I_{1}=I_{2}$ if each pair of corresponding pixels from $I_{1}$ and $I_{2}$ has the same intensity.

The convention in Figure 2.4, Definition 2.2.1, and Equation $2.1$ give us a means to write an equation, corresponding to the upper left pixel of Image D,
$$8=0 \alpha+4 \beta+8 \gamma$$
This equation has a very specific form: it is a linear equation. Such equations are at the heart of the study of linear algebra, so we recall the definition below.

## 数学代写|线性代数代写linear algebra代考|Exercises

1. 使用图像 A、B 和图像上的算术运算来表达图像 3C. （请注意，图像 3 中的像素强度都是 4,8 或 16。）
2. 使用图像 A、B 和 C 的算术运算来表达图像 4。（请注意，图像 4 中的像素强度都是 0 或 16。）
3. 在 OCTAVE/MATLAB 的命令窗口中输入以下代码行。请注意，以分号结束一行会抑制终端输出。如果要显示计算结果，请删除行尾的分号。简要描述每行代码产生的内容。
1. 一次输入以下几行代码，并说明每行的作用。
2. 编写您自己的代码行来检查您对生成图像 3 和/或 4 的猜想。这些与图像 3 和/或 4 有多接近？
3. 我们经常考虑将值为 0 的像素分配给黑色的显示比例。如果记录设备使用这样的比例，那么我们不希望它产生的任何图像包含具有负值的像素。但是，在我们对图像的定义中，我们不限制像素值。在这个问题中，您将探索 OCTAVE/MATLAB 如何显示具有负像素值的图像，并且您将探索不同灰度范围对图像的影响。

## 数学代写|线性代数代写linear algebra代考|Systems of Equations

( 图片 2)=(12)( 图片 一个)+(0)( 图片 乙)+(1)( 图片 C)

8=0一个+4b+8C

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## MATLAB代写

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